Contents

4 Multi-channel Modulation 290

4.1 Basic Multitone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
4.2 Parallel Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
4.2.1 The single-channel gap analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
4.2.2 A single performance measure for parallel channels - geometric SNR . . . . . . . . 299
4.2.3 The Water-Filling Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
4.2.4 Margin Maximization (or Power Minimization) . . . . . . . . . . . . . . . . . . . . 302
4.3 Loading Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
4.3.1 Computing Water Filling for RA loading . . . . . . . . . . . . . . . . . . . . . . . 305
4.3.2 Computing Water-Filling for MA loading . . . . . . . . . . . . . . . . . . . . . . . 308
4.3.3 Loading with Discrete Information Units . . . . . . . . . . . . . . . . . . . . . . . 311
4.3.4 Sorting and Run-time Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
4.3.5 Dynamic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
4.3.6 (bit-swapping) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
4.3.7 Gain Swapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
4.4 Channel Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
4.4.1 Eigenfunction Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
4.4.2 Choice of transmit basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
4.4.3 Limiting Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
4.4.4 Time-Domain Packets and Partitioning . . . . . . . . . . . . . . . . . . . . . . . . 336
4.4.5 Filters with Multitone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
4.5 Discrete-time channel partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
4.5.1 Optimal Partitioning Vectors - Vector Coding . . . . . . . . . . . . . . . . . . . . . 339
4.5.2 Creating the set of Parallel Channels for Vector Coding . . . . . . . . . . . . . . . 339
4.5.3 An SNR for vector coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
4.6 Discrete Multitone (DMT) and OFDM Partitioning . . . . . . . . . . . . . . . . . . . . . 343
4.6.1 The Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
4.6.2 DMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
4.6.3 An alternative view of DMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
4.6.4 Noise-Equivalent Channel Interpretation for DMT . . . . . . . . . . . . . . . . . . 354
4.6.5 Toeplitz Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
4.6.6 Gain Normalizer Implementation - The FEQ . . . . . . . . . . . . . . . . . . . . . 355
4.6.7 Isakssons Zipper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
4.6.8 Filtered Multitone (FMT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
4.7 Multichannel Channel Identication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
4.7.1 Gain Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
4.7.2 Noise Spectral Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
4.7.3 SNR Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
4.8 Stationary Equalization for Finite-length Partitioning . . . . . . . . . . . . . . . . . . . . 371
4.8.1 The innite-length TEQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
4.8.2 The nite-length TEQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
4.8.3 Zero-forcing Approximations of the TEQ . . . . . . . . . . . . . . . . . . . . . . . 382
288
4.8.4 Kwak TEQ Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
4.8.5 Joint Loading-TEQ Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
4.8.6 Block Decision Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
4.8.7 Cheongs Cyclic Reconstruction and Per-tone Equalizers . . . . . . . . . . . . . . . 385
4.9 Windowing Methods for DMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
4.9.1 Receiver Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
4.10 Peak-to-Average Ratio reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
4.10.1 PAR fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
4.10.2 Tone Reservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
4.10.3 Tone Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
4.10.4 Nonlinear Compensation of PAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
Exercises - Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
289
Chapter 4
Multi-channel Modulation
Multi-channel modulation methods are generally the best for data transmission channels with moderate
or severe intersymbol interference. The modulation and equalization methods studied in Chapter 3
have only mildly considered the channel pulse response - that is, the modulator basis functions were
xed. Chapter 3 studied equalization for mitigation of distortion (ISI) that might not have been of
concern in data transmission with carefully designed channel-dependent basis functions. In practice, a
particular modem may have to work with acceptable performance over a range of variation in specic
transmission channels. For instance, there are nearly a billion telephone lines worldwide that exhibit
signicant variation in channel response and noise power spectral density. Also, cable-television coaxial
links can also exhibit signicant variation from customer to customer. Large performance improvement
can be attained in these applications when the modulator basis functions can be designed as a function of
measured channel characteristics. Multi-channel modulation methods allow such modulator adaptation
to channel.
Figure 4.1: Basic concept of adaptive multi-channel (multitone) transmission.
Figure 4.1 illustrates the basic operation of DSL circuits, which use multitone transmission. The
basis functions can be viewed as a set of non-overlapping narrow QAM signals (i.e., tones). In the
upper portion of Figure 4.1, the channel response is shown to remove low-frequencies (DC notching of
transformer coupling to the telephone line) and also attenuates high-frequencies. The noise is presumed
290
AWGN. Note that the energy and/or amount of information carried on each QAM channel roughly
follows the channel transfer function. Multitone designs thus avoid need for equalization on any tone
by making each suciently narrow that the associated channel response in that band appears at
or satisfying the Nyquist criterion. The same DSL modem used on a dierent telephone line might
see the same DC notch and attenuation with increasing frequency, but perhaps also a notch caused
by a bridged tap (open-circuit unterminated extension-phone line that branches from the main line),
AM radio noise (RF noise), and possibly crosstalk noise (which increases with frequency) from other
DSLs sharing the same telephone cable. The transmit basis function are the same, but clearly carry a
very dierent allocation of energy and information for this second example. Again though, equalization
(which would have been very complex in this second case) is avoided. This chapter will develop a theory
to analyze and ultimately nd optimum the strategy basically outlined in Figure 4.1 on any stationary
linear channel with additive Gaussian noise.
The origins of multi-channel modulation date to antiquity. Shannons 1948 information-theory pa-
per constructs capacity bounds for transmission on a AWGN channel with linear ISI, eectively using
multitone modulation. The rst uses of multi-channel modulation appear to be in the 1950s with many
claiming invention. The 1958 Collins Kineplex modem, the very rst voiceband data modem, used a
rudimentary form of multitone modulation. This modem operated at data rates of about 3000 bps, more
than 10 time faster than other modems in existence for the subsequent decade an early testimonial to
the potential of multitone modulation. Holsingers 1964 MIT dissertation describes experiments on an-
other modem with optimized transmit distributions that approximated Shannons famous water-lling
best adaptive spectrum, which was also more completely developed within that same dissertation. Gal-
lager of MIT later added detailed mathematical rigor and popularized the term water-lling for the
shape of the optimized transmit spectrum as a function of the channel and noise characteristics. An
early problem was the stable implementation of multiple modulators in hardware, stiing wide use of
the method for nearly 20 years.
A number of increasingly successful attempts at the use of multichannel modulation occurred over a
30-year period, including attempts by Chang and Saltzberg of Bell Labs (1967), Weinstein and Eber of
Bell Labs (1973), Peled and Ruiz of IBM (1980), Keasler of U of Illinois (1970s, licensed by Gandolf),
Baran of Telebit (1990s), Mallory and Chaee of IMC (1990s), and Hirosaki of NEC (1987). Some of
the early implementers mentioned became frustrated in their analog-signal-processing implementations
and never recovered to later benet from nor understand the merits of the approach in digital-signal-
processing implementation. While the various wired-line
1
multichannel modems did often work better
than other single-channel modems, a variety of practical problems had earned multitone a reputation
slandered with cynicism and disdain by the early 1990s.
A nal thrust at popularizing multitone transmission, this time addressing many of the practical
problems inherent in the technique with digitally implementable solutions under the name Discrete
Multitone (DMT) emanated from Stanford University
2
. With anti-multitone hostilities at extreme
and ugly levels, the academic Stanford group was grudgingly allowed to compete with a traditional
equalized systems designed by three independent sets of recognized experts from some of the worlds most
reknown telecommunications companies in what has come to be known as the Bellcore Transmission
Olympics in 1993 for DSL. The remaining worlds transmission experts anticipated the nal downfall
of the multitone technique. In dramatic reversal of fortune, with measured 10-30 dB improvements
over the best of the equalized systems on actual phone lines, the gold-medalist DMT system ressurected
multitone transmission at 4:10 pm on March 10, 1993
3
, when a US (and now world-wide) standard
was set for transmitting data over nearly a billion phone lines using multichannel modulation this is
DSL
4
. At time of writing, over 200 million DMT components have been sold for DSL. Such credibility
1
Wireless uses of multitone transmission are somewhat dierent and more recent and do not adapt the transmitter to
the channel, instead using various codes that are pertinent to channels that vary so quickly that the transmitter cannot
be adapted as suggested here in Chapter 4. Wireless multitone systems will be mentioned more in Section 4.5.
2
Special thanks to former Stanford students and Amati employees Jim Aslanis, John Bingham, Jacky Chow, Peter
Chow, Toni Gooch, Ron Hunt, Kim Maxwell, Van Whitis and even an MIT, Mark Flowers, and a Utah, Mark Mallory.
3
An interesting and unexpected coincidence is that this formal decision time and the very rst demonstration of an
operating telephone (Watson, Come here I need you, said A.G. Bell) are coincident to the minute 117 years apart, with
Bells famous demonstration occuring on March 10, 1876, also at 4:10 pm.
4
It should be noted that a second Bellcore Olympics was held in 2003 for higher speed DSLs as the remnants of the old
291
established, it is now possible to develop the descriptions in this Chapter to the potential successful use
by the reader.
Section 4.1 illustrates the basic multitone concept generically and then specically through the use
of the same 1+.9D
1
channel that has been used throughout this text. Multichannel modulation forms
a set of parallel independent transmission channels that are used in aggregate to carry a data bit stream.
The generic parallel-channel situation is investigated in Section 4.2. The investigations include a review
of the SNR gap of earlier chapters as well as an elaboration of Chapter 6s water-lling concepts in
a design (rather than capacity theory) context. The water-ll concept is easily described mathemat-
ically and pictorially, but can require large complexity in implementation. Loading algorithms that
optimize the transmit bit and energy distribution for the parallel channels are introduced in Section 4.3.
Channel partitioning, the construction of parallel independent channels from a channel with intersymbol
interference, is studied in Section 4.4. With isolated transmit symbols, achieved by waiting for trailing
intersymbol interference to decay to zero, the optimum vector coding and suboptimum DMT meth-
ods are described and analyzed. These methods are found to converge to the same performance as the
number of channels increases. The identication of the channel is very important to vector coding and
DMT methods and specic channel identication methods are studied in Section 4.7. A more complete
discussion of channel identication occurs in Chapter 13. The isolation of transmit symbols can be un-
desirable and was one of the problems poorly addressed in early multichannel eorts (leading to some of
the cynicism). An improvement, but not the last improvement in the authors opinion, called a TEQ
or time-domain equalizer is discussed in Section 4.8. Some other methods for practical isolation are
suggested in Section 4.9, while some straightforward decoding and code-concatenation implementation
issues are addressed in Section 4.10.
single-carrier component suppliers attempted to overturn the earlier results. The results were again the same, dramatic
huge gains measured by a neutral lab of the DMT technique, and thus high speed DSLs known as VDSL are again
DMT-based
292
Figure 4.2: Basic multitone transmission.
4.1 Basic Multitone
Multitone transmission uses two or more coordinated passband (QAM or QAM-like) signals to carry a
single bit stream over a communication channel. The passband signals are independently demodulated in
the receiver and then remultiplexed into the original bit stream. The heuristic motivation for multitone
modulation is that if the bandwidth of each of the tones is suciently narrow, then no ISI occurs on any
subchannel. The individual passband signals may carry data equally or unequally. Usually, the passband
signal(s) with largest channel output signal-to-noise ratio carry a proportionately larger fraction of the
digital information, consistent with Section 1.6s gap approximation that

b
n
=
1
2
log
2
_
1 +
SNRn

_
.
Figure 4.2 shows the simplest multitone system to understand.

N QAM-like modulators, along with
possibly one DC/baseband PAM modulator, transmit

N+1 subsymbol components X
n
, n = 0, 1, ...,

N.
5
(

N

= N/2, and N is assumed to be even in this chapter.) This system is essentially the same as
the general modulator of Chapter 1. X
0
and X
N
are real one-dimensional subsymbols while X
n
,
n = 1, ...,

N 1 can be two-dimensional (complex) subsymbols. Each subsymbol represents one of
2
bn
messages that can be transmitted on subchannel n. The carrier frequencies for the corresponding
subchannels are f
n
= n/T, where T is again the symbol period. The baseband-equivalent basis functions
are
n
(t) =
1

T
sinc
_
t
T
_
n.
6
The entire transmitted signal can be viewed as

N + 1 independent
transmission subchannels as indicated by the frequency bands of Figure 4.3.
The multitone modulated signal is transmitted over an ISI/AWGN channel with the correspond-
ing demodulator structure also shown in Figure 4.2. Each subchannel is separately demodulated by
rst quadrature decoupling with a phase splitter and then baseband demodulating with a matched-
lter/sampler combination. With this particular ideal choice of basis functions, the channel output
basis functions
p,n
(t) are an orthogonal set. There is thus no interference between the subchannels.
Each subchannel may have ISI, but as N , this ISI vanishes. Thus, (sub)symbol-by-(sub)symbol
detection independently applied to each subchannel implements an overall ML detector. No equalizer
is then necessary (large N) to implement the maximum-likelihood detector. Thus, maximum-likelihood
5
Capital letters are used for symbol component elements rather than the usual lower case letters because these elements
are frequency-domain elements, and frequency-domain quantities in this book are capitalized.
6
The passband basis functions are actually the real and imaginary parts of
_
2/Tsinc(
t
T
) e
2fnt
with receiver scaling
tacitly moved to the transmitter as in Chapter 2.
293
Figure 4.3: Illustration of frequency bands for multitone transmission system.
294
detection is thus more easily achieved with multitone modulation on an ISI channel than it is on a single
QAM or PAM signal. Again, equalization is unnecessary if the bandwidth of each tone is suciently
narrow to make the ISI on that subchannel negligible. Thus a high-speed modulator and equalizer are
replaced by

N + 1 low-speed modulators and demodulators in parallel.
Multitone modulation typically uses a value for

N that ensures that the pulse response of the ISI
channel appears almost constant at H(n/T)

= H
n
= H(f) for [f n/T[ < 1/2T. In practice, this
means that the symbol period T greatly exceeds the length of the channel pulse response. The (scaled)
matched lters then simply become the bandpass lters
p,n
(t) =
n
(t) = 1/

T sinc(t/T) e
(2/T)nt
and the sampled outputs become
Y
n
H
n
X
n
+ N
n
. (4.1)
The accuracy of this approximation becomes increasingly exact as N . Figure 4.3 illustrates the
scaling of H
n
at the channel output on each of the subchannels. Each subchannel scales the input X
n
by the pulse-response gain H
n
. Each subchannel has an AWGN with power-spectral density
2
n
. Thus,
each subchannel has SNR
n
=

Ln]Hn]
2

2
n
.
Each subchannel in the multitone system carries b
n
bits per symbol or

b
n
bits per dimension (note
both n = 0 and n =

N are by denition one-dimensional subchannels with subchannel

N viewed as
single-side-band modulation)
7
. The total number of bits carried by the multitone system is then
b =

n=0
b
n
(4.2)
and the corresponding data rate is then
R =
b
T
=

n=0
R
n
, (4.3)
where R
n

= b
n
/T. Thus the aggregate data rate R is divided, possibly unequally, among the subchan-
nels.
With suciently large N, an optimum maximum-likelihood detector is easily implemented as

N +1
simple symbol-by-symbol detectors. This detector need not search all combinations of M = 2
b
possible
transmit symbols. Each subchannel is optimally symbol-by-symbol detected. The reason this maximum-
likelihood detector is so easily constructed is because of the choice of the basis functions: multitone
basis functions are generically well-suited to transmission over ISI channels. Broader bandwidth basis
functions, as might be found in a wider bandwidth single-carrier (for instance QAM) system, are simply
not well suited to transmission on this channel, thus inevitably requiring suboptimum detectors for
reasons of complexity. Chapter 5 investigates the concept of canonical transmission and nds a means
for generalizing QAM transmission so that it is eectively just as good using a device known as the
Generalized Decision Feedback Equalizer or GDFE, which can in some trivial cases look like the usual
DFE of Chapter 3, but in most cases signicantly diers in its generalized implementation, both at
transmitter and receiver.
This chapter later trivially shows that multitone modulation, when combined with a code of gap
on each AWGN subchannel, comes as close to its theoretical capacity as that same code would come to
capacity on an AWGN. That is, for suciently large N, multitone basis functions are indeed optimum
for transmission on an ISI channel.
EXAMPLE 4.1.1 (1 +.9D
1
) This book has repeatedly used the example of a channel
with (sampled) pulse response 1 + .9D
1
to compare performance of various ISI-mitigation
structures. Multitone modulation on this example, as well as on any linear ISI channel,
achieves the highest performance. In order to have a continuous channel for multitone, we
will write that H(f) = 1+.9e
2f
(T = N). This channel increasingly attenuates with higher
frequencies, so multitone modulation will be used with subchannel n = 4 silenced (X
4
= 0).
7
Typically, subchannel 0 and subchannel

N are not used in actual systems
295
1 +.9D
1
Gains and SNRs
n c
n
H
n
SNR
n
b
n
arg Q-func (dB)
0 8/7 1.9 22.8 (13.6 dB) 1.6 9.2
1 16/7 1 + .9e
/4
= 1.76
,
21.3
o
19.5 (12.9 dB) 2 1.5 9.2
2 16/7 1 + .9e
/2
= 1.35
,
42.0
o
11.4 (10.6 dB) 2 1.2 9.1
3 16/7 1 + .9e
3/4
= .733
,
60.25
o
3.4 (5.3 dB) 2 .5 10
Table 4.1: Subchannel SNRs for the 1 + .9D channel with gap at 4.4 dB.
N = 8 is not suciently large for the approximation in (4.1) to hold, but larger N would
make this example intractable for the reader. So, we assume N = 8 is suciently large for
easy illustration of concept.
Table 4.1 summarizes the subchannel gains and SNRs for each of the subchannels in this ex-
ample: For multitone, the phase choice of the channel is irrelevent to achievable performance,
and the example would be the same for 1 + .9D
1
as for 1 + .9D.
The SNRs in the above table are computed based on the assumption that
A0
2
= .181 that
is, SNR
MFB
= 10 dB if binary modulation is used (as in previous studies with this example).
Of course, SNR
MFB
is a function of the modulated signals and in this example has meaning
only in the context of performance comparisons against previous invocations of this example.
The energy per subchannel has been allocated so that equal energy per dimension is used on
the 7 available dimensions. With T = N = 8, this means that the transmit power of 1 for
previous invocations of this example (that is one unit of energy per dimension and symbol
period 1) is maintained here because the transmit power is

n
c
n
/T = 8/8 = 1. For this
simple example, 4 symbol-by-symbol detectors operating in parallel exceeded the performance
of a MMSE-DFE, coming very close to the MFB for binary PAM transmission. The data rate
remains constant at

b = 1, and the power also is constant at

c
x
= 1, so comparison against
the DFE of Chapter 3 on this example is fair. The above tables Q-function arguments
are found by using the QAM formula d
min
/2 =
_
3
M1
SNR for n = 1, 2, 3 and the PAM
formula d
min
/2 =
_
3
M
2
1
SNR with M = 2
bn
this approximation can be made very
accurate by careful design of the signal sets for each subchannel.
For this example, N = 8 is not suciently high for all approximations to hold and a more
detailed analysis of each subchannels exact SNR might lead to slightly less optimistic per-
formance projection. On the other hand, the energy per subchannel and the bit distribution
in this example are not optimized, but rather simply guessed by the author. One nds for a
large number of subchannels that performance will come down to 8.8 dB when the tones are
suciently narrow for all approximations to hold accurately. An innite-length MMSE-DFE,
at best, achieves 8.4 dB in Chapter 3 with Tomlinson precoding, the actual performance at
the same data rate of

b = 1 was 1.3 dB lower or 7.1 dB. Thus a MMSE-DFE would perform
about 1.7 dB worse than multitone on this channel. (The GDFE of Chapter 5 allows most
of this 1.7 dB to be recovered, but also uses several subchannels.)
This example illustrates that there is something to be gained from multitone. An interesting aspect
is also the parallelism inherent in the multitone receiver and transmitter, allowing for the potential of
a very high-speed implementation on a number of parallel processing devices. The 1 + .9D
1
channel
has mild ISI (but is easy to work with mathematically without computers) the gain of 1.7 dB can be
much larger on channels with severe ISI.
296
4.2 Parallel Channels
The multitone transmission system of Section 4.1 inherently exhibits

N + 1 independent subchannels.
Generally, multichannel systems can be construed as having N subchannels (where in the previous
multitone system, several pairs of these subchannels have identical gains and can be considered as
two-dimensional subchannels). This section studies the general case of a communication channel that
has been decomposed into an equivalent set of N real parallel subchannels. Of particular importance
is performance analysis and optimization of performance for the entire set of subchannels, which this
section presents.
The concept of the SNR gap for a single channel is crucial to some of the analysis, so Subsection
4.2.1 reviews a single channel and then Subsections 4.2.2 and 4.2.3 use these single-channel results to
establish results for parallel channels.
4.2.1 The single-channel gap analysis
The so-called gap analysis of Chapter 1 is reviewed here briey in preparation for the study of parallel
channels and loading algorithms.
On an AWGN channel, with gain P
n
and noise power spectral density
2
n
, the SNR
n
is
L]Pn]
2

2
n
. This
channel has maximum data rate or capacity
8
of
c
n
=
1
2
log
2
(1 + SNR
n
) (4.4)
bits/dimension. More generally,

b
n
=
1
2
log
2
(1 +
SNR
n

) (4.5)
Any reliable and implementable system must transmit at a data rate below capacity. The gap
is a convenient mechanism for analyzing systems that transmit with

b < c. Most practical signal
constellations in use (PAM, QAM) have a constant gap for all

b .5. For

b
n
< .5, one can construct
systems of codes to be used that exhibit the same constant gap as for

b .5.
9
For any given coding
scheme and a given target probability of symbol error P
e
, an SNR gap is dened according to

=
2
2 c
1
2
2

b
1
=
SNR
2
2

b
1
. (4.6)
To illustrate such a constant gap, Table 4.2.1 lists achievable

b for uncoded QAM schemes using square
constellations. For

b < 1, this gap concept allows mild additional hard coding in the uncoded system
to obtain small additional coding gain when b = .5.
For uncoded QAM or PAM and P
e
= 10
6
, the SNR gap is constant at 8.8 dB (this assumes use
of special constellations or coding for

b .5). With uncoded QAM or PAM and P
e
= 10
7
, the gap
would be xed instead at about 9.5 dB. The use of codes, say trellis or turbo coding and/or forward
error correction (see Chapters 10 and 11) reduces the gap. A very well-coded system may may have
a gap as low as .5 dB at P
e
10
6
. Figure 4.4 plots

b versus SNR for various gaps. Smaller gap
indicates stronger coding. The curve for = 9 dB approximates uncoded PAM or QAM transmission at
probability of symbol error P
e
10
6
. More powerful, but implementable, codes have smaller gaps, as
small as 1 dB. A gap of 0 dB means the theoretical maximum bit rate has been achieved, which requires
innite complexity and delay - see Chapter 8.) For the remainder of this section and most of these notes,
the same constant gap is assumed for the coding of each and every subchannel.
For a given coding scheme, practical transmission designs often mandate a specic value for b, or
equivalently a xed data rate. In this case, the design is not for

b
max
=
1
2
log
2
(1 + SNR/), but rather
for b. The margin measures the excess SNR for that given bit rate.
8
Capacity in Volume I is simply dened as the data rate with a code achieving a gap of = 1 (0 dB). This is the lowest
possible gap theoretically and can be approached by sophisticated codes, as in Volume II. Codes exist for which

Pe 0 if

b c.
9
Some of the algorithms in Section 4.3 do not require the gap be constant on each tone, and instead require only tabular
knowledge of the exact bn versus required En relationship for each subchannel.
297

b .5 1 2 3 4 5
SNR for P
e
= 10
6
(dB) 8.8 13.5 20.5 26.8 32.9 38.9
2
2

b
1 (dB) 0 4.7 11.7 18.0 24.1 30.1
(dB) 8.8 8.8 8.8 8.8 8.8 8.8
Table 4.2: Table of SNR Gaps for

P
e
= 10
6
.
2 4 6 8 10 12 14 16
0
0.5
1
1.5
2
2.5
3
SNR (dB)
b
i
t
s
/
d
i
m
e
n
s
i
o
n
Achievable bit rate for various gaps
0 dB
3 dB
6 dB
9 dB
Figure 4.4: Illustration of bit rates versus SNR for various gaps.
298
(repeated from Section 1.5)
Denition 4.2.1 The margin,
m
, for transmission on an AWGN (sub)channel with a
given SNR, a given number of bits per dimension

b, and a given coding-scheme/target-

P
e
with gap is the amount by which the SNR can be reduced (increased for negative margin in
dB) and still maintain a probability of error at or below the target P
e
.
Margin is accurately approximated through the use of the gap formula by

m
=
2
2

bmax
1
2
2

b
1
=
SNR/
2
2

b
1
. (4.7)
The margin is the amount by which the SNR on the channel may be lowered before performance degrades
to a probability of error greater than the target P
e
used in dening the gap. A negative margin in dB
means that the SNR must improve by the magnitude of the margin before the P
e
is achieved. The
margin relation can also be written as

b = .5 log
2
_
1 +
SNR

m
_
. (4.8)
EXAMPLE 4.2.1 (AWGN with SNR = 20.5 dB) An AWGN has SNR of 20.5 dB. The
capacity of this channel is then
c = .5 log
2
(1 + SNR) = 3.5 bits/dim . (4.9)
With a P
e
= 10
6
and 2B1Q,

b = .5 log
2
_
1 +
SNR
10
.88
_
= 2 bits/dim . (4.10)
With concatenated trellis and forward error correction (or with turbo codes see Chapters
10 and 11), a coding gain of 7 dB at P
e
= 10
6
, then the achievable data rate leads to

b = .5 log
2
_
1 +
SNR
10
.18
_
= 3 bits/dim . (4.11)
Suppose a transmission application requires

b = 2.5 bits/dimension, then the margin for the
coded system is

m
=
2
23
1
2
22.5
1
=
63
31
3 dB . (4.12)
This means the noise power would have to be increased (or the transmit power reduced) by
more than 3 dB before the target probability of error of 10
6
is no longer met. Alternatively,
suppose a design transmits 4-QAM over this channel and no code is used, then the margin is

m
=
2
22
1
2
21
1
=
15
3
7 dB . (4.13)
4.2.2 A single performance measure for parallel channels - geometric SNR
In a multi-channel transmission system, it is usually desirable to have all subchannels with the same P
e
.
Otherwise, if one subchannel had signicantly higher P
e
than others, then it would dominate bit error
rate. Constant P
e
can occur when all subchannels use the same class of codes with constant gap . In
this case, a single performance measure characterizes a multi-channel transmission system. This measure
is a geometric SNR that can be compared to the detection SNR of equalized transmission systems or to
the SNR
MFB
.
299
For a set of N (one-dimensional real) parallel channels, the aggregate number of bits per dimension
is

b = (1/N)
N

n=1

b
n
= (1/N)
N

n=1
1
2
log
2
_
1 +
SNR
n

_
(4.14)
=
1
2N
log
2
_
N

n=1
_
1 +
SNR
n

_
_
(4.15)

=
1
2
log
2
_
1 +
SNR
m,u

_
. (4.16)
Denition 4.2.2 (Multi-channel SNR) The multi-channel SNR for a set of parallel
channels is dened by
SNR
m,u

=
_
_
_
N

n=1
_
1 +
SNR
n

_
_
1/N
1
_
_
. (4.17)
The multi-channel SNR is a single SNR measure that characterizes the set of subchannels by an
equivalent single AWGN that achieves the same data rate, as in Figure 4.5. The multichannel SNR in
(4.17) can be directly compared with the detection SNR of an equalized single channel system. The bit
rate is given in (4.16) as if the aggregate set of parallel channels were a single AWGN with SNR
m,u
.
When the terms involving +1 can be ignored, the multichannel SNR is approximately the geometric
mean of the SNRs on each of the subchannels
SNR
m,u
SNR
geo
=
_

n
SNR
n
_
1/N
. (4.18)
Returning to Example 4.1.1, the multi-channel SNR (with gap 0 dB ) is
SNR
m,u
=
_
(22.8 + 1)(19.5 + 1)
2
(11.4 + 1)
2
(3.4 + 1)
2

.125
1 = 8.8(dB) . (4.19)
This multichannel SNR with = 0 is very close to the argument that was used for all the Q-functions.
This value is actually the best that can be achieved.
10
SNR
m,u
= 9.7 dB for = 8.8 dB, very close to the
value in the sample design of Example 4.1.1. This value is artically high only because the approximation
accruing to using too small a value of N for illustration purposes - if N were large, SNR
m,u
would be
close to 8.8 dB even with > 1, and also the gap approximation loses accuracy when

b < 1.
4.2.3 The Water-Filling Optimization
To maximize the data rate, R = b/T, for a set of parallel subchannels when the symbol rate 1/T is xed
requires maximization of the achievable b =

n
b
n
over b
n
and c
n
. The largest number of bits that can
be transmitted over a parallel set of subchannels must maximize the sum
b =
1
2
N

n=1
log
2
(1 +
c
n
g
n

) , (4.20)
where g
n
represents the subchannel signal-to-noise ratio when the transmitter applies unit energy to
that subchannel. (For multitone, g
n
= [H
n
[
2
/(
2
n
).) g
n
is a xed function of the channel, but c
n
can be
varied to maximize b, subject to an energy constraint that
N

n=1
c
n
= N

c
x
. (4.21)
10
Somewhat by coincidence because the continuous optimum bandwidth from Chapter 8 is .88. This example for-
tuitously can approximate .88 very accurately only using 7/8 subchannels to get .875. In other cases, a much larger
number of subchannels may need to be used to approximate the optimum bandwidth.
300
Figure 4.5: Equivalent single channel characterized by Multi-channel SNR.
301
Using Lagrange multipliers, the cost function to maximize (4.20) subject to the constraint in (4.21)
becomes
1
2 ln(2)

n
ln
_
1 +
c
n
g
n

_
+
_

n
c
n
N

c
x
_
. (4.22)
Dierentiating with respect to c
n
produces
1
2 ln(2)
1

gn
+ c
n
= . (4.23)
Thus, (4.20) is maximized, subject to (4.21), when
c
n
+

g
n
= constant . (4.24)
When = 1 (0 dB), the maximum data rate or capacity of the parallel set of channels is achieved.
The solution is called the water-lling solution because one can construe the solution graphically by
thinking of the curve of inverted channel signal-to-noise ratios as being lled with energy (water) to a
constant line as in Figure 4.6. (Figure 4.6 illustrates the discrete equivalent of water-lling for a set of
6 subchannels, which will be subsequently described further.) When ,= 1, the form of the water-ll
optimization remains the same (as long as is constant over all subchannels). The scaling by makes
the inverse channel SNR curve, /g
n
vs n, appear more steep with n, thus leading to a more narrow
(fewer used subchannels) optimized transmit band than when capacity ( = 1) is achieved. The number
of bits on each subchannel is then
b
n
= .5 log
2
_
1 +
c
n
g
n

_
(4.25)
For the example of multi-tone, the optimum water-lling transmit energies then satisfy
c
n
+

2
n
[H
n
[
2
= constant . (4.26)
Figure 4.6 depicts water-lling for a transmission system with 6 subchannels with g
n
=
]Hn]
2

2
n
. Equation
(4.26) is solved for the constant when

n
c
n
= N

c
x
and

c
x
0. Note in Figure 4.6 that 4 of the six
subchannels have positive energies, while 2 were eliminated for having negative energy, or equivalently
having a noise power that exceeded the constant line of water lling. The 4 used subchannels have
energy that makes the sum of normalized noise and transmit energy constant for all. For methods of
computing the water-ll solution, see the next section. The term water-lling arises from the analog
of the curve of /g
n
being a bowl into which water (energy) is poured, lling the bowl until there is no
more energy to use. The water will rise to a constant at level in the bowl. The amount of water/energy
in any subchannel is the depth of the water at the corresponding point in the bowl.
The water-lling solution is unique because the function being minimized is convex, so there is
a unique optimum energy distribution (and corresponding set of subchannel data rates) for each ISI
channel with multi-channel modulation.
4.2.4 Margin Maximization (or Power Minimization)
For many transmission systems, variable data rate is not desirable. In this case, the best design will
maximize the performance margin at a given xed data rate. To maximize xed-rate margin, the designer
equivalently minimizes the total energy (which is also called xed-margin loading)
N

n=1
c
n
(4.27)
302
Figure 4.6: Illustration of discrete water-lling for 6 subchannels.
subject to the data rate and margin being xed according to
N

n=1
1
2
log
2
_
1 +
c
n
g
n

_
= b . (4.28)
The best margin will then be

m
=
Nc
x

N
n=1
c
n
. (4.29)
The solution to the energy minimization problem after correct dierentiation is again the water-lling
solution given by
c
n
+

g
n
= constant . (4.30)
However, in this case water/energy is poured only until the number of bits per symbol (which is computed
at each subchannel according to (4.25) and then summed over all subchannels) is equal to the given
xed rate in (4.28). Then the margin is computed according to (4.29). Each subchannel has its energy
increased uniformly with respect to that which minimized the sum of energies by a factor of the maximum
margin. Sometimes in multi-user channels, see Chapter 12, the solution with minimum energy for a
given data rate and given margin is of interest in what is called iterative water-lling. This energy-
minimizing form of the loading problem is known in mathematics as the dual form of the rate-adaptive
formulation.
The energy minimization is equivalent to margin maximization because any other bit distribution
would have required more energy to reach the same given data rate. That greater energy would then
clearly leave less energy to be distributed (uniformly) among the remaining subchannels if the new dis-
tribution used the same or a larger number of subchannels. If the new distribution used less subchannels
than the water-ll, the amount by which the new energy distributions total energy increased still exceeds
the improvement caused by extra energy being distributed over the smaller number of subchannels for
303
if this were not the case, then

b could have been achieved more eciently with the new distribution (a
contradiction).
304
4.3 Loading Algorithms
Loading algorithms compute values for b
n
and c
n
for each and every subchannel in a parallel set of
subchannels. One example of a loading algorithm is the optimum water-lling algorithm of Section 4.2
that solves a set of linear equations with boundary constraints, as described further in this section. The
solution of these water-lling equations for large N may produce b
n
that have fractional parts or be very
small. Such small or fractional b
n
can complicate encoder and decoder implementation. Alternative
suboptimal loading algorithms approximate the water-ll solution, but constrain b
n
to integer values, as
also described later in this section.
There are two types of loading algorithms those that try to maximize data rate and those that try
to maximize performance at a given xed data rate. Both are studied in this section.
Denition 4.3.1 (Rate-Adaptive (RA) loading criterion) A rate-adaptive loading pro-
cedure maximizes (or approximately maximizes) the number of bits per symbol subject to a
xed energy constraint:
max
Ln
b =
N

n=1
1
2
log
2
_
1 +
c
n
g
n

_
(4.31)
subject to: N

c
x
=
N

n=1
c
n
(4.32)
Denition 4.3.2 (Margin-Adaptive (MA) loading criterion) A margin-adaptive load-
ing procedure minimizes (or approximately minimizes) the energy subject to a xed bits/symbol
constraint:
min
Ln
c
x
=
N

n=1
c
n
(4.33)
subject to: b =
N

n=1
1
2
log
2
_
1 +
c
n
g
n

_
(4.34)
The maximum margin is then

max
=
N

c
x
c
. (4.35)
4.3.1 Computing Water Filling for RA loading
The set of linear equations that has the water-ll distribution as its solution is
c
1
+ /g
1
= K (4.36)
c
2
+ /g
2
= K (4.37)
.
.
. =
.
.
. (4.38)
c
N
+ /g
N
= K (4.39)
c
1
+... + c
N
= N

c
x
. (4.40)
There are a maximum of N + 1 equations in N + 1 unknowns The unknowns are the energies c
n
, n =
1, ..., N and the constant K. The solution can produce negative energies. If it does, the equation with
the smallest g
n
should be eliminated, and the corresponding c
n
should be zeroed. The sets of equations
are solved recursively by eliminating the smallest g
n
and zeroing c
n
, until the rst solution with no
negative energies occurs.
305
In matrix form, the equations become
_

_
1 0 0 0 ... 1
0 1 0 0 ... 1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 1
1 1 1 ... 1 0
_

_
_

_
c
1
c
2
.
.
.
c
N
K
_

_
=
_

_
/g
1
/g
2
.
.
.
/g
N
N

c
x
_

_
, (4.41)
which are readily solved by matrix inversion. Matrix inversions of size N +1 down to 1 may have to be
performed iteratively until nonnegative energies are found for all subchannels.
An alternative solution sums the rst N equations to obtain:
K =
1
N
_
N

c
x
+
N

n=1
1
g
n
_
, (4.42)
and
c
n
= K /g
n
n = 1, ..., N . (4.43)
If one or more of c
n
< 0, then the most negative is eliminated and (4.42) and (4.43) are solved again
with N N 1 and the corresponding g
n
term eliminated. The equations preferably are preordered
in terms of signal-to-noise ratio with n = 1 corresponding to the largest-SNR subchannel g
1
= max
n
g
n
and n = N corresponding to the smallest-SNR subchannel g
N
= min
n
g
n
. Equation (4.42) then becomes
at the i
th
step of the iteration (i = 0, ..., N)
K =
1
N i
_
c
x
+
Ni

n=1
1
g
n
_
, (4.44)
which culminates with N

= N i for the rst value of i that does not cause a negative energy on c
i
to
occur, then leaving the energies as and
c
n
= K /g
n
n = 1, ..., N

= N i . (4.45)
The water-lling algorithm is illustrated by the ow chart of Figure 4.7.
The sum in the expression for K in (4.42) is always over the used subchannels. The following
simplied formulae, apparently rst observed by Aslanis, then determine the number of bits and number
of bits per dimension as
b
n
=
1
2
log
2
_
K g
n

_
=
1
2
log
2
_
1 +
c
n
g
n

_
. (4.46)
and

b =
1
2
log
2
_
K
N

/N
(

n=1
g
n
)
1/N

/N
_
=
1
N
N

n=1
b
n
(4.47)
=
1
2
log
2
_
1 +
SNR
m,u

_
, (4.48)
respectively. The SNR for the set of parallel channels is therefore
SNR
m,u
=
_
_
K

_
N

/N
_
N

n=1
g
1/N
n
_
1
_
. (4.49)
The water-lling constant K becomes the ratio
K =
_
2
2b

n=1
g
n
_
1/N

. (4.50)
306
Figure 4.7: Flow chart of Rate-Adaptive water-lling algorithm.
307
EXAMPLE 4.3.1 (1 +.9D
1
again) The water-ll solution (for = 1) can be found on
the 1 + .9D
1
example to compute capacity. First, the subchannels are characterized by
g
0
=
1.9
2
.181
= 19.94 (4.51)
g
1
=
1.7558
2
.181
= 17.03 (4.52)
g
2
=
1.3454
2
.181
= 10.00 (4.53)
g
3
=
.7329
2
.181
= 2.968 (4.54)
g
4
=
.1
2
.181
= .0552 (4.55)
Using (4.42) with all subchannels (N = 8 or actually 5 subchannels since 3 are complex with
identical gs on real and imaginary dimensions), one computes
K =
1
8
_
8 + 1 (
1
19.94
+
2
17.03
+
2
10.0
+
2
2.968
+
1
.0552
)
_
= 3.3947 , (4.56)
and thus c
4
= 3.3947 1/.05525 = 14.7 < 0. So the last subchannel should be eliminated
and the equations propagated again. For N = 7,
K =
1
7
_
8 + 1 (
1
20
+
2
17
+
2
9.8
+
2
3
)
_
= 1.292 , (4.57)
The corresponding subchannel energies,

c
n
, on the subchannels are 1.24, 1.23, 1.19, and
.96, which are all positive, meaning the solution with positive energy on all subchannels
has been found. The corresponding number of bits per each subchannel is computed as

b
n
= .5 log
2
(Kg
n
), which has values

b=2.3, 2.2, 1.83, and .98 respectively. The sum is very
close to 1.54 bits/dimension, almost exactly the known capacity of this channel. Figure 4.8
illustrates the water-ll energy distribution for this channel.
Note the capacity exceeds 1 bit/dimension, implying that codes exist that allow arbitrarily
small probability of error for the selected transmission rate in earlier uses of this example of
1 bit/dimension.
The accuracy of this computation of capacity can be increased somewhat by further increasing
N the eventual value is approximately 1.55 bits/dimension. Then, SNR
m,u
= 2
2(1.54)
1 =
8.8 dB. This is the highest detection-point SNR that any receiver can achieve. This is
better than the MMSE-DFE by .4 dB (or by 1.7 dB with error propagation/precoding eects
included). On more severe-ISI channels (1 + .9D
1
is easy for computation by hand but is
not really a very dicult transmission channel), the dierence between multitone and the
MMSE-DFE can be very large. Chapter 5 shows how to correct the MMSE-DFEs transmit
lter and symbol rate in some situations to improve the MMSE-DFEs detection SNR to the
level of multitone, but this is not always possible.
4.3.2 Computing Water-Filling for MA loading
MA water-lling will have a constant K
ma
that satises
c
n
= K
ma


g
n
(4.58)
for each used subchannel. The bit-rate constraint then becomes
b =
1
2
N

n=1
log
2
_
1 +
c
n
g
n

_
(4.59)
308
Figure 4.8: Water-lling energy distribution for 1 + .9D
1
channel with N = 8.
=
1
2
N

n=1
log
2
_
K
ma
g
n

_
(4.60)
=
1
2
log
2
_
N

n=1
K
ma
g
n

_
(4.61)
2
2b
=
_
K
ma

_
N

n=1
g
n
(4.62)
Thus, the constant for MA loading is
K
ma
=
_
2
2b

n=1
g
n
_
1/N

. (4.63)
MA water-ll algorithm
1. Order g
n
largest to smallest, n = 1,..., N, and set i = N.
2. Compute K
ma
=
_
2
2b

i
n=1
gn
_
1/i
(this step requires only one multiply for i N 1).
3. Test subchannel energy c
i
= K
ma

gi
< 0? If yes, then i i 1, go to step 2, otherwise continue
4. Compute solution with n = 1, ..., N

= i
c
n
= K
ma


g
n
(4.64)
b
n
=
1
2
log
2
_
K
ma
g
n

_
(4.65)
309
310
5. Compute margin
max
=
N

L
x

n=1
Ln
.
Figure 4.9 illustrates margin-adaptive water-lling. Figure 4.9 recognizes that the numerator and denom-
inator terms in Equation (4.63) could be quite large while the quotient remains reasonable, so logarithms
are used to reduce the precisional requirements inside the loop.
EXAMPLE 4.3.2 Returning to the 1 +.9D
1
example, the MA water-ll iterations for a
gap of = 8.8 dB for a P
e
= 10
6
are:
1. iteration i = 5 subchannels
K
ma
= 10
.88

_
2
16
19.945 17.032
2
10
2
2.968
2
.05525
_
.125
= 6.3228 (4.66)
c
4
= 6.3228 10
.88
/.0552 = 131 < 0 i i 4 (4.67)
2. step i = 4 subchannels
K
ma
= 10
.88

_
2
16
19.845 17.032
2
10
2
2.968
2
_
1/7
= 4.0727 (4.68)

c
3
= 4.0727 10
.88
/2.968 = 1.5169 > 0 c
3
= 3.0337 (4.69)
c
2
= 6.6283, c
1
= 7.2547 , c
0
= 3.6827.

max
=
8
3.0337 + 6.6283 + 7.2547 + 3.6827
= (1/2.57) = 4.1 dB (4.70)
The negative margin indicates what we already knew, this channel needs coding or more
energy (at least 4.1 dB more) to achieve a P
e
= 10
6
, even with multitone. However,
gap/capacity arguments assume that a code with nearly 9 dB exists. The amount of addi-
tional coding necessary is a minimum with multi-tone (when N ).
4.3.3 Loading with Discrete Information Units
Water-lling algorithms have bit distributions where b
n
can be any real number. Realization of bit dis-
tributions with non-integer values can be dicult. Alternative loading algorithms allow the computation
of bit distributions that are more amenable to implementation with a nite granularity:
Denition 4.3.3 (Information Granularity) The granularity of a multichannel trans-
mission system is the smallest incremental unit of information that can be transmitted, .
The number of bits on any subchannel is then given by
b
n
= B
n
, (4.71)
where B
n
0 is an integer.
Typically, , takes values such as .25, .5, .75, 1, or 2 bit(s) with fractional bit constellations being
discussed in both Chapters 10 and 11.
There are two approaches to nite granularity. The rst computes a bit distribution by rounding of
approximate water-ll results, and is called Chows algorithm. The second is based on greedy methods
in mathematics and rst suggested by Hughes-Hartog; however, considerable signicant improvements,
a complete and mathematically veriable framework, as well as an approach that circumvents many
drawbacks in the original Hughes-Hartog methods has recently been developed independently by Jorge
Campello de Souza
11
and Howard Levin and are now known as Levin-Campello (LC) Algorithms.
These algorithms solve the nite- loading problem directly, which is not a water-lling solution. This
section rst describes Chows algorithm, and then proceeds to the LC methods.
11
379C students take heart, it was his course project!
311
Chows Algorithm
Peter Chow studied a number of transmission media with severe intersymbol interference, in particular
telephone line digital transmission known generally as Digital Subscriber Line (DSL), in his 1993 Stanford
Dissertation. Chow was able to verify that an on/o energy distribution, as long as it used the same
or nearly the same transmission band as water-lling, exhibits negligible loss with respect to the exact
water-lling shape. The reader is cautioned not to misinterpret this result as saying that at energy
distribution is as good as water-lling, but rather on/o; where on means at energy distribution in
the same used bands as water lling and o means zero energy distribution where water-lling would
be zero.
Chows algorithms take as much or more computation than water-ll methods, rasing the issue of
their utility even if close to optimum. The reason for the use of on/o energy distributions is that
practical systems usually have a constraint on the maximum power spectral density along with a total
energy/power constraint. Such power-spectral-density constraints are typically at over the region of
bandwidth considered for transmission. Such on/o energy distribution comforts those concerned with
the eect of emissions from a transmission channel on other transmission systems or electronic devices.
Using this observation, Chow was able to devise algorithms that approximate water-lling with nite
granularity in the bit loading b
n
and equal energy c
n
=
N
N

c
x
. Both the RA and MA problems can be
solved with Chows algorithms, which in both cases begin with Chows Primer below:
Chows on/o Loading Primer:
1. Re-order the g
n
so that g
1
is the largest
2. Set b
temp
(N + 1) = 0, i = N. (b
temp
(i) and i are the tentative total bits and number of used
subchannels, respectively.)
3. Let c
n
= N

c
x
/i for n = 1, ..., i, and then SNR
n
= g
n
c
n
. (When power-spectral density limits
apply - the energy level is set at the psd limit and reset i to i + 1 or set all subchannels at the
limit ifi = 0, making one last pass through steps 4 and 5).
4. Compute b
temp
(i) =

i
n=1
1
2
log
2
(1 + SNR
n
/).
5. If b
temp
(i) < b
temp
(i + 1), then keep the bit distribution corresponding to b
temp
(i + 1) - otherwise
i i 1 and go to step 3.
The bit distribution produced by Chows Primer will likely contain many subchannels with non-
integer numbers of bits.
EXAMPLE 4.3.3 (1 +.9D
1
) The algorithm above can be executed for our example with
= 0 dB. The 3 passes necessary for solution are:
Pass i=4

c
n
= 1, n = 0, 1, 2, 3, 4.
SNR
0
=
1.9
2
.181
= 19.9448 (4.72)
SNR
1
=
1.7558
2
.181
= 17.032 (4.73)
SNR
2
=
1.3454
2
.181
= 10.000 (4.74)
SNR
3
=
.7392
2
.181
= 2.968 (4.75)
SNR
4
=
.1
2
.181
= .0552 (4.76)
b
temp
(4) = .5log
2
(1+19.9448)+log
2
(1+17.032)+log
2
(1+10)+log
2
(1+2.968)+.5log
2
(1.0552) = 11.8533 .
(4.77)
312
Pass i=3

c
n
= 8/7, n = 0, 1, 2, 3.
SNR
0
=
8 1.9
2
7 .181
= 22.794 (4.78)
SNR
1
=
8 1.7558
2
7 .181
= 19.4651 (4.79)
SNR
2
=
8 1.3454
2
7 .181
= 11.4286 (4.80)
SNR
3
=
8 .7392
2
7 .181
= 3.3920 (4.81)
b
temp
(3) = .5 log
2
(23.794) + log
2
(20.4651) + log
2
(12.42865) +log
2
(4.3920) = 12.4118 .
(4.82)
Pass i=2

c
n
= 8/5, n = 0, 1, 2.
SNR
0
=
8 1.9
2
5 .181
= 31.9116 (4.83)
SNR
1
=
8 1.7558
2
5 .181
= 27.2512 (4.84)
SNR
2
=
8 1.3454
2
5 .181
= 16.000 (4.85)
b
temp
(2) = .5 log
2
(32.9116) + log
2
(28.2512) + log
2
(17) = 11.428 < 12.4118 . (4.86)
STOP and use i = 3 result.
The algorithmhas terminated with the computation of b = 12.4118 or

b = 1.54 bits/dimension,
the known capacity of this channel for = 0 to almost 3 digit accuracy, conrming on this
example that on/o energy is essentially as good as water-lling energy shaping. The key
in both Chow and Water-lling methods is that the tone at Nyquist is not used - its use
corresponds to about a 10% loss in data rate on this example.
To illustrate that one could reorder the tones from smallest to largest in terms of channel
SNR, the 5 passes necessary for solution follow:
Pass 1 SNR = 160
b
temp
(1) = .5 log
2
(1 + 160/1) = 3.7

b
temp
= b
temp
/8 = .46
Pass 2 SNR
0
= 53.2
SNR
1
= 45.5
b
temp
(2) = 2.9 + 2(2.75) = 8.4

b
temp
= b
temp
/8 = 1.1
Pass 3 SNR
0
= 31.9
SNR
1
= 27.3
SNR
2
= 16.0
b
temp
(3) = 2.5 + 2(2.4) + 2(2.0) = 11.3

b
temp
= b
temp
/8 = 1.42
Pass 4 SNR
0
= 22.8
SNR
1
= 19.5
SNR
2
= 11.4
SNR
3
= 3.4
b
temp
(4) = 2.3 + 2(2.1) + 2(1.8) + 2(1.05) = 12.4

b = b
temp
/8 = 1.55
313
Figure 4.10: Illustration of sawtooth energy distribution characteristic of Chows Algorithm.
Pass 5 SNR
0
= 20.0
SNR
1
= 17.1
SNR
2
= 10.0
SNR
3
= 3.0
SNR
4
= .055
B = 2.2 + 2(2.1) + 2(1.7) + 2(1) = 11.8

b
temp
= b
temp
/8 = 1.48
Again, the correct solution was obtained with a larger number of steps. Designers sometimes
run the algorithm backwards if they expect less than half the subchannels to be used. Then,
computation is saved.
Chows RA Algorithm To solve the RA problem, the following 3 steps adjust the result of the
Chows Primer (with steps 6 and 7 being executed for all used subchannels):
6. If
bn

bn

| < .5, then round b

n
to the largest integer multiple of contained in b
n
,
B
n
= b
n
/|. Reduce energy on this subchannel so that the performance is the same
as on all other subchannels that is scale energy by
2
2

Bn
1
2
2

b
old,n
1
so that the new integer
value then satises b
n
=
1
2
log
2
(1 + SNR
n
/).
7. If
bn

bn

| > .5, then round to the next largest integer, B

n
= b
n
/|. Increase energy,
c
n
, by the factor
2
2

Bn
1
2
2

b
old,n
1
so that performance is the same as other subchannels and
then b
n
=
1
2
log
2
(1 + SNR
n
/).
8. Compute total energy c =

n
c
n
and then scale all subchannels by the factor N

c
x
/c.
This procedure produces a characteristic sawtooth energy distribution that may deviate from a at
distribution by as much as
6

N
dB, where

N is the dimensionality of the subchannel. Figure 4.10
314
illustrates the sawtooth energy produced by Chows algorithm for the SNR(f) curve simultaneously
displayed. The discontinuities are points at which the number of bits on a subchannel changes by the
smallest information unit . An increase in b
n
by means more energy is necessary, while a decrease
in b
n
by means less energy is necessary. The curve however, on rough average, approximates a at
straight line where energy is on.
EXAMPLE 4.3.4 Returning to the earlier 1+.9D
1
example, rounding the bit distribution
with = 1 bit leads to
2 2.28 = b
0
= .5 log
2
(23.794) (4.87)
4 4.355 = b
1
= log
2
(20.4651) (4.88)
4 3.6356 = b
2
= log
2
(12.4286) (4.89)
2 2.13 = b
3
= log
2
(4.3920) (4.90)
The maximum data rate is then 12 bits per symbol or 1.5 bits/dimension. The new energy
distribution is
.7520 c
0
=
8
7

2
22
1
2
22.28
1
(4.91)
1.7614 c
1
=
16
7

2
4
1
2
4.355
1
(4.92)
3.000 c
2
=
16
7

2
4
1
2
3.6356
1
(4.93)
2.0215 c
3
=
16
7

2
2
1
2
2.1349
1
(4.94)
The total energy is then c
x
= 7.5349, which means that the distribution of energies may all
be increased by the factor 8/7.5349 so that the total energy budget is maintained. The loss
in performance caused by quantization in this example is 10 log
10
_
(2
3
1)/(2
3.1
1

10
log
10
(8/7.5349) or -.08 dB, pretty small!
The examples use of = 1 for the complex subchannels was essentially equivalent to a number of
bits/dimension granularity of 1/2 bit. The granularity thus need not be the same on all subchannels as
long as each subchannel is handled consistently within the constraints of the granularity of its specic
encoding scheme.
EXAMPLE 4.3.5 (DSL rate adaption) A number of standards (ANSI T1.413, ITU G.992.1
(G.dmt) and G.992.2 (G.lite) use a form of multitone described more fully in Section
4.5. These sometimes are all generally known as Digital Subscriber Lines (DSLs) and allow
high-speed digital transmission on phone lines with one modem in the telephone company
central oce and the other at the customer. These all have a mode of operation for internet
access where rate-adaptive loading is used. The symbol rate is eectively 4000 Hz in these
methods, meaning the tone width is eectively 4000 Hz. The granularity in the standards
is = 1 bit per two-dimensional subchannel. The T1.413 and G.992.1 standards use 256
tones in one direction of transmission known as downstream (G.992.2 uses 128) and 32-
64 tones upstream, the asymmetry being a match to usually asymmetric le transfers and
acknowledgment packet-length ratios in IP trac for internet use.
Thus the data rate can be any multiple of 4kbps with RA loading. Usually in these systems,
the gap is intentionally increased by 6 dB - this is equivalent to forcing 6 dB of margin when
RA loading completes. With nominally 100-250 tones being used and average numbers of bits
per tone of 2-9 bits, data rates from 1 to 9 Mbps are possible. Because of a byte-oriented FEC
code, the RA output may be further rounded down on the subchannels otherwise rounded
up until the total number of bits is a multiple of 8 (so to ensure an integer number of bytes).
315
Chows MA Loading algorithm Chows MA algorithm rst computes a margin, from the data rate
given by the Chow Primer, with respect to the desired xed rate. It then recomputes the bit distribution
including that margin, which will make the number of bits exactly equal to the desired number. However,
the number of bits will still not be integer multiples of in general. The resultant bit distribution is
then rounded as in the RA algorithm, but the sum of the number of bits is monitored and rounding is
adjusted to make the sum exactly equal to the desired number of bits per symbol.
The steps that follow the Chow Primer to complete the MA solution are:
6. Compute
SNR
m,u
=
_
_
_
_
i

n=1
_
1 +
SNR
n

_
_
1/i
1
_
_
_
= 2
2

bmax
1 , (4.95)
7. Compute the tentative margin

temp,m
=
SNRm,u

2
2

b
1
n = 1, ..., N

, (4.96)
8. Compute the updated bit distribution
B
temp,n
=
1
2
log
2
_
1 +
SNR
n

temp,n
_
n = 1, ..., N

, (4.97)
9. Round B
temp,n
to the nearest integer (if more than one beta is being used for dierent subchannels,
then use the correct value for each subchannel) and recompute energy as
c
temp,n
=

_
2
Bn
1
_
g
n
n = 1, ..., i , (4.98)
10. Compute
B
temp
=
N

n=1
B
temp,n
; b
temp
= B
temp
(4.99)
and check to see if b
temp
= b, the desired xed number of bits per symbol. If not, then select
subchannels that were rounded (perhaps by choosing those close to one-half information granularity
unit) and round the other way until the data rate is correct, recomputing energies as in step 9.
11. Compute the energy scaling factor
f
e
=
N

c
x

n=1
c
n
(4.100)
and scale all subchannel energies to
c
n
f
e
c
n
, (4.101)
and the maximum margin becomes

max
=
temp,max
f
e
. (4.102)
EXAMPLE 4.3.6 (MA loading for 1 +.9D
1
channel) The rst step in Chows MA
procedure nds SNR
m,u
=8.8 dB and the margin with respect to transmission at

b = 1 is
then 10
.88
/(2
2
1) = 4.0 dB. The updated bit distribution computed from step 8 of Chows
MA loading is
b
0
=
1
2
log
2
_
1 +
(8/7) 19.948
10
.4
_
= 1.7 (4.103)
b
1
= log
2
_
1 +
(8/7) 17.032
10
.4
_
= 3.1 (4.104)
316
b
2
= log
2
_
1 +
(8/7) 10
10
.4
_
= 2.47 (4.105)
b
3
= log
2
_
1 +
(8/7) 2.968
10
.4
_
= 1.2 . (4.106)
For = 1, this bit distribution rounds to b
0
= 2, b
1
= 3, b
2
= 2, and b
3
= 1. The number of
bits sums to 8 exactly, and so the procedure concludes. The energy distribution then changes
to
c
0
1.7939 =
8
7

2
4
1
2
1.7
1
(4.107)
c
1
2.1124 =
16
7

2
3
1
2
3.1
1
(4.108)
c
2
1.5102 =
16
7

2
2
1
2
2.4
1
(4.109)
c
3
1.7618 =
16
7

2
1
1
2
1.2
1
(4.110)
Total energy is then 7.1783 and all subchannels have an additional margin of of 8/7.1783
leading to an additional margin of about .5 dB, so that the total is then 4.5 dB, when
a system with powerful code (say gap is 0 dB at P
e
= 10
6
) is used to transmit on the
1 +.9D
1
channel at

b = 1 and the same P
e
as the gap used in loading.
Had = .5 for these subchannels, then the bit distribution would have been b
0
= 1.5, b
1
= 3,
b
2
= 2.5, and b
3
= 1, which still adds to 8. However, had = .5 only for subchannel one,
then the rounding would have produced b = 8.5 and so subchannel 0 would then need to be
rounded down to 2 bits anyway.
Optimum Discrete Loading Algorithms
Optimum discrete loading algorithms recognize the discrete loading problem as an instance of what is
known as greedy optimization in mathematics. The basic concept is that each increment of additional
information to be transported by a multi-channel transmission system is placed on the subchannel that
would require the least incremental energy for its transport. Such algorithms are optimum for loading
when the information granularity is the same for all subchannels, which is usually the case.
12
Several denitions are necessary to formalize discrete loading.
Denition 4.3.4 (Bit distribution vector) The bit distribution vector for a set of
parallel subchannels that in aggregate carry b total bits of information is given by
b = [b
1
b
2
... b
N
] . (4.111)
The sum of the elements in b is clearly equal to the total number of bits transmitted, b.
Discrete loading algorithms can use any monotonically increasing relation between transmit symbol
energy and the number of bits transmitted on any subchannel. This function can be dierent for each
subchannel, and there need not be a constant gap used. In general, the concept of incremental energy
is important to discrete loading:
Denition 4.3.5 (Incremental Energy and Symbol Energy) The symbol energy c
n
for
an integer number of information units b
n
= B
n
on each subchannel can be notationally
generalized to the energy function
c
n
c
n
(b
n
) , (4.112)
12
The 1+.9D
1
channel example has a PAM subchannel at DC for which this chapter has previously used a granularity of
1 bit per dimension while the other subchannels were QAM and used granularity of 1 bit per two dimensions, or equivalently
1/2 bit per dimension this anomaly rarely occurs in practice because DC is almost never passed in transmission channels
and complex-baseband channels use a two-dimensional QAM subchannel at the baseband-equivalent of DC.
317
where the symbol energys dependence on the number of information units transmitted, b
n
, is
explicitly shown. The incremental energy to transmit b
n
information units on a subchannel
is the amount of additional energy required to send the B
th
n
information unit with respect to
the (B
n
1)
th
information unit (that is, one more unit of ). The incremental energy is
then
e
n
(b
n
)

= c
n
(b
n
) c
n
(b
n
) . (4.113)
EXAMPLE 4.3.7 (Incremental Energies) As an example, uncoded SQ QAM with =
1 bit-per-two-dimensions, and minimuminput distance between points on subchannel n being
d
n
, has energy function
c(b
n
) =
_
2
bn
1
6
d
2
n
b
n
even
2
bn+1
1
12
d
2
n
b
n
odd
. (4.114)
The incremental energy is then
e
n
(b
n
) =
_
2
bn
1
12
d
2
n
b
n
even
2
bn
+1
12
d
2
n
b
n
odd
. (4.115)
For large numbers of bits, the incremental energy for SQ QAM is approximately twice the
amount of energy needed for the previous constellation, or 3 dB per bit as is well known to
the reader of Chapter 1. A coded system with ,= 1 might have a more complicated exact
expression that perhaps is most easily represented by tabulation in general. For instance,
the table for the uncoded SQ QAM is
b
n
0 1 2 3 4 5 6 7 8
c
n
(b
n
) 0
d
2
n
4
d
2
n
2
5d
2
n
4
5d
2
n
2
21d
2
n
4
21d
2
n
2
85d
2
n
4
85d
2
n
2
e
n
(b
n
) 0
d
2
n
4
d
2
n
4
3d
2
n
4
5d
2
n
2
11d
2
n
4
21d
2
n
4
43d
2
n
4
85d
2
n
4
d in this table is selected for a given desired probability of error (and the gap approximation thus avoided
since it is an approximation.
13
Alternatively, an energy function for QAM with = 1 could also be dened via the gap approximation
as
c
n
(b
n
) = 2

g
n
_
2
bn
1
_
. (4.116)
The incremental energy is then
e
n
(b
n
) =

g
n
_
2 2
bn
2
bn
_
(4.117)
=

g
n
2
bn
(2 1) (4.118)
=

g
n
2
bn
(4.119)
= 2 e
n
(b
n
1) , (4.120)
which is again approximately 3 dB per bit. If = .25, then the gap-based incremental QAM energy
formula would be
e
n
(b
n
) =

g
n
2
bn+.75
_
2 2
.75
_
, (4.121)
and in general for gap-based QAM
e
n
(b
n
) =

g
n
2
bn+1
_
2 2
1
_
. (4.122)
13
The SNRm.u in this case of no gap is obtained by calculated the usual formula in (4.17) with = 1.
318
Once an encoding system has been selected for the given xed and for each of the subchannels in
a multichannel transmission system, then the incremental energy for each subchannel can be tabulated
for use in loading algorithms.
Clearly, there are many possible bit distributions that all sum to the same total b. A highly desirable
property of a distribution is eciency:
Denition 4.3.6 (Eciency of a Bit Distribution) A bit distribution vector b is said
to be ecient for a given granularity if
max
n
[e
n
(b
n
)] min
m
[e
m
(b
m
+)] . (4.123)
Eciency means that there is no movement of a bit from one subchannel to another that reduces the
symbol energy. An ecient bit distribution clearly solves the MA loading problem for the given total
number of bits b. An ecient bit distribution also solves the RA loading problem for the energy total
that is the sum of the energies c
n
(b
n
).
Levin and Campello have independently formalized an iterative algorithm that will translate any bit
distribution into an ecient bit distribution:
Levin-Campello (LC) Ecientizing (EF) Algorithm
1. m argmin
1iN
[e
i
(b
i
+)] ;
2. n arg max
1jN
[e
j
(b
j
)] ;
3. While e
m
(b
m
+ ) < e
n
(b
n
) do
(a) b
m
b
m
+
(b) b
n
b
n

(c) m arg min

1iN
[e
i
(b
i
+ )] ;
(d) n argmax
1jN
[e
j
(b
j
)] ;
By always replacing a bit distribution with another distribution that is closer to more ecient, and
exhaustively searching all single-information-unit changes at each step, the EF algorithm will produce
an ecient bit distribution when the energy function is monotonically increasing with information (as
always the case in practical systems). Truncation of a water-lling distribution produces an ecient
distribution.
EXAMPLE 4.3.8 (Return to 1 + .9D
1
for EF algorithm) By using the using the un-
coded gap approximation at P
e
= 10
6
, the PAM and SSB subchannels energy is given by
c
n
(b
n
) =
10
.88
g
n
_
2
2bn
1
_
(4.124)
and the QAM subchannels energies are given by
c
n
(b
n
) = 2
10
.88
g
n
_
2
bn
1
_
. (4.125)
The incremental energies e
n
(b
n
) then are:
n 0 1 2 3 4
e
n
(1) 1.14 .891 1.50 5.112 412.3
e
n
(2) 4.56 1.78 3.03 10.2
e
n
(3) 18.3 3.56 6.07 20.4
e
n
(4) 7.13 12.1
e
n
(5) 14.2 24.3
319
If the initial condition were b = [0 5 0 2 1], the sequence of steps would be:
1. b = [0 5 0 2 1]
2. b = [1 5 0 2 0]
3. b = [1 4 1 2 0]
4. b = [1 4 2 1 0]
5. b = [2 3 2 1 0].
The EF algorithm retains the total number of bits at 8 and converges to the best solution
for that number of bits from any initial condition.
The Levin-Campello RA and MA loading algorithms both begin with the EF algorithm.
Levin-Campello RA Solution An additional concept of E-tightness is necessary for solution of
the RA problem:
Denition 4.3.7 (E-tightness) A bit distribution with granularity is said to be E-tight
if
0 N

c
x

N

n=1
E
n
(b
n
) min
1iN
[e
i
(b
i
+ )] . (4.126)
E-tightness implies that no additional unit of information can be carried without violation of the total
energy constraint.
To make a bit distribution E-tight, one uses the Levin-Campello E-tightening (ET) algorithm below:
1. Set S =

N
n=1
c
n
(b
n
);
2. WHILE
_
N

c
x
S < 0
_
or
_
N

c
x
S min
1iN
[e
i
(b
i
+)]
_
IF
_
N

c
x
S < 0
_
THEN
(a) n argmax
1iN
[e
i
(b
i
)] ;
(b) S S e
n
(b
n
)
(c) b
n
b
n

ELSE
(a) m arg min
1iN
[e
i
(b
i
+ )] ;
(b) S S +e
m
(b
m
+)
(c) b
m
b
m
+
The ET algorithm reduces the number of bits when the energy exceeds the limit. The ET algorithm
adds additional bits in the least energy-consumptive places when energy is suciently below the limit.
EXAMPLE 4.3.9 (ET algorithm on EF output in example above) If the initial bit
distribution vector for the above example were the EF solution b = [2 3 2 1 0], the sequence
of ET steps lead to:
1. b = [2 3 2 0 0] with c 16.49
2. b = [1 3 2 0 0] with c 11.92
3. b = [1 2 2 0 0] with c 8.36
4. b = [1 2 1 0 0] with c 5.32
Thus, the data rate is halved in this example for E-tightness and the resultant margin is
8/5.32 = 1.8dB.
If the ET algorithm is applied to an ecient distribution, the RA problem is solved:
320
The Levin-Campello RA Algorithm
1. Choose any b
2. Make b ecient with the EF algorithm.
3. E-tighten the resultant b with the ET algorithm.
The Levin-Campello MA Solution The dual of E-tightness for MA loading is B-tightness:
Denition 4.3.8 (B-tightness) A bit distribution b with granularity and total bits per
symbol b is said to be B-tight if
b =
N

n=1
b
n
. (4.127)
B-tightness simply states the correct number of bits is being transmitted.
A B-tightening (BT) algorithm is
1. Set

b =

N
n=1
b
n
2. WHILE

b ,= b
IF

b > b
(a) n argmax
1iN
[e
i
(b
i
)] ;
(b)

b

b
(c) b
n
b
n

ELSE
(a) m arg min
1iN
[e
i
(b
i
+ )] ;
(b)

b

b +
(c) b
m
b
m
+
EXAMPLE 4.3.10 (Bit tightening on 1 + .9D
1
channel) Starting froma distribution
of b = [0 0 0 0 0], the BT algorithm steps produce:
1. b = [0 0 0 0 0]
2. b = [0 1 0 0 0]
3. b = [1 1 0 0 0]
4. b = [1 1 1 0 0]
5. b = [1 2 1 0 0]
6. b = [1 2 2 0 0]
7. b = [1 3 2 0 0]
8. b = [2 3 2 0 0]
9. b = [2 3 2 1 0]
The margin for this distribution has previously been found as -4.3 dB because it is the same
distribution produced in the Chow method. The total energy is 21.6.
The BT algorithm reduces bit rate when it is too high in the place of most energy reduction per unit
of information and increases bit rate when it is too low in the place of least energy increase per unit of
information. If the BT algorithm is applied to an ecient distribution, the MA problem is solved.
321
The Levin-Campello MA Algorithm
1. Choose any b
2. Make b ecient with the EF algorithm.
3. B-tighten the resultant b with the BT algorithm.
Some transmission systems may use coding, which means that there could be redundant bits. Ide-
ally, one could compute the incremental energy table with the eect of redundancy included in the
incremental-energy entries. This can be quite dicult to compute. Generally more redundant bits
means higher gain but also allocation to increasingly more dicult subchannel positions, which at some
number of redundant bits becomes a coding loss.
4.3.4 Sorting and Run-time Issues
Campello in pursuit of the optimized discrete loading algorithm, studied further the issue of computa-
tional complexity in loading algorithms. Mathematically, a sort of the vector b can be denoted
b
sort
= Jb (4.128)
where J is a matrix with one and only one nonzero unit value in each and every row and column. For
instance, the N = 3 matrix
J =
_
_
0 0 1
1 0 0
0 1 0
_
_
(4.129)
takes b
3
and moves it to the rst position 3 1, moves the rst position to the second position 1 2
and nally places the second position last 2 3. To unsort,
b = J

b
sort
(4.130)
since J
1
= J

is also a sorting matrix.

Water-lling and Chows methods begin with a sort of the channel SNR quantities g
n
. Finding a
maximum in a set of numbers takes N1 comparisons and then subsequently nding the minimum takes
N 2 comparisons to order the g
n
. Continuing in such fashion would require N
2
/2 comparisons. Binary
sort algorithms instead select an element at random from the set to be ordered and then partition this
set into two subsets containing elements greater than and smaller than the rst selected element. The
process is then repeated recursively on each of the subsets with a minimumcomputation of approximately
Nlog
2
(N) and a maximum still at N
2
/2. The average is order Nlog
2
(N).
Campello noted that for water-lling and Chows algorithms, absolute ordering is not essential. The
important quantity is the position of the transition from on to o in energy. Rather than sort
the SNRs initially, a guess at the transition point can be made instead. Then, with N 1 additional
comparisons, the quantities g
n
can be sorted into two subsets greater than or less than or equal to the
transition point. The water-lling or Chows algorithm can then test the transition point subchannel for
negative energy. If it is negative, the set of larger g
n
s again has contains the correct transition point.
If it is positive, then the point is the transition point or it is in the set of lesser g
n
s. Worst-case-BUT-
VERY-LOW PROBABILITY, this procedure will nd the transition point with N
2
comparisons. On
the average, the amount of computation is order N operations.
The Levin-Campello EF algorithm routinely must also know the position bit (information unit)
presently absorbing the most energy and the position of an additional bit that would require the least
energy. With monotonic energy versus bits on each subchannel, this again requires a sort of the sub-
channels, leading again to average order N complexity. Campello further studied supersets of tones,
which he notes is an instance of the computer-scientists concept of heaps in data bases. Essentially
the heap contains incremental energies that are larger than the smallest but not necessarily otherwise
ordered. In searching the heaps, one needs only the minimum incremental energy in each of two heaps
to be compared. The smallest is selected for next additional loaded information unit, and then than
322
heap searched for the next smallest. There may be some order within the heap (like all bits on the same
tone grouped together), but not complete order.
Generally, the sort is not an enormous complexity for initializing the multi-channel modem as it
may be much less computation than other early signal-processing procedures. However, if the channel
changes, an optimized design might need to repeat the procedure. This is the topic of the next subsection.
4.3.5 Dynamic Loading
In some applications, the subchannel SNRs, g
n
, may change with time. Then, the bit and energy dis-
tribution may need corresponding change. Such variation in the bit and energy distribution is known as
dynamic loading, or more specically dynamic rate adaption (DRA) or dynamic margin adaption
(DMA), the later which was once more colloquially called bit-swapping, by this author that later
name is in much wider use than DMA.
It is conceivable that an entire loading algorithm could be run every time there is a change in the
bit and/or energy distribution for a channel. However, if the channel variation has been small, then
reloading might produce a bit distribution with only minor variation from the original. Generally, it
is more ecient to update loading dynamically than to reload. Usually, dynamic loading occurs through
the transmission of bit/energy-distribution-change information over a reverse-direction reliable low-bit-
rate channel. The entire bit distribution need not be re-communicated, rather compressed incremental
variation information is desirable to minimize the relative data rate of the reverse channel. Dynamic
loading is not possible without a reverse channel.
The LC algorithms lend themselves most readily to incremental variation. Specically, to continu-
ously solve the MA problem, the previous bit distribution b can be input to the EF algorithm as an
initial condition for the new table of incremental energies, e
n
(b
n
), which is simply scaled by g
n,old
/g
n,new
for any reasonable constellation. Steps 3(a) and 3(b) of the EF algorithm execute a swap whenever
it is more ecient energy-wise to move a bit from one subchannel to another. As long as a sucient
number of swaps can occur before further variation in the channel, then the EF algorithm will continue
to converge to the optimum nite-granularity solution. The incremental information is the location of
the two subchannels for the swap and which is to be decremented. Such information has very low bit
rate, even for large numbers of subchannels. The energy gain/loss on each of the two subchannels must
also be transferred (from which the new incremental energy added/deleted from swapping subchannels
can be computed) with use of the EF algorithm. The quantization and maintenance of the tables of g
n
at both transmitter and receiver should be implemented exactly when this algorithm is used - in other
words, the receiver should quantize a new g
n
before execution of the algorithm and this exact value
must be transmitted through the reverse channel and maintained identically at the transmitter to avoid
anomalies. Then, no gain information need be transmitted over the reverse channel. In practice, DMT
modems actually send the index of the tone to add or delete a bit (or both when data rate is maintained
in MA loading) along with not the g
n
, but the gain increase or decrease factor for the corresponding
tones. The increase or decrease factor is relative so the need for exact maintenance of nite-precision
quantiles is not necessary (any dierent then consequently being noted in the subsequent adaptive esti-
mation of g
n
in the receiver to be described shortly). Alternatively, the incremental energy calculation
for both loading and dynamic loading can be based on a suboptimum on/o distribution over a given
number of subchannels like Chows algorithm.
The instance of continuous bit-rate variation to correspond to channel change is rare (usually some
higher-level communications authority prefers to arbitrate when bit rate changes are allowed to occur,
rather than leave it to the channel). Thus, bit-swapping DMA is used even after a RA initial loading has
occured (with some extra margin in the form of an increased gap to guard against drop in capacity of
the channel with time). Periodically, a higher-level entity can allow reloading, which is what sometimes
is called DRA.
However, true DRA can occur through the execution of the ET algorithm. That algorithm will
change the number of bits total over the subchannels to conform to the best use of limited transmit
energy. The eciency of the resultant bit distribution will be maintained by the execution of the ET
algorithm. The incremental information passed through the reliable reverse channel would be the index
of the currently to-be-altered subchannel and whether a bit is to be added or deleted.
323
Figure 4.11: A multichannel normalizer for one subchannel, n.
Synchronization of dynamic loading
For maintenance of the channel, both transmitter and receiver need to know on which symbol a bit/energy
distribution change has occurred. Thus two synchronized symbol counters must be maintained, one in
the transmitter and one in the receiver. They can be circular counters as long as the period of each
is the same and longer than the worst-case time that it takes for a bit-swap, bit-add, or bit-delete to
occur. Upon receipt of a request for a change, a transmitter then acknowledges the change and returns a
counter symbol number on which it intends to execute the change. Then both transmitter and receiver
execute the change simultaneously on the symbol corresponding to the agreed counter value.
The multichannel normalizer and computing the new subchannel SNRs
Figure 4.11 illustrates a multi-channel normalizer for one subchannel. There is a normalizer for each
and every subchannel. The multi-channel normalizer has no direct impact on performance because both
signal and noise are scaled by the same value W
n
. When multitone transmission is used, W
n
is complex
and often called a Frequency EQualizer (FEQ). Nominally, the multi-channel normalizer allows a
single decision device to be time-shared for decoding all the subchannel outputs. Such normalization of
distance satises:
d = [W
n
H
n
d
n
[ , (4.131)
where d
n
is the minimum distance between points on the n
th
subchannel constellation on the transmitter
side. The normalizer forces all subchannels to have output distance d, independent of the number of
bits carried or signal to noise ratio. Clearly,
W
n
=
d
H
n
d
n
. (4.132)
However, since the subchannel may change its gain or SNR, the value for W
n
is adaptively computed
periodically (as often as every symbol is possible). The error,

U
n
between the input and output of
decision device shown in the enlarged normalizer box estimates the normalized noise (as long as the
decision is correct). The updated value for W
n
can be computed in a number of ways that average
several successive normalized errors, of which the two most popular are the zero-forcing algorithm
W
n,k+1
= W
n,k
+

U
n
X

n
(4.133)
which produces an unbiased estimate of the signal at the normalizer output and converges when
0 <
c
x
2
. (4.134)
324
Figure 4.12: Illustration of reverse channel in bit-swapping.
equivalently E[

U
n
/X
n
] contains only noise, or the more common MMSE algorithm
W
n,k+1
= W
n,k
+

U
n
Y

n
, (4.135)
which instead produces a biased estimate and an optimistic SNR estimate of (Chapter 3 reader will
recall SNR
n
+ 1). The ZF algorithm is preferred because there is no issue with noise enhancement here
on this one-tap normalizer. The factor is a positive gain constant for the adaptive algorithm selected
to balance time-variation versus noise averaging.
The channel gain is estimated by

H
n
=
d
d
n
W
n
. (4.136)
The mean square noise, assuming ergodicity at least over a reasonable time period, is estimated by
time-averaging the quantity [

U
n
[
2
,

2
n,k+1
= (1
t
)
2
n,k
+
t
[

U
n
[
2
, (4.137)
where
t
is another averaging constant and
0
t
< 2 . (4.138)
Since this noise was scaled by W
n
also, then the estimate of g
n
is then
g
n
=
[H
n
[
2
n
2
]Wn]
2
=
d
2
d
2
n

1

n
2
. (4.139)
Since the factor d
2
/d
2
n
is constant and the loading algorithm need only know the relative change in
incremental energy, that is the incremental energy table is scaled by
g
n,old
g
n,new
, (4.140)
the new incremental-energy entries can be estimated by simply scaling up or down the old values by the
relative increase in the normalizer noise.
4.3.6 (bit-swapping)
If the loading algorithm moves or swaps a bit from one tone to another, then the multi-channel
transmitter and receiver need to implement this bit-swap on the same symbol. A bit-swap is usually
implemented by the receiver sending a pair of tone indices n (tone where bit is deleted) and m (tone
where bit is added) through some reverse channel to the transmitter, as in Figure 4.12. There is often
also an associated change in transmitted energy on each tone:
G
i
=
c
i
(new)
c
i
(old)
for i = m, n . (4.141)
The energy for the bit-swap tones i = m or i = n can be recomputed by summing the new increments
in the table,
c
i
(new) =
g
i,old
g
i,new
bi

j=1
e
i,old
(j) . (4.142)
325
Typically this factor is near unit value and usually less than a 3 dB change for 2-dimensional subchannels
like those of MT (and less than 6 dB change for one-dimensional subchannels). The near-unity-value
occurs because tone ns factor of decrease in channel SNR
gn,old
gn,new
is about 2 and nearly the reciprocal of
the constellation-energy decrease when halving the constellation size on that tone n. Similarly, tone ms
increase in channel SNR by about 2 is nearly the reciprocal of the doubling of energy caused when tone
ms constellation is doubled in size in the swap. The nearly, but not exactly, unit gain factor G
i
should
also be conveyed through the reverse channel to the transmitter. Some implementations thus ignore this
factor if it is always close to one.
Because the two gain factors may produce a total energy of the transmitter that is not exactly equal
to the allowed energy limit, total energy may be reset also. The total energy will always decrease if the
loading algorithm is functioning properly. After execution of a swap, both the transmitter and receiver
would know by previous agreement if the transmitter plans to increase total energy (which will be slightly
reduced by a swap) back to the maximum allowed c
x
. If so, then all the entries in the incremental energy
table also could be scaled by this same factor to be accurate (but it will not cause any future swaps to
be any dierent if the table is not scaled). Furthermore, all the multi-channel normalizer coecients
need to be reduced by the inverse square-root of this factor.
When a swap is executed, the values for W
m
and W
n
and the corresponding noise estimates need
scaling by an additional factor: The new values for W
n
and
2
n
are
W
n
(new) W
n
(old)
1

G
n
,
2
n
(new)
2
n
(old)
1
G
n
, (4.143)
and similarly for tone index m. Nominally, d
n
(new) scales by the energy increase (decrease) multiplied
by about
_
1/2 (or by about

2). However, the exact value depends on the constellation details. A
list of the exact factor of increases (decreases) is stored by the receiver for each possible increment or
decrement to the constellation. If dierent subchannels use dierent constellation types, then a set of
lists will need to be maintained. The successive entries in the list are the ratios of distance of successively
larger (smaller) constellations when the energy is held constant.
Systems using the Chow type at energy assumption will simply assume that energy remains the
same and squared distance changes by a factor of 2, perhaps eliminating the book-keeping on how much
energy is transmitted on each tone (unless the number of tone changes).
The accuracy of channel estimation is important and discussed in Section 4.5.
4.3.7 Gain Swapping
In addition to any energy changes associated with a bit-swap, gain may be changed in the absence of a
bit-swap and is known as gain swapping. While MA bit-swapping will continue to move bits from
channels of higher energy cost to those of lower energy cost, there is still a barrier to moving a bit that
is the dierence between the cost to add a bit in the lowest cost position of the incremental-energy table
relative to the savings of deleting that same bit on the highest-cost position in that same table. In the
worst case, this cost can be equal to the cost of a unit of information. On a two-dimensional subchannel
with = 1, this cost could be 3 dB of energy for a particular subchannel. This means that a probability
of error of 10
6
on a particular subchannel could degrade by as much as roughly 3 orders of magnitude
before the swap occurred. A particularly problematic situation would be a loss of for instance 2.9 dB
that never grew larger and the swap never occurred. With a number of subchannels greater than 1000,
the eect would be negligible, but as N decreases, the performance of a single subchannel can more
dramatically eect the overall probability of error.
There are two solutions to this potential problem. The rst is to reduce , but this may be dicult.
The second is alter the transmit energy to increase transmit energy on one subchannel and place it on
another subchannel. Furthermore, the presence of a dramatically increased or increase noise (for instance
a noise caused by another system turning on or o) might lead to a completely dierent distribution of
energy when loading. Thus, the energy transmitted on a subchannel (or equivalently the relative factor
applied to such a subchannel) may need to change signicantly. Gain swapping (sending of the factor of
increase/decrease of energy on a subchannel is thus commonly used. Usually by convention, a value of
b
n
= 0 on a subchannel means either no energy is to be transmitted or a very small value instead of the
326
nominal level. If a gain-swap occurs, then the two columns of the incremental energy table needed to be
scaled by the respective factors and indeed a bit-swap may subsequently occur because of this change.
327
4.4 Channel Partitioning
Channel Partitioning methods divide a transmission channel into a set of parallel, and ideally in-
dependent, subchannels to which the loading algorithms of Section 4.3 can be applied. Multi-tone
transmission (MT) is an example of channel partitioning where each subchannel has a frequency index
and all subchannels are independent, as in Section 4.1 and Figure 4.2. The MT basis functions
n
(t)
form an orthonormal basis while also exhibiting the desirable property that the set of channel-output
functions h(t)
n
(t) remains orthogonal for any h(t). Reference to Chapter 8 nds that the MT
basis functions, when combined with continuous-time water-lling, and N are optimum. For a
nite N, the MT basis functions are not quite optimum basically because the channel shaping induced
by h(t) in each MT frequency band is not quite at in general. Furthermore, the MT basis functions
exist over an innite time interval, whereas practical implementation suggests restriction of attention to
a nite observation interval. Thus, MT makes a useful example for the introduction of multi-channel
modulation, but is not practical for channel partitioning.
This section introduces optimum basis functions for modulation based on a nite time interval of T of
channel inputs and outputs. Subsections 4.4.1 and 4.4.2 show that the optimum basis functions become
channel-dependent and are the eigenfunctions of a certain channel covariance function. There are two
ways to accommodate intersymbol interference caused by a channel impulse response. Theoretically,
Subsection 4.4.2 investigates a conceptual method for computing best possibilities. Subsection 4.4.4
instead investigates the elimination of the interference between subchannels with the often-encountered
use of guard periods.
4.4.1 Eigenfunction Transmission
A one-shot multi-channel modulated signal x(t) satises
x(t) =
N

n=1
x
n

n
(t) , (4.144)
while a succession of such transmissions satises
x(t) =

k
N

n=1
x
n,k

n
(t kT) . (4.145)
The convolution of a channel response h(t) with x(t) produces
h(t) x(t) =

k
N

n=1
x
n,k

n
(t kT) h(t) =

k
N

n=1
x
n,k
p
n
(t kT) , (4.146)
which is characterized by N pulse responses p
n
(t) , n = 1, ..., N. Following a development in Section 3.1,
the set of sampled outputs of the N matched lters p

(t) at all symbol instants kT forms a sucient

statistic or complete representation of the transmitted signal as shown in Figure 4.13. There are N
2
channel-ISI-describing functions
q
n,m
(t)

=
p
m
(t) p

n
(t)
|p
m
| |p
n
|
(4.147)
that characterize the N N square matrix Q(t). The quantity q
n,m
(t) is the matched-ltered channel
from subchannel m input (column index) to subchannel n output (row index). The system is now
a multiple-input/multiple-output transmission system, sometimes called a MIMO, with sampled-time
representation
Q
k
= Q(kT) . (4.148)
328
Generalization of the Nyquist Criterion
Theorem 4.4.1 (Generalized Nyquist Criterion (GNC)) A generalization of the Nyquist
Criterion of Chapter 3 for no ISI nor intra-symbol interference (now both are called ISI to-
gether) is
Q
k
= I
k
, (4.149)
that is the sampled MIMO channel matrix is the identity matrix.
proof: Since Q
k
= I
k
= 0 if k ,= 0, there is no interference from other symbols, and since
q
n,m
(t) =
n,m
= 0 if n ,= m there is also no interference between subchannels. The unit
value follows from the normalization of each basis function, q
n,n
(0) = 1. QED
The earlier Nyquist Theorem of Chapter 3 is a special case that occurs when Q
k
= q
k
=
k
. Thus,
to avoid ISI, the designer could choose basis functions so that (4.149) holds. In general, there are many
choices for such functions. For a sub-optimum set of basis functions that do not satisfy the Generalized
Nyquist Criterion, vector equalization may be used as discussed in Section 5.7. This section and Chapter
deal only with those that do satisfy the GNC or the GNCs discrete-time equivalents.
4.4.2 Choice of transmit basis
The channel impulse response forms a channel autocorrelation function according to
r(t) = h(t) h

(t) , (4.150)
which is presumed to be nonzero only over a nite interval (T
H
/2, T
H
/2). This channel autocorrelation
function denes a possibly countably innite-size set of orthonormal eigenfunctions,
n
(t), nonzero
only over the time interval [(T T
H
)/2, (T T
H
)/2), that satisfy the relation
14

n

n
(t) =
_
T/2
T/2
r(t )
n
()d n = 1, ..., , t [(T T
H
)/2, (T +T
H
)/2) , (4.151)
that is a function
n
(t) that when convolved with the matched-ltered channel over the interval [T/2, T/2)
reproduces itself, scaled by a constant
n
, called the eigenvalue. The set of eigenvalues is unique to the
channel autocorrelation r(t), while the eigenfunctions are not necessarily unique.
15
Each eigenfunction
represents a mode of the channel through which data may be transmitted independently of the other
modes with gain
n
. The restriction of the time interval to be equal to the symbol period allows intersym-
bol interference to be zero without further consideration. The restriction is not absolutely necessary, but
then requires the additional constraint that successive translations of the basis functions at the symbol
rate lead to a Q
k
that satises the generalized Nyquist criterion. This problem is very dicult to solve
when r(kT) ,=
k
. A number of mathematicians have addressed the problem when r(kT) =
k
, and the
theory is known as wavelets, lter banks, polyphase lters, or reconstruction lters. However,
the theory is dicult to apply when there is ISI. Subsection 4.4.5 briey discusses this concept. Section
5.7 solves the problem with inter-block-symbol overlap directly with multidimensional block equalizers.
Modal Modulation (MM) in Figure 4.13 uses the channel-dependent orthonormal set of eigen-
functions (the N with largest eigenvalues) as a basis for modulation,
x(t) =
N

n=1
x
n

n
(t) . (4.152)
14
Most developments of eigenfunctions dene the functions over the causal time interval [0, T T
H
) - we instead
noncausaly center this interval about the origin at [(T T
H
)/2, (T + T
H
)/2) in this theoretically abstract case so that
taking the limit as T produces a true two-sided convolution integral, which will then have eigenvalues as the Fourier
transform.
15
The eigenfunctions are generally dicult to compute (unless the symbol period is innite) and used here for theoretical
purposes. An approximation method for their computation is listed at the end of this subsection.
329
Figure 4.13: Modal Modulation: optimum channel partitioning.
330
The receiver signal y(t) in Figure 4.13 is then
y(t) =
N

n=1
x
n
[r(t)
n
(t)] + n(t) (4.153)
where n(t) is the matched-ltered noise with autocorrelation function
A0
2
r(t). There is no information
loss in this signal because it is the output of the matched lters for each of the transmit bases. Through
Equation (4.151), this signal y(t) is rewritten
y(t) =
N

n=1
(
n
x
n
)
n
(t) , (4.154)
which is a sum of weighted channel input samples,
n
x
n
, times the orthonormal basis functions. This
summation has the same form as the modulated waveforms studied in Chapter 1. Each symbol element
x
n
is replaced by
n
x
n
.
The MAP/ML receiver is thus also the same as in Chapter 1 and recovers
n
x
n
by processing each
lter output by a matched-lter/sampler for each of the basis functions, as shown in Figure 4.149. The
noise samples at the output of these matched lters are independent for dierent n because the basis
functions are orthogonal, or equivalently,
E[ n
i
n

j
] =
^
0
2
_
T/2
T/2
_
T/2
T/2
r(t s)
i
(t)

j
(s)dtds (4.155)
=
^
0
2
_
T/2
T/2

i
(t)

j
(t)dt (4.156)
=
i
^
0
2

ij
. (4.157)
Thus, the joint-probability density function p
y/x
factors into N independent component distributions,
and an ML/MAP detector is equivalent to an independent detector on each sample output. Thus, the
MM transmission system generates a set of parallel channels to which the results of Sections 4.2 and 4.3
can then be applied. The SNRs are SNR
n
= (
2
n
c
n
)/(
n
A0
2
) =
n
c
n
/
A0
2
.
The functions p
n
(t) dened by modal modulation clearly satisfy the generalized Nyquist criterion
because of the independence of the subchannels created and because they are zero outside the interval
[T/2, T/2), thus averting intersymbol interference.
Theorem 4.4.2 (Optimality of Modal Modulation) Given a linear additive Gaussian
noise channel with ISI, modal modulation produces the largest SNR
m,u
of any N-dimensional
set of orthogonal basis functions exist only on the interval t [(T T
H
)/2, (T T
H
)/2).
Proof: The MM transmission system of Figure 4.149 can be optimally detected by observing
each of the independent subchannel outputs individually. Its performance thus can not be
improved. Any set of functions nonzero over the interval [(T T
H
)/2, (T T
H
)/2) that
satises the generalized Nyquist criterion are necessarily the eigenfunctions. The set of
eigenvalues are are unique, and MM has selected eigenfunctions corresponding to the N
largest. A well-known property of the eigenvalues is that no other set of N has a larger
product. Then,
N

n=1
g
n
(4.158)
is a maximum. For any given set of energies c
n
, the product
N

n=1
_
1 +
c
n
g
n

_
(4.159)
331
Figure 4.14: Parallel Subchannels of Modal Modulation.
332
is also a maximum because the function 1 +
Lngn

is linear (convex) in g
n
. Then since this
function is maximized for any set of energies, it is certainly the maximum for the water-lling
set of energies that are known to maximize for any set of g
n
. Thus, the equivalent-channel
SNR is the highest for any gap, and modal modulation achieves the highest possible SNR of
any partitioning scheme on the interval [(T T
H
)/2, (T T
H
)/2). As T for xed
T
H
, then the length of the autocorrelation function for any practical and thus nite-length
channel becomes negligible, then modal modulation is also optimum over the entire time axis
without restriction on the time interval to be shorter than the symbol period. QED.
The above theorem essentially states that MM is optimum given observation of a nite time interval
[(T T
H
)/2, (T T
H
)/2). It is then natural to expect the following lemma that states that MM
converges to MT modulation in performance as both allow the number of dimensions N to go to innity
while also allowing the time interval to span (, ):
Theorem 4.4.3 (Convergence of MT to Optimum) Multitone Modulation converges to
Modal Modulation as both N and [T/2, T/2) (, ).
Proof: The set of eigenvalues of any autocorrelation function is unique. The set of eigen-
values essentially determine the performance of MM through the SNR
m,u
for that channel.
By simple substitution of
n
(t) = e

2n
T
t
into the dening equation of eigenvalues shows
that these exponential functions are a valid choice of eigenfunctions as the symbol period
becomes innite, [T/2, T/2) (, ), and the corresponding eigenvalues are the values
of R() at frequencies of the exponentials. Thus, the eigenvalues are the values of the R()
at each frequency. As N , these frequencies become innitely dense, and equal to those
of the MT system, which then has no ISI on any tone. The corresponding MT receiver is
optimum (ML/MAP). Thus, both systems have the same SNR
m,u
and both have optimum
receivers, and by Theorem 4.4.2. The multitone system is thus optimum for innite-length
symbol period and an innite number of tones. QED.
The implication of the above lemma is that for suciently large N and T, the designer can use either
the MM method or the MT method and rest assured the design is as good as can be achieved on the
linear ISI channel with Gaussian noise.
4.4.3 Limiting Results
The spectrum used by the basis functions of modal modulation thus converges also to the set of used
tones in the innite-dimensional MT system. The interval may be a set of continuous frequencies or
several such sets and is denoted by . The measure of is the total bandwidth used
[[ =
_

d . (4.160)
An optimum bandwidth

will then be that corresponding to the subchannels used in water-lling as

N . If N

such subchannels are used in MT, then

lim
N
N

N
2 = [

[ . (4.161)
The data rate is
lim
T
b
T
. (4.162)
Continuous frequency can then replace the frequency index n according to
= 2 lim
N
n
NT
, n = N/2, ...., N/2 , (4.163)
and the width of a tone becomes
d = lim
N
2
NT
(4.164)
333
Figure 4.15: Capacity frequency interval for 1 + .9D
1
channel.
If 1/T
t
is suciently large to be at least twice the highest frequency that could be conceived of for use
on any given band-limited channel, then

becomes the true optimum band for use on the continuous

channel. The two-sided power spectral density at frequency corresponding to
n
NT
then is
S
x
() = lim
N

c
n
. (4.165)
Note for innite time and frequency as used here, there is no need for complex baseband equivalents and
thus all dimensions are considered real. The power then becomes
P
x
= lim
N
1
N T

N/2

n=N/2

c
n
=
1
2

_

S
x
()d . (4.166)
The water-lling equation then has continuous equivalent

c
n
+

g
n
S
x
() +

g()
= a constant (4.167)
subject to the total power constraint in (4.166) , which has corresponding solution for S
x
( > 0 for all

and S
x
() = 0 at all other frequencies. The data rate then becomes
R =
1
2
_

1
2
log
2
_
1 +
S
x
() g()

_
d (4.168)
=
1
2
_

1
2
log
2
_
g()

_
d (4.169)
EXAMPLE 4.4.1 (1 +.9D channel) The channel with impulse response h(t) = sinc(t) +
.9sinc(t 1) has the same performance as the 1 + .9D
1
channel studied throughout this
book, if the transmit lter is
1

T
sinc(t/T). An system with optimized MT basis functions of
innite length would have an optimum bandwidth

= [W, W] for some W /T as in

Figure 4.15 Then, continuous water-lling with = 1 produces
P
x
=
_
W
W
_

.181
1.81 + 1.8 cos()
_
d
2
(4.170)
where W is implicitly in radians/second for this example. If P
x
= 1 with
A0
2
= .181, the
integral in (4.170) simplies to
=
_
W
0
_

.181
1.81 + 1.8 cos()
_
d (4.171)
=
t
W .181
_
2

1.81
2
1.8
2
arctan
_

1.81
2
1.8
2
1.81 + 1.8
tan
_
W
2
_
__
(4.172)
334
At the bandedge W,

t
=
.181
1.81 + 1.8 cos(W)
. (4.173)
leaving the following transcendental equation to solve by trial and error:
=
.181W
1.81 + 1.8 cos(W)
1.9053 arctan(.0526 tan(W/2)) (4.174)
W = .88 approximately solves (4.174), and the corresponding value of is = 1.33.
The highest data rate with 1/T = 1 is then
( =
2
2
_
.88
0
1
2
log
2
_
1.33
.181
(1.81 + 1.8 cos )
_
d (4.175)
=
1
2
_
.88
0
log
2
7.35d +
1
2
_
.88
0
log
2
(1.81 + 1.8 cos ) d (4.176)
= 1.266 + .284 (4.177)
1.55bits/second . (4.178)
This exceeds the 1 bit/T transmitted on this channel in the examples of Chapter 3. Later
sections of this chapter show that the SNR
m,u
for this channel becomes 8.8 dB for large
N, .4 dB better than the best MMSE-DFE, and actually 1.7 dB better than the precoded
MMSE-DFE. The MT system has no error propagation.
In computing data rates as in water-lling with > 0 dB, the designer needs to remember that the
gap concept is only accurate when

b 1. Often water-lling may include regions of bandwidth for which

b in at least those bands is less than 1. The gap approximation becomes increasingly accurate as the
gap gets smaller, and indeed is accurate at all

b when the gap is 0 dB, meaning capacity results are all
exact.
Periodic Channels
A channel with periodic r(t) = r(t + T) will always have eigenfunctions
n
(t) = e

2
T
nt
for the interval
[T/2, T/2). Periodic channels do not exist in practice, but designers often use extra bandwidth in the
design of a transmission problem to make the nite-duration channel appear as if it were periodic. In
this case, MT and MM would again be the same.
Computing the Eigenfunctions
The eigenfunctions can be dicult to compute in closed form for most channels. At a sampling rate
1/T
t
suciently high to capture any signicant channel frequencies, the eigenfunction equation can be
sampled over the time interval (T T
H
)/2 = LT
t
to (T T
H
)/2 = LT
t
to yield

n

_

n
(LT
t
)

n
(LT
t
+ T
t
)
.
.
.

n
(LT
t
)
_

_
= T
t

_
r(2LT
t
) ... r(0)
r(2LT
t
+ T
t
) ... r(T
t
)
.
.
.
.
.
.
.
.
.
r(0) ... r(2LT
t
)
_

_
_

n
(LT
t
)

n
(LT
t
+ T
t
)
.
.
.

n
(LT
t
)
_

_
(4.179)

n

n
= R
n
. (4.180)
This equation is equivalent to nding the eigenvectors and eigenvalues of the matrix R. The largest N
eigenvalue magnitudes [
n
[ select the corresponding eigenvectors as the sampled basis functions. These
basis functions may be interpolated to create eigenfunction approximations. Clearly, the smaller the T
t
,
the more accurate the approximation.
In implementation, Section 4.5 illustrates a yet better way to approximate eigenfunctions in discrete
time, which is known as vector coding.
335
Figure 4.16: Illustration of guard period.
Figure 4.17: Filters with multitone.
4.4.4 Time-Domain Packets and Partitioning
Overlap, Excess Bandwidth, and the Guard Period
A realistic transmission channel has most or all its nonzero impulse response conned to a nite time
interval T
H
. Thus, convolution of the (necessarily causal) channel impulse response with nonzero input
basis functions existing over [0, T) produces an output with duration longer than the symbol period
T
16
as in Figure 4.16. The rst T
H
seconds of any symbol are thus possibly corrupted by intersymbol
interference from the last symbol if that previous symbol had nonzero transmit energy over (T T
H
, T).
Thus, the receiver then usually ignores the rst T
H
seconds of any symbol. If T
H
<< T, then this
overhead penalty, or excess bandwidth penalty, is small and no equalization need be used. For a xed
channel and thus xed T
H
, as T , this excess bandwidth becomes negligible. This time period
where the receiver ignores channel outputs is often called a guard period in multichannel modulation.
For nite symbol period, the multichannel system then has an excess bandwidth given by the factor
= T
H
/(T T
H
).
4.4.5 Filters with Multitone
Multitone partitioning almost satises the Generalized Nyquist criterion with any impulse response h(t),
but requires an innite-length basis function (t) =
1

T
sinc
_
t
T
_
and is not physically realizable even
when N is nite. The orthogonality of subchannels is ensured because the Fourier transforms of the basis
function translates by f =
n
T
do not overlap and thus are trivially orthogonal at the linear time-invariant
channel output. ISI is nearly averted by the narrow bands that are nearly at even at channel output.
Thus, many designers have investigated the possibility of approximating the multitone functions with
16
This section will now begin to pursue realizable transmission systems and not theoretical abstractions, thus the basis
functions will necessarily have to be causal in the remainder of this section.
336
physically realizable basis functions, as generally depicted in Figure 4.17. There is no guard band in the
time domain, but instead an excess bandwidth in the frequency domain.
The basic idea is to avoid the use of a T
H
-second guard period in the time domain by instead
having very sharp basis functions that approximate (t) =
1

T
sinc
_
t
T
_
. One method might be to simply
truncate this basis function to say 100 symbol periods, thus introducing a delay of approximately
101T in realization (and introducing a high complexity), and of course necessarily another 101T in the
receiver. Contrast this with a delay of just T with the use of the guard period and one sees the method
is not attractive. However, shorter length approximations may suce. One method is to introduce
again the raised cosine functions of Chapter 3 with excess bandwidth of 10 to 20%. Sometimes this level
of excess bandwidth can be less for certain types of channels (or noise-equivalent channels) that have
sharp band-edges, notches, or narrow-band noises, all of which result in long impulse responses and thus
large T
h
in the guard band of Section 4.4.4. Other functions may also achieve the same time of low
side bands for the Fourier transform of the basis function. Of course, any such realization requires in
reality some type of over-sampled digital signal processing. Section ??, Subsection 4.6.8 studies discrete
realizations of a few methods that have been proposed to date.
337
Figure 4.18: Digital realization of channel partitioning.
4.5 Discrete-time channel partitioning
While MM restricts attention to a nite time interval for construction of the transmit basis functions,
these functions remain continuous time and a function of the channel. Such functions would be dicult
to implement exactly in practice. Discrete-time channel partitioning partitions a discrete-time
description of the channel. This description is a linear matrix relationship between a nite number
of output samples (presumably sampled at a rate 1/T
t
exceeding twice the highest frequency to be
transmitted) and a corresponding nite set of channel input samples at the same rate that constitute
the transmitted symbol. Such a description can never be exact in practice, but with a sucient number
of input and output samples included, this description can be very close to the exact action of the ISI
channel. Figure 4.18 illustrates general N-dimensional discrete-time channel partitioning. Discrete-time
basis vectors, m
n
replace the continuous basis functions of MM or MT. These basis vectors have a
nite length of N + samples over a duration of time T. The sampled pulse response, p(kT
t
), of the
channel is assumed to span a time interval of less than + 1 sample periods, where ( + 1)T
t
= T
H
.
Each basis vector, m
n
, multiplies a sub-symbol element X
n
before being summed to form the transmit
symbol vector x.
17
Then, x is output through a digital-to-analog converter. Some transmit ltering
of the output DAC occurs (typically analog lowpass anti-aliasing) before the signals pass through the
ISI channel. The combined post-DAC ltering, channel impulse response, and pre-ADC ltering in the
receiver form a pulse response p(t). The sampling rate 1/T
t
= (N+)/T is higher than twice the highest
frequency that the designer hopes to use for the transmitted signals. The modulator attempts to use
basis vectors, m
n
, that will remain orthogonal after undergoing the dispersive eect of the ISI channel,
just as orthogonality was maintained with MM.
The receiver low-pass lters with gain 1/

T
t
(that is also included in the pulse response) to maintain
noise per dimension at
A0
2
), and then samples at rate 1/T
t
the received modulated signal. The matched
lters are discrete-time and also nite length. The matched lters are only complex if the entire set of
modulated signals had been passband modulated by an external passband modulator that is not shown
in Figure 4.18, and would in this case correspond to a complex baseband-equivalent channel. There are
N + input samples that lead to N matched-lter output samples, with samples lost to the guard
period to avoid intersymbol interference. The samples can be real or complex, which in the latter case
17
Xn (upper case) is used to avoid confusion with x
k
, a transmitted sample that is the direct input to the discrete-time
channel.
338
tacitly will imply a doubling of the number of real dimensions.
4.5.1 Optimal Partitioning Vectors - Vector Coding
The last N ADC channel-output symbol samples in Figure 4.18 have vector representation, with time
indexed within a symbol from k = to N 1,
_

_
y
N1
y
N2
.
.
.
y
0
_

_
=
_

_
p
0
p
1
... p

0 ... 0
0 p
0
.
.
. p
1
p

.
.
. 0
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 0
0 ... 0 p
0
p
1
... p

_
_

_
x
N1
.
.
.
x
0
x
1
.
.
.
x

_
+
_

_
n
N1
n
N1
.
.
.
n
0
_

_
, (4.181)
or more compactly,
y = Px + n . (4.182)
The N (N + ) matrix P has a singular value decomposition
P = F
_

.
.
. 0
N,
_
M

, (4.183)
where F is a N N unitary (FF

= F

F = I) matrix, M is a (N + ) (N + ) unitary (MM

=
M

M = I)matrix, and 0
N,
is an N matrix of zeros. is an N N diagonal matrix with singular
values
n
, n = 1, ..., N along the diagonal. The vector x is a data symbol vector. The notational use
of P for the channel matrix suggests that any anti-alias analog lters at transmitter and receiver have
been convolved with the channel impulse response h(t) and included in the discrete-time response of the
matrix channel.
Vector Coding creates a set of N parallel independent channels by using the transmit basis vectors,
m, that are the rst N columns of M - in other words, the transmit vector x is obtained from the N
vector components X
n
, n = 1, ..., N according to
x = M
_

_
X
0
.
.
.
0
_

_
= [m
N1
m
N2
... m
1
m
0
... m

]
_

_
X
N1
X
N2
.
.
.
X
0
0
.
.
.
0
_

_
=
N1

n=0
X
n
m
n
. (4.184)
4.5.2 Creating the set of Parallel Channels for Vector Coding
The last columns of M that do not aect P because they are multiplied by zero in (4.184), these
columns cannot contribute to the channel output and can be ignored in implementation. The corre-
sponding vector-coding receiver uses discrete matched lters as the rows of F

, forming
Y = F

y =
_
_
f

N1
y
...
f

0
y
_
_
. (4.185)
The mathematical description of the resultant N parallel channels is
Y = X +N , (4.186)
or, for each independent channel or entry in Y ,
Y
n
=
n
X
n
+ N
n
. (4.187)
339
Since F is unitary, the noise vector N is also additive white Gaussian with variance per real dimension
identical to that of n, or
A0
2
.
Theorem 4.5.1 (Optimality of Vector Coding) Vector Coding has the maximum SNR
m,u
for any discrete channel partitioning.
Proof: The proof of this theorem is deferred to Chapter 5 (Lemma 5.4.1) where some not-
yet-introduced concepts will make it trivial.
The optimality of VC is only strictly true when capacity-achieving codes with Gaussian distri-
butions are used on each of the subchannels. The restriction to independent symbols, requiring the
receiver to ignore the rst samples of any symbol also restricts the optimality. If this restriction were
relaxed or omitted, then it is possible that a better design exists. For N >> , such a potentially better
method would oer only small improvement, so only the case of > .1N would be of interest in further
investigation of this restriction (see Section 5.7).
Comment on correlated noise:
In the discrete-time transmission system of Figure 4.18, the noise is unlikely to be white. Section 1.7
previously indicated how to convert any channel into an equivalent white-noise channel, which involved
receiver preprocessing by a canonical noise-whitening lter and the equivalent channel frequency response
becoming the ratio of the channel transfer function to the square-root of the power-spectral density of
the noise. Since this lter may be dicult to realize in practice, then the discrete-time processing also
described in Section 1.7 of factoring the noise covariance to
E[nn

] = R
nn

2
= R
1/2
nn
R
1/2
nn

2
, (4.188)
can be used. Then a discrete white-noise equivalent channel becomes
y R
1/2
nn
y =
_
R
1/2
nn
P
_
x + n . (4.189)
The extra matrix multiply is absorbed into the receiver processing as F

y F

R
1/2
nn y. The preceding
development the continues with P replaced by the matrix R
1/2
nn
P, the later of which then undergoes
SVD. The matrix P will not be Toeplitz any longer, but will be very close to Toeplitz if the duration
of the noise whitening lter is much less than the symbol period. In any case, vector coding does not
require P to be Toeplitz.
4.5.3 An SNR for vector coding.
The number of bits/symbol for vector coding is achieved on the channel using a coding method with
constant gap (regardless of b
n
) on all subchannels. This b is
b = (1/k)
N

n=1
log
2
_
1 +
SNR
n

_
, (4.190)
where k = 2 for real subchannels and k = 1 for complex subchannels. When = 1, the data rate on
each subchannel is as high as possible with the given energy c
n
and is sometimes called the mutual
information. (See Chapter 5 for more on mutual information). If the water-ll energy distribution is
also used, then b is the capacity b = c or highest possible number of bits per channel that can be reliably
transmitted on the discrete-time channel. The best energy distribution for vector coding and > 1 is a
water-ll with SNR
n
SNR
n
/, as in Section 4.3:
c
n
+
A0
2

2
n
= K (4.191)
N

n=1
c
n
= (N +)

c
x
(4.192)
340
The SNR for vector coding is then
SNR
vc
=
_
N

n=1
_
1 +
SNR
n

_
_1/(N+)
. (4.193)
Note that (N +)

c
x
is now the total energy. The exponent depending on N + is used because vector
coding uses N + channel input samples
18
. When = 1, for complex passband systems,

b =
b
2(N + )
= .5 log
2
_
1 +
SNR
vc

_
, (4.194)
and

b =
b
(N +)
= .5 log
2
_
1 +
SNR
vc

_
(4.195)
for real (PAM) baseband systems. Thus, the geometric SNR of vector coding can be compared against
the detection SNR of an AWGN, a MMSE-LE, or a MMSE-DFE because SNR
vc
has the same relation
to achievable data rate with the same class of modulation codes of constant gap . Further, we have the
interesting result
SNR
vc
= 2
2

I
1 , (4.196)
and
SNR
vc
= 2
2

C
1 (4.197)
when water-lling is used with = 1. Thus the SNR is maximum when the data rate is maximized at
capacity using water-lling.
EXAMPLE 4.5.1 (Vector Coding for 1 +.9D
1
Channel) For this channel, a guard
period of one sample = 1 is sucient to prevent overlap between symbol transmissions at
the channel output. We will choose N = 8 to be consistent with previous invocations of this
channel. The total energy is then c = (8 + 1) 1 = 9. This energy may be distributed over a
maximum of 8 subchannels. The channel-description matrix is
p =
_

_
.9 1 0 0 0 0 0 0 0
0 .9 1 0 0 0 0 0 0
0 0 .9 1 0 0 0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 0 0 0 0 .9 1
_

_
. (4.198)
Singular value decomposition (implemented by the svd command in matlab) produces the
following 8 nonzero singular values
[1.87 1.78 1.64 1.45 1.22 .95 .66 .34] , (4.199)
and corresponding subchannel SNRs
g
n
=

2
n

2
= [19.3 17.6 15.0 11.7 8.3 5.0 2.4 .66] . (4.200)
Execution of water-lling with = 0 dB on the subchannels nds that only 7 can be used
and that
K =
1
7
_
9 +
6

n=0
1
g
n
_
= 1.43 . (4.201)
18
The number of samples is N + dimensions for real channels, but 2N + 2 dimensions for complex channels. In the
complex case, each SNR factor would be squared and then (4.193) holds in either the complex or real case without further
modication
341
The corresponding subchannel energies are:
[1.38 1.37 1.36 1.34 1.30 1.23 1.01 0] , (4.202)
and the SNRs are
[26.6 24.2 20.4 15.8 10.8 6.2 2.4 0] , (4.203)
resulting in an SNR of
SNR
V C
=
_
6

n=0
SNR
n
+ 1
_
1/9
1 = 6.46 = 8.1 dB . (4.204)
The capacity for N = 8 and = 1 is then
c =
1
9
6

n=0
1
2
log
2
(1 + SNR
n
) = 1.45bits/dim. (4.205)
The capacity is close to the true capacity of 1.55 bits/dimension, which would be determined
by executing this example as N . Note this set of subchannels are exactly independent
and the SNR and capacity are exact - no approximations of no-ISI on a subchannel were
made. The SNR is then 8.8 dB and exceeds the previous best equalized SNR on this channel,
achieved by the MMSE-DFE. And, there is no error propagation. The overhead of = 1
becomes zero as N increases.
342
4.6 Discrete Multitone (DMT) and OFDM Partitioning
Discrete MultiTone (DMT) and Orthogonal Frequency Division Multiplexing (OFDM) are
two very common forms of Vector Coding that add a restriction to reduce complexity of implementation.
Both DMT and OFDM have the same channel partitioning they dier in the loading algorithm: OFDM
puts equal bits on all subchannels, rather than optimizing b
n
and c
n
, as in DMT. OFDM is used on
broadcast (i.e., one-way channels) for which the receiver cannot tell the transmitter the bits and energies
that are best. DMT is used on slowly time-varying two-way channels, like telephone lines. OFDM is
used in wireless time-varying channels with codes that allow recovery of lost subchannels caused by
time-varying notches in multipath intersymbol interference.
The DMT/OFDM channel partitioning forces the transmit symbol to have x
k
= x
Nk
for k = 1, ....
Such repeating of the last samples at the beginning of the symbol is called a cyclic prex. Tremendous
processing simplication occurs with the cyclic prex.
With cyclic prex, the matrix description of the channel has a square N N circulant P matrix:
_

_
y
N1
y
N1
.
.
.
y
0
_

_
=
_

_
p
0
p
1
... p

0 ... 0
0 p
0
.
.
. p
1
p

.
.
. 0
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 0
0 ... 0 p
0
p
1
... p

0 ... 0 p
0
... p
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
p
1
... p

0 ... 0 p
0
_

_
_

_
x
N1
.
.
.
x
0
_

_ +
_

_
n
N1
n
N1
.
.
.
n
0
_

_
(4.206)
y =

Px +n . (4.207)
Singular value decomposition produces unique singular values if those values are restricted to be
nonnegative real values (even when P is complex). For transmission, this restriction is superuous and
a variant of SVD will be simpler to compute. For the case of the cyclic prex, the SVD may be replaced
by the eigendecomposition or spectral factorization of the circulant matrix P,
P = MM

(4.208)
where MM

= M

M = I and is a square diagonal matrix with eigenvalues on the diagonal

n
.
The magnitude of eigenvalues are equal to the singular values. Further, eectively, M and F become
the same matrix for this special case. The product SNR for this alternative factorization in (4.208) is
the same as for SVD-factorization because the magnitude of each eigenvalue is equals a corresponding
singular value.
19
This type of transmission could be called Eigenvector Coding, analogous to modal
modulations use of eigenfunctions. While the subchannel SNRs are the same, there are many choices for
the transform matrices M and F in partitioning. Since all have the same performance, this subsection
investigates a very heavily used choice that has two advantages (1) the matrices M and F are not a
function of the channel, and (2) they are eciently implemented.
4.6.1 The Discrete Fourier Transform
A review of the DFT is helpful for DMT channel partitioning.
Denition 4.6.1 (Discrete Fourier Transform) The Discrete Fourier Transform(DFT)
of a N-dimensional input symbol x is
X

=
_

_
X
N1
X
N2
.
.
.
X
0
_

_
(4.209)
19
A caution to the reader using Matlab for eigenvalue/vector computation the eigenvalues are not ususally ordered as
they are in singular-value decomposition.
343
where
X
n
=
1

N
N1

k=0
x
k
e

2
N
kn
n [0, N 1] , (4.210)
and
x =
_

_
x
N1
x
N2
.
.
.
x
0
_

_
. (4.211)
The corresponding Inverse Discrete Transform (IDFT) is computed by
x
k
=
1

N
N1

n=0
X
n
e

2
N
kn
k [0, N 1] . (4.212)
In matrix form, the DFT and IDFT are
X = Qx (4.213)
x = Q

X , (4.214)
where Q is the orthonormal matrix given by
Q =
1

N
_

_
e

2
N
(N1)(N1)
... e

2
N
2(N1)
e

2
N
(N1)
1
e

2
N
(N1)(N2)
... e

2
N
2(N2)
e

2
N
(N2)
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
e

2
N
(N1)
... e

2
N
2
e

2
N
1
1 ... 1 1 1
_

_
. (4.215)
Both x and X can be presumed to be one period of a periodic sequence to generate DFT/IDFT
values for indices k and/or n outside the interval (0, N1). The matrix Q

appears the same

as Q in (4.215) except that the exponents all have a plus sign where the minus now occurs.
Further, we can dene some vectors q
n
according to
Q

= [q
N1
, ..., q
0
] , (4.216)
which are useful in the following theorems proof.
The IDFT, X
n
, and DFT, x
k
, values can be complex. For real time-domain signals, the restriction
X
Nn
= X

n
for n = 1, ..., N 1 holds. This restriction implies that there are

N = N/2 complex
dimensions when N is even
20
. The IDFT generally corresponds to N input samples whether complex
or real, while this textbook has previously used N to be the number of real dimensions and

N to be the
number of tones. Thus, there is a slight inconsistency in notation. This inconsistency is resolved simply
by considering N in the IDFT and DFT denitions to be the number of tones used if complex, and twice
the number of tones with the congjugate symmetry on the IDFT input enforced (or real time-domain
signals). The FFT size should then be equal to the number of real dimensions only in the latter case
of real baseband signals. When complex baseband signals are used, the number of real dimensions is
twice the FFT size. With this understanding, matlab can be used to easily generate the Q matrix here
exactly according to the commands
J=hankel([ zeros(1,N-1) 1])
Q=(1/sqrt(N))*J*fft(J)
20
Actually, there are

N 1 complex dimensions and two real dimensions corresponding to n = 0 and n =

N. The extra
two real dimensions are often grouped together and considered to be one complex dimension, with possibly unequal
variances in the 2 dimensions. Usually, practical systems use neither of the real dimensions, leaving

N 1 subchannels
344
Lemma 4.6.1 (DFT is M for circularly prexed VC) The circulant matrix P = MM

has a eigendecomposition with M = Q

- that is the eigenvectors are the columns of the IDFT

matrix and not dependent on the channel. Furthermore, the diagonal values of are
n
= P
n
- that is, the eigenvalues of this circulant matrix are found through the DFT.
Proof: The eigenvalues and corresponding eigenvectors of a matrix are found according to

n
q
n
= Pq
n
, (4.217)
which has matrix equivalent Q

= PQ

. By investigating the eect of the bottom element

of the product Pq
n
, where q
n
is given above in (4.216),
1

_
p
0
+ p
1
e

2
N
n
+ ... + p
N1
e

2(N1)
N
n
_
= P
n
. (4.218)
The next row up is then
1

_
p
0
e

2
N
n
+p
1
+... + p
N1
e

2(N2)
N
n
_
= P
n
e
+
2
N
n
. (4.219)
Clearly, repeating this procedure, one nds
Pq
n
= P
n
q
n
, (4.220)
making q
n
an eigenvector of P. Thus, a choice of the eigenvectors is simply the columns of
the IDFT matrix Q

. For this choice, the eigenvalues must exactly be the DFT values of the
rows of P. QED.
4.6.2 DMT
Discrete Multi-tone (DMT) transmission uses the cyclic prex and M = FQ

in vector coding for

the constructed cyclic channel. These matrices are not a function of the channel because of the use of the
cyclic prex. Orthogonal Frequency Division Multiplexing (OFDM) also uses these same vectors
(but does not use optimum loading, and instead sends equal energy and information on all subchannels).
Thus, the equivalent of the F and M matrices for VC with a cyclic prex become the well-known
DFT operation. The DFT can be implemented with N log
2
(N) operations, which is less than the full
matrix multiplies of vector coding that would require N
2
equivalent operations. For large N, this is a
very large computational reduction. Clearly, since an additional restriction of cyclic prex is applied
only to the case of DMT/OFDM channel partitioning, then DMT/OFDM can only perform as well as
or worse than VC without this restriction. When N >> , the dierence will be small simply because
the dierence in P is small. The designer should take care to note that the energy constraint for DMT
when executing loading algorithms is lower by the factor N/(N +) because of the wasted energy in
the cyclic prex, the price for the computational simplication.
Figure 4.19 illustrates the basic structure of the DMT transmission-system partitioning. Determina-
tion of the subsymbol values X
n
, energy c
n
and number of bits b
n
is determined by a loading algorithm.
The partitioning then uses the FFT computationally ecient algorithms for implementation of the DFT
and IDFT.
For the case of N , the cyclic prex becomes negligible and vector coding performs the same as
DMT, and all the individual subchannels of DMT also have the same SNR
n
as VC as N . This is
consistent with the earlier nding that eigenvector coding in the limit of large symbol size and innite
number of subchannels becomes MT. VC is a discrete form of eigenvector coding while DMT is a discrete
form of MT. In the limit, with suciently high sampling rate, DMT, MT, VC and eigenvector coding all
perform the same. Only DMT is easily implemented. No other channel partitioning can perform better
as long as N is suciently large.
345
Figure 4.19: DMT and OFDM partitioning block diagram.
n
n
= [P
n
[ g
n
=
]Pn]
2
.181
c
n
SNR
n
0 1.90 20 1.24 24.8
1 1.76 17 1.23 20.9
2 1.76 17 1.23 20.9
3 1.35 9.8 1.19 11.7
4 1.35 9.8 1.19 11.7
5 .733 3 .96 2.9
6 .733 3 .96 2.9
7 .100 .05525 0 0
Table 4.3: Table of parameters for 1 + .9D
1
channel example with DMT with = 0 dB.
EXAMPLE 4.6.1 (DMT for 1 +.9D
1
Channel) Continuing the example for the 1 +
.9D
1
channel for DMT, the circulant channel matrix for N = 8 and = 1 becomes
P =
_

_
.9 1 0 0 0 0 0 0
0 .9 1 0 0 0 0 0
0 0 .9 1 0 0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1 0 0 0 0 0 0 .9
_

_
. (4.221)
Table 4.6.1 summarizes the results of waterlling with 8 units of energy (not 9, because 1
unit is wasted in the cyclic prex on average). Only 7 of the 8 subchannels were used - note
that 6 of the subchannels are eectively two-dimensional QAM subchannels.
The SNR is
SNR
DMT
=
_
6

n=0
(1 + SNR
n
)
_
1/9
1 = 7.6 dB < 9.2 dB from Example 4.1.1. (4.222)
346
The DMT SNR is less than that found earlier for MT, but is exact and not approximated.
It is also slightly worse than the SNR of vector coding for the same number of dimensions.
However, DMT can increase the number of dimensions signicantly before the complexity of
FFTs exceed that of the orthogonal matrix multiplies. One nds that for N = 16, this SNR
increases to its maximum value of 8.8 dB.
Let us compare the complexity of this DMT system with its exact SNR now correctly com-
puted with that of the DFE studied earlier and also against the VC system. To get 7.6
dB, the MMSE-DFE required 3 feedforward taps and 1 feedback tap for a complexity of 4
multiply-accummulates per sample. VC requires 7(8)/9 multiply-accumulates in the receiver
to get the slightly higher SNR of 8.1 dB, or a complexity 6.2/sample. DMT requires a size-
8 FFT, which nearly exactly requires 8 log
2
(8) multiply accumulates when the real-output
constraint is exploited. DMT then gets 7.6 dB also, but with 2.7 multiplies/sample. A factor
of 2-3 improvement in complexity for a given level of performance with respect to the DFE
and VC is common. For the size 16 DMT with SNR=8.8 dB, the complexity is still only
3.8/sample while for instance a DFE requires 10 feedforward taps and 1 feedback to get only
8.4 dB, then requiring 11/sample.
Matlab programs
The following matlab functions have been written by teaching assistants Atul Salvekar, Wonjong Rhee,
and Seong-Taek Chung, and are useful for computing DMT performance and parameters quickly. See
class web site for complete programs.
--------- rate-adaptive water-fill DMT ------------------------------------
function [gn,en_bar,bn_bar,Nstar,b_bar]=waterfill(P,SNRmfb,Ex_bar,Ntot,gap)
% Written by Atul Salvekar, Edited by Wonjong Rhee
%
% function [gn,en_bar,bn_bar,Nstar,b_bar]=waterfill(P,SNRmfb,Ex_bar,N,gap)
%
% P is the pulse response
% SNRmfb is the SNRmfb in dB [Ebar ||p||^2 / (N0/2)]
% Ex_bar is the normalized energy
% Ntot is the total number of real/complex subchannels, Ntot>2
% gap is the gap in dB
%
% gn is channel gain
% en_bar is the energy/dim in the nth subchannel
% bn_bar is the bit/dim in the nth subchannel
% Nstar is the number of subchannel used
% b_bar is the bit rate
-------------------------------------------------------------------------------
--------- margin-adaptive water-fill DMT ------------------------------------
function [gn,en_bar,bn_bar,Nstar,b_bar_check,margin]=MAwaterfill(P,SNRmfb,Ex_bar,b_bar,Ntot,gap)
% Written by Seong Taek Chung
%
% P is the pulse response
% SNRmfb is the SNRmfb in dB
% b_bar is the normalized bit rate
% Ex_bar is the normalized energy
% Ntot is the total number of real/complex subchannels, Ntot>2
347
% gap is the gap in dB
%
% gn is channel gain
% en_bar is the energy/dim in the nth subchannel
% bn_bar is the bit/dim in the nth subchannel
% Nstar is the number of subchannel used
-------------------------------------------------------------------------------------------------
--------- rate-adaptive Levin-Campello DMT ------------------------------------
function [gn,En,bn,b_bar]=LC(P,SNRmfb,Ex_bar,Ntot,gap)
%
% Taek Chung
%
% Levin Campellos Method
%
% P is the pulse response
% SNRmfb is the SNRmfb in dB
% Ex_bar is the normalized energy
% Ntot is the total number of real/complex subchannels, Ntot>2
% gap is the gap in dB
%
% gn is channel gain
% En is the energy in the nth subchannel (PAM or QAM)
% bn is the bit in the nth subchannel (PAM or QAM)
% Nstar is the number of subchannel used
% b_bar is the bit rate
%
% The first bin and the last bin is PAM, the rest of them are QAM.
---------------------------------------------------------------------------------
--------- margin-adaptive Levin-Campello DMT ------------------------------------
function [gn,En,bn,b_bar_check,margin]=MALC(P,SNRmfb,Ex_bar,b_bar,Ntot,gap)
%
% Taek Chung
%
% Levin Campellos Method
%
% P is the pulse response
% SNRmfb is the SNRmfb in dB
% Ex_bar is the normalized energy
% Ntot is the total number of real/complex subchannels, Ntot>2
% gap is the gap in dB
% b_bar is the bit rate
% gn is channel gain
% En is the energy in the nth subchannel (PAM or QAM)
% bn is the bit in the nth subchannel (PAM or QAM)
% b_bar_check is the bit rate for checking - this should be equal to b_bar
% margin is the margin
% The first bin and the last bin is PAM, the rest of them are QAM.
---------------------------------------------------------------------------------
348
Figure 4.20: ADSL use example.
EXAMPLE 4.6.2 (ADSL) By far the most heavily used broadband internet delivery
method in the world is asymmetric digital subscriber lines (ADSL) in Figure 4.20, allow-
ing nominally 1.5 Mbps (range is 500 kbps to 12 Mbps) to customers on telephone lines
downstream and roughly about 1/3 to 1/10 that rate from the customer to the telephone
company upstream (where it is sent to the internet service provider).
ADSL is world-wide standardized in ITU Standard G.992.1 and uses DMT. The symbol rate
is T = 250s. The downstream modulation uses 256 4.3125 kHz wide subchannels and a
sampling rate of 1/T
t
=2.208 MHz. The cyclic prex is then necessarily 40 samples. Each
time-domain symbol contains then 512+40 or 552 samples. Hermetian symmetry is used to
create a real signal for transmission over the band from 0 to 1.104 MHz. Typically, the rst 2-3
tones near DC and DC are not used to prevent interference into voiceband telephony (POTS
= plain old telephone service), which shares the same telephone line as ADSL. Upstream
transmission uses instead 32 tones to frequency 138 kHz. Tone 256 is also not used. Tone
64 (276 kHz) is reserved for pilot signal (known point in 4 QAM sent continuously) that is
used to recover the symbol and sampling rates. The sampling rate upstream is 276 kHz and
the cyclic prex is 5 samples for a block size of 69 samples. Upstream is then exactly 1/8
downstream. Downstream tones may or may not share the upstream band (when they do
share, a device called an echo canceller is used to separate the two directions of transmission).
Usually, the upstream and downstream bands are placed for further adaptive renement by
loading as shown in Figure 4.21. Some typical ADSL channels (taken from the American
National Standards Institute) appear in Figure 4.22. The lines have signicant variation with
length, and also with impairments like bridged-taps (open circuited extension phones) that
cause the ripple in the characteristics in the second graph in Figure 4.22. Also noises are
highly frequency selective and could be nearby radio stations captured by the phone-line or
other DSLs with varying bandwidths that are suciently close to crosstalk into the ADSL
line. All this channel variation among possible lines for deployment of ADSL creates a huge
349
Figure 4.21: ADSL spectrum use - further adaption by loading within each band.
Figure 4.22: ADSL example lines.
350
need for adaptive loading, which will optimize the DMT performance for each situation.
A maximum power of 20.5 dBm is permitted in ADSL downstream and 14.5 dBm upstream.
The maximum number of bits permitted to be loaded on to any single tone is b
n
15. While
antognists once said ADSL would fail as being too complex to implement, components that
implement a complete modem (analog and digital) now sell for less than $10 and over 30
million customers will be broadband on-line with ADSL by the end of 2002. That number will
go to 100s of millions in the coming years. While the design may be conceptually complex,
most customers simply call the phone company, procure a modem, plug and play in less than
15 minutes a testimonial to the engineer a conceptually complex design is acceptable if
it keeps the customer use very simple (the opposite is almost never true) and the cost down.
Wireless transmission often uses OFDM transmission because no optimization at the transmitter of
b
n
is possible if the channel varies too rapidly with time. Instead, over a wide bandwidth in wireless
transmission, multipath ISI typically results in notches in the transmission band. Part of the signal (some
tones) is lost. Thus, with coding methods not discussed until later chapters of this book, it is possible to
recover the information of several lost tones as long as the code is suciently powerful, no matter where
those lost tones are located (that is, no matter which tone indices n are aected). Also, in broadcast
wireless transmission, there may be several dierent channels to each customer of a service, and no
feedback channel, so then the b
n
can not be optimized as a function of the identied channel (which
necessary must occur at the receiver, which is the only place the channel can be known practically and
estimated). The use of OFDM in wireless transmission with codes is often called COFDM. COFDM is
the most powerful method for handling the time-varying channel, even though OFDM might have been
greatly inferior to DMT if the channel could have been known at the transmitter. COFDMs greatest
asset is simply the practical advantage that the transmitter is basically xed (an IFFT operation with
xed bit loading and energy on all tones) actually the gain and phase of each subchannel is estimated
and used, but only at the receiver. Rapid determination of a full DFE-like equalizer is dicult and
loses performance greatly in wireless transmission, leaving multicarrier a superior method coincidentally
for wireless transmission as well as wireline. However, the key is the additional code, as nominally an
un-loaded or uncoded OFDM signal will not be very eective. This heuristic explanation then leads
to the next couple of examples of OFDM partitioning use.
EXAMPLE 4.6.3 (802.11a Wireless LAN) The IEEE has standardized a wireless local-
area-network known as IEEE 802.11(a). This method transmits up to 54 Mbps symmetrically
between wireless users in relative close proximity (nominally, within a building or complex),
nominally of 100 meters or less with one of 3 transmit power levels (assuming 6 dB antenna
gain) of 16, 23, or 29 dBm. The system is complex baseband (unlike ADSL, which is real
baseband) with

N = 64 (so up to 128 real dimensions). The carrier frequencies are in the
5 GHz range and are exactly (in MHz) 5180 + i(20) (16 dBm) , 5260 + i(20) (23 dBm), or
5745 + i(20) (29 dBm) where i = 0, 1, 2, 3.
At baseband, the IEEE 802.11(a) system numbers the tones from -31 to +31 (the 64th
carrier at the band edge is not used). The sampling frequency is exactly 20 MHz (complex,
so two 20 MHz real converters conceptually) and the carrier spacing is then 20MHz/64=312.5
kHz. The cyclic prex has 16 complex samples, and thus the symbol consists of 64+16=80
samples. The symbol rate is then 250 kHz (or T = 4s, while T
t
= 50ns). The guard period
(or maximum length of ISI for zero inter-tone distortion) is 0.8s.
Of the 63 possible carriers cited, the carrier in the middle of the band carries a xed phase
(not data) for carrier recovery. Also, tones -31 to -27 and 27 to 31 at the band edges are not
used to allow analog-lter separation of adjacent carrier bands of dierent 802.11(a) signals.
Furthermore, carriers, -21, -7, 7, and 21 are pilots for symbol-timing recovery. This leaves 48
tones for carrying data. The total bandwidth used is 15.5625 MHz. Data rates with integer
numbers of bits per tone will thus be multiples of
R = k(1 bit /2 dimensions) (48 tones) 250kHz or 6 kMbps. (4.223)
351
R (Mbps) constellation code rate b
n

b
n
b
6 BPSK 1/2 1/2 1/4 24
9 BPSK 3/4 3/4 3/8 36
12 4QAM 1/2 1 1/2 48
18 4QAM 3/4 3/2 3/4 72
24 16QAM 1/2 2 1 96
36 16QAM 3/4 3/2 3/4 144
48 64QAM 1/2 3 3/2 192
54 64QAM 3/4 9/2 9/4 216
Table 4.4: Table of IEEE802.11(a) data rates and associated parameters.
Table 4.4 summarizes the options for transmission. The coding rate is a fraction of the actual
bits that carry information while the remaining bits are redundant and used for overhead as
described in Chapters 10 and 11 (the extra bits help reduce the gap).
Table 4.4 also shows that while all (used) tones carry the same b
n
= b/48, no tone carries
the full number of messages possible in the QAM constellation because of the coding used
without this coding, OFDM would not be a wise transmission method for wireless. The
spectrum mask of the composite analog signal uses at power spectral density (power divided
by 16 MHz) over the used band, and must be 20 dB attenuated at frequencies further than
11 MHz from carrier, 28dB attenuated at 20 MHz from carrier, and 40dB attenuated at
30 MHz from carrier.
Two new IEEE standards IEEE 802.11(g) and IEEE802.11(aw) are appearing shortly where
(g) allows carrier frequencies near 2.4 GHz but otherwise looks like (a) and (aw) allows use
of multiple input and output antennas with (a) to create several spatial paths between users
and allow between 108 Mbps and 600 Mbps (perhaps 1 Gbps), and possibly simultaneous
use of multiple carrier frequency 20 MHz slots by a single user/network. The potential
multiple antenna use being suggested by this instructor/author in a 1994 version of this
class (then called EE479) and then successfully pursued by several former students. Chapter
5 illustrates a general theory of how to generate the spatial or extra channels in various
multiple-input/multiple-output channels.
There is very little adaptive spectra in any of the IEEE 802.11 standards, other than a
selection of data rate. The future may see uplink power control used to at least set the power
levels of signals to the base station so that sidebands of a nearby signal do not unduly disrupt
the main spectrum of a distant signal.
EXAMPLE 4.6.4 (Digital Video Broadcast) Digital terrestrial television broadcast has
been very successful outside the USA (where analog still prevails outside of cable systems and
satellite broadcast). This successful system is known generally as Digital Video Broadcast
(DVB). A reason for the success elsewhere is the use of a COFDM system with something
called SFN (single frequency network), as illustrated in Figure 4.23. SFN basically uses
multiple broadcast sites to deliver exactly the same signal to customers so that buildings,
structures, and hills blocking one path to customer likely do not block another path. How-
ever, when more than one signal reaches the customer, likely with dierent delays, the eect
is equivalent to intersymbol interference or multipath distortion. This delay can be quite
large (10s of s) if transmitters are say 10s of km apart. Over the 6-7.6 MHz wide bands
of Television, such delay would overwhelm even the best of Decision Feedback Equalizers as
there could be 100s of deep notches in the transmission band, leading to huge length lters
and numerical/training problems. However, for COFDM, codes are used to mitigate any
loss of tones at frequencies disabled by such notches. The SFN basically solves one problem
(blocking of signal), but forces the use of multicarrier or OFDM.
352
Figure 4.23: Single Frequency Network example for television-broadcast channel partitioning.
The DVB system is also complex-baseband and uses conventional TV carrier frequencies
(basically above 50 MHz and 7.61 MHz spacing outside US). The DVB system uses either

N=2048 or 8192 tones, of which either 1705 or 6817 are used. The number of pilots used for
recovery of symbol rate is 45 or 177, leaving 1650 or 6650 tones available for carrying infor-
mation. The exact parameters depend on the cyclic prex length, which itself is programmed
by the broadcaster in DVB to be one of 1/4, 1/8, 1/16, or 1/32 of the symbol length. The
tone width is xed at approximately 1.116 kHz (896s) for 8192 tones and 4464 kHz (224s)
for 2048 tones. The symbol period T thus varies with cyclic prex and can be any of 1120s,
1008s, 952s, or 924s for 8192 tones and any of 280s, 252s, 238s, or 231s for 2048
tones. The sampling rate is xed at 1/T
t
9.142 MHz (T
t
= 10.9375 ns) for each of two
real sampling devices.
Code rates for DTV (see course web site for DTV standard document from ETSI with all the
details) and there is a concatenated coding system (see Chapter 11). Basically, used tones
have 4QAM, 16QAM, or 64QAM transmission with various code rates of rst code rate
(172/204) second code rate 1/2, 2/3, 3/4, 5/6, and 7/8, and produces data rates between
approximately 4.98 Mbps and 31.67 Mbps (selected by broadcaster, along with number of
TV channels carried from 2 to 8. Note this system at least doubles the number of channels
of analog TV, and often increases by a factor of 8, enabling greater choice by consumer and
higher revenue by broadcaster.
4.6.3 An alternative view of DMT
If the input sequence is periodic with period N samples, then the corresponding output sequence from
an FIR channel is also periodic. The output of the channel may be computed by taking the IDFT of the
sequence Y
n
= P
n
X
n
(where P
n
are the samples of the DFT of the channel pulse response samples).
When the cyclic prex is used, however, the input appears periodic as far as the last N samples of any
symbol, which are all the receiver processes. DMT thus can also be viewed a circular convolution trick,
although this obscures the optimality and connections to VC.
353
4.6.4 Noise-Equivalent Channel Interpretation for DMT
The equivalence of an additive (colored) Gaussian noise channel to an AWGN was established in
Section 1.7 and used in Chapters 3 and 4. The colored Gaussian noise generally has power spectral
density
A0
2
R
n
(f). The channel pulse response of the equivalent AWGN is the pulse response of the
original channel adjusted by the inverse square-root autocorrelation of the Gaussian noise:
P(f)
P(f)
_
R
n
(f)
. (4.224)
The receiver uses a noise-whitening ltering 1/
_
R
n
(f) as a preprocessor to the matched-lter and
sampling device. The noise-equivalent viewpoint construes this whitening lter as part of the channel,
thus altering the channel pulse response according to (4.224).
For DMT/OFDM, the noise equivalent view discussed earlier leads to a set of subchannels with signal
gains

c
x
]P(n/T)]
2
Rn(n/T)
and AWGN variance
A0
2
, as long as R
nn
is also circulant. In practice, R
n
(n/T) is
not circulant unless N ; however, for reasonable noises, the autocorrelation matrix can be close to
circulant for moderate values of N. A more common view, which avoids the use of the whitening lter,
is that each of the subchannels has gain

c
x
[P(n/T)[
2
and noise
A0
2
R
n
(n/T). The set of SNRs remains
the same in either case, but the need for the whitening lter is eliminated in the second viewpoint. The
second viewpoint has the minor aw that the noise variances per subchannel are not completely correct
unless the noise n(t) is cyclostationary with period T or NT
t
is large with respect to the duration of
the noise autocorrelation function. In practice, the values used for N are almost always more than
suciently large that implementations corresponding to the second viewpoint are ubiquitously found.
Furthermore, noise power is measured as in Section 4.6.
4.6.5 Toeplitz Distribution
The rigorous mathematical justication for convergence of vector coding to DMT derives from Toeplitz
Distribution arguments:
Theorem 4.6.1 (Toeplitz Distribution) Given a square Toeplitz matrix R with rst row
[r
0
r
1
... r
N1
], which extends to N = by allowing both ends of the rst row to grow as
[... r
2
r
1
r
0
r
1
r
2
...] while maintaining Toeplitz structure, the Fourier Transform is dened
by
R(f) =

k=
r
k
e
2fk
. (4.225)
Any N-dimensional Toeplitz submatrix of R along its diagonal has eigenvalues
(N)
k
. Then
for any continuous function g[]
lim
N
1
N
N

k=1
g[
(n)
k
] =
_
.5
.5
g[R(f)]df . (4.226)
No proof given:
Thus, for the square toeplitz matrix PR
xx
P

that has eigenvalues c

n
[
n
[
2
, and as N , then

b =
1
N
N

n=1
.5 log
2
_
1 +
c
n
[
n
[
2
A0
2
_

_
.5
.5
.5 log
2
_
R
x
(f)[P(f)[
2
A0
2
_
.df (4.227)
This expression is the mutual information for the discrete-time channel with sampling rate 1/T
t
, as
derived in Chapter 8. This integral for data rate in bits per dimension maximizes to capacity,

b = c,
when a water-ll distribution is used for R
x
(f),
R
x
(f) + [P(f)[
2
/
^
0
2
= constant . (4.228)
354
Figure 4.24: The FEQ - frequency-domain equalizer.
So, Toeplitz distribution also relates that vector-coding converges (in performance) to MT and to the
optimum transmission system when water-ll is used. A caution to note is that the sampling rate 1/T
t
needs to be selected suciently high for this discrete-time system to converge to the true optimum of
capacity over all sampling rates. When equivalent noise channels are used,
[P(f)[
2

[P(f)[
2
A0
2
R
n
(f)
(4.229)
in the above integral.
4.6.6 Gain Normalizer Implementation - The FEQ
For DMT and/or VC, the multichannel normalizer is often called an FEQ (Frequency Domain Equalizer),
which is shown in Figure 4.24.
A ZFEE is applied to each complex DMT/OFDM subchannel with target X
n
for each subchannel.
From Section 3.7, with the matched lter absorbed into the equalizer setting W
n
, the lter is a single
complex multiply that adjusts the gain and phase of each subchannel independently to minimize the
mean-square error between W
n
Y
n
and X
n
. The FEQ setting for the n
th
subchannel of DMT is
W
n
=
1
P
n
. (4.230)
The algorithm that converges to this setting as a function of time was provided earlier in Section 4.3 in
Equation (4.133).
Because it is a single tap and cannot change (unbiased) SNR, the FEQ simply scales the DFT outputs
so that the DMT/OFDM receiver outputs estimate X
n
, and on each subchannel
SNR
ZFE,n
= SNR
n
. (4.231)
With a common code class, (, and constant gap , a common decoder can be reused to decode all sub-
channels. The FEQ can be implemented adaptively as described in Section 3.7 just as more complicated
multiple tap equalizers are implemented.
4.6.7 Isakssons Zipper
Often DMT partitioning is used on a bi-directional system. That means both directions of transmission
share a common channel. The signals may be separated by echo cancellation (a method of synthesizing
355
Figure 4.25: Illustration of alignment problem in DMT with only cyclic prex.
356
Figure 4.26: Use of cyclic sux to align DMT symbols at both ends of transmission.
from the known transmitted signal a replica of any reected transmit energy at the opposite-direction
signals co-located receiver and subtracting. The signals may also be separated by assignment of indepen-
dent tone sets for each direction of transmission, known as frequency-division multiplexing. Both echo
cancellation and frequency-division multiplexing have enormous simplication of their signal processing
if the DMT signals in the two directions have common sampling rates and symbol rates. Alignment of
the symbol boundary at one end is simple because the choice of the transmitted symbol boundary can
be made to align exactly with the received DMT signals symbol boundary. However, it would appear
that alignment at the other end is then impossible with any realistic channel having delay.
This basic problem is illustrated in Figure 4.25 for DMT systems with N = 10, = 2, and = 3
(the delay of the channel). LT and NT represent dierent ends of a transmission line at which each such
end has both a transmitter and a receiver. The ds and us appearing in Figure 4.25 refer to downstream
(LT to NT) and upstream (NT to LT) transmission. Alignment at the LT leads to misalignment at the
NT. Figure 4.26 shows a solution that adds a 6 sample cyclic sux after the DMT signal, repeating the
beginning of the symbol (that is the beginning after the cycli prex). There are now 6 valid positions
for which a receiver FFT to process the recieved signal without interference among tones. One of those
6 positions at the NT downstream is the same position as one of the 6 positions upstream, allowing
alignment at the NT and LT simultaneously for the price of an additional 6 wasted samples. The value 6
is clearly in general 2 . As N , the cyclic sux overhead penalty can be made negligible. Figure
4.27 shows a method to reduce the smallest cyclic sux length to by advancing the LT downstream
signal by samples, allowing one of 3 valid downstream positions at the LT to corresponding to one of
357
Figure 4.27: Use of both cyclic sux and timing advance to align DMT signals at both ends of trans-
mission.
3 valid upstream positions, and correspondingly one overlap also at the NT. This is the minimum cyclic
sux length necessary for alignment at both ends simultaneously. This method was introduced in 1997
by Michael Isaksson, then of Telia Research in Sweden and is standardized for use in what is known as
VDSL.
EXAMPLE 4.6.5 (VDSL) VDSL is a recent ospring of ADSL that uses the cyclic sux
and timing advance of Zipper. VDSL, like ADSL, is baseband real, except up to 16 times
wider bandwidth than ADSL and intended for use on shorter phone lines where speeds as
high as 100 Mbps (more typically 10 Mbps) symmetric can be achieved with VDSL. In VDSL,
both upstream and downstream transmission may use up to 4092 tones, so N = 8192 with
congugate symmetry. The symbol period remains 4000 Hz and the tone spacing is 4.3125
kHz. The cyclic prex is now 16 40 = 640 samples long. The sampling rate is 35.328 MHz.
The cyclic prex period is actually shared with cyclic sux of minimum-length, determined
on a per-line basis adaptively, and DMT symbols are aligned at both ends. The upstream
and downstream tones are allocated currently via frequency-division mulitiplexing (above
138 kHz, and can be both directions below that) with frequencies determined as best needed
as a function of line, channel, and service-provider desires. The Zipper duplexing leaves no
need for concern with complicated analog lters to separate up and down transmissions
this is entirely achieved digital as is sometimes also known as digital duplexing. Somewhat
suprisingly also in retrospect, the digital duplexing allows simple power-minimizing-for-given-
rate/margin LC loading of this chapter as essentially optimum if run simultaneously on a set
of crosstalking telephone lines to nd a best trade in terms of spectrum among all the lines.
This is sometimes called iterative water-lling (see Chapter 12) and can lead to enormous
358
data rates on telephone lines.
4.6.8 Filtered Multitone (FMT)
Filtered Multi-Tone (FMT) approaches use excess bandwidth in the frequency domain (rather than or in
addition to excess dimensions in a time-domain guard period) to reduce or eliminate intersymbol inter-
ference. There have been many approaches to this FMT area under the names of wavelets (DWMT
, W=wavelet), lter banks , orthogonal lters, polyphase by various authors. This subsection
pursues a recent approach, specically given the name FMT, pioneered by Cherubini, Eleftheriou, and
Olcer of IBM. This latter FMT approach squarely addresses the issue of the channel impulse responses
(h(t)s) usual destruction of orthogonality that was basically ignored in all the other approaches. The
basic idea in all approaches though is to nd a discrete-time realizable implementation of a lter that
basically approximates the GNC-satisfying brick-wall frequency-tone characteristic of the sinc(t) func-
tion of ideal multi-tone. Essentially, realizable basis functions with very low frequency content outside
a main lobe of the fourier transform (i.e., the main lobe is centered at the carrier frequency of the
tone and there are second, third, fourth, etc. lobes that decay in any realizable approximation to the
ideal band-pass lter characteristic of a modualted sinc function). Very low frequency content means
the functions after passing through any channel should be almost orthogonal at the output. This al-
most orthogonal unfortunately is the catch in most approaches, which ultimately fail on one channel or
another. The FMT approach essentially provides a way of quantifying the amount of miss and working
to keep it small on any given channel. There will be a trade-o similar to the trade-o of versus N
in terms of dimensionality lost in exchange for ensuring realizable and orthogonal basis functions, which
will be characterized by essentially the excess bandwidth need in FMT.
The original modulated continuous-time waveform is again
x(t) =

j=
N1

n=0
X
n,j

n
(t jT) , (4.232)
where j is a symbol index. This waveform in discrete construction is sampled at some sampling period T
t
where T = (N+)T
t
, but there is no cyclic prex or time-domain guard period. In the FMT approach,
is sometimes set to zero (unlike DMT). The discrete-time FMT modulator implementation is illustrated
in Figure 4.28. The up-arrow device in Figure 4.28 is an interpolator that simply inserts (N + 1)
zeros between successive possibly nonzero values. The down-arrow devices selects every (N+)
th
sample
from its input and ignores all others. The down-arrow devices are presumably each oset successively
by T
t
sample periods from one another. In FMT approaches, each of the discrete-time basis functions
is a modulated version of a single low-pass function

n
(t) = (t) e

2
T

N+
N
nt
. (4.233)
Each FMT tone when > 0 has excess bandwidth in that the tone spacing is (1 +

N
)
1
T
like DMT but
without cyclic prex. Said excess bandwidth can be used to obtain a realizable approximation to an
ideal bandpass lter for all tones (subchannels).
A sampling time index is dened according to
m = k (N + ) + i i = 0, ..., N + 1 , (4.234)
where i is an index of the sample oset within a symbol period and k measures as usual symbol instants.
The samples x
m
= x(mT
t
) of the continuous time waveform are then (with
n
(mT
t
) =
n,m
)
x
m
=

j=
N1

n=0
X
n,j

n,mj(N+)
(4.235)
=

j=
N1

n=0
X
n,j

n,k(N+)+ij(N+)
i = 0, ..., N + 1 (4.236)
359
Figure 4.28: FMT partitioning block diagram.
Each of the N + basis lters is also indexed by the sample oset i and essentially decomposes into

n,k(N+)+ij(N+)
=
(i)
n,kj
, each of which can be then implemented at the symbol rate (rather than
the original single lter at the sampling rate). Further substitution of the expression in (4.233) nds

(i)
n,kj
=
(i)
kj
e

2
N
n[(kj)(N+)+i]
(4.237)
=
(i)
kj
e

2
N
n[(kj)(N+)]
e

2
N
n[i]
. (4.238)
Substitution of (4.238) into 4.236 provides for all i = 0, ..., N + 1
x
k(N+)+i
=

j=
N1

n=0
X
n,j

(i)
kj
e

2
N
n[(kj)(N+)]
e

2
N
n[i]
(4.239)
=

j=

(i)
kj

N1

n=0
X
n,j
e

2
N
n[(kj)(N+)]
e

2
N
n[i]
. (4.240)
=

j=

(i)
kj
x
i,j
([k j]) (4.241)
where x
i,j
([k j]) is the N-point inverse discrete Fourier transform of X
n,j
e

2
N
n(kj)
in the i
th
sampling oset.
21
Nominally, this IDFT would need to be recomputed for every (k j) - however one
notes that the term e

2
N
n(kj)
that depends on (k j) is simply a circular shift of the time-domain
output by (k j) samples. Figure 4.29 illustrates the use of the IDFT (or IFFT for fast computation)
and notes that the lter inputs may be circularly shifted versions of previous symbols IDFT outputs
when ,= 0. If the FIR lter
(i)
k
has length 2L + 1, say indexed from time 0 in the middle to L,
then previous (and future - implement with delay) would use IDFT output time-domain samples that
correspond to (circular) shifts of increasing multiples of samples as the deviate from center. When
= 0, no such shifting of the inputs from preceeding and succeeding symbols is necessary. This shifting
is simply a mechanization of interpolation to the sampling rate associated with the excess bandwidth
that is (1 +/N) times faster than with no excess bandwidth.
21
Since i = 0, ..., N + 1, the samples of x
i,j
([k j]) will be interpretted as periodic in i with period N.
360
Figure 4.29: FMT partitioning using DFTs and symbol-rate lters.
parameter DMT FMT
f
1+

N
T
1+

N
T
symbol length N + samples N + samples
cyc. pre. length 0
lost freq dimensions 0
excess bandwidth

N

k

k
lowpass lter
delay N + integer (N +)
implementation N-point FFT N-point FFT plus symbol-rate lter
performance likes small- channels likes sharp bandpass channels
Table 4.5: Comparison of DMT and FMT
Clearly if = 0 and
k
=
k
, then FMT and DMT are the same (assuming same 1/T = 1/(NT
t
)).
However, the two systems dier in how excess bandwidth is realized (DMT in time-domain cyclic prex,
FMT in essentially wider tones than would otherwise be necessary without cyclic prexes by same ratio
/N). FMT also allows a shaping lter to be used to try to approximately synthesize ideal-multitone
bandpass subchannels. DMT and FMT are compared in Table 4.5.
Any FMT system will have at least some residual intersymbol interference and interchannel inter-
ference that is hopefully minimized by good choice of
k
. However, equalizers may be necessary on
each subchannel and possibly between a few adjacent subchannels as shown in Figure 4.30. Usually, the
functions have such low out-of-band energy that only intersymbol interference on each subchannel are
of concern, so the receiver reduces to the FFT followed by a bank of symbol-rate DFEs as shown in
Figure 4.30.
Filter Selection for FMT
There are an enormous number of possible choices for the function (t) in FMT. Some authors like to
choose wavelet functions that have very low sidelobes but these often introduce intersymbol interference
while eliminating crosstalk. While wavelets may apply nicely in other areas of signal processing, they
are largely ineective in data transmission because they ignore the channel. Thus, pursuit of a large
361
Figure 4.30: FMT receiver block diagram assuming negligible or no inter-channel interference.
number of wavelet functions is probably futile and a digression from this text.
Instead, IBM has suggested a couple of functions for DSL applications that appear promising and
were designed with cognizance of the class of channels that are seen in those applications:
EXAMPLE 4.6.6 (Cherubinis Coaster) Cherubinis coaster functions use no excess
bandwidth and have = 0. The coaster is actually a set of functions parameterized by
0 < < 1 and the rst-order IIR lowpass lter
(D) =
_
1 +
2

1 +D
1 + D
. (4.242)
As 1, the functions become very sharp and the impulse response
k
is very long. Figure
4.31 illustrates these functions for a = .1. Note the rapid decay with frequency, basically
ensuring in practice that the basis functions remain orthogonal at the channel output. Note
also that there will be intersymbol interference, but as 1, this ISI becomes smaller and
smaller and the functions approach the ideal multitone functions (except this IIR lter is
stable and realizable). An DFE is typically used in the receiver with Cherubinis coaster.
The hope is that there is very little intersymbol or inter-channel interference and N can be
reduced (because the issue of /N excess bandwidth being small is inpertinent when = 0).
The discrete-time response is given by

k
=
_
1 +
2

_

k
+ (1 )()
k1
u
k1

. (4.243)
In this gure, zero-order-hold (at interpolation) is used between T
t
-spaced samples to pro-
vide a full Fourier transform of a continuous signal (such zero-order hold being characteristic
of the behavior of a DAC). This eect causes the frequency rippling in the adjacent bands
362
Figure 4.31: Cherubinis Coaster basis functions in frequency domain with = .1.
that is shown, but is quite low with respect to what would occur if
k
=
k
where this rip-
pling would follow a T sinc(fT) shaping with rst side lobe down only 13.5 dB from the
peak as in Figure 4.38 of Section 4.8 instead of the 55 dB shown with Cherubinis Coaster.
The second example uses excess bandwidth and raised-cosine shaping.
EXAMPLE 4.6.7 (Raised-Cosine with > 0) Raised cosine functions with excess band-
wisth (/N) can also be used. In this case the function
(i+)
k
can be simplied from the
square-root raised cosine in Section 3.2 to

(i)
k
=
4
_
1 +
N

T
t

cos
_

_
k +
i
N
_
+
1+
N

4[k+
i
N
]
sin
_

_
k +
i
N

N
N+
_
1 16
_
k +
i
N

2
. (4.244)
Warning - expression above rst-time evaluated by professor and not checked by plotting or
otherwise.
This function with suciently large number of taps in FIR approximation (for given excess
bandwidth, smaller /N means longer FIR lter in approximation) can be made to have
suciently low out-of-band energy and clearly avoids intersymbol interference if the tones
are suciently narrow.
363
4.7 Multichannel Channel Identication
Generally, channel identication methods measure the channel pulse response and noise power spectral
density, or their equivalents (that is direct measurement of SNRs without measuring pulse response and
noise separately). Multi-channel channel identication directly estimates signal and noise parameters for
each of the subchannels. Many channel identication methods exist, but only a DMT-specic method for
channel identication appears in this Chapter. Other methods are discussed in Chapter 7. The method of
this section separately identies rst the pulse-response DFT values at the subchannel center frequencies
used by a DMT system and then second the noise-sample variances at these same frequencies. Channel
identication then nishes by computing the SNRs from the results of gain and noise estimation. The
measurement of the pulse-response DFT values is called gain estimation
22
, while the measurement
of the noise variances is called noise estimation. This sections particular method is in common use
because it reuses the same hardware/software as steady-state transmission with DMT; however, it is
neither optimum nor necessarily a good choice in terms of performance for limited data observation. It
is easy to implement. Fortunately, a lengthy training interval is often permitted for DMT-transmission
applications, and this method will become very accurate with sucient observation. The presented
method is usually not appropriate for wireless OFDM systems.
The channel noise is an undesired disturbance in gain estimation, and this noise needs to be averaged
in determining subchannel gains as in Subsection 4.7.1. When gain estimation is complete, the estimated
(noiseless) channel output can be subtracted from the actual channel output to form an estimate of the
channel noise. The spectrum of this channel noise can then be estimated as in Section 4.7.2. In gain
estimation, care is taken to ensure that any residual gain-estimation error is suciently small that it
does not signicantly corrupt the noise-spectral-estimation process. Such small error can require large
dynamic range in estimating the subchannel gains.
Equation (4.139) shows that the channel SNR g
n
can be computed from the known subchannel input
distance, d
n
, the known multichannel normalized output distance d, and the noise estimate
2
n
. During
training, it is possible to send the same constellation on all subchannels, thus essentially allowing d
n
= d,
and thus making
g
n
=
1

2
n
. (4.245)
Two questions were not addressed, however, by Subsection ??, which are invidually the gain constants
and
t
for the updating loops for the multichannel normalizer W
n
in (4.133) and for the noise-variance
estimation in (4.137), respectively. These values will be provided shortly, but a more direct consideration
of computation of g
n
will be considered for the special case of training rst. The choice of is an example
of the general problem of gain estimation in Subsection 4.7.1, which is the estimation of the subchannel
signal gain. The choice of the second gain constant
t
is an example of the general problem of noise
estimation in Subsection 4.7.2, which is the estimation of the unscaled subchannel noise.
4.7.1 Gain Estimation
Figure 4.32 illustrates gain estimation. The known training sequence, x
k
, is periodic with period N
equal to or greater than the number of coecients in the unknown channel pulse response p
k
(periodic
can mean use of the cyclic prex of Section 4.4). The channel output sequence is
y
k
= x
k
p
k
+u
k
(4.246)
where u
k
is an additive noise signal assumed to be uncorrelated with x
k
. The channel distortion sequence
is denoted by u
k
, instead of n
k
, to avoid confusion of the frequency index n with the noise sequence and
because practical multichannel implementations may have some residual signal elements in the distortion
sequence u
k
(even though independence from x
k
is assumed for simplication of mathematics).
Gain estimation constructs an estimate of the channel pulse response, p
k
, by minimizing the mean
square of the error
e
k
= y
k
p
k
x
k
. (4.247)
22
This gain is usually a complex gain meaning that both subchannel gain and phase information are estimated.
364
Figure 4.32: Channel-identication basics and terminology.
Ideally, p
k
= p
k
.
The Discrete Fourier Transform, DFT, of x
k
is
X
n
=
1

N
N1

k=0
x
k
e

2
N
kn
. (4.248)
When x
k
is periodic
23
with period N samples, the DFT samples X
n
are also periodic with period N
and constitute a complete representation for the periodic time sequence x
k
over all time. The N-point
DFTs of y
k
, u
k
, p
k
, p
k
, and e
k
are Y
n
, U
n
,

P
n
, P
n
, and E
n
, respectively.
Since x
k
is periodic, then
Y
n
= P
n
X
n
+U
n
. (4.249)
The corresponding frequency-domain error is then
E
n
= Y
n

P
n
X
n
n = 0, ..., N . (4.250)
(Recall that subchannels 0 and N are one-dimensional in this development.) A vector of samples will
contain one period of samples of any of the periodic sequences in question, for instance
x =
_

_
x
N1
.
.
.
x
1
x
0
_

_
. (4.251)
Then the relations
X = Q

x x = QX (4.252)
23
The periodicity can also be emulated by a cyclic prex of length samples at the beginning of each transmitted block.
365
follow easily, where Q

is the matrix describing the DFT (q

N1i,N1k
= e

2
N
ki
) and QQ

= Q

Q =
I. Similarly, y, Y , e, E, p, P, p,

P, u and U as corresponding time- and frequency-domain vectors.
For the case of the non periodic sequences u
k
and thus y
k
and e
k
, a block time index l dierentiates
between dierent channel-output blocks (symbols) of N samples, i.e. y
l
.
The MSE for any estimate p is then
MSE = E
_
[e
k
[
2
_
(4.253)
=
1
N
E
_
|e|
2
_
=
1
N
N1

k=0
E
_
[e
k
[
2
_
(4.254)
=
1
N
E
_
|E|
2
_
=
1
N
N

n=0
E
_
[E
n
[
2
_
. (4.255)
The tap error is
k

= p
k
p
k
in the time domain and
n

= P
n


P
n
in the frequency domain. The
autocorrelation sequence for any discrete-time sequence x
k
is dened by
r
xx,k
= E
_
x
m
x

mk
_
(4.256)
with corresponding discrete power spectrum
R
xx,n
= Q r
xx,k
. (4.257)
The norm tap error (NTE) is dened as
NTE

= E
_
|p p|
2
_
= E
_
||
2
_
(4.258)
= E
_
|P

P|
2
_
= E
_
||
2
_
(4.259)
=
N

n=0
[P
n

P
n
[
2
. (4.260)
When the estimate p equals the channel pulse response p, then
e
k
= u
k
(4.261)
and
r
ee,k

= E
_
e
m
e

mk

= r
uu,k
k = 0, ..., N 1 , (4.262)
where r
ee,k
is the autocorrelation function of e
k
, and r
uu,k
is the autocorrelation function of u
k
. The
MSE is then, ideally,
MSE = r
ee,0
=
2
u
, (4.263)
the mean-square value of the channel noise samples.
2
u
=
A0
2
for the AWGN channel. In practice,
p
k
,= p
k
and the excess MSE is
EMSE

= MSE
2
u
. (4.264)
The overall channel SNR is
SNR =
r
xx,0
|p|
2
r
uu,0
(4.265)
and the SNR of the n
th
channel is
SNR
n
=
R
xx,n
[P
n
[
2
R
uu,n
. (4.266)
The gain-estimation SNR is a measure of the true channel-output signal power on any subchannel to
the excess MSE in that same channel (which is caused by p ,= p), and is

=
R
xx,n
[P
n
[
2
R
ee,n
R
uu,n
. (4.267)
366
EXAMPLE 4.7.1 (Accuracy)
n
should be 30-60 dB for good gain estimation. To un-
derstand this requirement on
n
, consider the case on a subchannel in DMT with SNR
n
=
Rxx,n]Pn]
2
Ruu,n
= 44 dB, permitting 2048 QAM to be transmitted on this subchannel. The noise
is then 44 dB below the signal on this subchannel.
To estimate this noise power to within .1 dB of its true value, any residual gain estimation
error on this subchannel, R
ee,n
R
uu,n
, should be well below this mean-square noise level.
That is
R
ee,n
< 10
.1/10
R
uu,n
approx(1 + 1/40) R
uu,n
(4.268)
Then,

n
=
R
xx,n
[P
n
[
2
R
ee,n
R
uu,n
=
SNR
n
(1/40)
= 40 snr (4.269)
is then 60 dB (= 44+16 dB). Similarly, on a comparatively weak subchannel for which we
have SNR
i
=14 dB for 4-point QAM, one would need
n
of at least 30 dB (= 14+16).
In general, for x dB accuracy

n

_
10
x/10
1
_
1
SNR
n
. (4.270)
An Accurate Gain Estimation method
A particularly accurate method for gain estimation uses a cyclically prexed training sequence x
k
with
period, N, equal to or slightly longer than , the length of the equalized channel pulse response. Typi-
cally, N is the same as in the DMT transmission system. The receiver measures (and possibly averages
over the last L of L + 1 cycles) the corresponding channel output, and then divides the DFT of the
channel output by the DFT of the known training sequence.
The channel estimate in the frequency domain is

P
n
=
1
L
L

l=1
Y
l,n
X
l,n
. (4.271)
Performance Analysis The output of the n
th
subchannel on the l
th
symbol produced by the receiver
DFT is Y
l,n
= P
n
X
l,n
+U
l,n
. The estimated value for

P
n
is

P
n
= P
n
+
L

l=1
U
l,n
L X
l,n
, (4.272)
so

n
=
L

l=1
U
l,n
L X
l,n
. (4.273)
By selecting the training sequence, X
l,n
, with constant magnitude, the training sequence becomes X
n
=
[X[e
l,n
.
The error signal on this subchannel after

P
n
has been computed is
E
n
= Y
n

P
n
X
n
=
n
X
n
+U
n
(4.274)
= U
n
+
1
L
L

l=1
U
l,n
e
(nl,n)
. (4.275)
The phase term on the noise will not contribute to its mean-square value. U
n
corresponds to a block
that is not used in training, so that U
n
and each of U
l,n
are independent, and
E
_
[E
n
[
2
_
= R
ee,n
= R
uu,n
+
1
L
R
uu,n
= (1 +
1
L
)R
uu,n
. (4.276)
367
The excess MSE on the i
th
subchannel then has variance (1/ L)R
uu,n
that is reduced by a factor equal
to the the number of symbol lengths that t into the entire training interval length, with respect to the
mean-square noise in that subchannel. Thus the gain-estimation SNR is

n
= L
R
xx,n
[P
n
[
2
R
uu,n
= L SNR
n
, (4.277)
which means that the performance of the recommended gain estimation method improves linearly with
L on all subchannels. Thus, while noise may be relatively higher (low SNR
n
) on some subchannels than
on others, the extra noise caused by misestimation is a constant factor below the relative signal power
on that same subchannel. In general then for x dB of accuracy
L
_
10
x/10
1
_
1
. (4.278)
For an improvement in gamma 16 dB, as determined earlier in an example provides .1 dB error or
less no matter what the subchannel SNR
n
, gain estimation requires L = 40 symbols of training.
For instance, the previous ADSL and VDSL examples of Section 4.3 used symbol periods of 250s.
Thus, gain estimation to within .1 dB accuracy is possible (on average) within 10ms using this simple
method.
Windowing When the FFT size is much larger than the number of signicant channel taps, as is
usually the case, it is possible to signicantly shorten the training period. This is accomplished by
transforming (DFT) the nal channel estimate to the time domain, windowing to +1 samples and then
returning to the frequency domain. Essentially, all the noise on sample times + 2...N is thus removed
without disturbing the channel estimate. The noise on all estimates, or equivalently the residual error
in the SNR, improves by the factor N/ with windowing. For the ADSL/VDSL example again, if the
channel were limited to 32 nonzero successive samples in duration, then windowing would provide a
factor of (32/512) reduction or 12 dB of improvement, possibly allowing just a few DMT symbols to
have an (on average) accuracy of .1 dB.
Suggested Training Sequence
One training sequence for the gain estimation phase is a random or white sequence
24
L blocks are used
to create

P
n
, which is computed recursively for each of the subchannels, one symbol at a time, by
accumulation, and then divided by L. The L blocks can be periodic repeats or cyclic prex can be used
to make the training sequence appear periodic. For reasons of small unidentied channel aberrations,
like small nonlinearities caused by drivers, converters, ampliers, it is better to use a cyclic prex because
any such aberrations will appear in the noise estimation to be subsequently described.
In complex baseband channels a chirp training sequence sometimes appears convenient
x
k
= e

2
N
k
2
. (4.279)
This chirp has a the theoretical minimum peak-to-average (one-dimensional) power ratio of 2 and is a
white sequence, with r
xx,k
=
k
, with
k
being the Kronecker delta (not the norm tap error). While
white is good for identifying an unknown channel, a chirps low PAR can leave certain small nonlinear
noise distortion unidentied, which ultimately would cause the identied SNR
n
s to be too optimistic.
The value for in the FEQ The EMSE for the rst-order zero-forcing update algorithm in Equation
(4.133) can be found with some algebra to be
EMSE =
c
n
2 c
n
R
uu,n
. (4.280)
24
Say the PRBS of Chapter 10 with polynomial 1 + D + D
16
) with bn = 2 on channels n = 1, ..., N 1 and bn = 1 on
the DC and Nyquist subchannels. One needs to ensure that the seed for the PRBS is such that no peaks occur that would
exceed the dynamic range of channel elements during gain estimation. Such peaks are usually compensated by a variety
of other mechanisms when they occur in normal transmission (rarely) and would cause unnecessarily conservative loading
if their eects were allowed to dominate measured SNRs.
368
Thus, for the multichannel normalizer to maintain the same accuracy as the initial gain estimation in
Equation (4.276)
1
L
=
c
n
2 c
n
(4.281)
or
=
2
(L + 1) c
n
, (4.282)
so for L = 40 and c
n
= 1, the value would be = 1/42.
4.7.2 Noise Spectral Estimation
It is also necessary to estimate the discrete noise power spectrum
25
, or equivalently the mean-square noise
at each of the demodulator DFT outputs, computation of the SNRs necessary for the bit-distribution.
Noise estimation computes a spectral estimate of the error sequence, E
n
, that remains after the channel
response has been removed. The training sequence used for gain estimation is usually continued for noise
estimation.
Analysis of Variance
The variance of L samples of noise on the n
th
subchannel is:

2
n
=
1
L
L

l=1
[E
l,n
[
2
. (4.283)
For stationary noise on any tone, the mean of this expression is the variance of the (zero-mean) noise.
The variance of this variance estimate is computed as
var
_

2
n
_
=
1
L
2
_
3 L
4
n
L (
2
n
)
2
_
=
2
L

4
n
(4.284)
for Gaussian noise. Thus the excess noise in the estimate of the noise is therefore
_
2/L
2
n
and the
estimate noise decreases with the square-root of the number of averaged noise terms. To make this
extra noise .1 dB or less, we need L = 3200. Recalling that the actual computed value for the noise
estimate will have a standard deviation on average that leads to less than .1 dB error, there is still
a 10% chance that the statistics of the noise will lead to deviation greater than 1 standard deviation
(assuming Gaussian distribution, or approximating the more exact Chi-square with a Gaussian). Thus,
sometimes a yet larger value of L is used. 98% condence occurs at L = 12, 800 and 99.9% condence
at L = 28, 800.
EXAMPLE 4.7.2 (ADSL) The ADSL example with symbol rate of 4000 Hz has been
discussed previously. The training segment for SNR estimation in G.992.1 initially is called
Medly and consists of 16000 symbols (with QPSK on all subchannels) for 4 seconds of train-
ing. This exceeds the amount given as a guideline above - this extra time is provided with
the cognizance that the standard deviation of the measured noise and channel should lead to
no more than .1 dB error with a smaller training interval, but that there are still statistical
deviations with real channels. Essentially, with this increased value, about 99% of the time
the computed value will deviate by this amount of .1 dB or less. The gain estimation error
clearly can be driven to a negligible value by picking L to a few hundred without signicantly
aecting training time. (The long training interval also provides time for SNR computation
and loading algorithms after gains and noises are estimated.)
25
Clearly the noise can not have been whitened if it is unknown and so this step precedes any noise whitening that might
occur with later data transmission.
369
The value for
t
in the FEQ The standard deviation for the rst-order noise update in the FEQ
can be found to be
std(

2
n
) =

t
2
t

2
n
. (4.285)
Thus, to maintain a constant noise accuracy, the step size should satisfy
2
L
=

t
2
t
(4.286)
or

t
=
4
L + 2
(4.287)
so when L = 6400, then
t
6.25 10
4
. In reality, the designer might want high condence (not just
an average condence) that the accuracy is .1 dB, in which case a of perhaps 10
4
would suggest the
chance of missing would be the same as noise occuring with 4 times the standard deviation (pretty small
in Gaussian and Xi-square distributions).
4.7.3 SNR Computation
The signal to noise ratio is then estimated for each subchannel independently as
SNR
n
= c
n
[

P
n
[
2

2
n
. (4.288)
Equivalently, using the FEQ, the SNR is given by (4.245). Loading can be executed in the receiver
(or after passing SNR values, in the transmitter) and then the resulting b
n
and c
n
(or relatively in-
crease/decrease in c
n
with respect to what was used last) are sent to the transmitter via the reverse link
always necessary in DMT. This step is often called the exchange in DMT modems and occurs as a
last step prior to live transmission of data, often called showtime!
370
4.8 Stationary Equalization for Finite-length Partitioning
A potential problem in the implementation of DMT/OFDM and VC channel-partitioning methods is the
use of the cyclic prex or guard period of an extra samples, where +1 is the length in sampling periods
of the channel pulse response. The required excess-bandwidth is then /N. On many practical channels,
can be large - values of several hundred can occur in both wireless and wired transmission problems,
especially when channels with narrowband noise are considered. To minimize the excess bandwidth, N
needs to be very large, potentially 10,000 samples or more. Complexity is still minimal with DMT/OFDM
methods with large N, when measured in terms of operation per unit time interval. However, large
N implies large memory requirement (to store the bit tables, energy tables, FEQ coecients, and
intermediate FFT/IFFT results), often dominating the implementation. Further, large N implies longer
latency (delay from transmitter input to receiver output) in processing. Long latency complicates
synchronization and can disrupt certain higher-level data protocols used and/or with certain types of
carried applications signals (like voice signals).
One solution to the latency problem, invented and rst studied by J. Chow (not to be confused
with the P. Chow of Loading Algorithms), is to use a combination of short-length xed (i.e. stationary)
equalization and DMT/OFDM (or VC) to reduce , and thereby the required N, to reach the highest
performance levels with less memory and less latency. The equalizer used is known as a time equal-
izer (TEQ), and is studied in this section. Chows basic MMSE approach is shown here to result in
an eigenvector-solution problem that can also be cast in many other generalized eigenvector forms.
Various authors have studied the TEQ since J. Chow, and have claimed improvements: Actually, the
basic theory developed by Chow is better than any of the subsequent generalized-eigenvector work
26
.
Al-Dhahir and Evans have developed yet better theories for the TEQ, but the solution will not be easily
obtained in closed form and requires advanced numerical optimization methods for non-convex optimiza-
tion Chows TEQ usually performs very close to those methods. Essentially these methods attempt
to maximize the overall SNR
dmt
by optimizing equalizer settings and energy/bit distributions jointly. A
simpler approach using cyclic reconstruction appears later in this chapter.
Subsection 4.8.1 studies the general data-aided equalization problem with innite-length equalizers.
The TEQ is there seen to be a generalization of the feedforward section of a DFE. Subsection 4.8.2 then
reviews performance calculation of the nite-length TEQ for a target that is potentially less than the
channel pulse-response length. Subsection 4.8.3 illustrates the zero-forcing white-input specialization
of Chows TEQ, which is found to in that special lower-performance situation is equivalent to many
developed by Melsa and Rohrs, by Gatherer, by Bingham, by Pal, and by Ilani using what are called
windows and walls (or alternatively a series of projections). An example introduced in Subsection
4.8.2 and revisited in Subsection 4.8.3 illustrates why the latter approaches might be avoided. Subsection
4.8.5 investigates the joint loading/equalizer design problem and then suggests a solution that could be
implemented.
The very stationarity of the TEQ renders it suboptimum because the DMT/OFDM or VC system is
inherently cyclostationary with period N + samples. There are 3 simplications in the development
of the TEQ that are disturbing:
1. The equalizer is stationary and should instead be periodic with period N +.
2. The input sequence x
k
will be assumed to be white or i.i.d. in the analysis ignoring any loading
that might have changed its autocorrelation.
3. A scalar mean-square error is minimized rather than the multichannel SNR SNR
m,u
.
Nonetheless, TEQs are heavily used and often very eective in bringing performance very close to
the innite-length multitone/modal-modulation optimum. An optimum equalizer is cyclostationary or
periodic and found in Chapter 5, but necessarily more complex. When such an cyclic-equalizer is used,
then problems 2 and 3 are eliminated. If the stationary xed equalizer is used, Al-Dhahir and Evans
have considered solutions to problems 2 and 3, and also a newer solution is posed in Subsection 4.8.5.
26
Most of the latter generalized-eigenvector work focuses its criticism on an algorithm given by Chow to be implemented
in the frequency-domain, and thus fails to observe that his early theory that the algorithm tried to approximate in actual
implementation. His implementation algorithmto approximate the theory is not the best to use today, but the basic theory
still provides a better equalizer.
371
Figure 4.33: The Maximum Likelihood Innite-Length Equalization problem.
4.8.1 The innite-length TEQ
The general baseband complex decision-aided, or in our study input-aided, equalization problem ap-
peared in Figure 4.33. The channel output y
p
(t) is
y
p
(t) =

k
x
k
p(t kT) + n
p
(t) . (4.289)
The channel output signal y
p
(t) is convolved with a matched (or anti-alias) lter

p
(t) (where a su-
perscript of denotes conjugate), sampled at rate 1/T, and then ltered by a linear equalizer. The
matched-lter output is information lossless. When the equalizer is invertible, its output is also in-
formation lossless with respect to the channel input. Thus, a maximum-likelihood detector applied to
the signal at the output of the equalizer w
k
in Figure 4.33 is equivalent in performance to the overall
maximum-likelihood detector for the channel.
Heuristically, the function of the linear equalizer is to make the equalizer output appear like the
output of the second lter, which is the convolution of the desired channel shaping function b
k
and the
known channel input x
k
. The target b
k
has a length + 1 samples that is usually much less than the
length of the channel pulse response. The minimum mean-square error between the equalizer output
and the desired channel shaping is a good measure of equalizer performance. However, the implication
that forcing the channels response to fall mainly in + 1 sample periods only loosely suggests that
the geometric-SNR for the set of parallel subchannels will be maximized. Also, an imposition of a
xed intersymbol interference pattern, rather than one that varies throughout a symbol period is also
questionable. Nonetheless, this method will be eective in a large number of practical situations. Further
development requires some review and denition of mathematical concepts related to the channel.
Minimum Mean-Square Error Equalization
The error sequence in Figure 4.33 is
e
k
= b
k
x
k
w
k
y
k
(4.290)
where denotes convolution. The signal is dened as the sequence b
k
x
k
, which has signal energy
c
s
= |b|
2
c
x
. The D-Transformof a sequence x
k
remains X(D)

=

k
x
k
D
k
. The notation x

(D

) =

k
x

k
D
k
represents the time-reversed conjugate of the sequence. Thus,
E(D) = B(D)X(D) W(D)Y (D) . (4.291)
The minimum mean-square error is

2
MMSE
= min
wk , bk
E
_
[e
k
[
2

. (4.292)
372
A trivial solution is W(D) = B(D) = 0 with
2
MMSE
= 0. Aversion of the trivial solution requires
positive signal power so that c
s
> 0 or, equivalently, |b|
2
= constant = |p|
2
> 0. For at white
excitation (i.e., S
x
(D) =

c
x
) of the channel dened by B(D),
SNR
MFB
=
c
s

2
MMSE
. (4.293)
When |b|
2
= [|p|
2
> 0 is constrained constant, SNR
MFB
is independent of |b|
2
. Maximizing SNR
MFB
is then equivalent to minimizing
2
MMSE
. While this problem formulation is the same as the formulation
of decision feedback equalization, the TEQ does not restrict B(D) to be causal nor monic.
The orthogonality principle of MMSE estimation states that the error sequence values e
k
should be
orthogonal to the inputs of estimation, some of which are the samples y
k
. Thus, E[e
k
y

l
] = 0 k, l,
which is compactly written
R
ey
(D) = 0 (4.294)
= E
_
E(D) Y

(D

(4.295)
= B(D)

R
xy
(D) W(D)

R
yy
(D) . (4.296)
then,
W(D) = B(D)
R
xy
(D)
R
yy
(D)
. (4.297)
Then,
Y (D) = |p|Q(D)X(D) + n(D) , (4.298)
where Q(D)

=
1
2|p|
2
_

n
[P
_
2n
T
_
[
2
d. Then,

R
xy
(D) =

c
x
|p| Q(D) (4.299)

R
yy
(D) =

c
x
|p|
2
Q(D)
_
Q(D) +
1
SNR
MFB
_
. (4.300)
Recalling the canonical factorization from Chapter 3 DFE design,
Q(D) +
1
SNR
MFB
=
0
G(D) G

(D

) . (4.301)
This factorization is sometimes called the key equation.
The error autocorrelation sequence is
R
ee
(D) = B(D)
_
R
xx
(D)
R
xy
(D)
R
yy
(D)
R
yx
(D)
_
B

(D

) , (4.302)
where the reader may recognize the autocorrelation function of the MMSE linear equalizer error in the
center of the above expression. We simplify this expression further using the key equation to obtain
R
ee
(D) =
_

c
x

0
SNR
MFB
_

B(D) B

(D

)
G(D) G

(D

)
. (4.303)
Dening the (squared) norm of a sequence as
|x|
2

=

k=
[x
k
[
2
, (4.304)
the mean square error from (4.303) is
MSE =
_

c
x

0
SNR
MFB
_
|
b
g
|
2
. (4.305)
373
From the Cauchy-Schwarz lemma, we know
|b|
2
= |
b
g
g|
2
< |
b
g
|
2
|g|
2
(4.306)
or
|
b
g
|
2
>
|b|
2
|g|
2
. (4.307)
Thus, the MMSE is

2
MMSE
=
_

c
x

0
SNR
MFB
_
, (4.308)
which can be achieved by a multitude of B(D) if the degree of G(D) is less than or equal to
27
, only
one of which is causal and monic.
The MMSE solution has any B(D) that satises
B(D) B

(D

) = G(D) G

(D

) . (4.309)
From (4.297) to (4.301), one nds W(D) from B(D) as
W(D) =
1

0
|p|
B(D)
G(D) G

(D

)
, (4.310)
which is always invertible when
A0
2
> 0. The various solutions dier in phase only. The alert reader
may note that such a solution is biased and the equalized channel is not exactly B(D) = b

i D
i
+... +
b
i+
D
i+
and in fact will be reduced by by the scale factor w
0
/b
i
1. Such a bias can be removed
by the inverse of this factor without changing performance and the channel will then be exactly B(D)
with an exactly compensating increase in error variance.
Solutions
As mentioned earlier, there are many choices for B(D), each providing then a corresponding TEQ W(D)
given by Equation (4.310).
The MMSE-DFE The Minimum-Mean-Square Error Decision Feedback Equalizer (MMSE-DFE)
chooses causal monic B(D) as
B(D) = G(D) so W(D) =
1

0
|p| G

(D

)
, (4.311)
is anti-causal.
In ML detection that is DMT or VC detection with the MMSE-DFE choice of TEQ lter W(D),
the feedback lter B(D) is never implemented avoiding any issue of error propagation or need for
precoders and their correspodning loss. That ML detector is easily implemented in the case of DMT
or VC (but requires exponentially growing complexity, see the Viterbi MLSD method of Chapter 9, if
PAM/QAM is used).
The MMSE-AR Equalizer The MMSE Auto Regressive Equalizer (MMSE-ARE) corresponds to
the anti-causal monic B(D) choice
B(D) = G

(D

) so W(D) =
1

0
|p| G(D)
, (4.312)
is causal. This is another possible solution and the MMSE level and the SNR
MFB
are the same as all the
other optimum solutions, including the MMSE-DFE. One would not make the choice in (4.312) if sample-
by-sample decisions are of interest because neither future decisions are available in the receiver nor can
a Tomlinson precoder be implemented (properly) with a noncausal B(D). However, such causality is
not an issue with the TEQ and ML detection via either DMT or VC.
27
If the degree is greater than , then the methods of Section 4.8.2 must instead be used directly.
374
Comparison of MMSE-DFE and MMSE-ARE
Both of the MMSE-DFE and the MMSE-ARE equalizer have the same performance (when ML detection
is used). Essentially the causality/noncausality conditions of the feedforward lters and feedback lters
have been interchanged. Because B(D) is never used in the receiver (only for our analysis), either
structure could be used in a practical situation. A multicarrier modulation method used with W(D)
equalization would also perform the same for either choice of W(D) (or any other optimum choice).
Thus, it would appear that since all structures are equivalent, we might as well continue with the
use of the MMSE-DFE, since it is a well-known receiver structure. However, when nite-length lters
are used, this can be a very poor choice in attempting to maximize performance for a given complexity
of implementation. As a trivial example, consider a channel that is maximum phase, the feedforward
lter for a MMSE-DFE, in combination with the matched lter, will try to convert the channel to
minimum-phase (at the output of W(D), leading to a long feedforward section when (as in practice) the
matched-lter and feedforward lter are implemented in combination in a fractionally spaced equalizer.
On the other hand, the MMSE-ARE equalizer feedforward section (combined with matched lter) is
essentially (modulo an anti-alias lter before sampling) one nonzero tap that adjusts gain. The opposite
would be true for a minimum-phase channel.
In data transmission, especially on cables or over wireless transmission paths, the channel character-
istic is almost guaranteed to be of mixed-phase if any reections (multipath, bridge-taps, gauge-changes,
slight imbalance of terminating impedances) exist. It is then natural (and correct) to infer that the best
input-aided equalization problem is one that chooses B(D) and W(D) to be both mixed-phase also. The
problem then becomes for a nite-length feedforward section (W(D)) of M + 1 taps and a nite-length
channel model of + 1 taps (B(D)), to nd the best B(D) and W(D) for an ML detector designed for
B(D), the signal constellation, and additive white Gaussian noise (even if the noise is not quite Gaussian
or white). This is the problem addressed in Section 4.8.2 and the solution is often neither MMSE-DFE
nor MMSE-ARE ltering.
4.8.2 The nite-length TEQ
Like the DFE of Section 3.6, the nite-length MMSE-TEQ problem is characterized by the error
e
k
() = b

x
k
w

y
k
(4.313)
where
x
k

=
_

_
x
k
x
k1
.
.
.
x
k
_

_
; (4.314)
b

= [b

0
b

1
... b

] is not necessarily causual, monic nor minimum-phase; and w

= [w

0
w

1
... w

L1
],
L is the number of equalizer coecients and may be at another sampling rate as in Section 3.6 for the
fractionally spaced case
28
; and
y
k

=
_

_
y
k
y
k1
.
.
.
y
kL+1
_

_
. (4.315)
The length is typically less than the length of the channel pulse response and equal to the value
anticipated for use in a multichannel modulation system like DMT/OFDM or VC.
The input/output relationship of the channel is
y
k
= Px
k
+ n
k
. (4.316)
28
The equalizer output should be decimated to some integer multiple of the channel-input sampling rate. Usually that
integer multiple is just unity in DMT and vector coding designs because that sampling rate is already greater than twice
the highest frequency likely to be used. However, as Pal has noted, with very short equalizers, sometimes an oversampling
factor greater than 1 can improve performance for xed complexity, depending on channel details.
375
P need not be a convolution matrix and can reect whatever guard bands or other constraints have been
placed upon the transmit signal. However, in most developments, this would render the channel output
cyclostationary, so we will assume P is a convolution matrix of the exact form discussed for DFEs in
Chapter 3.
The MSE is minimized with a constraint of |b|
2
= c a constant, with c usually chosen as c = |p|
2
to force the at-input SNR
MFB
of the channel to be the at-input SNR
MFB
of the TEQ problem.
However, this non-zero constant produces the same TEQ shape and does not change performance but
simply avoids the trivial all-zeros solution for w. Minimization rst over w forces
E[e()y

k
] = 0 . (4.317)
The solution forces
w

= b

R
xy
()R
yy
1
(4.318)
for any b. Then, e() becomes
e() = b

(x
k
R
xy
()R
yy
1
y) . (4.319)
(4.319) has a vector of the minimized MSE nite-length LE errors within the parentheses. This MMSE-
LE error vector has an autocorrelation matrix that can be computed as
R
LE
() = c
x
I R
xy
()R
yy
1
R
yx
() . (4.320)
Tacit in (4.320) is the assumption that R
yy
is a xed L L matrix, which is equivalent to assuming
that the input to the channel is stationary as is the noise. In the case of transmission methods like
DMT and VC, this stationarity is not true and the input is instead cyclostationary over a symbol
period. Furthermore, the initial solution of the TEQ is often computed before transmit optimization
has occurred so that R
xx
is often assumed to be white or a diagonal matrix as will also be the case in
this development. With these restrictions/assumptions, the overall MMSE is then the minimum of
MMSE = min
b,
b

R
LE
()b . (4.321)
with the constraint that |b|
2
= c. A biased matched-lter bound SNR for the TEQ-equalized channel
is then
SNR
MFB
(bias) =

c
x
|b|
2

min
|b|
2
= 1 + SNR
MFB
(unbias) , (4.322)
as with all MMSE equalizers, there is a bias that is removed by scaling the equalizer output by
SNR
MFB
(bias)
SNR
MFB
(unbias)
. (4.323)
However, such bias removal is not necessary in the DMT systems because of the ensuing FEQ that will
remove it automatically in any case. The solution of this problem is easily shown to be the eigenvector
of R
ee
that corresponds to the minimum eigenvalue. Thus
b =

c q
min
() . (4.324)
Thus, the TEQ setting is determined by computing R
ee
() for all reasonable values of and nding
the MMSE () with the smallest value. The eigenvector corresponding to this is the correct setting
for b (with scaling by

c) and w, the TEQ, is determined through Equation (4.318 ).
An alternative notes that equation (4.320 can also be written
R
LE
() = c
x
I

c
2
x
[010] P

R
yy
1
P [010] , (4.325)
where the position of the 1 determines . Then, the best delta corresponds to the maximumeigenvalue
of the matrix P

R
yy
1
P.
376
0 1 2 3 4 ...
c
lost
4.26 3.45 2.99 2.26 1.83 ... 0
dB loss =
5.26Llost
5.26
-7.2 -4.6 -3.6 -2.4 -1.9 ... 0
Table 4.6: Table of energy loss by truncation (rectangular windowing) of p
k
= .9
k
u
k
.
The L L matrix R
yy
need only be inverted once, but the eigenvectors must be computed several
times corresponding to all reasonable values of . This computation can be intensive. For FIR channels,
the matrix R
yy
is given by
R
yy
= PR
xx
P

+R
nn
general non-white input case (4.326)
=

c
x
PP

+R
nn
usual white-input case. (4.327)
Any stationary R
xx
can be assumed. If a nonstationary (perhaps an R
xx
representing some known
input optimization and cyclostationarity) R
xx
is used, then R
yy
becomes cyclostationary also and a
function of both position within a symbol. The cross correlation matrix is

R
xy
(D)() = E[x
k
y
k
] = E[x
k
x

k
] P

. (4.328)
The cross-correlation matrix will have zeros in many positions determined by unless again an optimized
input is being used and then again this matrix would also become a function of both and the position
within a symbol. Because it is hard to know which position within a symbol to optimize or to know in
advance the optimized spectrum used on the channel (which usually is aected by the TEQ), the TEQ
is set rst prior to channel identication, partitioning, and loading. It could be periodically updated.
EXAMPLE 4.8.1 (IIR Channel 1/1 .9D) A discrete-time channel might have an IIR
response so that no nite is ever completely sucient. An example is the single-pole IIR
digital channel
P(D) =
1
1 .9D
, (4.329)
with time-domain response easily found to be
p
k
= .9
k
k 0 . (4.330)
The gain of the channel is |p|
2
=
1
1.9
2
= 5.2632. Let the AWGN variance per real dimension
be
2
= .1 and the input energy is

c
x
= 1. If PAM were used on this channel, then
SNR
MFB
= 17.2 dB. It is interesting to investigate the amount of energy with guard period
of samples as varies. The total energy outside of a guard period of length is then
|p|
2

k=0
1 (.81)
+1
1 .81
. (4.331)
This lost energy, which becomes distortion for the DMT or VC system is in Table 4.6
To have only .1 dB loss, then > 16, meaning N > 500 for small loss in energy and
bandwidth. Alternatively, one can use a TEQ, which will reshape the channel so that most
of the energy is in the rst coecients. Because of the special nature of a single-pole IIR
channel, it is pretty clear that a 3-tap equalizer with characteristic close to 1 .9D will
reduce substantially while altering the noise. Thus, it is fairly clear that an equalizer with
L = 3 taps and delay = 0 should work fairly well. We select = 1 also for this example.
For these values, some algebra leads to

R
xy
(D)(0) =
_
1 0 0
.9 1 0
_
(4.332)
377
and
R
yy
=
_
_
5.3636 .9(5.2636) .81(5.2636)
.9(5.2636) 5.3636 .9(5.2636)
.81(5.2636) .9(5.2636) 5.3636
_
_
. (4.333)
The error autocorrelation matrix is
R
ee
=
_
.1483 .0650
.0650 .1472
_
, (4.334)
with eigenvalues .2128 and .0828 and corresponding eigenvectors [.7101 .7041] and [.7041 .7101]
respectively. Then R
ee
has smallest eigenvalue .0828 with corresponding MMSE=.4358,
much less than the
2
+ 3.45 = .1 + 3.45 = 3.55 (3.45 is the entry for energy loss in Table
4.6 above when = 1) for no equalization. Obviously the equalizer reduces distortion signif-
icantly with respect to no use of equalizer. The corresponding setting for b from Equation
(4.324) is
B(D) = |p| [.7041 .7101] =

5.2632 [.7041 .7101] = 1.6151 [1 + 1.0084D] . (4.335)
B(D) is not minimum phase in this example, illustrated that the TEQ is not simply a
DFE solution. The corresponding value for w from Equation (4.318) leads to W(D) =
1.4803 (1 + .1084D .8907D
2
) and a new equalized channel
z(D) = 1.4803
_
1 + .1084D .8907D
2
_

X(D)
1 + .9D
+W(D) N(D) (4.336)
= 1.4803 [1 + (.1084 + .9)D] + 1.4803 (.81 +.9 .1084 .8907)
D
2
X(D)
1 + .9D
+ W N
= 1.4803 [1 + (.1084 + .9)D]
. .
w
0
b
0
B(D)
+1.4803(.01686)
X(D) D
2
1 + .9D
+W(D) N(D)
. .
w
0
b
0
E(D)
.
The bias in estimating B(D) is w
0
/b
0
and can be removed by multiplying the entire received
signal by the constant b
0
/w
0
without altering performance this step is usually not neces-
sary in DMT systems because the multichannel normalizer automatically includes any such
adjustment. The PSD of the distortion can be determined as
R
EE
(D)
_
w
0
b
0
_
2
= 1.4803
2
(.01686)
2
1
1 .9
2
+ .1 W(D) W

(D

) (4.337)
= 1.4803
2

_
(.001496) + .1
_
1 + .1084
2
+.8907
2
_
(4.338)
r
EE
(0)
_
w
0
b
0
_
2
= .3988 = .4538
_
w
0
b
0
_
2
. (4.339)
The ratio of signal energy in the rst two terms to distortion is then 1.4803
2
(1+1.0084
2
)/.3988
or 10.4 dB. This signal to noise ratio cannot be signicantly improved with longer equalizers,
which can be vered by trial of larger lengths, nor is > 1 necessary as one can determine
by comparing with an innite-length MT result for this channel.
This new channel plus distortion can be compared against the old for any value of N in VC
or DMT, with an obvious clear improvement, as in the homework. The equalized channel is
shown in Figure 4.34 where it is clear the rst two positions contain virtually all the energy.
Figure 4.35 shows the power spectrum of the error, which in this case is almost at (see the
vertical scale) so one could apply loading directly to the subchannels as if the noise were
white and of variance .3988 without signicant error. Without this TEQ, the loss in
performance nearly 10 times higher.
378
Figure 4.34: Equalized response for 3-tap TEQ with 1/(1 .9D) channel.
Figure 4.35: Distortion Spectrum for 3-tap TEQ on 1/(1 .9D) channel with white noise at
2
= .1.
379
The TEQ performance increase over no equalizer can indeed be much greater than 4.2 dB on channels
with more severe responses. (See for instance Salvekars Marathon in Problem 4.25.) The TEQ is not
a DFE and does not maintain white noise nor channel magnitude in the nite-length case.
The issue of bias can return in MMSE design and is somewhat more involved than in Chapter 3. It
was removed as a part of Example 4.34. To calculate the correct unbiased distortion energy in general
E

(D) = B(D) X(D) D

W(D) Y (D) (4.340)

= B(D) X(D) D

W(D) P(D) X(D) + W(D) N(D) (4.341)

By noting that the equalizer output component W(D) P(D) X(D + W(D) N(D) will in general be
scaled by from the desired B(D) D

X(D), then
E

(D) = B(D) X(D) D

B(D) D

X(D) E
U
(D) (4.342)
= (1 ) X(D) B(D) D

E
U
(D) . (4.343)
Bias removal then would occur by multiplying the equalizer output by 1/, which would more than
likely be absorbed into the equalizer as in the example. The bias factor itself is obtained by rst forming
the convolution C(D) = W(D) P(D), which is the noise-free equalized output. The bias factor diers
slightly with B(D) ,= 1 from the simple SNR/(SNR + 1) of Chapter 3. The bias will then be the ratio
= c

/b
0
. (4.344)
To compute the r
u,E
(4.343) leads to
MMSE = (1 )
2

c
x
|b|
2
+
2
r
u,E
(4.345)
= (1 )
2

c
x
|p|
2
+
2
r
u,E
(4.346)
=
min
|p|
2
. (4.347)
Thus,
r
u,E
=
|p|
2
_

min
(1 )
2

c
x

2
(4.348)
provides the value for r
u,E
. The unbiased SNR
MFB
for the equalized channel with included to remove
bias is then
SNR
MFB
=

c
x
|p|
2
r
u,E
. (4.349)
The following second example shows a near equivalence of a DFE and an equalized DMT system
when both use exactly the same bandwidth, a concept explored and explained further in Chapter 5.
EXAMPLE 4.8.2 (Full-band TEQ) A 7-tap FIR channel is shortened to = 3 (4 taps)
with an 11-tap equalizer in this example and then DMT applied. The channel is such that
all 128 tones of the DMT system are used. This is compared against a MMSE-DFE with
20 total taps (14 feedforward, 6 feedback). The matlab commands are inserted here and the
reader can then follow the design directly:
>> [snr,w]=dfecolor(1,p,14,6,13,1,[.1 zeros(1,13)])
snr = 16.8387
w = -0.0046 0.0056 0.0153 -0.0117 -0.0232 0.0431 0.0654 -0.0500 -0.0447
0.2263 0.2748 -0.0648 0.1642 -0.1479
feedback is -0.0008 -0.7908 0.0001 0.0473 0.0001 0.1078
------------------------------------------------------------------------------------------------
>> p=[-.729 .81 -.9 2 .9 .81 .729 ];
>> np=norm(p)^2
np = 7.9951
>> P=toeplitz([p(1) , zeros(1,10)],[p, zeros(1,10)]);
380
some entries in P are:
P = Columns 1 through 8
-0.7290 0.8100 -0.9000 2.0000 0.9000 0.8100 0.7290 0
0 -0.7290 0.8100 -0.9000 2.0000 0.9000 0.8100 0.7290
0 0 -0.7290 0.8100 -0.9000 2.0000 0.9000 0.8100
0 0 0 -0.7290 0.8100 -0.9000 2.0000 0.9000
0 0 0 0 -0.7290 0.8100 -0.9000 2.0000
0 0 0 0 0 -0.7290 0.8100 -0.9000
>> ryy=P*P+.1*eye(11);
The delay after some experimentation was found to be = 10 for an 11-tap TEQ on this
particular channel. This example continues then in Matlab according to:
>> rxy=[ zeros(4,10) eye(4) zeros(4,3)]*P
rxy = Columns 1 through 8
0 0 0 0 0.7290 0.8100 0.9000 2.0000
0 0 0 0 0 0.7290 0.8100 0.9000
0 0 0 0 0 0 0.7290 0.8100
0 0 0 0 0 0 0 0.7290
Columns 9 through 11
-0.9000 0.8100 -0.7290
2.0000 -0.9000 0.8100
0.9000 2.0000 -0.9000
0.8100 0.9000 2.0000
>> rle=eye(4)-rxy*inv(ryy)*rxy
rle=0.0881 0.0068 -0.0786 -0.0694
0.0068 0.1000 -0.0045 -0.1353
-0.0786 -0.0045 0.1101 0.0464
-0.0694 -0.1353 0.0464 0.3666
>> [v,d]=eig(rle)
v =-0.2186 -0.5939 -0.7658 -0.1144
-0.3545 0.2811 -0.2449 0.8575
0.1787 0.7319 -0.5695 -0.3287
0.8914 -0.1806 -0.1710 0.3789
d = 0.4467 0 0 0
0 0.1607 0 0
0 0 0.0164 0
0 0 0 0.0410
>> b=sqrt(np)*v(:,3)
b = -2.1653 -0.6925 -1.6103 -0.4834
>> w=b*rxy*inv(ryy)
w = 0.0101 0.0356 -0.0771 -0.1636 0.0718 0.1477 -0.4777 -0.7924 -0.0078 -0.2237 0.1549
The is .89836 and the SNR
MFB
= 17.7868 dB corresponding to a biased error energy
381
of .1288 and unbiased at .1331. A TEQ program written by 2005 student Ryoulhee Kwak
executes this procedure and appears at the web page. By running the program a few times,
one can nd that by using = 3 and 14 taps in the equalizer with delay 9, the SNR
MFB
increases to 18.9 dB. By then running water-lling, one nds an SNR in the range of 15 to
16 dB for various TEQs of dierent length on this channel.
Instead, if no TEQ is used with FFT size 1024, then all tones are used and
>> [gn,en_bar,bn_bar,Nstar,b_bar]=waterfill(b,19.2,1,1024,0);
>> Nstar = 1024
>> b_bar = 2.8004
>> 10*log10(2^(2*b_bar) -1)
ans = 16.8
The performance is the same as the DFE system earlier. Thus, this second example illustrates
that the TEQ is not wisely used and perhaps a larger symbol size is better. Note the no-
TEQ DMT receiver has total complexity log
2
(1024) = 10 per sample, while the DFE has
complexity 20. One might note that the transmitter of DMT also requires 10 per sample.
Both get the same performance for about the same total complexity (transmitter and receiver)
in the case that entire set of tones is used (a curious result developed more in Chapter 5).
4.8.3 Zero-forcing Approximations of the TEQ
Several designers take an alternative approach to design of the TEQ that they claim is superior to Chows
method. This is incorrect. The method does not consider MMSE and instead looks at the equalized
response W(D) P(D) within a window of +1 samples and outside that window in the other N 1
samples, which are called the wall. The heurestic explanation is that energy in the equalizer response
in the wall should at a minimum relative to the energy within the window. First, this window/wall
approach tacitly assumes that the input power spectral density is white S
x
(D) =

c
x
(or else this
criterion makes no sense). Furthermore, noise is clearly ignored in the approach. For nite-length TEQ
(or any equalizer) design, zero-forcing solutions cannot exactly force a specic b. The MMSE approach
to equalizer design becomes zero forcing for all lengths when the SNR parameter is set to an innite
value. In this case, the mean-square energy of the wall is then necessarily minimized relative to the
total energy of

c
x
c = |p|
2
in the MMSE-TEQ when the input is white. As Ishai astutely noted, the
problem of maximizing window to wall energy is the same as maximizing total equalized pulse energy
to wall energy. Thus, all these approaches (which eventually result in a generalized eigenvalue problem
that must be solved) are the same as the MMSE-TEQ when the designer assumes white input spectrum
S
x
(D) =

c
x
and sets
A0
2
= 0.
In fact, one notes that Chows method is at least as computationally ecient as even the best of the
other methods (introduced by Ilani) in that the matrix for which an eigenvector is determined is only
size ( +1) ( +1) and indeed Chows method need only invert the matrix R
yy
once. The eigenvector
must determined for all reasonable values of in all methods.
EXAMPLE 4.8.3 (Return to 1/(1 +.9D) channel example of Section 4.8.2) A de-
signer might decide to use one of the various zero-forcing TEQ designs instead on the
1
1+.9D
channel. Clearly if noise is ignored, then a best equalizer is w(D) =

5.2636(1+.9D), which
attens the channel so that even = 0 is sucient. In fact all zero-forcing equalizers are given
by the class W(D) = (a+bD)(1+.9D) where a
2
+b
2
= |p|
2
. The rst zero-forcing equalizer
has smallest enhanced noise given by .9527 relative to |p|
2
= 5.2636 for a ratio of 7.4 dB,
about 3 dB worse than including the eect of the white noise. A better choice of ZF-TEQ
(since there are many in this special case) would be W(D) = (1.65 1.59 D) (1 +.9D).
29
,
29
The author found this by minimizing output noise power over choices for a and b, which results in a quadratic equation
for b with solutions b = 1.59 or .626 (a = 2.21).
382
and has ratio instead of 10 dB, which is still less than 10.4 dB. In fact, no higher ratio than 10
dB can be found for the ZF case. Thus, all ZF-TEQs would all have worse SNR. However,
the astute reader might observe loading should be executed with the correct noise power
spectral density (which is not close to white in this example) and indeed is very high-pass,
so the set of g
n
might be better. This observation that it is really SNR
dmt
that is important
then leads to the next subsection.
4.8.4 Kwak TEQ Program
Former student Ryoulhee Kwak has provided the TEQ program at the web site that works for both FIR
and (all pole) FIR channel inputs. The program is
function [P,ryy,rxy,rle,v,d,w,b,mmse,SNRmfb_in_dB,c]=teq(p,L,mu,delta,sigma,Ex_bar,filtertype)
% programmed by Ryoulhee Kwak
%
% filtertype 1=FIR , 2= IIR(just for one pole filter)
% p=channel impulse response
% in case of FIR, p= 1+0.5D+2.3D^2->p=[1 0.5 2.3]
% in case of IIR, p=1/(2.4-0.5D)-> p=[2.4 -0.5]
% sigma =>noise power in linear scale
% delta =>delay
% L => number of taps
% c=> w*p
% v=>eigen vectors
% d=>eigent values
%mu => for b
if filtertype==1% FIR
norm_p=norm(p);
[temp size_p]=size(p);
P=toeplitz([p(1), zeros(1,L-1)],[p,zeros(1,L-1)]);
ryy=Ex_bar*P*P+sigma*eye(L,L);
rxy=[zeros(mu+1,delta) eye(mu+1) zeros(mu+1,L+size_p-2-mu-delta)]*P*Ex_bar;
rle=eye(mu+1)*Ex_bar-rxy*inv(ryy)*rxy;
[v d]=eig(rle);
d_temp=diag(d);
[m,n ]= min(d_temp);
b=norm_p*v(:,n);
w=b*rxy*inv(ryy);
conv(w,p);
mmse=b*rle*b;
%by using easy way to get error energy
c=conv(w,p)
alpa=c(delta+1)./b(1)
biased_error_energy=norm_p^2*(m-Ex_bar*(1-alpa)^2)
unbiased_error_energy=norm_p^2*(m-Ex_bar*(1-alpa)^2)/alpa^2
SNRmfb=norm_p^2*Ex_bar/unbiased_error_energy;
SNRmfb_in_dB=10*log10(SNRmfb)
else %IIR
%in case of IIR filter P matrix is infinite dimesion but there is a trick to get exact rxy, ryy
norm_p=sqrt(1/(p(1)^2*(1-(p(2)/p(1))^2 )));
383
[temp size_p]=size(p);
ptemp=[(-p(2)/p(1)).^(0:1:L-1)]/p(1); %-1 factor!
P=toeplitz([ptemp(1) zeros(1,L-1)],ptemp);
Ptemp=toeplitz(ptemp,ptemp);
ryy=Ex_bar*Ptemp*norm_p^2+sigma*eye(L,L);
rxy=[zeros(mu+1,delta) eye(mu+1) zeros(mu+1,L-1-mu-delta)]*P*Ex_bar;
rle=eye(mu+1)*Ex_bar-rxy*inv(ryy)*rxy;
[v d]=eig(rle);
d_temp=diag(d);
[m,n ]= min(d_temp);
b=norm_p*v(:,n);
w=b*rxy*inv(ryy);
c=conv(w,p);
sum(conv(w,p).^2)-norm(b)^2*(w(1)/b(1))^2;
mmse=b*rle*b;
%by using easy way to get error energy
alpa=c(delta+1)./b(1)
biased_error_energy=norm_p^2*(m-Ex_bar*(1-alpa)^2)
unbiased_error_energy=norm_p^2*(m-Ex_bar*(1-alpa)^2)/alpa^2
SNRmfb=norm_p^2*Ex_bar/unbiased_error_energy;
SNRmfb_in_dB=10*log10(SNRmfb);
end
4.8.5 Joint Loading-TEQ Design
Usually TEQs are designed (or trained) before loading in DMT or VC systems, and an assumption of
initially at input spectrum S
x
(D) =

c
x
is then made. For OFDM, this assumption is always true, but
when loading is used in DMT, then it will likely not be true once loading is later computed. In fact, an
optimum design would maximize SNR
dmt
over the TEQ settings as well as the c
n
for each subchannel.
Al-Dhahir was the rst to approach this problem and attempt solution. A yet better solution recently
emanated from Evans (see December 2001 IEEE transactions on DSP). While Evans beautifully sets up
the problem, he notes at the end that it requires very dicult and advanced optimization procedures
to be exactly solved, and he further notes that such solution is not feasible in actual implementation.
Evans then reverts to the MMSE solution and shows it is often close to the MAX-SNR
dmt
solution.
This subsection poses the following solution:
1. Initially design MMSE-TEQ for at or white input.
2. Execute loading (water-lling or LC) and compute SNR.
3. Redesign the MMSE-TEQ using the new R
xx
.
4. Re-loading the channel and see if SNR
dmt
has signicantly improved.
(a) If improved, return to step 3 with the latest R
xx
from loading.
(b) If not, use the TEQ with best SNR
dmt
so far determined.
Clearly with dynamic loading, the movement of energy could cause the TEQ to be somewhat de-
cient. This would be particularly true if narrow-band noise occurred after initial training. The procedure
can then be repeated. The TEQ change would need to occur on the same symbol that synchronizes the
change of energy and bit tables in the corresponding modems.
For those interested in extra credit, this method might be executed for the 1/(1 + .9D) example
above and compared with MMSE approach as well as the method given by Evans.
384
Figure 4.36: Kasturias Block DFE.
4.8.6 Block Decision Feedback
The overhead of the extra dimensions of vector coding and and DMT/OFDM is negligible for large N.
When large N leads to unacceptable delay in the transmission path, smaller N may have to be used. In
this case, the extra samples between symbols (blocks of samples) may be eliminated by what is known
as Kasturias Block Decision Feedback Equalizer.
Block Decision Feedback uses the decisions from the last block of samples to reconstruct and cancel
the interference from previous symbols. The pulse response is known in the receiver so that the eect of
the last columns of P can be subtracted as in Figure 4.36. Assuming previous decisions are correct,
the resultant vector channel description is
y =

Px +n , (4.350)
where

P is the N N matrix containing the left-most N columns of P. P must be converted to
a minimum-phase (MMSE sense) channel by a conventional MMSE-DFE feedforward lter for best
performance. Vector coding is applied to this matrix channel with SVD performed on the matrix

P.
While simple in concept, the subtraction of previous symbols in practice can be complex to implement.
4.8.7 Cheongs Cyclic Reconstruction and Per-tone Equalizers
K. W. Cheong in his 2000 dissertation and earlier results used the concept of cyclic reconstruction (rst
introduced for echo cancellers by Ho and Cio) to simplify the concept of a full N N block equalizer
that would be equivalent to a separate TEQ for each tone.
30
Van Acker and Mooenen (and van de
Wiel and Pollet) later renamed this matrix equalization a per-tone equalizer and used the equivalent
of cyclic reconstruction to simplify the amount of computation (apparently unaware of Cheongs earlier
work and the equivalence to cyclic reconstruction). The method requires a slightly higher complexity
than a TEQ in real-time, but does not require eigen-vector computation. This method actually then
leads to an easy adaptive implementation that maximizes product SNR.
The key observation in cyclic reconstruction is that the cyclic prex could be faked to look longer if
at each sampling instant that is not already part of the cyclic prex the value from the previous symbol
30
Clearly having a separate equalizer for each tone allows the individual SNRs to be maximized and then also the
product SNR to be maximized given a certain input energy distribution, essentially decoupling the loading problem from
the equalization problem.
385
Figure 4.37: Cheongs Cyclic Reconstruction (sometimes called a per-tone equalizer).
were somehow replaced by the sample from the current symbol. Essentially, the cyclic prex becomes
innite so then the FEQ would be sucient for all equalization on the resultant periodic system.
If the receiver saves the last symbol of samples and they were correct, then the ISI from the last
sample could be subtracted and then an ISI equivalent to the repeated sample could be added. Cheong
used this in a basic Block DFE structure to allow simplication and retain use of the FFTs (unlike
Kasturias earlier block DFE). At some further increase in complexity, it is possible (and obvious) that
this can be implemented also in the time domain using instead the channel outputs. A truly cyclic
channel would have outputs that are also cyclic. Any deviations from such cyclic behavior would be
completely represented by the dierences between points separated by N samples.
Thus a set of dierences can be computed by the receiver or
The basic observation is the nominal FFT has N values
(y
N+i1
y
i1
) i = 1, ..., N . (4.351)
Any deviation from perfect cyclic channels must be a linear combination of these dierences on
a linear channel. Thus, one implementation is shown in Figurerefcheong The coecients are adapted
according to the same algorithmas in (4.133) using the same error signal

U
n
but with each FEQ coecient
augmented by a set of N taps that have the time-domain dierences. This remains the complexity of
N N. However, dierences far away from the cyclic prex will have negligible contribution. If the
(FIR) TEQ for any (or all) tones were L taps long, then clearly it has only L degrees of freedom and
could be aected by at most 2L dierences.
Thus the complexity is essentially double that of an L-tap TEQ but also has the advantage of
essentially being a dierent equalizer for each tone output. Moonen and Acker developed what they call
a per-tone equalizer well after the Kasturia and Cheong work, but did not realize the equivalent use
of cyclic reconstruction, thus the apparent dierent name. However, the concept is clearly and obviously
equivalent when viewed from the perspective of cyclic reconstruction. These 2L samples are the same
386
for all tones but do depend on the delay chosen. In a system with no delay, they would be the 2L
samples closest to the previous symbol. With a delay delta, the 2L samples closest to the previous
symbol and the samples at the end of the symbol would be used. Such a system requires collection of
the next symbol to compute the dierences (or at least output samples from the next symbol) and so
has between samples and one symbol of additional delay required for implementation.
An adaptive implementation would use the ZF algorithm used for the FEQ on each tone with the
single complex coecient now being replaced by 1 + 2L dierence terms.
387
4.9 Windowing Methods for DMT
While cyclostationary
31
DMT/OFDM signals are not deterministically periodic over all time simply
because the transmitted input message is not the same in each symbol period. Deterministically for any
tone, the corresponding output signal for a given symbol period is
x
n
(t) = X
n
e

2
N

N+
T
nt
w
T
(t) , (4.352)
where w
T
(t) is a window function. In the DMT/OFDM partitioning of Section 4.6, the window function
was always
w
T
(t) =
_
1 t [0, T)
0 t , [0, T)
. (4.353)
Letting

f

=
_
1 +

N
_

1
T
=
1
NT
t
, (4.354)
the product in (4.352) corresponds to convolution in the frequency domain of the impulse X
n
(f

f)
with W
T
(f). The fourier transform of the rectangular window in (4.353) is
W
T
(f) = sinc
_
f

f
_
e
fT
, (4.355)
which has the magnitude illustrated in Figure 4.38. While there is no interference between tones of DMT
because of the zeros in this function, at intermediate frequencies (f ,= n

f), there is non-zero energy.
Thus the DMT spectrum is not a brick-wall bandpass system of tones, but decays rather slowly with
frequency from any tone. Such slow decay may be of concern if emissions from the present channel
into other transmission systems
32
are of concern. Even other DMT transmissions systems that might
sense crosstalk from the current DMT system would need exact synchronization of sampling rate 1/T
t
to avert such crosstalk. A slight dierence in DMT sampling clocks, or just another modulation method,
on the other transmission system might lead to high interference. For this reason, some DMT/OFDM
modulators use a dierent window than in (4.353) to reduce the sidelobes with respect to those in
Figure 4.38.
Non-rectangular windowing of the transmit signal in DMT could re-introduce interference among
the tones, so must be carefully introduced. Figure 4.39 illustrates two methods for introducing windows
without reintroduction of intra-symbol interference through the use of extra cyclic prex and sux. The
cyclic sux used for Zippered duplexing in Section 4.6 is also useful in windowing. Extra prex samples
beyond those needed for ISI (that is larger than the minimum required) and extra sux samples
(beyond either 0 with no Zippering, or larger than the minimum required with Zippering) are used
for the shaped portion of the window. The minimum N + + samples within the symbol remain
unwindowed (or multiplied by 1, implying no change) so that DMT or OFDM partitioning works as
designed when the receiver FFT processes only N of these samples. The remaining samples are shaped
in a way that hopefully reduces the sidelobes of the window function, and thus makes the frequency
response of the modulated signal fall more sharply. This sharper fall is achieved by having as many
continuous derivatives as is possible in the window function. The windows from the extra prex of the
current symbol and the extra sux of the last symbol can overlap without change in performance if the
sum of the two windows in the overlapping region equals 1. Such overlap of course reduces the increase
in excess bandwidth introduced by the window.
The most common choice of window in practice is a raised-cosine window of 2L+1 samples of extra
cyclic extension (extra prex plus extra sux). This function is given in the time domain by
w
T,rcr
(t) =
1
2

_
1 + cos
_

_
t
LT
t
___
t (LT
t
, LT
t
) (4.356)
31
DMT/OFDM signals are stationary at symbol intervals, but cyclic over a symbol period.
32
That may share the same channel or unfortunately sense radiation from it.
388
Figure 4.38: Rectangular window fourier transform versus
f

f
.
Figure 4.39: Illustration of extra prex and sux and non-overlapped and overlapped windows.
389
where time t = 0 is in the middle of the extra extension. The actual implementation occurs with samples
w
k
= w
T,rcr
(kT
t
) for k = L, ..., 0, ..., L. Letting
=
L
N + +
, (4.357)
the fourier transform of this window becomes
W
T,rcr
(f) =
L

sinc
_
f

f
_

cos
f

f
1
_
2 f

f
_
2
. (4.358)
Figure 4.40 illustrates the sidelobes of the raised cosine function for 5% (blue) and 25% (red) along with
those of W
T
(f) from Figure 4.38 (this time in decibels). Emissions into other systems can be signicantly
reduced for between 5% and 25% excess bandwidth. The FMT systems of Section 4.6 oer even greater
reduction (or can be combined) if required.
4.9.1 Receiver Windowing
Also of interest are any non-white noises that may have autocorrelation of length greater than T
t
.
Extremely narrow band interference noise, sometimes called RF interference is an example of such
a noise. DMT block length N may not be selected suciently large and the TEQ of section 4.8 may
not adapt quickly enough to notch or reduce such noise, or the TEQ may not be suciently long
to reduce the noise level. The same extra prex and sux interval in the receiver can again be used
to apply a window (perhaps the same as used in the transmitter). Windowing at the receiver again
corresponds to frequency-domain convolution with the transform of the window function. Low sidelobes
can reduce any narrow-band noise and improve all the g
n
(and particularly those close to the narrowband
noise). Whether used for transmitter, receiver, or both, Figure 4.40 shows that the rst few sidelobes
are about the same no matter how much excess bandwidth (extra cyclic extension) is used. However, for
somewhere between 5% and 25% excess bandwidth, 20 dB and more rejection of noise energy coupling
through sidelobes is possible, which may be of considerable benet in certain applications. Of course,
this extra bandwidth is the same as any excess bandwidth used by the transmitter (so excess bandwidth
does not increase twice for transmitter and receiver windowing).
390
Figure 4.40: Illustration of transforms of rectangular window (green) and raised-cosine with 5% (blue)
and 25% (red) excess bandwidth versus (f

f)/

f.
391
4.10 Peak-to-Average Ratio reduction
The peak-to-average ratio of an N-dimensional constellation was dened in Chapter 1 as the ratio
of a maxium magnitude constellation point to the energy per dimension. That denition is useful for
N = 1 and N = 2 dimensional constellations. For the higher dimensionality of DMT and OFDM signals
(or vector coding or any other partitioning), the maximum time-domain value is of practical interest:
x
max
= max
k , k=0,...N+1
[x
k
[ . (4.359)
This value determines the range of conversion devices (DACs and ADCs) used at the interfaces to the
analog channel. The number of bits needed in the conversion device often determines its cost and also
its power consumption. Thus, smaller is usually better (especially at high speeds). A higher ratio
x
max
/

c
x
leads to more bits in such a device. Also, subsequent analog circuits (ampliers, lters, or
transformers) may exhibit increased nonlinearity as x
max
increases. Finally, the power consumption of
linear ampliers depends on the peak power, not the average power. In the analog portion of the channel
between conversion devices, actually x
max
should be redened to be
x(max) = max
t[0,T)
[x(t)[ , (4.360)
which can exceed the discrete-time x
m
ax. High-performance transmission systems necessarily increase
the peak voltages in theory, so the PAR for the transmitted signal x(t) (and/or x
k
) can become important
to designers.
This section investigates this peak value and shows that additional digital signal processing can
render the nominally in theory large peak inconsequential in practice.
Subsection 4.10.1 begins with some fundamentals on the probability of peaks and their relationship to
data rate, showing that indeed very little data rate need be lost to eliminate large peaks in transmission.
Subsection 4.10.2 then progresses to describe the most widely used and simplest PAR-reduction method
known as tone reservation (developed independently by former 379 students and their colleagues!)
where a few tones are used to carry peak annihilating signals instead of data. This method essentially
requires no coordination between transmitter and receiver and thus is often preferred. The loss of
tones is somewhat objectionable, so for more complexity and some receiver-transmitter understanding,
the equally eective tone-injection method appears in Subsection 4.10.3. This method causes no
data rate loss and instead cleverly inserts signals invisible to the data connection, but that annhilate
peaks. Subsection 4.10.4 discusses the most complicated nonlinear-equalization methods that take the
perspective of allowing clipping of signals to occur and then essentially remove the clips at the receiver
through an ML detector that includes the clips. All the approaches presented in this section reduced
the probability of nonlinear distortion to a level that can be ignored in practice.
33
A series of coded approaches have also been studied by various authors, where points in the N-
dimensional constellation that would lead to time-domain large x
max
are prohibitted from occuring by
design. Unfortunately, these approaches while elegant mathematically are impractical in that they lose
far too much data rate for eective performance, well above theoretical minimums. This book will not
consider them further.
4.10.1 PAR fundamentals
This book chooses to focus on real signals x
k
so that X

Ni
= X
i
for each symbol. For complex signals,
the theory developed needs to be independently applied to the real dimension and to the complex
dimension.
34
33
That is a level at which outer forward error-correction codes, see Chapters 10 and 11, will remove any residual errors
that occur at extremely infrequent intervals.
34
In which case, the signal processing is independently applied to the even part of X, X
e,i

= .5 (X
i
+ X

Ni
) that
generates the real part of x
k
and then again to the odd part of X, X
o,i

= .5 (X
i
j X

ii
) that generates the imaginary
part of x
k
.
392
First, this subsection considers the discrete-time signal x
k
and generalizes to over-sampled versions
that approximate x(t). The time-domain signal x
k
is again
x
k
=
N1

n=0
X
n
e

2
N
nk
, (4.361)
which is a sum of a large number of random variables. Losely speaking with the basic Central Limit
Theorem or probability, such a sum is Gaussian in distribution and so has very small probability of very
large values. For the Gaussian distribution, a value with 14.2 dB of peak-to-average ration (Q(14.2dB))
has a probability of occurrence of 10
7
and a PAR of 12.5 dB has occurrence 10
5
. When such a peak
occurs, with high-probability large clip noise occurs if the conversion devices or other analog circuits
cannot handle such a large value. So, with probabilities on the order of the probability of bit-error,
entire DMT/OFDM symbols may have many bit errors internally unless the PAR in the design of such
circuits accommodates very high PAR. Typically, designers who use no compensation set the PAR at 17
dB to make the occurrence of a peak clip less likely than the life-time of the system (or so small that it
is below other error-inducing eects).
One might correctly note that the same high-peak problem occurs in any system that uses long
digital lters (i.e., see the transmit-optimizing lters for QAM and PAM systems in Chapter 5) will have
each output sample of the lter equal to a sum of input random variables. For eective designs, the
lter is long and so the Gaussian nature of actual transmitted signal values is also present. Furthermore
aside from lters or summing eects, best coding methods (particularly those involving shaping) provide
transmit signal and constellation distributions that approximate Gaussian (see Chapter 8). Thus, one
can nd any well-designed transmission system having the same high PAR problem. However, solutions
to this problem are presently known only for the DMT/OFDM systems (which with the solutions,
actually then have a lower PAR than other systems).
At rst glance, it might seem prospects for high performance in practice are thus dim unless the
designer can aord the higher dyanmic range and power consumption of conversion and analog circuits.
However, let us assume a PAR of say 10 dB is acceptable (even sinusoids in practice require 3-4 dB
so this is say one bit of range
35
more than a non-information-bearing signal would require. Such a
peak in a Gaussian distribution occurs with probability 10
3
. Let us suppose that internal to the data
transmission stream, a small amount of overhead data rate was allocated to capturing the size of the clip
that occurs in analog circuits and sending a quantized version of that size to the receiver. No more than
20 bits per clip would be required to convey a very accurate description, and perhaps just a few bits if
designs were highly ecient. Those twenty bits would represent on average only 2% of the data rate.
Clearly not much is lost for imposing maximums on the transmitted range of quantities, at least from
this trivial fundamental perspective. Such an approach might have signicant delay since the overhead
bit stream would be very low with respect to other clocks in the design. Nonetheless, it fundamentally
indicates that signicant improvement is possible for a small loss in overall data rate.
This is exactly the observation used in the three following approaches of this section.
4.10.2 Tone Reservation
Tone reservation allows the addition of a peak annihilator c to the transmit signal x on each sym-
bol. Thus, x + c is cyclically extended and transmitted. The peak-annihilator attempts to add the
negative of any peak so that the signal is reduced in amplitude at the the sampling instant of the
peak. The peak-annihilation vector c is constructed by frequency-domain inputs on a few tones in set
1 = i
1
, i
2
, ..., i
r
, ..., N i
r
, ..., N i
1
. Typically r << N/2. The determination of this set is deferred
to the end of this subsection.
Thus,
x + c = Q

[X +C] (4.362)
and
C
n
=
_
0 n , 1
mboxselected n 1
. (4.363)
35
Conversion devices basically provide 6 dB of dynamic range increase for each additional bit used to quantize quantities.
393
Figure 4.41: Probability of peak above specied PAR level for N = 128 OFDM/DMT systems with 5%
and 20% of tones reserved. The extra FFT curve can be ignored and represents a double-pass of the
transmitter IDFT where the phase of some signals has been selectively altered on the second pass to
reduce PAR.
Thus, X
n
,= 0 when n , 1, leaving N r tones available for data transmission and the others reserved
for selection and optimization of the peak annihilator.
The problem can be formulated as
min
C
max
k
[x + Q

C[ (4.364)
with the constraints that C
n
= 0, n 1 and |x + Q

C|
2
N

c
x
. This problem can be recognized
with some work as equivalent to a linear-programming problem for which algorithms, although complex,
exist for solution. This subsection shows the performance of such exact solution to see potential benet,
but rapidly progress to an easily implemented algorithm that approximates optimum and is used in
practice. Figure 4.42 illustrates the reduction in probability of a peak exceeding various PAR level. For
reservation of 5% of the tones, a 6 dB reduction (savings of one bit is possible). For 20% reservation
(perhaps an upper limit on acceptable bandwidth loss), a 10 dB reduction is possible. Indeed, the PAR
achieved for this OFDM/DMT system is well below those of systems that perform worse. However,
the complexity of the solution used would be burdensome. Thus, the alternative method of minimizing
squared clip noise is subsequently derived.
The alternative method makes use of the function
clip
A
(x
k
) =
_
x
k
[x
k
[ < A
Asgn(x
k
) [x
k
[ A
(4.365)
where A > 0 and x
k
and A are presumed real. The clip noise for a symbol vector x is then

2
x
= |x clip
A
(x)|
2
=
N1

k=0
[x
k
clip
A
(x
k
)]
2
. (4.366)
394
With the peak-annihilation in use

2
x+c
= |x clip
A
(x = Q

C)|
2
. (4.367)
The gradient of the clip-noise can be determined when a peak-annihilator is used:

2
x+c
= 2

k]xk+ck]>A&i1
sgn(x
k
+ c
k
) [[x
k
+c
k
[ A] Q

1
q

k
(4.368)
where Q

1
is the N 2r matrix of columns of Q

that correspond to the reserved tones and q

k
is an
2r 1 column vector that is the hermetian transpose of the k
th
row of Q

1
. The N 1 vectors
p
k
= Q

1
q
k
(4.369)
can be precomputed as they are a function only of the IFFT matrix and and the indices of the reserved
tones.
36
These vectors are circular shifts of a single vector. They will have a peak in the k
th
position
and thus changing the sign of this peak and adding it to the original x to be transmitted reduces the
PAR. Such an operation could occur for each and every k such that [x
k
+ c
k
[ > A. That is essentially
what the gradient above computes a direction (and magnitude) for reducing the clip noise in those
dimensions where it occurs.
A steepest-descent gradient algorithm for estimating the best c then sums a (small) scaled sum of
gradients successively computed. Dening

k
= sgn(x
k
+ c
k
) [[x
k
+ c
k
[ A] k [x
k
+ c
k
[ > A (4.370)
as a type of successive error signal, then the gradient algorithm becomes
c
j+1
= c
j

k]xk+ck]>A

k
(j) p
k
, (4.371)
where j is an algorithm index reecting how many iterations or gradients have been evaluated. This
algorithm converges to close to the same result as the optimum linear-programming solution. Figure ??
shows the results of successive iterations of this gradient algorithm.
After approximately 10 iterations, the PAR has been reduced by 6 dB. After 40 iterations, the
algorithm is within .5 dB of best performance with 5% of tones used. These calculations were for
N = 128 and represent resulting from averaging over a variety of data vectors(and of course peaks only
occur for a small fraction of those random data vectors).
For most applications, the indices in 1 can be randomly chosen among those tones that have non-zero
energy allocated in loading (use vectors that have zero energy at the channel output usually leads to
a peak reproduced at the output of the channel even if the input had no such peak, which means the
ADC will saturate in the receiver). Tellado has studied methods for optimizing the set 1 but came to
the conclusion that simple random selection the indices of to get 5% of the used tones is sucient. (And
if the resultant vector to be circularly shifted does not have a peak in it, a second random selection is
then made.)
This method, largely due to Tellado, but also studied at length by Gatherer and Polley, is the
predominant method used for PAR reduction in practice today. Its lack of need for coordination between
transmitter and receiver (the transmitter simply forces the receiver to o-load the tones it wants by
sending garbage data on those tones that forces swaps away from them). Thus, there is no need to
specify the method in a DMT standard. However, in OFDM standards the set of used tones may need
agreement before hand. Indeed since peaks do not occur often, the tones used for pilots in OFDM may
also carry the very infrequent peak-reduction signals without loss of performance of the (well-designed)
receiver PLLs.
36
These terms arise in the gradient vector because of the restriction to zero of the non-reserved tones and would be
deleted from the gradient altogether if there were no restrictions on C or c.
395
Figure 4.42: Probability of peak above specied PAR level for successive iterations of the gradient
algorithm.
4.10.3 Tone Injection
5% or more tone reservation for PAR is considerably in excess of the minimums required by theory to
reduce PAR substantially. Tellado, and independently Kschichang, Narula, and Eyuboglu of Motorla,
found means of adding peak-annihilation signals to the existing data-carrying tones.
In tone injection, no tones are reserved for only peak-reduction signals. Instead, signals are added
to the data symbols already present on some tones in a way that is transparent to the receiver decision
process. The basic concept is illustrated in Figure 4.43. There a selected constellation point called A
can be replaced by any one of 8 equivalent points if the 16QAM constellation shown were repeated in
all directions. Each of these 8 points has greater energy and increases the transmitted energy for that
particular tone if one of these is selected. The basic idea of tone injection is to search over points like
the 8 shown for several tones to nd an added signal that will reduce the peak transmitted. Because
peaks do not occur often and because the extra energy only occurs on a few tones, the penalty of tone
injection is a very small increase in transmit power, typically .1 dB or less on average. A knowledgeable
receiver design would simply decode the original point or any of the 8 possible replacement points as the
same bit combination. The oset in either horizontal or vertical direction is

d units.
Thus, peaks in time domain are traded for peaks in the frequency domain where they have little
aect on the transmitted signal energy, and in particular a very large power-reducing eect on the
consumed power of the analog driver circuits that often have consumed power dependent on the peak
signal transmitted.
More formally, the time domain DMT or OFDM signal (avoiding DC and Nyquist, which are simple
extensions if desired and letting R
n

= 'X
n
| and I
n

= X
n
|) is
x(t) =
2

N/21

n=1
R
n
cos(
2
NT
t
nt) I
n
sin(
2
NT
t
nt) (4.372)
396
Figure 4.43: Illustration of constellation equivalent points for 16QAM and Tone Injection.
397
Figure 4.44: Illustration of successive iterations of tone injection.
over one symbol period ingnoring the cyclic prex. x(t) may be sampled at sampling period T
t
=
T/(N + ) or any other higher sampling rate, which is denoted x
k
= x(kT
tt
) here. Over a selected set
of tones n 1, x
k
can be augmented by injecting the peak-annihilation signal
x
k
= x
k

2

d

N
cos(
2
NT
t
nkT
tt
)
2

d

N
sin(
2
NT
t
nkT
tt
) n 1 . (4.373)
It is not necessary often to add both the cos and sin terms, as one is often sucient to reduce the peak
dramatically. Algorithms for peak injection then try to nd a small set of n indices at any particular
symbol for which an injected peak signal is opposite in sign and rougly equal in magnitude to the
largest peak found in the time-domain symbol. An exhaustive search would be prohibitively complex.
Instead, typical algorithms look for indices n that correspond to large points on the outer ranges of the
constellation used on tone n
0
having been selected by the encoder and for which also the cos or sin is
near maximum at the given peak location k
0
. Then, roughly,
[ x
k0
[ = [x
k0
[ 2

d/

N = [x
k0
[
2

6 2
2

bn
0
2
2

bn
0
1
. (4.374)
On each such iteration of the algorithm, a peak is reduced at location k
i
, i = 0, ... until no more reduction
is possible or a certain PAR level has been achieved. Figure 4.44 illustrates successive steps of such tone
reduction applied to an OFDM system with 16 QAM on each tone. Note that about 6 dB of PAR
reduction is obtained if enough iterations are used. The transmit power increase was about .1 dB.
398
4.10.4 Nonlinear Compensation of PAR
Tellado also studied the use of receivers that essentially store the knowledge of nonlinear mapping caused
by peaks that exceed the linear range of the channel in a look-up table. He then proceeded to implement
a maximum likelihood receiver that searches for most likely transmitted signal. This method is very
complex, but has the advantage of no bandwidth nor power loss, and is proprietary to recievers so need
not be standardized as would need to be the case for at least the silenced tones in tone reservation and
the modulo-reception-rules for tone injection.
399
Exercises - Chapter 4
4.1 Short Questions
a. (1 pt) Why is the X
0
subsymbol in Figure 4.2 chosen to be real?
b. (1 pt) Why is the X
N
subsymbol in Figure 4.2 chosen to be one-dimensional?
c. (3 pts) Given ideal sinc transmit lters,
n
= (1/

T) sinc (t/T), why are the channel output basis

functions of multitone,
p,n
, orthogonal for all possible impulse responses h(t)?
4.2 A simple example on Multitone
(3 pts) Redo Example 4.1 with H(f) = 1+0.5e
j2f
(Suppose T = N). The rest of the specications
are :

L
x
|h|
2

2
= 10dB, design target arg. of Q-function = 9dB,

E
x
= 1, N = 2

N = 8. As in the
example, the last sub-channel is silenced. Also, attempt a design with N = 2

N = 16. The bit rate
of your new design will go down if correctly designed. Why?
4.3 GAP Analysis
a. (1 pt) While the SNR gap may be xed for often-used constellations, this gap is a function of two
other parameters. What are these two parameters?
b. (2 pts) Based on your answer to part (a), how can the gap be reduced?
c. (2 pts) For an AWGN channel with SNR = 25dB and QAM transmission,

P
e
= 10
6
, and b = 4.
What is the margin for this transmission system?
d. (2 pts) Let b = 6 for part (c), how much coding gain is required to ensure

P
e
= 10
6
?
4.4 Geometric SNR
a. (1 pt) For the channel in Problem 4.2, calculate the SNR
m,u
using the exact formula and the
approximate one, SNR
geo
(for a gap, use = 8.8dB).
b. (1 pt) Are these two values closer than for the channel 1 + .9D
1
? Explain why.
c. (3 pts) Compare the dierence of the SNR
MFB
to the exact SNR
m,u
for the 1 + .5D
1
and
1 +.9D
1
. Which one is closer? Why?
4.5 Water-Filling Optimization
Repeat Problem 4.2 using water-lling optimization. Given spec is H(f) = 1 +.5e
j2f
, SNR
MFB
=
10dB, = 1(0dB). (Note that all the subchannels may be used in this problem)
a. (2 pts) Calculate the optimal distribution of energy for the sub-channels and the maximum bit
rate assuming that the gap, = 1(0dB).
b. (1 pt) Calculate the gap for PAM/QAM that produces an arg. of the Q-function equal to 9dB.
(The gap for

b 1 is the dierence between the SNR derived from capacity and the argument of
the Q-function for a particular probability of error)
c. (2 pts) Calculate the optimal distribution of energy for the sub-channels and the maximum bit
rate using the gap found in (b).
4.6 Margin Maximization
For the channel with H(f) = 1+0.5e
j2f
, as in Problem 4.2 maximize the margin using water-lling.
(All the subchannels may be used in this problem). Let N=8 for this problem.
a. (2 pts) Is transmission of uncoded QAM/PAM at P
e
= 10
7
at data rate 1 possible? (i.e. is the
margin 0)? If so, what will the margin be?
400
b. (1 pt) For data rate 1, what gap provides zero margin? For uncoded PAM/QAM, what

P
e
corre-
sponds to zero margin?
c. (2 pts) What is the margin if = 1(0dB)?
d. (1 pt) What is the margin if = margin in part (c) (dB)?
4.7 Creating a DMT Water-lling Program
Attach the appropriate program for this problem. On plots versus N, you may elect to plot only for
even N values in this program. The sampling period is presumed to be T
t
= 1.
a. (4 pts) Write a program to do DMT-RA water-lling on an arbitrary FIR channel with the pulse
response given as a row vector (ex. h = [1111] for 1+D+D
2
D
3
or 1+D
1
+D
2
D
3
channel).
will be determined as the length of your FIR channel-input vector. The input parameters should
be
A0
2
,

c
x
, the number of subchannels and the gap, . (Hint, try using the ow charts in Figures
(4.7) - (4.9)). What are

b and the energy distributions for RA loading on the 1 + .9D
1
channel?
(N = 8, = 1 (0dB),

E
x
= 1 and
A0
2
= .181)?
b. (2 pts) plot

b vs. N for the channel 1 + D D
2
D
3
. ( = 1 (0dB),

E
x
= 1 and
A0
2
= .1)
c. (2 pts) Modify your program (i.e. write another program) to do MA water-lling on the same
FIR channel. What is the margin and the energy distribution for MA loading on the 1 + .9D
1
channel? (N = 8, = 1 (0dB),

b = 1,

c
x
= 1 and
A0
2
= .181)?
d. (2 pts) plot margin vs. N for the channel 1 +DD
2
D
3
with

b = 1. ( = 1 (0dB),

c
x
= 1 and
A0
2
= .1)
4.8 Levin-Campello Algorithms
Please show work. This problem executes the Levin-Campello loading algorithm for the 1 + 0.5D
1
channel with PAM/QAM,

Pe = 10
6
, SNR
MFB
= 10dB, N=8, and

E
x
= 1. Use = 1.)
a. (2 pts) Create a table of e(n) vs. channel. (You may truncate the table sucient to answer part
(b) below).
b. (3 pts) Use the EF algorithm to make

b=1.
c. (1 pt) Use the ET algorithm to nd the largest

b.
d. (1 pt) If the design were to reduce the b by 2 bits from part (c), use the EF and BT algorithms to
maximize the margin. What is this maximum margin?
4.9 Creating a DMT Levin-Campello Program
Attach the appropriate program for this problem. On plots versus N, you may elect to plot only for
even N values in this program. The sampling period is presumed to be T
t
= 1.
a. (4 pts) Write a programs to do RA LC (EF and ET) on an arbitrary FIR channel with the pulse
response given as a row vector (ex. h = [111 1] for 1 + D + D
2
D
3
or 1 + D
1
+ D
2
D
3
channel). The input parameters should be
A0
2
,

c
x
, the number of subchannels and the gap, .
What are

b and the energy distributions for RA loading on the 1 + .9D
1
channel? (N = 8, =
1 (0dB),

E
x
= 1 and
A0
2
= .181)?
b. (2 pts) plot

b vs. N for the channel 1 + D D
2
D
3
. ( = 1 (0dB),

E
x
= 1 and
A0
2
= .1)
c. (2 pts) Modify your program (i.e. write another program) to do MA LC (EF and BT) on the same
FIR channel. What is the margin and the energy distribution for MA loading on the 1 + .9D
1
channel? (N = 8, = 1 (0dB),

b = 1,

c
x
= 1 and
A0
2
= .181)?
d. (2 pts) plot margin vs. N for the channel 1 +DD
2
D
3
with

b = 1. ( = 1 (0dB),

c
x
= 1 and
A0
2
= .1)
401
4.10 Sensitivity Analysis
This problem looks at the sensitivity of E
n
and P
e,n
(P
e
for each sub-channel), to variations in the
values of g
n
for water-lling.
a. (1 pt) S
Eij
=
Ei/Ei
gj/gj
. Show that S
Eij
=
(ij 1/N

)
gj Ei
, where N

is the number of sub-channels

being used and
ij
is the Kronecker delta. (Hint. Solve for the closed form expression for E given
that N

is known and then dierentiate it.)

b. (2 pts) Calculate the worst case sensitivity for the 1 + 0.9D
1
and the 1 + 0.5D
1
for multitone
with N=16 and = 3dB. Which one do you expect to be more sensitive? Why?
c. (2 pts) Now, calculate bounds for the sensitivity calculated in (a). Show that
(ij1/N

)/gMAX
N

E/N

+(1/gharm1/gMAX)
S
Eij

(ij1/N

)/gMIN
N

E/N

+(1/gharm1/gMIN)
where g
harm
is the harmonic mean of gs.
(Hint: Replace c
i
by its closed form expression and manipulate the expression.)
d. (1 pt) Show that A
n
/g
n
A
n
/g
n
, where A
n
is the argument of the Q-Function for the n
th
sub-channel assuming that the bit-rate b
n
is held constant. Interpret the result.
(Hint. Work with closed form expression for g
n
E
n
instead.)
e. (2 pts) Find the sub-channel which has the worst sensitivity in terms of P
e,n
for 1 + 0.9D
1
and
1 + 0.5D
1
channels. Use = 3dB, N=16, SNR
MFB
=10dB for multitone.(Assume A
n
/g
n
=
A
n
/(g
n
)).
4.11 Complexity of Loading
This problem uses 100 parallel channels(N=100) and evaluates dierent algorithm complexities:
To compute a water-lling energy/bit distribution, we must solve (??) with the additional constraint of
positive energies. An operation is dened as either an add, multiply, or logarithm. The word complexity
denotes the number of operations.
For parts (a) and (b), this problem is primarily interested in the complexity as an order of N and
N

, (i.e. is it O(N
2
)?) Also, comparison of part (a) with part (b) provides an idea of how much less
complex Chows algorithm is, so detailed complexities are also of interests, although an exact number is
not necessary. Please use a set of reasonable assumptions (such as sorting takes O(N log N)). Chows
algorithm here refers to Chows on/o primer. For (c), assume that we start with b = [000...0], then
do E-tight. The complexity would depend on the total number of bs, and . So, you can express your
answer in terms of b and . It is dicult to compare (c) with (a) or (b), as they really solve dierent
problems. Nevertheless, some reasonable estimates would be helpful.
a. (2 pts) Formulate an ecient algorithm that computes the water-lling energy/bit distribution.
Compute the number of operations for water lling. How does this complexity depend on N

?
b. (2 pts) What is the complexity for Chows algorithm?
c. (2 pts) What is the complexity for LC algorithm?
d. (1 pt) Compare the dierent complexities calculated in (a),(b),(c).
4.12 Dynamic Loading
An ISI-channel with AWGN has a multitone transmission system that uses MA LC loading changes
from 1 + D D
2
D
3
to 1 + D .1D
2
.1D
3
. The energy per dimension is

c
x
= 2 and

b = 1 with
A0
2
= .1. Use = 1 and a gap corresponding to uncoded PAM/QAM with P
e
= 10
6
.
a. (2 pts) Find the number of bit-swaps that occur because of this change when N = 8 and N = 12.
(You may assume innite-length basis functions and that the SNR is approximated by using the
fourier transform value in the center of each band as in the examples in Sections 4.1-4.3.)
402
b. (4 pts) Find the information that is transmitted from the receiver to the transmitter for each of
the swaps in part a. Specically, nd the tone indices and the gain-factor changes associated with
the add and delete bit positions.
4.13 VC for the 1 + 0.5D
1
For the 1 + .5D
1
channel, N = 8,

E = 1,
A0
2
= .125, T
t
= 1, and = 3dB
a. (2 pts) Solve for the VC subchannels, g
n
.
b. (2 pts) Find the optimal RA bit Water-lling distribution. What is the bit rate?
c. (2 pts) Find the optimal MA bit Water-lling distribution for

b = 1.25. What is the margin?
4.14 DMT for the 1 + 0.5D
1
For the 1 + .5D
1
channel, N = 8,

E = 1,
A0
2
= .125, T
t
= 1, and = 3dB
a. (2 pts) Solve for the DMT gains, P
n
, and the optimal water-lling RA bit distribution. What is
the bit rate?
b. (2 pts) Compare the gains P
n
to the gains
n
of vector coding for the same channel. Which one
is better in terms of the ratio of largest to smallest bit assigned? Why is that?
c. (3 pts) Plot the Fourier spectra (i.e. the t with appropriate zero-padding) of the columns of M for
VC and for DMT in case of N=8,16,32. What is the change of pattern in spectra as N increases?
4.15 Loading - 15 pts - Midterm 2000
A DMT design is used on the channel with discrete-time response P(D) = 1D
2
with additive white
Gaussian noise that has psd (discrete-time variance)
2
= .181 and the transmit energy per dimension is

c
x
= 1. Loading will be executed on this channel for N = 16 and = 2. (Hint think carefully if anything
about this channel looks familiar and might save you considerable time in executing the computations -
for instance, what do you know about even and odd sample times at the output?)
a. Sketch the magnitude of the Fourier Transform of the channel, [P(e
T
)[ and nd the channel
SNRs, g
n
, at all frequencies of interest to DMT loading. (3 pts)
b. Execute waterll rate-adaptive loading for this channel with a 0 dB gap. (3 pts)
c. Round the distribution that you obtained in part b so that the granularity is = 1 bit per two
dimensions on complex subchannels and one-bit per dimension on one-dimensional subchannels
and nd the new data rate. (1 pt)
d. Now use the LC algorithm to compute a RA data rate for = 1. Be sure to show your incremental
energy table. (4 pts)
e. Find a MA bit distribution with = 1 for transmission with 8.8 dB gap at a data rate corresponding
to

b = 1, and the corresponding margin. (4 pts)
4.16 Generalized Nqyuist
a. (2 pts) For what N 1 does multitone transmission satisfy the generalized Nyquist criterion and
symbol rate 1/T on any linear ISI additive white Gaussian noise channel?
b. (2 pts) What could the receiver do to attempt to correct the situations (that is, the N values) for
which the GNC in part (a) is not satised.
c. (2 pts) How does the N necessary for a channel depend on the channel characteristics?
4.17 Periodic Channels
The simplest possible case of nding eigenfunctions is for the periodic channel; for which this problem
seeks to prove various interesting facts. These facts will be useful for the investigation of Discrete
Multitone Transmission. r(t) is the channel autocorrelation function
403
a. (2 pts) Suppose r(t) = r(t + T), show
n
(t) = e
j
2
T
nt
are eigenfunctions of the channel autocor-
relation function. (Hint: express r(t) in a Fourier series.) By eigenfunction, this problem means
that if (t), t [0, T), is used for a channel with periodic r(t), the output y(t), t [0, T), will be
a scaled version of (t). The scaling factor is the eigenvalue. Note that here, both the input and
output signals are restricted to the interval [0, T).
b. (1 pt) What are the associated eigenvalues?
c. (2 pts) If the channel were not periodic, but had nite length (), how could the designer create
a new set of functions

n
(t) from
n
(t) such that part of the output signal looks periodic? (hint:
think of cyclic prex in DMT.)
Parts (a) and (b) show the eigenfunctions are independent of the channel, which is a very desirable
property. However, this only happens with periodic channel. The question is, for r(t) non-periodic,
can the designer modify the input so that the output appears is as if the channel is periodic. More
precisely, let r
r
(t) denote the autocorrelation of a periodic channel, r(t) denote the non-periodic
version (i.e. r(t) = r
r
(t) for t [0, T), r(t) = 0 outside of [0, T)). The relation (t) r
r
(t) = k(t)
is true, where k is the eigenvalue. Can the designer modify (t) so that (t) r(t) = k(t), for
t [0, T)? If so, the transmission system can then use the channel-independent eigenfunction. By
part of output signal looks periodic, we mean the output signal looks as if it were created by a
periodic channel. Hint: think guard period and extensions.
d. (1 pt) If the receiver only looks at the periodic part of the output signal, what would the associated
eigenvalues be?
e. If a transmission system uses the channel-independent eigenfunctions developed in this problem,
and thus provides the appearance of a periodic channel, how much more bandwidth does the design
have to use, or in other words, how much faster does the sampling rate have to be?
4.18 Vector Coding
This problem concerns channel partitioning and Vector Coding.
a. (2 pts) Consider the following vector channel :
The system uses a matrix transmit lter, M, and a matrix receiver lter, F

, such that FF

=
MM

= I i.e. the matrix lter doesnt change the energy.

Moreover, we want that Y
n
=
n
X
n
+ N
n
Given that the channel is 1 + 0.5D, calculate P, M, F

and
t
s. Assume
A0
2
= .125,

c
x
= 1, and
N = 8.
b. (3 pts) Using the value of s, calculate the optimal energy and bit distribution using a loading
algorithm with = 3dB.
c. (1 pt) What is the SNR
vc
for this vector-coded system?
404
4.19 Theoretical questions on Vector Coding
a. (1 pt) Show that N in vector coding is indeed white and Gaussian with variance N
0
/2.
b. (3 pts) Calculate R
yy
and show that [R
yy
[ =

N
n=1
(
A0
2
+ c
n

[
n
[
2
).
c. (2 pts) Verify 1 + SNR
vc
=
_
]R
yy
]
]R
nn
]
_
1/N+
assuming that the gap is 0 dB.
d. Prove that F

=
_

_
1 +
1
SNR1
0 0
1 +
1
SNR2
0
. .
0 .
0 0 . . 1 +
1
SNRN
_

_
R
Xy
R
1
yy
4.20 Grasping Vector Coding DMT
Assume the channel is 1 + aD
1
. This problem varies a and N to determine their eects on F,
and M.
a. (1 pt) For N=4 and a=0.9, compute F, and M. Plot the modulus of the spectra of the eigenvectors
of M, that is, take the columns of M as time sequences and calculate their abs(t). Use at least
32 points for the ts. Turn in this plot. What shape do these lters have?
b. (2 pts) Repeat part (a) for N = 4 and a = 0.1. Do you notice any dierences in the resulting
lters ts compared to (a)? What explains this behavior?
(Hint: Part (c) might be helpful)
c. (2 pts) Set a = 0.9 and N = 8. Calling
i
the diagonal elements of the matrix, calculate

0
,(
1
+
2
)/2, (
3
+
4
)/2, (
5
+
6
)/2 and compare to the values of H
n
in Example 4.1.1.
Explain the similarity.
d. For all the examples tested, matrix F seems to have some special structure. Describe it. How can
you use this to reduce the number of operations in the receiver?
4.21 Partitioning - 2002 Midterm
A wireless transmission system is shown in Figure 4.45. Uncoded QAM with gap 8.8 dB at P
e
= 10
6
is used on each of 3 transmit antennae with the same carrier frequency but independent input data
streams. Each has symbol rate 100 kHz and the energy per dimension at that rate is unity (c
x
= 1).
There are also 3 receive antennae. All 9 paths shown are AWGN channels with independent noise of
two-sided psd .01 on each. The gains are as shown. The other signals may be considered as noise for
each of the 3 main signal paths from xmit antenna n to receive antenna n or all 3 signals may be jointly
transmitted and received. This problem investigates the dierence in approaches.
405
Figure 4.45: Wireless MIMO channel.
a. (2 pts) If the other two signals are considered to be additive Gaussian noise, what is the possible

b and b as well as data rate R, according to the gap formula ? (2 pts)

b. (3 pts) Consider the possibility of co-generation of all three xmit antennae signals and co-reception
of all 3 receive antennae signals. Show this can be described by a 3 3 channel matrix P. This is
sometimes called a MIMO channel (multiple-input, multiple-output)
c. (1 pt) How many real dimensions are there in this MIMO channel? How many complex dimensions?
d. (5 pts) Use your answer in (b) to design a transmitter and a receiver that will provide 3 independent
parallel channels. Show the transmit and receiver processing and specically provide the entries
in any matrix used. Also show subchannel gains g
n
.
e. (1 pt) What is the best water-lling energy and corresponding number of bits for each of your
channels in the answer to part d?
f. (3 pts) Determine

b and b as well as R for your system in parts (d) and (e) and compare to your
answer in part (a). What does this tell you about partitioning in space?
4.22 Channel ID
A DMT system uses the channel ID method of section 4.7. Slobhouse Designs Inc. has decided that
1dB accuracy in SNR measurement is sucient.
a. (1 pt) Slobhouse decided to measure gain so as to induce less than 1dB error in SNR. How many
training symbols (L) does SDI need?
b. (1 pt) If Slobhouse uses the same number (L) of symbols as in part a to do noise estimation,
comment on SDIs noise estimation.
c. (3 pts) If L
1
is the number of measurements to estimate the gain, and L
2
is the number of
measurements to estimate the noise, approximately estimate the minimum L
1
+ L
2
such that the
overall SNR error is less than 1 dB. There may be several solutions that are close in answer for
this part.
4.23 Channel ID - Midterm 2000
Channel identication of Section 4.7 estimates channel SNRs for a system with symbol rate 4 Hz on
a linear bandlimited AWGN channel. A periodic training signal is used. This problem estimates training
time.
a. How many symbols of training are necessary for an average gain-estimation error of .25 dB (so
that the variance of the gain estimation error is less than .25 dB away from the correct answer)?
How long will this be? (2 pts)
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b. Suppose the FFT size is N = 8192, but that the channel length is less than = 640 sample
periods. If the gain-estimation distortion is distributed evenly in the time domain over all 8192
time samples, but the result was windowed to 640 time-domain samples, what would be the new
training time in symbol periods and in absolute time? (2 pts)
c. Suppose .25 dB noise estimation error is tolerable on the average (so that the standard deviation
of the noise variance estimate is less than .25 dB away from the true noise). What is the step
size of the steady-state FEQ-error based noise estimator,
t
, with unit energy transmitted on the
corresponding subchannel? (1 pt)
4.24 Avoiding SVD in Vector Coding.
Vector coding is used on the AWGN channel.
The receiver uses W = R
Xy
R
1
yy
, a MMSE estimate for X given y. Then
X = Wy U EUU

= R
UU
a. (5 pts) Find W in terms of
n
,
n
,
2
and F. Look at your expression for W. What is the equalizer
W doing to each sucbchannel? Why is this the same as minimizing MSE on each subchannel
individually?
b. (1 pt) Show how the VC receiver W could be implemented adaptively, without explicit knowledge
of P or F

(pictorially and in few words).

Now, the hard part, implementing the transmitter without explicit knowledge of P or M.
c. (2 pts) Pretend x is a channel output and Y is a channel input. Write a vector AWGN expression
for x in terms of Y and u.
d. (2 pts) What is the MMSE matrix equalizer, call it H, for Y given x?
e. (2 pts) If the receiver could, during training, communicate Y and SNR
n
over a secure feedback
path to the transmitter, show how the vector coding transmit matrix could be determined adap-
tively. (pictorially and in few words)
f. (1 pt) Do we need to do SVD to implement VC (is there an adaptive way to do this perhaps)?
4.25 TEQ Analysis in a DMT system - Salvekars Marathon
A causal IIR channel has sampled-time response D-transform
1
1.9D
.
The transmission system is an ISI-free DMT system only if goes to innity. The goal is to reduce
the ISI, while making sure that the noise is not enlarged greatly. In general, it may be dicult to derive
the R
xy
and R
yy
matrices for a general IIR channel. However, we may have a program which solves the
TEQ problem for arbitrary FIR channels. So, instead of simply doing all the calculations by hand, we
will use the aforementioned programs. If the decay in the IIR channel is fast enough, we just need to
take enough taps to ensure that our channel is OK.
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a. (3 pt) Check if the results of your program are the same as the results of the example (choose an
FIR equivalent of the IIR channel, i.e truncate IIR channel into FIR channel with enough large
number of taps such that FIR approximation is VERY close to IIR channel). Report the output
of your programs in the form of R
xy
, R
yy
, w,and b.
(b) (h) Convert your program to nd the PSD of the distortion. Note, that while your old
program may not have computed which frequency bins were actually used, in this case, we need to
know which bin is used. This means that the program best uses the t instead of eig subroutines of
matlab, since we are not sure which frequency bin corresponds to which eigenvalue. Furthermore,
taking the eigenvalue decomposition of large matrices is very time consuming, especially when we
know the eignevectors. For the following problems = 0.
b. (1 pt) First, write an expression for the output of the channel after the teq. This should be in
terms of P(D), W(D), X(D) and N(D).
c. (2 pt) Now, suppose the rst + 1 are the ones that the DMT system is synchronized with(i.e.
the rest of the taps are ISI). Write an expression for the ISI+noise. Call the rst + 1 taps of
P(D) W(D) as H
t
(D). The nal form of the answer should be: Err(D) = I(D)X(D) + N
t
(D),
where N
t
(D) is the noise through the system, and I(D) is a function of H
t
(D).
d. (2 pt) Prove that for a single pole P(D), P(D) = (
1
1aD
), the I(D) will eventually go as c a
n
.
When does this behavior begin?
e. (2 pt) If your noise and ISI are uncorrelated, what is the PSD of the error (write it in terms of D
transforms)? What is this in the frequency domain, S
ee
(e
jw
)?
f. (2 pt) In a DMT program, we need to specify the noise at each discrete frequency. We want to
approximate the noise power in each bin by the sampled PSD. Write a program that will do this in
matlab. The inputs into this program should be

E
x
,
2
, I(D),and W(D). As a check, let

E
x
= 1,

2
= .5, N=10, I(D) = .3D
4
+.1D
5
, W(D) = 1+.9D. Using t can make this very simple. Please
present your values.
g. (3 pt) Now, change your DMT program so that it takes in the (sampled) error spectrum which we
can get from the program in (f) as an argument. The DMT program will now do waterlling Find

b for an

E
x
= 1, N=10, P(D)=1+.5D, and S
ee
(e
jw
)[ 2pik
N
= [P[k][
2
, where S
ee
(e
jw
) is the power
spectrum of the error sequence and P[k] is the t of the pulse response(i.e, what is

b if the PSD of
error is the same as the PSD of pulse response (=P(e
jw
)[ 2pik
N
)? ).
h. (2 pt) Putting (f) and (g) together:
For the following parameters, what is the

b in the DMT system? Choose the same setup as in
example 10.6.1. Now, with = 0, let L=4, = 3. Sweep N until we reach asymptotic behavior.
Plot

b vs. N. Also, do the same with the unequalized channel, remembering to include the error
(i.e, we do not have TEQ, but still have the same for cyclic prex!).
i. (1 pt.) Do (h) with

E
x
=1000. What does this tell you about the teq?
4.26 DMT Spectra
This problem concerns the spectrum of a DMT signal. The transmit signal will be assumed to be
periodic with period T.
a. (2 pts) Show that the frequency spectrum of this transmit DMT signal is a sum of impulses.
b. (3 pts) Show that multiplying the periodic transmit signal x(t) by a window function w(t) that has
the value 1 for 0 t T and is zero elsewhere generates the non-periodic DMT signal studied in
Section 4.4. Compute the spectrum of this new signal from the spectrum of the original periodic
signal and the spectrum of the window.
408
c. (1 pt) Compare the spectrum of part (b) for two situations where > 0 and = 0.
d. (2 pts) Suppose a duplex channel has signals traveling in opposite directions on the same media.
The designer wishes to separate the two directions by allocating dierent frequency bands to the
two DMT signals, and separating them by lowpass and highpass ltering. Can you see a problem
with such a duplexing approach? If yes, what is the problem?
4.27 Isakssons Zipper
For the EPR6 channel (1 + D)
4
(1 D),
a. (1 pt) Find the minimum length cyclic prex to avoid inter-tone interference in DMT.
b. (2 pts) Find the minimal length of the cyclic sux in terms of both channel length and channel
end-to-end delay in sampling periods to ensure that both ends have symbols aligned for both the
case of no timing advance and with timing advance.
c. (2 pts) Do the receiver or transmitter need any lowpass, bandpass, or highpass analog lters to
separate the two directions of transmission (assuming innite precision conversion devices)? Why
or Why not?
d. (2 pts) Suppose two dierent systems with at least one side co-located (i.e., the transmitter and
receiver for one end of each of the channels are in the same box). The two signals are added together
because the channel wires touch at the co-located end. Could Zipper have an advantage in this
situation? If so, what?
4.28 Kasturias Block DFE
Subsection 4.6.7 describes the so-called Block DFE for channel partitioning. This problem investi-
gates the improvement for this guardband-less transmission method for the 1 + .9D
1
channel of the
notes. Use all parameter values as always for this channel, except that we will assume this channel is
causal and 1 + .9D.
a. (2 pts) Find the 8 8 matrix

P for N = 8 with the BDFE, and its associated SVD for vector
coding.
b. (2 pts) Using water-lling for loading, what is the SNR for this system of parallel channels?
Compare to VC and DMT.
c. (2 pt) Draw a picture of the receiver for the BDFE in this case and indicate where the extra
complexity enters.
d. (2 pts) Kasturia found a way in 1988 to derive a Tomlinson-like precoder for this channel. Explain
essential features of such a precoder and why it might be dicult to implement. Can you state
why the BDFE might be undesirable for partitioning?
4.29 TEQ Design for the 1/(1 + 0.5D) Channel
For the 1/(1 + .5D) channel, with white input

c
x
= 1, AWGN with
A0
2
= .04, and T
t
= 1. Force
B(D) to have squared norm 4/3 with a 3-tap TEQ.
a. (4 pts) Find the best B(D) with MMSE design for = 1 and = 2.
b. (2 pts) Find the corresponding MMSEs in part a.
c. (2 pts) Find the corresponding TEQs (W(D)) for part a.
d. (2 pts) Compute the unbiased channel for each W(D) and corresponding distortion energy per
sample.
e. (4 pts) Using = 3 dB, nd SNR
dmt
for this equalized channel with N = 8. Repeat for large N
to nd the limiting value of the SNR
dmt
for the = 1 case. Find also then the largest possible
margin for

b = 1.
409