Lecture 7

Multicarrier Modulation and OFDM

 Update/Key Lecture Materials
 Have Considered
Basic components of digital commun. Systems
Source coding
Digital modulation and demodulation
Convolutional coding
Focused on AWGN channel
» We will now consider
 Frequency selectivity fading channel
 OFDM basics
 Implementation issues
 Performance enhancement
 Applications
2
 Introduction
 OFDM is short for Orthogonal Frequency
Division Multiplexing.
 Converts a wideband frequency selective fading
channel into a parallel collection of narrowband
frequency flat sub-channels
 Reduces the computation complexity associated
with high data-rate transmission over frequency-
selective channel

3
 Motivation
 High-data-rate wireless communications
 Large bandwidth
 Limitations caused by the radio environment 
multipath propagation
 OFDM can overcome and take advantage of
multipath fading and thus eliminate inherent data
rate limitations
 OFDM is good for high-data-rate systems

4
Physical limitations/Radio environment
 Path loss
 Doppler spread
 Delay spread caused by multipath fading

5
Wireless Channel

 Path loss
 Large-scale fading
 Small-scale fading

 Check Ch 13 of Proakis

6
Time and Frequency Domain Description of Multipath

7
 Inter-symbol Interference (ISI)
Direct
Path
0 T t
TX Indirect RX
Path

t
Received
Power 2

2
Delay t

Two-ray equal gain profile

/T small  negligible ISI
 is rms delay spread
/T large  severe ISI
8
Delay Spread and Coherence Bandwidth
 Delay Spread 
 It is the amount of time that elapses between the 1st
arriving path and the last arriving path
 A large  implies a highly dispersive channel in time and
a long channel impulse response
 Coherence Bandwidth Bc
 It is the frequency dual of the delay spread
 It indicates the range of frequencies over which the
channel stays constant
 Bc~1/

9
 Frequency-Selective Fading – ISI
 Frequency-flat fading
 If T   or B  Bc (T: symbol duration, B: system bandwidth)
 Flat in the transmission bandwidth
 Frequency-selective fading
 If T   or B  Bc
 Brings ISI, the larger  relative to T, the more severe the
ISI
 ISI brings an error floor for communication
 For high-data rate, broadband wireless system
 Large B, and small T
 Severe ISI – one of the major issues for broadband
wireless

10
 Techniques for ISI Suppression
 Spread spectrum (CDMA)
 Require bandwidth much larger than the data rate
 Unviable for broadband wireless
 Time-domain Equalization
 High overhead and complexity
 Check Ch 9 of Proakis
 OFDM
 Efficient implementation through IFFT/FFT
 Provide frequency diversity, multiuser diversity gain
 Better support for MIMO transmission

11
 History of OFDM
 The basic principles of OFDM was proposed in
several publications in the 1960’s.
 Since 1966 FDM systems with overlapping spectra
were proposed
 The next step is a proposal to realize an FDM
system with DFT
 Finally, in 1971 Weinstein and Ebert proposed a
complete OFDM system, which included
generating the signal with an FFT and adding a
guard interval in the case of multipath channels

12
The Multicarrier Concept – Time Domain
 Basic idea: to make T>
 Multicarrier modulation
 It divides the high-rate transmit stream into L low-rate
substreams
 L is chosen such that each subcarrier has effective
symbol time TL>

13
The Multicarrier Concept – Frequency Domain
 Basic idea: to make B<Bc
 Multicarrier modulation
 It divides the wideband incoming data stream into L
narrow-band sumstreams
 L is chosen such that each subcarrier has bandwidth
B/L<Bc

14
Example

15
 OFDM Basics
 In conventional multi-carrier modulation, the
whole bandwidth is divided into many narrow
sub-channels which are spaced apart and not
overlapped.  Low spectral efficiency
 In OFDM, by using
orthogonal carriers with
nulls at the center of the
other carriers, the sub- f

channels are overlapped.

 increase spectral efficiency
16
 Orthogonal Signals
 For each sub-band (or sub-channel), we associate a
sinusoidal carrier signal

 To have orthogonal sub-channels, we need

 This is satisfied when (Ts is the symbol interval)

 So the minimum sub-channel spacing is

17
Block Transmission with Guard Intervals
 Group L data symbols into a block – OFDM symbol
 Multicarrier modulation deals with intra-symbol ISI
 Guard time Tg
 Eliminate ISI between subsequent OFDM symbols
 As long as Tg > 

Block Transmission + Guard Interval

At transmitter

At receiver

 Two different representations

 DFT implementation and circular convolution
 Channel matrix representation
18
 OFDM Transmitter
f0
d0
d1

{dn} f1 Add s(t)

MOD
MOD S/P P/S Cyclic Mixer
fN-1 Prefix
dN-1

1 1
For orthogonality, f  
NT TS

Main idea: block transmission + CP

19
 Review - DFT and FFT
 The discrete Fourier transform (DFT) of the sequence x0,
x1, …,xN-1 is given as

 The inverse DFT of the sequence X0, X1, …, XN-1 is given

as

 To directly calculate the complete DFT, N2 complex

multiplications are required
 Fast Fourier transform (FFT) algorithms can reduce the
computation to only approximately Nlog2N complex
multiplications
20
 DFT Implementation
 Equivalent baseband notation of OFDM signal:
N 1
s (t )   d n exp  j 2  f n  t , 0  t  TS IDFT
n 0

 At a sample rate of Ts/N (Nyquist rate)

 kTS  N 1  TS 
s (k )  s     dn exp  j 2  nk  f  N , 0  k  N  1
 N  n 0  
 Since f  TS  1,
N 1 j 2 
nk
s (k )   d n e N
 IDFT dn , 0  k  N  1
n 0

 Then we can get s(t) via DAC

21
 DFT Implementation
Channel

h(k)

 (I)DFT can be much more efficiently implemented by (I)FFT

22
 DFT Implementation
 Matrix representation
s  Fd
 Each d n , n  0, 2, N  1 is a modulated
frequency domain sample
 Each sn , n  0, 2, N  1 is a sample of the
OFDM symbol, i.e. time domain sample

23
 Guard Interval – CP
 OFDM deals with ISI within one OFDM symbol
(OFDM block)
 Inter-block interference still exists if not inserting
guard interval longer than max
 Guard interval can consist of no signal. In this
case, however the problem of inter-carrier
interference (ICI) would arise, since sub-carriers
are no longer orthogonal.
 By cyclic prefix in OFDM symbol, ISI and ICI
can be eliminated completely

24
 Inter-block Interference
 When the length of the cyclic prefix Ns>P, after
the cyclic prefix is eliminated, there is no inter-
block interference

t
Ns N Ns

t
Ns N Ns N

TX:
No inter-block interference
RX:
25
 Inter-carrier Interference
 Let  and  denote linear and circular convolution
respectively
 At the receiver after DFT,

d n  DFT s ( k )  h ( k )  DFT n( k )
 DFT IDFT d n  h ( k )  DFT n( k )
  d n  items consisted of {d 0 ...d n 1 , d n 1 ...d N 1}  DFT n ( k )

 This causes inter-carrier interference (ICI)

26
 Inter-carrier Interference
 In order to cancel ICI, we want
IDFT d n  h ( k )  IDFT d n  h ( k )

 Then, d n  DFT IDFT d n  h ( k )  DFT n ( k )
 d n  DFT h ( k )  DFT n ( k ) , no ICI
 To realize the circular convolution equal to linear
convolution, the OFDM symbol is cyclically extended in
the guard interval to create cyclic prefix, cyclic postfix, or
both of them

27
 A Different View
-- Matrix Representation of the ISI Channel
 Assume channel impulse response length is P
P 1
yn   hk sn  k  nn
k 0
 Matrix representation
 y0   h0 0     0   s0   n0 
y   h h0 0    0   s1   n1 
 1  1    
 y2          s2   n2 
      
     hp 1         
  0           
      
       0     
y   0
    0 hp 1  h1 h0       

H is Toeplitz S 28
 Circulant Matrix
 A Circulant matrix is an n  n matrix whose rows are
composed of cyclically shifted versions of a length-n
list. For example, the 4  4 circulant matrix on the list
is given by l  1, 2,3, 4
4 1 2 3
3 4 1 2 

2 3 4 1
 
1 2 3 4
 One important property: a circulant matrix can be
diagonalized by the Fourier transformation matrix

29
 Cyclic Prefix
 s
In order to form a circulant matrix, instead of transmitting ,
transmit
 Assume P=2, then
 h1 h0 0   0  h0 0  0 h1 
0 h h   sN 1     s0 
 1 0       h1 h0   0   
s s
Hs           0    0       1   Hs 
       
     0       0 
 0   
0 h1 h0   sN 1 
   sN 1 
 0  0 h1 h0 
 An effective circulant matrix H  is created using cyclic prefix
 Efficiency: N/(N+Ns) with Ns>P, since a vector of length N+Ns
will be transmitted for a length-N data vector
 When N increases, efficiency increases
30
 Diagonization of Circulant Matrix
 Circulant matrix can be diagonalized as
 D
F H HF Gain of sub-channel
H
where N 1
 mn 
Fmn 
1 
exp  j 2
mn  (H )nn   hm exp   j 2 
  N
N  N  m0

 Apply a transmitter filter F, a receiver filter FH

 Similar to the implementation on Slide 23
 N parallel flat fading sub-channels are created
 No ISI within each OFDM symbol
 Note, the transmitter can diagonalize H without
knowing any information about H
 Different from MIMO, as we will see next lecture
31
 OFDM Tx-Rx Structure

Frequency domain Time domain

 Advantages of OFDM
 With cyclic prefix, intra and inter OFDM symbol ISI
can be eliminated completely
 An effective circulant matrix can be created using
cyclic prefix, as a result, ICI can be eliminated
completely
 Implementation complexity is significantly lower than
that of a single carrier system with an equalizer
 Provide frequency diversity
 Forward error correcting code such as convolutional code
with interleaver is needed as some sub-carriers will be in
deep fade

33
 Implementation Issues
 Coding and modulation
 Channel estimation
 Synchronization
 Peak power problem

34
 Coding and Modulation
 Coding across sub-channels brings frequency
diversity
 RS code, convolutional code, concatenated code and
turbo code
 Interleaving can be applied to randomize the
occurrence of bit errors prior to decoding
 QAM, especially rectangular constellations, is the
most popular type of modulation in combination
with OFDM
 Coded modulation: TCM, coded QAM, space-
time code
 Channel information is needed at receivers
35
 Channel Estimation
 Coherent and differential detection
 Pilot symbol aided channel estimation
 Decision-directed channel estimation

36
 Differential Detection
 The transmitter has to apply differential encoding
 Differential detection in frequency domain
 Each sub-carrier is compared with the adjacent sub-carrier
within the same OFDM symbol
 Sensitive to delay spread
 Differential detection in time domain
 Each sub-carrier is compared with the same sub-carrier of the
previous OFDM symbol
 Sensitive to Doppler spread
 Normally, differential detection has 3dB SNR
degradation relative to coherent detection
37
 Coherent Detection
 Require channel state information (CSI)
 The main issue is how to find CSI without
introducing too much training overhead
 Use estimates of the CSI to determine the best
possible decision boundaries for the constellation
of each sub-carrier
 Suffers an SNR loss because of imperfect channel
estimates, training overhead, which typically
reduces the difference between differential and
coherent detection from 3dB to about 1 to 2dB

38
 Pilot Symbol Aided Estimation
 Channels are time varying and frequency
selective
 Have to estimate time-varying amplitudes and
phases of all sub-carriers
 First estimate channel values at the known pilot
symbol positions
 Then based on these pilots, CSI at all other sub-
carriers and times can be estimated by
performing a two-dimensional interpolation

39
 Pilots Pattern
 The pilot spacing has to fulfill the Nyquist
sampling theorem. By choosing spacing
much smaller than minimum requirements,
good estimation occurs Carrier
 Pilot freq spacing- p  N index
2 / T

 For N=512
 If delay =τ/T=67 then spacing is 3 OFDM carriers
 if delay=11 then 23 sub-carrier spacing
Time index
 To reduce pilot overhead, time and frequency
spacing can be the maximum distance of less than
coherent time and coherent bandwidth, respectively
 Tradeoff between channel estimation performance
and pilot overhead
Reference: An analysis of two-dimensional pilot-symbol assisted
modulation for OFDM, R. Nilsson, et. al., 1997 40
 Decision-Directed Estimation
 To start decision-directed estimation, at least one
known OFDM symbol must be transmitted. This
enables the receiver to attain CSI of all sub-carriers,
which are then used to detect data in the following
OFDM symbols
 Data decisions from previous symbols are used to
predict the channel in the current OFDM symbol.
Channel correlation in time and frequency domains
are used in the prediction
 Avoid the power loss due to transmission of the
pilots

41
 Synchronization

 Frequency offset
 Timing offset
 Frequency offset estimation
 Timing offset estimation

42
 Frequency Offset
 Frequency offset is a critical factor in OFDM, since it violates
the orthogonality of sub-carriers and results in inter-carrier
interference (ICI)
 As there is no guard band, very small frequency offset can
lead to large inter-carrier interference
 For a frequency offset f between the receiver and transmitter,
N 1 k f
j 2  ( n  )
s( k )   d n e N f
, 0  k  N 1
 Thus, n 0

f
  f  j
f
d n , f  DFT s(k )  d n  sinc  e  ICI terms.
 f 
43
 Timing Offset
 Sampling frequency needs to be correct, but
sampling instance offset smaller than guard interval
does not violate the orthogonality of sub-carriers, it
only leads to a linear phase shift in the sub-channels’
gains
~ j 2  f n t
d n , t  d n  e .
 Otherwise, additional interference is generated

44
 OFDM vs Single-carrier
 Frequency synchronization
 The requirement of OFDM is more stringent compared to
single-carrier
 The orthogonality of the data symbols is reliant on their
being individually discernible in the frequency domain
 Timing synchronization
 The requirement of OFDM is somewhat relaxed
 OFDM symbol structure (CP) naturally accommodate a
reasonable degree of synchronization error

45
 Frequency Offset Estimation
 Frequency offset should be corrected before
the receiver FFT.
 OFDM-based pilot symbols can be used to
estimate the frequency offset for coarse and
fine synchronization.

46
 Timing Offset Estimation Method 1

 Redundancy in the cyclic prefix can be used to

estimate the timing offset
 Compute the correlation between two intervals
separated by NTs seconds, because the first NsTs
seconds part of each OFDM symbol is identical to the
last part identical

 Only effective when a large number of sub-carriers are

used
 Long synchronization time, suitable for tracking or
blind synchronization.
47
 Timing Offset Estimation Method 2

 Design special OFDM training symbols to do

estimation (Moose, Warner, and Schmidl)
 The entire received training signal is used to achieve
synchronization
 Short synchronization time
 Non-OFDM pilot based timing offset estimation:
Using a null signal inserted at the beginning of
each group of OFDM blocks.
Null Null
OFDM OFDM Symbols OFDM OFDM …. OFDM Symbols

48
 PAP Ratio Reduction
 Properties of peak-to-average power (PAP)
ratio.
 Clipping and peak windowing
 PAP ratio reduction codes
 Symbol scrambling
 Selective mapping
 Partial transmit sequences

49
 PAP Ratio Properties
 In the time domain, a multicarrier signal is the sum of many
narrowband signals
 At some times, the sum is large
 At other times, the sum is small
 The peak value is substantially larger than the average
 High peak-to-average ratio (PAR)
 OFDM signal has a Rayleigh amplitude, while the power has a
central chi-square distribution with two degrees of freedom and
zero mean
 The distribution of the PAP ratio is given by
P  PAPR  r   1  e  
N
r

 Large peak PAP ratios occur only infrequently

50
 Disadvantages of Large PAP Ratio
 RF power amplifier non-linearity
 Increased complexity of A/D and D/A converters
 Amplifier back-off: Clipping and windowing
 Deliberate clipping can reduce peak value, but will
bring in-band noise (increase BER) and spectral
spreading (cause adjacent channel interference)
 Peak windowing can be used to minimize spectral
spreading: multiply large signal peaks with a certain
non-rectangular window

51
 PAP Ratio Reduction Codes
 Only a small fraction of all possible OFDM symbols
has a bad PAP ratio, which suggests that PAP ratio
can be reduced by using a code that only produces
OFDM symbols for which the PAP ratio is below
some desirable level
 Good performance with little overhead
 A large part of these codes found are Golay
complementary sequences, which leads to the
systematic generation of PAP ratio reduction codes
with some error correction capacity

52
 Symbol Scrambling
 Can be seen as special case of PAP ratio reduction codes. The
difference is that it does not combine error correction and PAP
reduction
 The basic idea is that for each OFDM symbol, the input
sequence is scrambled by a certain number of scrambling
sequences. The output signal with smallest PAP ratio is
transmitted
 Scrambling were proposed under the names selective mapping
and partial transmit sequences. The first applies independent
scrambling rotations to all sub-carriers, while the latter one
only applies scrambling rotations to groups of sub-carriers

53
 Disadvantages of OFDM
 Overheads
 Cyclic Prefix: can be reduced by increasing N
 Power to transmit cyclic prefix: can be lowered by
increasing N
 Implementation issues
 Sensitivity to frequency offsets
especially when N is large and the sub-carrier spacing is
small
 Require highly linear power amplification
high peak-to-average-power ratio, especially when N is
large, which tends to reduce the power efficiency of the
RF amplifier
 Trade-off: efficiency and sensitivity
54
 Performance Enhancement
 Adaptive sub-carrier, bits, power allocation
 The OFDM transmitter can adapt its signaling strategy
to match the channel if the channel state information
(CSI) is available at the transmitter
 MIMO-OFDM
 Combine with antenna arrays at the transmitter and
receiver, results in a MIMO-OFDM system
 Further enhance the performance of OFDM systems
 Increase diversity gain and/or multiplexing gain

55
 Single-user Adaptive OFDM
 Basic idea: adaptive coding and modulation
 Different subcarriers are independent, with gains
sufficiently different across the channel bandwidth
 Subcarriers that yield a higher SNR due to a lower
attenuation can use a higher order modulation
 Subcarriers with lower SNR employ lower order
modulation
 This assignment of different constellation sizes to
different subcarriers is generally done in practice.
 See 11.2-7 (Proakis) for example

56
 Multi-user Adaptive OFDM
 When OFDM with adaptive bit and power allocation is
applied in a frequency selective fading channel, a
significant portion of the sub-carriers may not be used
 Since it is not power efficient to carry any information bit on
sub-carriers which experience deep fade
 Multi-user diversity can be exploited
 Channels of all users are mutually independent
 It is very unlikely that all users suffer deep fade in the same
sub-channel
 Adaptive multi-user sub-carrier allocation allows all the sub-
carriers to be used more effectively

57
 Multi-user Adaptive OFDM

MAO—Multi-user Adaptive OFDM

C. Y. Wong, R. S. Cheng, K. B. Letaief and R. D. Murch, “Multiuser OFDM with adaptive
subcarrier, bit, and power allocation,” IEEE J. Select. Areas Commun., vol.17, No.10, Oct.
1999 58
 MIMO-OFDM
 Recent developments in MIMO techniques promise
a significant boost in performance for OFDM
systems.
 By using OFDM, a frequency-selective MIMO
channel is transformed into a collection of N flat-
fading MIMO channels, one for each sub-carrier,
with each having dimension nR  nT
 Additional degree of freedom in space domain can
be exploited as a result of employing MIMO system
 More details will be covered about MIMO in the
next lecture
59
 Adaptive MIMO-OFDM
 Multiple users can be supported within the same
sub-carrier since they can be separated in the spatial
domain
 Co-channel interference (CCI) is introduced
 Adaptive sub-carrier allocation is needed, since co-
channel users should be carefully selected such that
severe ICI could be avoided
 Good multi-user MIMO technique is needed to fully
utilize the spatial diversity
 Significant gain can be achieved

60
 Applications
 Fixed / Wired-line
 Asymmetric Digital Subscriber Line (ADSL)
 Mobile /Radio
 Digital Audio Broadcasting (DAB)
 Digital Video Broadcasting – Terrestrial (DVB-T)
 Wireless LANs
 IEEE802.11
 HIPERLAN Type II
 Dedicated short-range communications (DSRC)
 LTE, Mobile WiMAX

61
 Summary
 We have covered
 Frequency selectivity fading channel
 OFDM basics
 Implementation issues
 Performance enhancement
 Applications
 Reading assignment
 Chapter 11, 13 of Proakis
 References
 T. Hwang, C. Yang, G. Wu, S. Li, and G. Li, “OFDM and its wireless
applications: A survey,” IEEE Trans. Vehicular Technology, vo. 58, no. 4, May
2009
 S. B. Weinstein, “The history of orthogonal frequency-division multiplexing,”
IEEE Commun. Magazine, Nov. 2009.

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