CONTRIBUTED
PAPER
Single Carrier Modulation With
Nonlinear Frequency Domain
Equalization: An Idea Whose
Time Has ComeVAgain
In high-speed single-carrier digital communication systems,
processing blocks of signals using Fast Fourier Transforms is an efficient way to
equalize (compensate) for interference between transmitted symbols.
By Nevio Benvenuto, Senior Member IEEE, Rui Dinis, Member IEEE,
David Falconer,
Life Fellow IEEE, and Stefano Tomasin, Member IEEE
ABSTRACT
|
In recent years single carrier modulation (SCM) has
again become an interesting and complementary alternative to
multicarrier modulations such as orthogonal frequency division
multiplexing (OFDM). This has been largely due to the use of
nonlinear equalizer structures implemented in part in the
frequency d omain by mea ns of fast Fou rier tran sforms,
bringing the complexity close to that of OFDM. Here a nonlinear
equalizer is formed with a linear filter to remove part of
intersymbol interference, followed by a canceler of remaining
interference by using previous detected dat a. Moreover, the
capacity of SCM is similar to that of OFDM in highly dispersive
channels only if a nonlinear equalizer is adopted at the receiver.
Indeed, the study of efficient nonlinear frequency domain
equalization techniques has further pushed the adoption of
SCM in various standards. This tutorial paper aims at providing
an overv iew of nonlinear equalization methods as a key
ingredient in receivers of SCM for wideband transmission. We
review both hybrid (with filters implemented both in time and
frequency domain) and all-frequency-domain iterative struc-
tures. Application of nonlinear frequency domain equalizers to
a multiple input multiple output scenario is also investigated,
with a comparison of two architectures for interference
reduction. We also present methods for channel estimation
and alternatives for pi lot insertion. T he impact on SCM
transmission of impairments such as phase noise, frequency
offset and saturation due to high power amplifiers is also
assessed. The comparison among the considered frequency
domain equalization techniques is based both on complexity
and performance, in terms of bit error rate or throughput.
KEYWORDS
|
Decision-feedback equalizers; digital modulation;
discrete Fourier transforms; multiple antennas
I. INTRODUCTION
EqualizationVthe c ompensation of the linear distortion
caused by channel frequency selectivityVis an essential
component of digital communications systems whose data
symbol rate is higher than the coherence bandwidth of
typically encountered channels. Intersymbol interference
that afflicts serial data transmission has traditionally been
mitigated by e qualization implemented in the time domain
with line ar filtering, usually with a transversal structure,
hence the designation linear equalizer [1]. Due to the
tradeoff between equalization of the channel impulse re-
sponse to remove intersymbol interference (both pre-
cursors and postcursors) and noise enhancement at the
decision point, a linear equalizer yields less than ideal
performance in terms of bit error rate, especially in
Manuscript received March 25, 2008; revised January 29, 2009. Current version
published December 23, 2009.
N. Benvenuto and S. Tomasin are with the Department of Information Engineering,
University of Padova, Italy (e-mail: nb@dei.unipd.it; tomasin@dei.unipd.it).
R. Dinis is with IT (Instituto das Telecomunicaço
˜
es) and FCT-UNL
(Faculdade de Cie
ˆ
ncias e Te
´
cnologia da Universidade Nova de Lisboa), Lisbon, Portugal
(e-mail: rdinis@ist.utl.pt).
D. Falconer is with the Department of Systems and Computer Engineering,
Carleton University, Ottawa, Canada (e-mail: ddf@sce.carleton.ca).
Digital Object Identifier: 10.1109/JPROC.2009.2031562
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dispersive channels. Other types of equalizers have there-
fore been proposed, especially ones with a nonlinear
structure denoted as decis ion feedback equalizer (DFE),
where, after a first transversal filter aiming at reducing the
precursors of the equivalent pulse at the detection point, a
linear feedback filter, whose input is the sequence of past
detected data symbols, removes by cancellation the inter-
symbol interference due to postcursors. Hence, the struc-
ture is nonlinear with respect to the received signal.
Indeed, due to the feedback of detected data symbols, the
DFE is hard to analyze. However in general, its perfor-
manceismuchbetterthanthatofalinearequalizerand
can come close to that of an optimum sequence detector,
e.g., implemented by the Viterbi algorithm, for a much
lower complexity [2] .
The signal processing complexity (number of arithme-
tic operations per data symbol) in time domain equaliza-
tion, exemplified by the number of transversal filter tap
coefficients, increases at least l inearly with the number of
data symbol intervals spanned by t he channel impulse
response. Frequency domain processing of blocks of signals,
using discrete Fourier transforms (DFT), provides lower
complexity per data symbol, and has therefore recently
emerged as the preferred mitigation approach to channel
frequency selectivity, for next-generation broadband wire-
less systems with bit rates of tens or hundreds of megabits/s.
In this overview paper, we survey frequency domain equ al-
ization structures, mo stly based on the DFE principle, for
single carrier wireless di gital transmissions.
Serial or single carrier modulation (SCM), in which
data symbols are transmitted in serial fashion, has been
the traditional digital communications format since the
early days of telegraphy. An alternative is multicarrier
transmission, where multiple data streams, each modu-
lating a narrowband waveform, or tone, are transmitted in
parallel, thus allowing each tone to be separately equalized
by a simple gain and phase factor. Multicarrier transmis-
sion has become popular and widely used within the last
two decades, due mainly to its excellent complexity/
performance tradeoff for data symbol rates far above
coherence bandwidths, and also for its flexible link
adaptation ability [3]–[5]. Among the first militar y and
commercial multicarrier systems were the Collins Kineplex
and General Atronics KATHRYN HF radio systems [6], [7]
of the 1950s and 1960 s. T he KATHRYN system used DFT
signal processing at the transmitter and receiver. With the
realization that the eigenvectors of a linear system are
sinusoids, multicarrier transm ission was recognized as an
optimal format for frequency selective channels in the early
1960s [8] , [9]. Generation and block pr ocessing of
multicarrier signals in the frequency domain, are enor-
mously simplified by implementing the DFTs by fast
Fourier transforms (FFTs), as was recognized by Weinstein
and Ebert in 1971 [10], yielding a signal processing
complexity that grows only logarithmically with the channel
impulse response length. This realization, and the ever-
growing demand for higher data rates on wireless and wired
systems propelled the application of multicarrier transmis-
sion to i) digital subscriber line transmission standards,
where it is generally known as discrete multitone transmission,
ii) IEEE 802.11a wireless LAN and iii) digital audio and video
broadcast standards, where it is known as ort hogonal
frequency division multiplexing (OFDM), or o rt hogona l
frequency divi sion multi ple access (OFDMA). The early
success of OFDM in standards after more then twenty years
since the pioneerin g implementations, has been marked by
Bingham in his landmark paper: Multicarrier modulation for
data transmission: an idea whose time has come, [3].
A related development in the early 1970s was the
realization that frequency domain processing techniques
could also be used to facilitate and simplify equalization of
SCM systems [11]. More recently, as an alternative to the
first OFDM applications in wireless standards, Sari et al.
[12]–[14] pointed out that traditional SCM could enjoy an
implementation simplicity/performance tradeoff similar to
that of OFDM for highly frequency selective channels with
the inverse DFT moved at the receiver. (A simpler struc-
ture, with applications to diversity reception, was proposed
by Clark [15] a few years later.) Indeed, this is true on ly for
a nonlinear frequency domain equalizer. In fact, only the
performance of a DFE can come close to or even exceed
that of OFDM [16]. SCM waveforms have the additional
advantage that for a given signal power their range of
amplitude, measured by the peak-to-average rat io,issignif-
icantly less than that of multicarrier signals. As a result,
their transmitted spectra and performance are less affected
by transmitter power amplifie r nonlinearities. This allows
cheaper and more efficient high power amplifiers to be
used for transmitting SCM signals. A further benefit of
SCM is its greater robustness to frequency offset and p hase
noise than that of OFDM [17] (see also [18]).
These features of robustness to radi o frequency hard-
ware impairments make single carrier with frequency
domain equalization an attractive alternative to OFDM,
especially for cost- and power consumption-sensitive next-
generation wireless user terminals which transmit uplink
to base stations [19]. T hus frequency domain implementa-
tions of SCM receivers can be said to be an idea whose time
has come again after a hiatus of about 20 y ears. However
the status of SCM now is n ot that of a potential replacement
of OFDM, but rather of a compleme nt to it. As we will see,
traditional SCM can morph to a special form of multicarrier
transmission, which can be called DFT-precoded OFDM. As
such, it is a form of generalized multicarrier transmission
[20] (see also [21] and [22]).
SCM in the form of DFT-precoded OFDM has been pro-
posed by the European 6th framework program Wireless
INitiative NEw Radio (WINNER) project as the uplink trans-
mission format for wide area cellular scenarios, mainly on
the basis of its radio frequency impairment robustness
properties. WINNER downlink and local area uplink trans-
missions rely on OFDMA, mainly because of its flexibility
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and transmission channel adaptability properties [23]. The
Third Generation Partnership ProjectVLong Term Evolution
(3GPP-LTE) and now LTE-Advanced standards group also
propose DFT-precoded OFDM, which they call single carrier
frequency division multiple access for the uplink of next-
generation wide area cellular broadband wireless systems,
again with OFDMA used in the downlink [24], [25]. These
initiatives and standards activities are contributing to the
International Mobile Telecommunications Advanced (IMT-
Advanced) initiative of the International Telecommunica-
tions Union. The 802.16m Task Group of the IEEE 802.16
Wireless metropolitan area network standards group has
recently been formed to contribute to IMT-Advanced. At the
time of writing, its proposed standard has not been finalized,
but versions of single carrier frequency domain equalization,
as well as OFDM, have been considered for uplinks. The
earlier 802.16a standard, which led to the WiMAX wireless
metropolitan area concept, has three transmission modes:
two based on versions of OFDM and one based on SCM.
The rest of the paper is organized as follows. In Section II
we provide the basic principles and signal structure of SCM
frequency domain nonlinear equalization. In Section III, we
present various nonlinear equalization techniques imple-
mented in the frequency domain for a single antenna system
and using the direct knowledge of the channel frequency
response. These structures will be extended to the case of
transmitters and receivers with multiple antennas in
Section IV, where we also describe an iterative equalizer
fully implemented in the frequency domain. Channel
estimation methods for the proposed structures are investi-
gated in Section V. Impacts of phase noise and other
disturbances on implementations of the nonlinear frequency
domain equalizers are considered in Section VI. Section VII
compares SCM with OFDM, with a focus of the considered
nonlinear frequency domain equalization structures. Lastly,
conclusions are outlined in Section VIII.
Notation:
denotes the complex conjugate,
T
denotes
the transpose,
H
denotes the Hermitian (transpose and
complex conjugate) operator. The DFT of sequence fs
n
g,
n ¼ 0; 1; ...; P 1, is
S
p
¼
X
P1
n¼0
s
n
e
j2
np
P
; p ¼ 0; 1; ...; P 1: (1)
The inverse DFT (IDFT) of sequence fS
p
g, p ¼ 0;
1; ...; P 1, is
s
n
¼
1
P
X
P1
p¼0
S
p
e
j2
np
P
; n ¼ 0; 1; ...; P 1: (2)
I
N
denotes the N N identity matrix. Circular convolution
among signals x and y is denoted as ðx yÞ.
II. SYSTEM DEFINITIONS AND THE
FINGERPRINT OF SINGLE CARRIER
FREQUENCY DOMAIN EQUALIZER:
TRANSMISSION FORMAT
A wireless mobile transmission is characterized by a slowly
time-varying multipath channel between each pair of
transmit and receive antennas in a multiple input-multiple
output (MIMO) scenario. For a system w ith N
T
transmit
and N
R
receive antennas, we denote the impulse response
of the time-invariant channel from antenna i to antenna j
as h
ðj;iÞ
Ch
ðÞ, i ¼ 1; 2; ...; N
T
, j ¼ 1; 2; ...; N
R
.Upontrans-
mission of signal s
ðiÞ
ðtÞ from antenna i, the received signal
at antenna j can be written as (baseband equivalent model)
r
ðjÞ
ðtÞ¼
X
N
T
i¼1
Z
h
ðj;iÞ
Ch
ðÞs
ðiÞ
ðt Þd þ w
ðjÞ
ðtÞ (3)
where w
ðjÞ
ðtÞ is the noise term, which we assume to be
complex Gaussian with zero mean and power spectral
density N
0
.
Traditionally, a SCM signal is generated as a
sequential stream of data symbols, at regular time instants
nT ,forn ¼ ...; 0; 1; 2; ...,whereT i s t he data symbol
interval, and 1=T is the symbol rate. Although generally
receivers perform oversampling, for the sake of a simpler
notation, we assume also that the received signal is
filtered and sampled with rate 1=T. Hence we describe the
transmission system by an equivalent discrete-time model
where the channel is characterized by the impulse
response h
ðj;iÞ
‘
, ‘ ¼ 0; 1; ...; N
h
1, obtained by sampl ing
the cascade of the transmit filter, the channel and the
receive fil ter. By indicating with s
ðiÞ
n
the symbol transmit-
ted from the ith antenna, the received signal after samp-
ling can be written as
r
ðjÞ
n
¼
X
N
T
i¼1
X
N
h
1
‘¼0
h
ðj;iÞ
‘
s
ðiÞ
n‘
þ w
ðjÞ
n
(4)
where w
ðjÞ
n
is the noise term with variance
2
w
.
In order to allow frequency domain block equalization
of the received signal, the convolutions in (4) must be
circular and this can be achieved in different ways.
As we will first consider the single input-single output
case, we drop the antenna index for sake of a simpler
notation. The MIMO case is considered in S ection IV.
A. Circular and Linear Convolution
The transmitted signal fs
n
g depends on the informa-
tion s ign al fd
n
g but, in general, the two m ay not coincide.
We examine conditions such that e ach linear convolution
in (4) appears as a circular convolution between the
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channel impulse response and t he information d ata
signals d
n
.
Let us consider the sequence of data symbols i n blocks
of M, fd
n
g, n ¼ 0; 1; ...; M 1; and the N
h
-size sequence
fh
n
g, n ¼ 0; 1; ...; N
h
1, with M > N
h
.Wedefinethe
periodic signals of period P, d
rep
P
;n
¼ d
ðn mod PÞ
,and
h
rep
P
;n
¼ h
ðn mod PÞ
, n ¼ 0; 1; ...; P 1, where in order to
avoid time aliasing, P M and P N
h
.
Now, th e circul ar convolution between fd
n
g and fh
n
g
is a periodic sequen ce of period P defined as
x
ðcircÞ
n
¼ðh dÞ
n
¼
X
P1
‘¼0
h
rep
P
;n‘
d
rep
P
;‘
: (5)
Then, if we indicate with fD
p
g, fH
p
g and fX
ðcircÞ
p
g,
p ¼ 0; 1; ...; P 1, the P-point DFT of sequences fd
n
g,
n ¼ 0; 1; ...; P 1, fh
n
g,andfx
ðcircÞ
n
g, n ¼ 0; 1; ...; P 1,
respectively, we obtain
X
ðcircÞ
p
¼ H
p
D
p
; p ¼ 0; 1; ...; P 1: (6)
The linear convolution with support n ¼ 0; 1; ...; M þ
N
h
2is
x
ðlinÞ
n
¼
X
N
h
1
‘¼0
h
‘
d
n‘
: (7)
By comparing (7) with (5), it is easy to see that only if
P M þ N
h
1, then
x
ðlinÞ
n
¼ x
ðcircÞ
n
; n ¼ 0; 1; ...; P 1: (8)
To compute the convolution between the two finite-length
sequences fd
n
g and fh
n
g, (8) requires that both sequences
be completed with zeros (zero padding) to get a length of
P ¼ M þ N
h
1 samples. Then, taking the P-point DFT of
the two sequences, performing the product (6), and taking
the inverse transform of the result, one obtains the desired
linear convolution.
However, there are other conditions, some of which are
listed below, that yield a partial equivalence between the
circular convolution fx
ðcircÞ
n
g and the linear convolution
x
n
¼
X
N
h
1
‘¼0
h
‘
s
n‘
; (9)
where fs
n
g depends on fd
n
g.
Overlap and Save: We consider as the transmitted signal
s
n
¼ d
n
, n ¼ 0; 1; ...; M 1 and assume P ¼ M.Weverify
that (9) coincides with (5) only for the instants n ¼N
h
1,
N
h
; ...; M 1, [26]. In other words, the equivalence
between the linear and the circular convolution holds
always on a sub set of the computed points.
Cyclic Prefix: An alternative to overlap and save is to
consider, instead of the transmission of the data sequence
fd
n
g,anextendedsequencefs
n
g that is obtained by
partially repeating fd
n
g withacyclicprefixofL N
h
1
samples, [26]:
s
n
¼
d
n
n ¼ 0; 1; ...; M 1
d
Mþn
n ¼L; ...; 2; 1.
(10)
Moreover, assume P ¼ M. It is easy to prove that (9)
coincides with (5) for n ¼ 0; 1; ...; M 1. Moreov er, the
equivalence (6) in the frequency dom ain holds for DFTs of
size P ¼ M, the data block size. This arrangement is used
also in multicarrier communications [11].
Pseudo Noise (PN) Extension: Consider a sequence fs
n
g,
obtained by fd
n
g with the addition of a fixed sequence p
n
,
n ¼ 0; 1; ...; L 1, of L N
h
1 samples, i.e.,
s
n
¼
d
n
n ¼ 0; 1; ...; M 1
p
nM
n ¼ M; ...; M þ L 1.
(11)
The first data block is also preceded by the sequence fp
n
g.
Moreover, now P ¼ M þ L.Thesequencefp
n
gcan contain
any symbol sequence, including all zeros (zero padding)
[27], [28], or a PN symbol sequence, denoted PN extension
or unique word. The choice of the extension is also influ-
enced by other factors, such as channel estimation [29]. It
can be easily proved, that (9) coincides w ith ðh sÞ
n
for
n ¼ 0; 1; ...; P 1, where now the circular convolution is
on s
n
instead of d
n
.
With reference to the noisy MIMO scenario (4), we can
organize the transmitted signal fs
n
g into blocks of size P,
each obtained by extending with a PN sequence a data block
of size M. Moreover, at the beginning a PN sequence is
transmitted first. Let fs
nþkP
g, n ¼ 0; 1; ...; P 1 be the kth
block and let fH
ðj;iÞ
p
g be the P-size DFT of the channel
impulse response fh
ðj;iÞ
‘
g. Then we obtain
R
ðjÞ
p
ðkÞ¼
X
N
T
i¼1
H
ðj;iÞ
p
S
ðiÞ
p
ðkÞþW
ðjÞ
p
ðkÞ;
p ¼ 0; 1; ...; P 1 (12)
where W
ðjÞ
p
ðkÞ is the noise term in the frequency domain,
which according to the hypothesis on fw
n
g is i.i.d. wi th
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variance
2
W
¼ P
2
w
. Note that i n this arrangement the
DFTs are of size P ¼ M þ L inste ad of size M as in the cyclic
prefix arrangement. Moreo ver, in this arrangement, an
additional PN extension is required before the first data
block. Among the advantages of this format are a simple
channel estimation, by using the PN sequence [29], a nd the
possibility of implementing an efficient frequen cy domain
(FD) nonlinear equalizer, as detailed in Section III. Gen-
erally, the PN extension yields a reduced bit error rate with
respect to the cyclic prefix, since in the latter case data
detection errors affect both the information data and the
cyclic prefix, thus reducing the intersymbol interference
cancellation capabilities of the nonlinear equalizer. In the
following we will consider operations on a single data block
andwewilldroptheindexk from FD signals.
B. Signal Generation
As described in the previous section, the data symbol
sequence may be organized into DFT blocks, which m ay
include PN extensions, or to which cyclic prefixes are
appended, thus facilitating D FT processing and FD
equalization at the receiver. The resulting data sequences,
with or without extensions and prefixes, are low pass
filtered for bandlimiting and spectrum-shaping purposes,
before being up-converted to the carrier frequency.
Fig. 1 shows a generalized multicarrier transmitter archi-
tecture [19], [20], [22], which can be adapte d to generate a
wide variety of signals, including SCM sig nals, as well as
OFDM, OFDMA, multicarrier code division multiple access
(CDMA), etc. Because its processing occurs in the FD, it is
easy to generate signals with arbitrary spectra, and to insert
FD pilot tones for channel estimation (see Section V). Com-
plexity is not a major issue since processing is done with
DFTs and IDFTs, implemented by FFTs. In the figure, the
IDFT block is preceded by a general pre-matrix operation,
which may include a DFT, spreading, a selection mechanism
and/or an allocation to multiple transmitting antennas in a
MIMO or space-time code. Recognition of this generalized
structure can also be found in [30]–[32].
Generation of a S CM signal block proceeds as follows.
After coding and serial to parallel (S/P) conversion, blocks
of N coded data symbols are mapped to the FD by a N-point
DFT. The resulting FD data components are mapped by the
pre-matrix tim e-frequency- space selector to a set of M N
data-carrying subcarrie rs, and then p rocessed by a
M–point inverse DFT to convert back to the time domain
(TD). The resulting samples are parallel-to-serial (P/S)
converted and appended with a prefix or extension for
transmission. The simplest frequency mapping is to N
contiguous subcarrier frequencies, with the remaining
M N being p added with zeroes. In this case, the output
samples are expressed as
s
n
¼
1
M
X
N1
‘¼0
d
‘
X
N1
p¼0
e
j2
pn‘
M
N
ðÞ
M
¼
X
N1
‘¼0
gn ‘
M
N
d
‘
; n¼0; 1; ...; M1(13)
where
gðnÞ¼e
j2
ðN1Þn
M
1
M
sin
Nn
M
sin
n
M
(14)
while fs
n
g, n ¼L; L þ 1; ...; 1, contains the cyclic
prefix.
This is recognized as a block of data symbols serially
transmitted at intervals of M=N samples. The sam pled
pulse waveform given by (14) is a circular version of a sinc
pulse with zero excess bandwidth, limited to a bandwidth
N=MT . SCM signals generated in this way are called DFT-
precoded OFDM signals by the WINNER project [23], and
local single carrier FDMA (SC-FDMA) by the 3GPP-LTE
standards body [24], [25]. For (13), s
mM=N
¼ N=Md
m
,thus
the DFT-precoded OFDM waveform at data symbol inter-
vals depends only on a single data symbol, and therefore
has a significantly lower peak to average power ratio than
that of a corresponding OFDM waveform, whose sample
Fig. 1. Generalized multicarrier transmitter (from [22]).
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![Fig. 1. Generalized multicarrier transmitter (from [22]).](/figures/fig-1-generalized-multicarrier-transmitter-from-22-3fsh4qi7.webp)