TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIES

Trans. Emerging Tel. Tech. (2012)

Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.2518

RESEARCH ARTICLE

Partial response DFT-precoded-OFDM modulation

Kiran Kuchi*,†
Indian Institute of Technology, Hyderabad, India

ABSTRACT
A low peak-to-average-power-ratio (PAPR) modulation technique is proposed for discrete Fourier transform precoded
orthogonal frequency division multiple access (DFT-precoded-OFDMA) systems. This technique reduces the PAPR by
introducing a phase rotation between successive modulation symbols along with partial response (PR) precoding before
feeding the data to a DFT-precoded-OFDMA modulator. The PAPR reduction is shown to be quite significant for
amplitude-shift-keying systems based on real constellations employing =2 phase rotation. In particular, for the special
case of binary modulation, the combination of phase rotation and PR precoding produces a signal with low amplitude
variations. We show that the class of PR precoders obtained by sampling the linearised Gaussian-minimum-shift-keying
pulse provides low PAPR and a small degradation in bit-error-rate (BER) performance. In particular, the widely linear
minimum-mean-square-error (WL-MMSE) estimation and WL MMSE decision feedback equaliser (WL MMSE-DFE)
methods that jointly filter the signal and its complex conjugate are shown to be useful in mitigating the additional inter-
symbol interference introduced by the PR precoder. The BER performance is comparable with that of conventional
DFT-precoded-OFDMA systems employing conventional equalisers.
The proposed technique is also suitable for reducing the PAPR of Q-ary phase-shift-keying systems based on complex
constellations employing a phase rotation of =Q. Introduction of Type A-2 PR precoder that is obtained from the lin-
earised Gaussian-minimum-shift-keying pulse reduces the PAPR by 3.0 dB for quadrature phase-shift keying, and 2:5 dB
reduction is observed for Q-ary phase-shift keying with Q > 8. The intersymbol interference created by the PR precoder
causes bit-error-rate degradation in the range of 2.0–2.5 dB when conventional MMSE-DFE receiver is used for symbol
detection. Copyright © 2012 John Wiley & Sons, Ltd.
KEY WORDS
Equalizers; MMSE; MMSE-DFE
*Correspondence
Kiran Kuchi, Indian Institute of Technology, Hyderabad, India.
E-mail: kkuchi@iith.ac.in
†
Part of this work is funded by the DIT project Cyber Physical Systems

Received 30 July 2011; Revised 6 January 2012; Accepted 27 January 2012

1. INTRODUCTION (PAPR). To mitigate the intersymbol interference (ISI)

induced by the DFT precoding, the SC-FDMA system
Orthogonal frequency division multiple access (OFDMA) uses frequency domain equalisation. The main advantage
has been widely adopted in both imminent and emerg- of lower PAPR is the less power amplifier back-off com-
ing wireless standards. The third generation partnership pared with OFDMA. Moreover, the SC-FDMA systems
project long-term-evolution standard has adopted OFDMA use the standard DFT and inverse DFT (IDFT) circuits that
in the downlink and discrete Fourier transform-precoded- are widely used in conventional OFDMA systems resulting
OFDMA (DFT-precoded-OFDMA)[1] (also known as in low implementation complexity.
single-carrier frequency-division multiple access (SC- Traditionally, partial response (PR) precoding is used
FDMA)) [2] in the uplink. SC-FDMA applies DFT to design Nyquist class of pulses that are less sensitive
precoding on the baseband modulation symbols before to sampling timing errors [3]. The duo-binary precoder
transmitting the data on the subcarriers. Different users is one such example. PR precoding is also used to intro-
share the available frequency resources in the frequency duce a controlled amount of ISI to reduce out of band
domain. Because of DFT precoding, the resultant modula- emissions. This principle is used in Gaussian-minimum-
tion signal exhibits single-carrier properties such as sinc- shift keying (GMSK) [4]. The continuous phase, con-
like pulse shaping [3] and low peak-to-average power ratio stant envelope GMSK signal can be approximated as a

Copyright © 2012 John Wiley & Sons, Ltd.

 K. Kuchi

pulse-amplitude-modulation (PAM) signal using the lin- up to 3.0 dB is observed for quadrature PSK (QPSK)
earised GMSK representation that includes only the first and 2.5 dB for Q-PSK for Q > 8. Introduction of con-
term in Laurent’s decomposition [5]. Linearised GMSK stellation rotation is also shown to be useful for QPSK,
has 90°phase rotation between successive binary mod- but this technique offered only a small amount of gain
ulation symbols and uses a specific pulse that intro- for higher-order PSK. For the case of 16 quadrature
duces ISI that spans a few symbols. The combination amplitude modulation (QAM), we observed only 0.5 dB
of constellation rotation and the pulse employed pro- PAPR reduction.
duces a signal with near constant envelope with PAPR For PSK systems employing circularly symmetric
close to unity. However, linearised GMSK is a nonband- complex-valued modulation alphabets, WL equalisation
limited signal whose spectral occupancy exceeds beyond does not provide any benefit over conventional equalisers
the Nyquist cut-off frequency. For Nyquist class of sig- in a white noise channel with ISI [13]. Therefore, in this
nals, the PAPR can also be reduced by designing pulses paper, we consider only conventional MMSE [14], MMSE-
with certain excess bandwidth (BW) [3]. The well-known DFE receivers for PSK detection. Simulation shows that
square-root-raised-cosine pulses have low PAPR depend- both methods are not able to mitigate the ISI introduced
ing on the excess BW allowed. However, spectrum being by the PR precoder fully. A signal-to-noise ratio (SNR)
a scarce resource, allowing additional BW only to reduce penalty in the range of 2.0–2.5 dB is observed for QPSK
PAPR, is generally not preferred. Therefore, there is a and 8-PSK systems. However, in coded systems, advanced
need for designing waveforms with low PAPR without receiver algorithms such as turbo equalisation may be able
BW expansion. reduce this penalty. This aspect needs further investigation.
In this paper, we propose a modification to conventional We remark here that the proposed technique, which uses
DFT-precoded-OFDM modulator where PR precoding and a combination of PR precoding and constellation rota-
modulation-specific constellation rotation operations are tion in DFT-precoded-OFDM systems, is a new method.
applied on the input data to reduce the PAPR. It is To the best of our knowledge, alternative methods that
shown that PR precoders obtained from a certain class reduce PAPR of DFT-precoded-OFDM without BW expan-
of linearised GMSK pulses offer a significant reduction sion have not been proposed in the literature. Considering
in PAPR. The proposed technique offers considerable the trade-off between PAPR reduction and the increase in
PAPR reduction for amplitude-shift-keying (ASK) systems the BER, the proposed technique is most useful for DFT-
employing real constellations using a constellation rotation precoded-OFDMA systems employing binary phase-shift
of =2. For the special case of binary signalling, we obtain keying (BPSK). In particular, this technique can be used
a PAPR of approximately 2.0 dB, which is 4.5 dB less in power-limited systems where the additional power gain
compared with the PAPR of conventional systems. Note obtained by low PAPR is useful in increasing the link
that standard PAPR reduction methods such as selective budget and the range.
mapping and partial sequencing cannot be applied after The organisation of the paper is as follows. In
DFT precoding of data because it violates the single-carrier Section 2, we introduce the proposed transmitter structure.
property and affects the PAPR properties. The techniques In Section 2.1, the PR precoder coefficients are derived
proposed in this paper are improvements upon standard from a linearised GMSK pulse, and the PAPR character-
DFT-precoded-OFDMA methods. istics are provided for BPSK, 4-ASK, QPSK, 8-PSK and
For the case of real-valued signalling, the additional 16-QAM modulation formats. In Section 3, we analyse the
ISI introduced by the PR precoder is mitigated using a WL MMSE receiver performance for the case of real con-
widely linear (WL) equaliser [6–13] that jointly equalises stellations. In Section 4, design details of WL MMSE-DFE
the signal and its complex conjugate. We propose WL receiver is given. The complexity analysis is provided in
minimum-mean-square-error (MMSE) estimation and WL Section 5. In Section 6, we present BER simulation results
MMSE-decision feedback equaliser (WL MMSE-DFE) for BPSK, QPSK and 8-PSK modulation formats followed
algorithms that are specifically tailored for DFT-precoded- by conclusions in Section 7.
OFDMA systems. We show that both methods are able to
deal with the additional ISI created by the PR precoder
in addition to the ISI inherently present in the propaga- 2. TRANSMITTER
tion channel. In typical wireless channels, the performance
degradation due to the introduction of the PR precoder is A conventional DFT-precoded-OFDMA transmitter (see
shown to be small compared with conventional systems. Figure 1) sends a block of M zero-mean, independent and
We note here that for the case of real-valued constellations, identically distributed real/complex modulation symbols
the WL receiver offers additional degrees of freedom that with E.ja.k/j2 / D 1, where k denotes the discrete time
are used to mitigate the ISI introduced by the PR precoder. index. The data stream a.k/ is precoded using a DFT as
Therefore, WL filtering plays a crucial role in enabling low
PAPR waveform design.
We also show that the proposed technique can be applied M
X 1
j 2lk
to symmetric Q-ary phase-shift-keying (PSK) systems A.l/ D a.k/e M ; l D 0; : : : ; M  1 (1)
that use complex constellations. A PAPR reduction of kD0

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

input output
Sub−carrier
M−point DFT Mapping N−point IDFT

Figure 1. Conventional discrete Fourier transform (DFT) precoded orthogonal frequency division multiple access transmitter.

Figure 2. Discrete Fourier transform precoded orthogonal frequency division multiple access (DFT-precoded-OFDMA) transmitter
with partial response (PR) precoding.

The precoded data are mapped to a set of contiguous‡ sub- For the class of PR precoders considered in this paper,
carriers among the available set of N subcarriers. The time the angle  is optimised using simulation to minimise the
domain signal s.t / is obtained using an inverse discrete PAPR for a given modulation type. Note that introduction
time Fourier transform of phase rotation reduces the number of zero-crossings in
the modulated signal and as a result reduces the PAPR.
N1 CM
X 1
1 For systems employing real constellations, introduction of
sQ .t / D A.l  N1 /e j 2lf t ; t 2 Œ0;TOFDM ; 90°phase rotation reduces the PAPR considerably. How-
M
lDN1 ever, for the case of complex-valued modulation, phase
0 6 N1 6 N  M and N >M rotation is most useful PSK systems with low modulation
(2) order such as QPSK. For higher-order modulation sizes,
where l denotes the subcarrier index, N1 is the index of phase rotation provides little benefit because the constel-
the starting point for subcarrier mapping, f is the sub- lation is already dense (i.e., a constellation point and its
carrier spacing, and the signal spans over the finite time phase rotated version are close to each other). For the spe-
interval TOFDM , which is denoted as the OFDM symbol cial case of binary modulation, it is possible to obtain a
duration, and A.l/ D DFT.a.k//. A cyclic prefix is added signal with very low amplitude variations using a suitable
to the signal before transmission. For large values of M , choice of PR precoder. In the following, we propose a set of
the signal sQ .t / exhibits similar PAPR characteristics as that PR precoders that provide a reduction in PAPR compared
of conventional single-carrier modulation employing sinc with conventional DFT-precoded-OFDMA signal sQ .t / for
[3] pulse shaping. all cases considered in this paper.
The proposed transmitter modifies the baseband data
sequence to further reduce the PAPR using the following
2.1. Partial response precoders obtained
steps (see Figure 2):
from linearised Gaussian-minimum-shift
keying pulse
 A constellation rotation of  radians is applied
between successive data symbols to obtain a sequence
p Gaussian-minimum-shift keying is a continuous phase
c.k/ D e j k a.k/, where j D 1 and a.k/ is the
modulation signal with constant envelope. We begin with
input data sequence.
the aim to synthesise a linearised GMSK-like signal with
 A controlled amount of ISI is introduced into the
low PAPR using the DFT-precoded-OFDMA framework,
constellation rotated modulation sequence c.k/ by
without using excess BW. To this end, we begin with the
circularly convolving the sequence c.k/ with a PR
Laurent’s decomposition [5] to approximate the differen-
precoder p.k/ to obtain x.k/ D c.k/ ˇ p.k/, where
tially encoded GMSK signal in linearised form
ˇ denotes circular convolution operation. The PR
DFT-precoded-OFDMA signal defined as X
sl .t / D j k b.k/p0 .t  kT /; t 2 .1; 1/ (4)
N1 CM
X 1 kD0
1
s.t / D X .l/e j 2lf t ;
M
lDN1 (3) where b.k/ takes values from a BPSK constellation, 1=T
t 2 Œ0; TOFDM ; 0 6 N1 6 N  M is the data rate and the pulse p0 .t / is the principal pulse
in Laurent’s decomposition [5, 15] that is given in the
and N >M
Appendix. A set of pulses can be obtained by controlling
the free variable BT .
where X .l/ D DFT.x.k//.
In the proposed framework, the linearised GMSK pulse
p0 .t / is sampled at symbol rate to obtain the discrete time
‡
The discrete Fourier transform precoded data can also be mapped samples that are used as PR precoder coefficients. The set
to distributed subcarriers with equal subcarrier spacing spanning the of samples that are obtained by sampling p0 .t / at t D kT
entire bandwidth. where k 2 .: : : ; 2; 1; 0; 1; 2; : : :/ is denoted as the PR

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

Table I. Type A partial response precoder coefficients. fundamentally distinct from GMSK; linearised GMSK
Number BT p.k /
pulse is a nonband-limited pulse whose spectral occupancy
exceeds beyond the Nyquist cut-off frequency, whereas
Type A-2 BT D 0:3 [0.0007 0.2609 0.9285 0.2609 0.0007] the proposed system synthesises a different pulse shap-
Type A-3 BT D 0:2 [0.0104 0.3514 0.86 0.3514 0.0104] ing filter (a digital sinc filter convolved with PR precoder
response) that does not cause BW expansion compared
Table II. Type A partial response precoder coefficients. with conventional DFT-precoded-OFDMA. Moreover, lin-
earised GMSK has near unity (0 dB) PAPR, whereas
Number BT p.k / the PAPR of the proposed system will be shown to be
Type B-1 BT D1 [0.707 0.707] in the range of 1.0–2.0 dB. Although linearised GMSK
Type B-2 BT D0:3 [0.0316 0.7070 0.7070 0.0316] uses binary modulation, we show that the PR precoders
Type B-3 BT D0:2 [0.0004 0.0917 0.695 0.695 0.0917 0.0004] obtained from the linearised GMSK pulse also reduce the
PAPR for higher-order modulation alphabets employing
real/complex constellations when a suitable choice of  is
used. Simulation showed that  D =2 is a suitable choice
precoder Type A and the set obtained for t D kT C .T =2/ for real constellations whereas  D =Q provides a low
is denoted as Type B precoder. More specifically, let PAPR for Q-ary PSK employing symmetric constellations.
( To calculate the PAPR, we first generate a discrete time
p0 .t /jtDkT Type A
p0 .k/ D (5) version of s.t /. We calculate the peak power, average sig-
p0 .t /jtDkT C T Type B nal power of the discrete time signal for each OFDM
2
symbol separately. The PAPR is recorded for each OFDM
denote T -spaced samples of the linearised GMSK pulse. symbol over 10 000 realisations. For all the cases shown in
Note that the elements of the sequence p0 .k/ are trun- Table I, the complementary cumulative distribution func-
cated to a suitable length by eliminating the coefficients tion of PAPR is plotted in Figures 3, 4 and 5. In the rest
that take very small values. Because the sequence of the of the discussion, we consider the PAPR measured at 0.01
proposed PR precoder coefficients p.k/ does not take neg- complementary cumulative distribution function point.
ative values, it is obtained by mapping the elements of p0 For BPSK modulation, referring to Figure 3, for M D
starting from left to right. The proposed PR precoders are 1200, Type B-3 precoder has a PAPR of 1.52 dB, whereas
explicitly defined in Tables I and II. Sampling the pulse at Type A-2 and Type B-1 have 2.39 and 2.09 dB PAPR,
t D kT and setting BT D 0:3, we obtain Type A-2 pre- respectively. Referring to Figure 4, for M D 12, all
coder, whereas setting BT D 0:2 gives Type A-3 precoder. the proposed PR precoders have similar PAPR char-
Further, sampling the pulse at t D kT C .T =2/ and setting acteristics, and the PAPR variation is in the range of
BT D 1 give Type B-1 precoder, and for BT D 0:3 and 1.0–2.0 dB. Compared with conventional DFT-precoded-
0.2, we obtain Types B-2 and B-3 precoders, respectively. OFDMA employing QPSK modulation, which has a PAPR
For BPSK signalling, we observe that introduction of of 7.6 dB, the reduction in PAPR of the proposed set of
e j k.=2/ D j k constellation rotation and PR precod- precoders is quite significant. Next, the influence of con-
ing produces a linearised GMSK-like signal with low stellation rotation and PR precoding is shown separately in
amplitude variations. Although the proposed system shares Figure 5. It can be seen that both PR precoding and con-
some commonalities with linearised GMSK/MSK, it is stellation rotation features contribute to PAPR reduction

1
0.9
0.8
0.7
0.6
cdf

0.5
0.4 Type A−2
Type B−1
0.3 Type B−3
Type B−2
0.2 Type A−3
Conv. QPSK DFT−p−OFDMA
0.1
0
1 2 3 4 5 6 7 8 9
PAPR (dB)

Figure 3. Complementary cumulative distribution function (cdf) of peak-to-average power ratio (PAPR) for binary phase-shift keying,
M D 1200. QPSK, quadrature phase-shift keying; DFT-p-OFDMA, discrete Fourier transform precoded orthogonal frequency division
multiple access.

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

1
0.9
0.8
0.7
0.6
cdf

0.5
0.4 Type A−2
Type B−1
0.3 Type B−3
Type B−2
0.2 Type A−3
Conv. QPSK DFT−p−OFDMA
0.1
0
1 2 3 4 5 6 7 8 9
PAPR (dB)

Figure 4. Complementary cumulative distribution function (cdf) of peak-to-average power ratio (PAPR) for binary phase-shift keying,
M D 12.

100

10−1
cdf

10−2

BPSK, zero deg

BPSK, 90 deg
BPSK, Type A−2, zero deg
10−3
BPSK, Type A−2, 90 deg

10−4−1 0 1 2 3 4 5 6 7 8
PAPR (dB)

Figure 5. Peak-to-average power ratio (PAPR) comparison with and without partial response recoding, M D 12.

when they are used independently; however, the maximum For 8-PSK modulation, referring to Figure 8, we see that
benefit is obtained when both techniques are used together. constellation rotation provides a small benefit only. This is
For 4-ASK, referring to Figure 6, we see that with- generally true for high-density complex constellations for
out using Type A-2 PR precoding, constellation rotation Q > 8. However, Type A-2 PR precoder offers approxi-
( D =2) alone reduces the PAPR by 2.5 dB. A total of mately 2.4 dB PAPR reduction. Similar results are obtained
4.4 dB PAPR reduction is obtained when both PR precod- for a PSK system of arbitrary modulation size employing
ing and constellation rotation methods are used together. circularly symmetric constellation. However, the proposed
For the case of QPSK modulation, simulation shows that PR precoding methods (and/or constellation rotation) are
the optimum constellation rotation is  D =4. In this not very effective in reducing the PAPR for higher-order
case, combined use of Type A-2 PR precoding and con- QAM systems. For the case of 16-QAM, we observed only
stellation rotation offers nearly 3.0 dB PAPR reduction 0.5 dB PAPR reduction (see Figure 9). In general, for the
(see Figure 7). Comparing 4-ASK and QPSK, the proposed case of higher-order QAM, the overall PAPR appears to be
system offers a PAPR of 4.4 dB for QPSK, whereas 4-ASK dominated by the amplitude variations of the constellation
has 5.5 dB PAPR. points themselves. Introduction of PR precoding (and/or

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

100

10−1

cdf

10−2

4ASK, w/o PR precoder, zero deg

4ASK, w/o PR precoder, 90 deg
4ASK, Type A−2, 0 deg
10−3 4ASK, Type A−2, 90 deg

10−4
3 4 5 6 7 8 9 10 11
PAPR (dB)

Figure 6. Complementary cumulative distribution function (cdf) of peak-to-average power ratio (PAPR) for 4ASK, M D 120.

100
QPSK, w/o PR precoder, 0−deg
QPSK, w/o PR precoder, 45−deg
QPSK, Type A−2, 0−deg
QPSK, Type A−2, 45−deg

10−1
cdf

10−2

10−3

10−42 3 4 5 6 7 8 9
PAPR (dB)

Figure 7. Complementary cumulative distribution function (cdf) of peak-to-average power ratio (PAPR) for quadrature phase-shift
keying (QPSK), M D 120.

constellation rotation) did not provide much benefit in this for real-valued signalling. For detection of complex-
case. Therefore, the proposed methods are effective for valued constellations, we use conventional MMSE and
constellations that do not have amplitude variations. MMSE-DFE receiver techniques proposed in [16, 17]. In
Next, we examine the BER performance of some of the the following, we discuss implementation of WL receivers,
proposed modulation formats. In particular, we develop whereas details of conventional receivers are omitted
WL MMSE and WL MMSE-DFE receiver algorithms for brevity.

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

100
8PSK, w/o PR precoder, zero deg
8PSK, w/o PR precoder, 22.5 deg
8PSK, Type A−2, zero deg
10−1
8PSK, Type A−2, 22.5 deg

cdf
10−2

10−3

10−42 3 4 5 6 7 8
PAPR (dB)

Figure 8. Complementary cumulative distribution function (cdf) of peak-to-average power ratio (PAPR) for 8-PSK, M D 120.

100
16QAM, w/o PR precoding
16QAM, Type A−2

10−1
cdf

10−2

10−3

10−41 2 3 4 5 6 7 8 9
PAPR (dB)

Figure 9. Complementary cumulative distribution function (cdf) of peak-to-average power ratio (PAPR) for 16-QAM, M D 120.

3. WIDELY LINEAR FREQUENCY  The receiver applies a constellation de-rotation step to

DOMAIN obtain a signal with real-valued modulation alphabets.
MINIMUM-MEAN-SQUARE-ERROR Because we deal with signals in frequency domain,
EQUALISER the constellation de-rotation is equivalent to circular
frequency shifting operation.
The receiver front end operations such as sampling,  Conventionally, a time domain WL receiver [13]
synchronisation, channel estimation and subcarrier de- jointly filters the signal and its complex conjugate.
mapping operations are similar to conventional DFT- Equivalently, in frequency domain, the WL receiver
precoded-OFDMA system. The WL receiver exploits the filters the signal and its complex-conjugated and time-
real-valued nature of the modulation alphabets to miti- reversed copy.
gate the ISI introduced by the PR precoder. We consider
detection of ASK that employs real-valued modulation We begin with the equivalent baseband frequency domain
alphabets that employ 90° constellation rotation. Before signal of interest
applying WL filtering, the receiver goes through the fol-
lowing steps: Y .l/ D H .l/X .l/ C N .l/; l D 0; : : : ; M  1

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

where H .l/ D DF T Œh.k/ is the frequency domain chan- we have

nel and h.k/ is the impulse response of the time domain
   
propagation channel, N .l/ is the thermal noise contained YO .l/ HO .l/PO .l/
in lth subcarrier with variance of N0 =2 per dimension  D A.l/
O
Y .M  l/ H .M  l/PO  .M  l/
O 
and X .l/ D P .l/C .l/ where P .l/ D DF T Œp.k/ is  
the frequency response of the PR precoder, and C .l/ D NO .l/
C (10)
DF T Œj k a.k/. Writing j k as e j 2k.l0 =M / , where l0 D NO  .M  l/
M =4, corresponds to a frequency shifting operation by
M =4 samples in frequency domain. Therefore, C .l/ can With the use of compact vector notation, Y.l/ N D
be represented as N
H.l/A.l/ N
C N.l/. N
Let Z.l/ N Y.l/
D W.l/ N represent the
N
decision variable. The WL filter W.l/ jointly filters the
M
X 1 
ll0
 frequency domain signal YO .l/ and its complex-conjugated
j 2k
C .l/ D a.k/e M
(6) and frequency-reversed copy YO  .M  l/ to obtain the
kD0 N
scalar decision variable Z.l/. An estimate of the input data
D A.l M l0 / (7) is obtained by taking the IDFT as a.k/
N N
D IDF T ŒZ.l/.
With the use of MMSE estimation [18], the vector-valued
where M represents modulo-M subtraction operation. In MMSE filter is given by
Equation (7), C .l/ is represented as a frequency-shifted
version of A.l/ with a frequency shift of M =4 samples. N  .l/
H
N
W.l/ D (11)
Furthermore, because a.k/ is a real-valued data sequence, N  .l/H.l/
N
N0 C H
the DFT of a.k/ exhibits conjugate symmetry, that is,
A .M  l/ D A.l/. The receiver obtains an estimate
of H .l/ using conventional methods. For instance, pilot
N
where W.l/ D ŒWN .l/; WN  .M  l/. Note that the sec-
sequences embedded in the middle of each data packet ond feed-forward filter (FFF) is a complex-conjugated and
can be used to enable channel estimation. In the rest of frequency-reversed copy of the first filter. The MMSE for
the paper, it is assumed that ideal knowledge of H .l/ is this case is expressed as
available at the receiver. Because P .l/ is a deterministic
filter, the total channel response H .l/P .l/ can be recon- M 1
1 X N0
structed at the receiver. The first step in the WL receiver MSE D 
(12)
M N N
is to de-shift the received signal by M =4 samples to ren- lD0 N0 C H .l/H.l/
der the modulation data to be a real-valued data sequence.
After shifting Y .l/ by l0 samples and using the fact that Note that
C .l ˚M l0 / D A.l/, we obtain
N  .l/ D ŒHO  .l/PO  .l/; HO .M  l/PO .M  l/
H (13)
YO .l/ D HO .l/PO .l/A.l/
N  .l/H.l/
H N DjHO .l/PO .l/j2
C NO .l/ (8)
C jHO  .M  l/PO  .M  l/j2 (14)

With the use of this result, the MSE can be expressed as

M 1
1 X N0
MSE D (15)
M
lD0
N0 C jH .l/P .l/j C jHO  .M  l/PO  .M  l/j2
O O 2

where YO .l/ D Y .l ˚M l0 /, HO .l/ D H .l ˚M l0 /, NO .l/ D

N .l ˚M l0 / and PO .l/ D P .l ˚M l0 /. Applying complex 3.1. Mean square error in additive white
conjugation and frequency reversal operations on Y .l/ and Gaussian noise channel
using the property A .M  l/ D A.l/, we obtain
For the special case of Type B-1 PR precoder, we have
YO  .M  l/ D HO  .M  l/PO  .M  l/A.l/ h i
1 j 2l
C NO  .M  l/ (9) P .l/ D DF T Œp.k/ D p 1Ce M
2
1 h j 2l
i
Observing the signals represented in Equations (8) and (9), PO .l/ D p 1  je M

we note that two copies of the frequency domain modu- 2

1 h j 2l
i
lation signal A.l/ are obtained with distinct channel coef- O 
P .M  l/ D p 1 C je M
ficients. Combining Equations (8) and (9) in vector form, 2

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

Table III. Post-SNR in decibel. N

where b.k/, k D 1; 2; : : : ; N b, is the time domain FBF
Type B-1 10 of length N b taps. The MSE is given by MSEDFE D
Type B-2 10.00 E je.k/j2 . Symbol decisions are obtained using a slicer
Type B-3 9.88 after eliminating the ISI caused by N b taps using the past
Type A-2 9.984 decisions. To obtain the FFF and FBF coefficients, we
Type A-3 9.76 adopt a WL generalisation of the MMSE-DFE receiver
algorithm proposed in [17] where the FFF is imple-
mented in frequency domain and the FBF is implemented
Substituting these expressions in Equation (15), we obtain

M 1
1 X N0
MSEType B1 D  l
 
M O 2 O 2 j 2 M
jHO .l/j2  jHO .M  l/j2
lD0 N0 C kH .l/j C jH .M  l/j C < je

where <./ denotes the real part of a complex num-

ber. If the propagation channel is nonfrequency selective,
in time domain. Detailed expressions for FFF and FBF
that is, HO .l/ D HO .M  l/ D h, the MSE becomes
are provided in [17]. Alternatively, the FFF can also be
MSEWL flat D .N0 =2/=Œjhj2 C .N0 =2/, which is inde-
obtained using the approach presented in [16]. It is impor-
pendent of the coefficients of the considered Type B-1
tant to note that in typical frequency-selective channels, the
PR precoder. The unbiased [18] post-processing SNR at
time domain channel impulse response of the propagation
the output of the MMSE receiver defined as post-SNR D
channel, that is, h.k/, k D 0; 1; : : : ; M  1, generally has
.1=MSE/  1 is given in Table III for other choice of pre-
length of M taps due to the rectangular windowing oper-
codes at SNRD10 dB. The degradation is within 0.25 dB
ation inherent in DFT-precoded-OFDMA. However, only
for all the considered cases. However, in the presence of
the first few taps carry dominant energy, and the rest take
ISI caused by the frequency-selective channel, the PR pre-
very small values. To reduce implementation complexity,
coder causes certain degradation in performance compared
it is desirable to limit N b to a small value. Appropriate
with the case without PR precoding. The performance is
choice of N b depends on the propagation channel type and
assessed using simulation in Section 6.
the value of M .
Note that the FBF b.l/ N is circularly convolved with
4. FREQUENCY DOMAIN the data a.k/. In detecting the first data symbol a.1/, the
WIDELY LINEAR received signal is corrupted by the ISI caused by the last
MINIMUM-MEAN-SQUARE-ERROR N b data symbols of a.k/. These decisions are not available
DECISION FEEDBACK EQUALISER to DFE unless the transmitter appends N b known symbols
at the end of the packet. Such an operation is impractical
In the proposed WL MMSE-DFE receiver, the received because it increases the overhead significantly for large val-
signal and its conjugated-time-reversed replicas are filtered ues of N b. This problem is alleviated using a two-stage
using two FFFs receiver. In the proposed implementation, we use a WL
MMSE to obtain hard decisions for the last N b symbols of
O
Z.l/ D WN .l/YN .l/ C WN  .M  l/YO  .M  l/ a.k/ in the first stage, and these estimates are used to ini-
N Y.l/;
D W.l/ N l D 1; 2; : : : ; M tialise the DFE in the second stage. Because the FFF of WL
MMSE is already contained in the FFF expression for the
where we use compact vector notation to represent the WL MMSE-DFE [17, 19] and because these coefficients
N
vector-valued FFF as W.l/ D ŒWN .l/; WN  .M  l/. The are already available to the receiver, the additional WL
ISI is eliminated using a feedback filter (FBF) as MMSE receiver module can be implemented with a moder-
ate increase in overall complexity. Similar approach is used
Q
Z.l/ O
D Z.l/ N
 B.l/A.l/ in [17] to implement conventional MMSE-DFE receivers.
N
where B.l/ is the frequency domain FBF. The error signal
is defined as 5. IMPLEMENTATION COMPLEXITY
 
N D Z.l/
E.l/ O N
 1 C B.l/ A.l/ The proposed method introduces additional complexity
compared with conventional DFT-precoded-OFDMA. At
In time domain, the transmit side, there is increased complexity due to
(a) constellation rotation operation and (b) PR precod-
Nb
X ing. Note that PR precoding operation requires circu-
e.k/ D z.k/
O  N ˇ a.k  l/  a.k/
b.l/ lar convolution of data with a PR precoder. Among the
lD1 proposed precoders, Type B-1 precoder, which has two

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

equal taps, has low implementation complexity because it complex-conjugated and frequency-reversed copy of
can be implemented using additions only. The amplitude the first FFF WN .l/ to reduce complexity.
scaling by factor 0:707 is common to the entire signal and  Computation of FFF incurs additional complex-
therefore can be implemented in later stages as part of ity because it involves sum of two terms (see
transmit power scaling. Other proposed precoders require Equation (14)) corresponding to the signal and its
both multiplication and addition operations. The receiver complex-conjugated copies.
requires the following additional operations compared with  In case of DFE, computation of FBF does not require
a conventional receiver: extra computational complexity compared with the
conventional system.

 Construction of equivalent channels H .l/P .l/ and

H .l/P .l/: Note that construction of H .l/P .l/
6. BIT ERROR RATE RESULTS
involves M multiplications one for each value of l.
The BER performance is reported for a DFT-precoded-
Alternatively, if the propagation channel is estimated
OFDM system with a subcarrier spacing of 15 KHz.
using time domain techniques, the estimated channel
Ideal channel knowledge is assumed at the receiver. The
impulse response h.k/ can be circularly convolved
BER is computed over 3000 independent channel real-
with the PR precoder p.k/ before taking the DFT of
isations. DFT-precoded-OFDMA without PR precoding
the equivalent time domain channel h.k/ ˇ p.k/.
with conventional MMSE receiver is used as the base-
 Constellation de-rotation of both received data Y .l/
line for all comparisons. Results are reported for additive
as well the received estimated equivalent channel
white Gaussian noise and the Ped-B [20] channel model,
H .l/P .l/ is implemented through appropriate fre-
which represents a typical urban channel model with high
quency de-shifting operations to obtain YO .l/ and
amount of frequency selectivity. We report results for
HO .l/PO .l/.
two subcarrier sizes M D 12 and 128 that represent low
 Complex conjugation and frequency reversal of
and high frequency selective cases. In all the figures, the
the received signal YO .l/ and HO .l/PO .l/ to obtain
BER comparisons are made with modulation size fixed to
YO  .M  l/, HO .l/PO .l/.
same value.
 In case of both WL MMSE LE and WL MMSE
DFE receivers, the received signal and its conjugated-
frequency-reversed replicas are filtered using two 6.1. Binary phase-shift keying
FFFs, whereas a conventional equaliser uses a sin-
gle FFF. However, the coefficients of the second This section discusses the performance of proposed meth-
FFF WN  .M  l/ can be obtained by taking the ods for the case of BPSK signalling. Figure 10 shows

100

WL LE, w/o PR precoder

Conv LE, w/o PR precoder
WL LE, Type A−2
WL LE, Type A−3
WL LE, Type B−1
WL LE, Type B−2
WL LE, Type B−3
10−1
BER

10−2

10−3
0 5 10 15
SNR

Figure 10. Bit error rate (BER) in PED-B channel with minimum mean square error, M D 12. SNR, signal-to-noise ratio; PR,
partial response.

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

100
WL LE w/o PR precoder
Conv LE w/o PR precoder
WL LE, Type A−2
WL LE, Type B−1
WL LE, Type B−2
BER 10−1 WL LE, Type B−3
WL LE, Type A−3

10−2

10−30 1 2 3 4 5 6 7 8 9 10
SNR

Figure 11. Bit error rate (BER) in PED-B channel with minimum mean square error, M D 128. SNR, signal-to-noise ratio; PR,
partial response.

100

10−1
BER

WL DFE w/o PR precoder

Conv DFE w/o PR precoder
−2 WL DFE Type A−2
10 WL DFE Type A−3
WL DFE Type B−1
WL DFE Type B−2
WL DFE Type B−3

10−3
0 1 2 3 4 5 6 7 8 9 10
SNR

Figure 12. Bit error rate (BER) in PED-B channel with minimum-mean-square-error decision feedback equaliser (DFE), M D 128.
SNR, signal-to-noise ratio; PR, partial response.

results for M D 12 that corresponds to 180 KHz BW. In causing high performance penalty. Compared with WL
this case, the channel has low frequency selectivity over the MMSE without PR precoding, the loss due to the introduc-
band of interest. Because of this, the performance differ- tion of precoding is less than 0.8 dB. Note that in all cases,
ence between WL MMSE and WL MMSE-DFE is found conventional MMSE with PR precoding performs poorly
to be very small. Therefore, we present results comparing because of excessive noise enhancement. Despite the pres-
conventional and WL MMSE receivers only. We see that ence of ISI caused by the frequency-selective channel, the
WL MMSE with Type A-2 precoder performs nearly same performance of WL MMSE with PR precoding is similar
as the conventional MMSE receiver without PR precod- to conventional MMSE without PR precoding.
ing. The performance loss of remaining precoders is well Without PR precoding, WL MMSE shows an advantage
within 0.2–0.4 dB of Type A-2 with WL MMSE receiver. over conventional MMSE. This gain stems from the fact
However, WL MMSE without PR precoding outperforms that WL MMSE inherently has lower noise enhancement
Type A-2 with WL MMSE by 0.35 dB. and lower MSE than conventional MMSE [13]. Remark-
In Figures 11 and 12, results are shown for subcarrier ably, even with PR precoding, WL MMSE performs very
allocation of M D 128. In this case, the channel becomes close to the baseline receiver. In frequency-selective chan-
highly selective over the band of interest. In all consid- nels, Type A-2 PR precoder has the least BER compared
ered cases, the WL MMSE-DFE provides a significant with all other considered choices. The BER of Type B-3
gain over WL MMSE receiver as well as the conventional precoder is approximately 0.8 dB worse than that of Type
MMSE/MMSE-DFE methods. The proposed receiver is A-2. Note that Type B-1 precoder does not cause noise
able handle the ISI generated by the PR precoder without enhancement in flat channels; however, its BER becomes

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

100

LE

MMSE−DFE

MMSE−DFE ideal

MFB

BER

10−1

10−2
0 1 2 3 4 5 6 7 8 9 10
SNR

Figure 13. Bit error rate (BER) in additive white Gaussian noise channel for quadrature phase-shift keying, M D 12. SNR,
signal-to-noise ratio; LE, linear equaliser; MMSE-DFE, minimum-mean-square-error decision feedback equaliser; MFB, matched
filter bound.

worse than other precoders in a frequency-selective chan- that can handle the ISI introduced by the PR precoder
nel. Overall, Type A-2 PR precoder appears to be a good should be investigated further. Techniques such as reduced
choice for reducing PAPR and in minimising the BER. state sequence estimation [21] and turbo equalisation are
Similar performance differences are observed for higher two candidates in this direction.
values of M . We also remark here that the WL equalis-
ers provide similar performance benefits for general Q-ary
ASK systems employing real constellations. Additional Remarks
For higher-order modulation, standard Q-ary PSK con-
stellations have higher minimum distance compared with
6.2. Bit error rate performance of ASK with equal number of bits per symbol. Therefore,
quadrature phase-shift keying and Q-ary PSK provides higher spectral efficiency compared
8-phase-shift keying with Q-ary ASK. However, in low SNR regime, BPSK,
which uses a real constellation, can be useful. For instance,
In Figures 13 and 14, we compare the BER of QPSK and the long-term evolution standard, which uses a link adap-
8-PSK system employing Type A-2 PR precoder in addi- tation algorithm, assigns QPSK modulation with a low rate
tive white Gaussian noise channel without fading. In both channel code (with bit-level data repetition) when the SNR
cases, the MMSE linear equaliser has up to 1.0 dB SNR of the link degrades certain threshold, for example, 0 dB
loss over MMSE-DFE, whereas MMSE-DFE has 2.0 dB or less. For such low-operating SNR, the system can alter-
degradation compared with a ISI free receiver modelled natively use the proposed method using BPSK modulation,
using the matched filter bound. For QPSK, the use of PR whereas the channel code rate can be adjusted to meet the
precoding and constellation rotation provides nearly 3.0 dB required target SNR. Because QPSK and BPSK methods
PAPR reduction over conventional DFT-precoded-OFDM require the same energy per bit, the additional power gain
but losses 2.0 dB SNR because of the ISI introduced by the obtained through PAPR reduction can be used to increase
PR precoder. For 8-PSK, the reduction in PAPR is nearly the cell coverage.
2.5 dB, whereas the loss caused by the ISI is close to 2.0 dB
compared with ISI free case. In Figure 15, the BER for
8-PSK employing Type A-2 PR precoder is shown for 7. CONCLUSIONS
Ped-B case. The degradation compared with the case with-
out PR precoding is in the range of 2.0–2.5 dB. For both Partial response DFT-precoded-OFDMA is proposed. This
QPSK and 8-PSK case, we observe that MMSE-DFE is technique reduces the PAPR by using a combination of
unable to mitigate the ISI caused by the PR precoder fully. constellation rotation and PR precoding. The class of PR
Therefore, to exploit the low PAPR properties of PSK sys- precoders obtained from linearised GMSK pulse is shown
tems employing PR precoding, more advanced receivers to provide considerable reduction in PAPR compared with

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

100
LE
MMSE−DFE
MMSE−DFE ideal
MFB

BER

10−1

10−2
0 5 10 15
SNR

Figure 14. Bit error rate (BER) in additive white Gaussian noise channel for 8-PSK, M D 12. PSK, phase-shift keying; SNR,
signal-to-noise ratio; LE, linear equaliser; MMSE-DFE, minimum-mean-square-error decision feedback equaliser; MFB, matched
filter bound.

100
MMSE
MMSE DFE
MMSE DFE ideal
MMSE DFE ideal w/o PR precoder
MMSE DFE w/o PR precoder

10−1
BER

10−2

10−3
0 2 4 6 8 10 12 14 16 18 20
SNR

Figure 15. Bit error rate (BER) in PED-B channel for 8-PSK, M D 128. PSK, phase-shift keying; SNR, signal-to-noise ratio; MMSE DFE,
minimum-mean-square-error decision feedback equaliser; PR, partial response.

conventionally used methods. In particular, for BPSK mod- system, which employs conventional MMSE/MMSE-DFE
ulation, the Type A-2 PR precoder, which is obtained using equalisers without PR precoding. The proposed techniques
BT D 0:3 and  D 0, is shown to be useful in reduc- are also shown to be useful in reducing the PAPR of Q-ary
ing the PAPR and in minimising the BER in frequency- ASK constellation that employs real constellations.
selective channels. Widely linear equalisers, which jointly A constellation rotation of =4 combined with Type
filter the received signal and its complex conjugate, play an A-2 PR precoder reduces the PAPR of QPSK by nearly
important role in low-complexity receiver design. The BER 3.0 dB. However, the ISI caused by the PR precoder led
degradation due to the introduction proposed PR precoder to 2.0 dB loss in BER when a conventional MMSE-DFE
is shown to be acceptable. In typical wireless channels, receiver used. In case of 8-PSK modulation, constella-
performance of the proposed method with low-complexity tion rotation has little effect on PAPR, whereas Type
WL MMSE/MMSE DFE equalisers is close to a baseline A-2 PR precoder reduces the PAPR by approximately

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
 K. Kuchi

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Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett