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WIRELESS CHANNEL EQUALISATION
by
Desmond P. Taylor1, Giorgio M. Vitetta
2, Brian D. Hart
3and Aarne Mmmel
4
1. Electrical and Electronic Eng., University of Canterbury, Christchurch, New Zealand.
2. Dept. of Information Eng., University of Pisa, 56126 Pisa, Italy.
3. Research School of Information Sciences and Engineering, Australian National
University, Canberra ACT, Australia.
4. VTT Electronics, Oulu, Finland (on leave at the University of Canterbury, Electrical
and Electronic Eng. Dept., Christchurch, New Zealand).
Abstract
Equalisation techniques for wireless channels, in particular for those encountered in
mobile wireless communications, are examined. Equalisation is broadly defined to
include reception techniques which estimate the state or response of the channel and
then attempt to compensate for its effects. The paper considers equalisation techniques
for fading dispersive channels which include both time and frequency selectivity. In
addition, brief consideration is given to the problems of blind equalisation, techniques
for dealing with fast fading channels and to the problem of joint equalisation and
decoding. The paper does not attempt to provide in-depth analysis or performance
results. Rather, the interested reader is referred to the extensive list of references.
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1. Introduction
In this paper, we review channel compensation or equalisation for wireless
channels. Although the history of equalisation goes back many years to the early work ofLucky and others summarised in [1], their work was aimed almost entirely at the
telephone channel which may be characterised as an essentially linear time-invariant,
intersymbol interference (ISI) channel. Later work examined the line of sight microwave
channel which may be considered as a very slowly time-varying channel to which most
of the early theory on equalisation could be directly applied albeit at much higher
transmission rates. This work is discussed in detail in [2] and its references. Finally,
other work, [e.g., 2, 19, 49, 133] considered the ionospheric and tropospheric channels,
both of which are time-varying wireless channels that have had a significant influence
on the development of equalisers for the mobile wireless channel.
The mobile digital wireless channel presents some different challenges [42], due
mainly to the fact that the transmitter and receiver are mobile with respect to each other.
When coupled with multipath propagation, fading results. The channel impulse response
may then have an appreciable rate of variation ranging from slow to fast with respect to
the signalling rate [49]. Multipath propagation is due to reflections and scattering and
may cause frequency-selective fading and hence ISI. For low transmission rates, there is
often significant time variability but little frequency selectivity. At higher rates, the
channel is typically frequency selective but usually varies significantly more slowly
with respect to the transmission rate. In the first instance, the main effect is a time-
varying attenuation which affects all frequency components equally; this is known as flat
fading or time-selectivity. In the second case, the channel response varies with frequency
across the bandwidth of the transmitted digitally modulated signal and causes ISI
between adjacent symbols. In all cases the channel may be considered to be linear.
Equalisation in general consists of estimating the response or state of the channel
and using the estimate to compensate the channel effects so as to improve transmission
system performance. Usually, equalisation is carried out at the receiver is based only on
observation of the received signal. In the time-selective case, equalisation consists ofestimating the time-varying attenuation and phase of the channel and using the estimate
to compensate their effects. In the frequency-selective case, it consists of estimating the
response of the channel and then using this information to adjust the parameters of some
form of filter to compensate for the frequency-selective effects. The filter may be linear
(e.g. the transversal equaliser [2]) or nonlinear (e.g. a decision feedback equaliser [2,
173] or a maximum likelihood sequence estimator [105]).
Equalisers may be either per-symbol or sequence based and are usually based on
optimal receiver theory [32]. The basic theory of optimal reception over randomly timevarying dispersive channels is to a large extent captured in the work of Kailath [15-18].
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He developed basic theoretical structures for per-symbol receivers. The work is readily
extendible to sequence based receivers and many modern receivers are based on the
maximum likelihood sequence estimation approach first enunciated by Forney [105] in
the context of channel equalisation. The optimal receiver structures are based on the
theory of statistical signal processing which is summarised in [72, 334].
Many digital wireless systems utilise forward error correction coding techniques
in order to obtain adequate error performance [177]. There is currently considerable
interest in the use of trellis or signal space encoded signalling in order to maintain
spectral efficiency. The receiver must then perform both equalisation and decoding. The
paper briefly considers the joint equalisation and decoding process.
The paper is organised as follows. In Section 2, we discuss modelling of the
wireless channel. We also consider some general aspects of receiver modelling and
classifications of equalisers. Section 3 focuses on equalisation or compensation of time-
selective or flat fading channels. Section 4 considers the equalisation of frequency-
selective fading channels. Section 5 briefly considers the problem of joint equalisation
and decoding and finally Section 6 provides some conclusions.
2. The Communications System
Here we develop a mathematical description of the physical communication
system. We consider the transmitter, channel and receiver, with particular emphasis on
characterising the channel. A diagram of the basic system is shown in Fig. 1.
2.1 The Transmitter
We consider two classes of transmitted signals, ( )s t : (a) linearly modulated
signals [19]; and (b) Continuous Phase Modulated (CPM) signals [20]. Linearly
modulated signals may be expressed in complex baseband form as
s t c p t kT kk
N
( ) ( )= =
1
(2.1)
where ck is the k-th transmitted symbol; cN Nc c c= [ , , , ]1 2 is the symbol sequence oflength N; T is the signalling interval or symbol period; and p t( ) is the transmitter
impulse response or pulse shape. The symbols, ck, are taken from an M-ary complex
constellation, whereMis normally a power of two. Possible constellations include M-
ary amplitude shift keying (ASK), where { }c Mk 0 1 2 1, , , ,( ) , M-ary phase shift
keying (PSK), wherec jM
jM
Mk
12 2 1
,exp( ), ,exp(( )
)
, andM-ary quadrature
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amplitude modulation (QAM). For square QAM constellations, Mis a power of 4 and
the symbols have the form { } { }{ }c M j M k + 1 3 1 1 3 1, , , , , , .
CPM signals may be expressed in complex baseband form on 0 t NT as[20]
s tE
Tj d
s
N
t
( ) exp ( , )=
2
0
c , (2.2)
where Es is the transmitted energy per symbol interval, ( , )t Nc is the information
bearing phase and the k-th transmitted symbol, ck, belongs to the M-ary real
alphabet,{ } 1 3 1, , , ( ) M . We primarily consider linearly modulated signals, but
the most of the concepts apply to CPM signals.
We note that while we are primarily concerned with equalisation, some
modulation methods are robust to delay spread [212,213]. Also, noncoherent detection
of orthogonal FSK is relatively unaffected by frequency-selectivity [200].
2.2 Wireless Channels
In wireless communications, the transmitted signal is modified by three physical
mechanisms: inverse distance power loss, shadowing and multipath propagation. It is
also corrupted by additive noise. Inverse distance power loss causes the received signal
strength to decrease with increasing distance from the transmitter, typically according toan inverse second to fourth power law [163]. Shadowing accounts for slow bulk signal
strength variations, as when the receiver is obscured from the transmitter by buildings,
hills, or tunnels [117, 121]. It is typically modelled by representing the envelope of the
line-of-sight component of the received signal as a random variable having a log-normal
probability density function [117, 121, 163]. Both power loss and shadowing merely
attenuate the received signal and have little influence on equaliser design.
On the other hand, multipath propagation, due to the presence of multiple paths
between transmitter and receiver, can severely distort the transmitted signal. Moreover,it is usually time-varying and causes fading. The distortion due to multipath changes
appreciably over one wavelength, a distance that is at least two orders of magnitude
smaller than the distance over which either inverse power loss or shadowing effects
change significantly [132, 210].
High performance equalisers can be designed only if adequate models of the
channel are provided [19, 134] to represent the distortion of the signal due to multipath
propagation. Given a channel model an equalisation strategy can be developed. Different
channel models lead to significantly different equaliser structures. In the following
subsections, we consider multipath channel characterisation and modelling.
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2.2.1 Multipath Channels
For simplicity, we assume a stationary transmitter (base station) and a mobile
receiver. The transmitted signal is reflected and diffracted by scatterers, such as hills,
buildings, trees and vehicles. Some of the signal reaches the receivers antenna, usually
via several paths. The paths exhibit differing attenuations and have different lengths, so
that the receiver observes several relatively delayed and attenuated versions of the
signal. Each path delay may be conceptually divided into two parts: the so-called cluster
delay [3], which is on the order of a symbol interval, and the fine delay, which is on the
order of the carrier period. The former depends on the relative positions of the large
scale scatterers and is preserved in the channel model. The latter can be modelled as a
random variable, affecting only the carrier phase. Hence, path attenuation and fine delay
are lumped together as a complex gain. Due to change in fine delay, this gain changes
markedly over a carrier wavelength (0.3m at 1GHz). The superposition of the arriving
paths at any value of delay induces destructive and constructive interference, varyingaccording to position. As an antenna moves through this interference pattern, its spatial
variation appears as a time-variation in the received signal. In addition, due to the
motion of the antenna, the signal on each path undergoes a Doppler shift that depends on
the path arrival angle but does not exceed some maximum,fD . We call fD the one-
sided Doppler spread. It equals the maximum relative speed of the transmitter, channel
scatterers and receiver divided by the carrier wavelength. Thus the received signal is the
sum of many Doppler shifted, scaled and delayed versions of the transmitted signal [19,
42, 49, 152].
Due to the linearity of the channel, the received signal can be modelled as [134]
r t z t n t ( ) ( ) ( )= + (2.3)where
z t s t a t d( ) ( ) ( , )=
. (2.4)
Here ( )tz is the noiseless received signal, ( ),ta is the instantaneous time-varying
channel impulse response and ( )tn is additive white Gaussian noise. Physically, the
channel can be visualised as a densely tapped delay line, with delay index , so ( ),tarepresents the time-varying tap gain at delay . In terms of the various paths, ( ),ta isthe sum of the complex gains of all paths with delay , measured at the current locationof the receivers antenna. Since the antenna is moving, the sum is time-varying.
A linear time varying channel may also be characterised by the Fourier transform
H f t( , ) of ( ),ta with respect to the delay variable . This function is known as thetime-variant transfer function and allows the use of frequency domain techniques [134].
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2.2.2 Characterising Multipath Fading Channels
The system functions ( ),ta and H f t( , ) describe any time-varying channel. In the
multipath fading case, they represent realisations of a stochastic process since, in
practice, the receiver has no knowledge of the instantaneous scatterer geometry. Thus,
statistical characterisation is necessary [134]. If we model the channel as clusters of
many independent scatterers [3], the Central Limit Theorem applies and we can assume
Gaussian statistics [49]. The channel is then characterised by the mean and correlation
functions of one of the time varying system functions. When the complex gains due to
the different scatterers have similar amplitudes, the functions ( ),ta and H f t( , ) have
zero mean and their envelope obeys a Rayleigh distribution. When there is a dominant
path (e.g. a line-of-sight path), or more generally, a dominant path per delay value, they
have a non-zero mean and their envelopes have a Rician distribution [19]. These
models are justifiable mathematically, but other distributions, such as the Nakagami-m
distribution [154, 201], fit some experimental results more closely. For equaliser design,the Rician channel model is usually sufficiently general. Many designs consider
Rayleigh fading only, since an equaliser designed for Rayleigh fading generally
performs better in a Rician fading channel. As a first order statistical description of the
fading, we can decompose ( ),ta into a specular and diffuse component, corresponding
to the dominant paths and the remainder. The specular component, defined by
a t E a t s ( , ) { ( , )} , (2.5)
is known as the channel mean. It may include a Doppler shift. The diffuse component isgiven by
a t a t E a t d( , ) ( , ) { ( , )} (2.6)
and is Rayleigh faded.
A sufficient second order statistical description of the channel process ( ),ta , under
the assumption of Gaussianity, is given by the correlation function [19, 134, 186]
{ }aaR t t E a t a t ( , , , ) ( , ) ( , )*1 1 2 2 1 1 2 2 = (2.7)
This is known as the tap gain cross-correlation function, as it represents the cross-
correlation between scatterers at different delays. If the channel can be modelled as a
collection of Wide Sense Stationary (WSS) scatterers, it simplifies to
{ }aaR E a t a t( , , ) ( , ) ( , )* 1 2 1 2 + (2.8)
In addition if the WSS scatterers are assumed to be uncorrelated (uncorrelated scatteringor US) the channel is said to be WSSUS and aaR ( , , ) 1 2 can be rewritten as
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aa aaR P( , , ) ( , ) ( ) 1 2 1 1 2= , (2.9)
where aaP ( , ) is called the tap gain correlation function. Of physical importance is
the function aaP ( , ) 0 which is proportional to the average power received fromscatterers at delay . Knowledge of aaP ( , ) 0 allows evaluation of the multipath delayspreadd which is the interval in over which aaP ( , ) 0 is effectively non-zero.
Another function commonly used to characterise a fading dispersive channel is
the scattering function aaS ( , ) which is defined as the Fourier transform of the tap
gain correlation function aaP ( , ) with respect to the correlation lag , that is
( )aa aaS P j d( , ) ( , ) exp
2 (2.10)
It is meaningful only for WSSUS channels, and is proportional to the power scattered by
the medium at delays ( ) , + d in the Doppler shift interval ( ) , + d . The width of
aaS ( , ) in is the multipath spread, and its width in is the two-sided Dopplerbandwidth DB (orDoppler spread) and is equal to 2fD .
Finally, another function used to characterise a WSSUS channel is the time
frequency correlation function aaQ ( , ) which is defined as the Fourier transform of
the tap gain correlation function with respect to the delay variable , as
aa aaQ P j d( , ) ( , ) exp( )
2 (2.11)
It represents the cross-correlation between received frequencies spaced by Hz.
Two other parameters, often used to provide a description of the time- and
frequency-selective properties of the channel, are the coherence time c and the
coherence bandwidth cB . The first represents the interval over which the received signal
can be considered coherent and is roughly equal to the inverse of the Doppler spread.
The second represents the frequency band over which the multipath fading can be
considered frequency-flat and is approximately the inverse of the multipath spread.
2.2.3 Properties of the Channel
Channel delay spread arises from the variations in path length and can produce
deep notches in the time varying frequency response H f t( , ) . Since the signal
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bandwidth is usually on the order1
of 1/T, the normalised delay spread, d/T, is ameasure channel frequency-selectivity. When d/T 1 /T;however, many of the concepts considered here may be directly extended.
2Deeper fades occur but more rarely.
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S ff
f
f f
f f
a
D
D
D
( ) =
1, the channel is overspread, and cannot be estimated3.
A channel may also loosely be regarded as belonging to one of four channel
classes, according to the values ofd/TandBDT. In the first, the channel is essentiallyboth time- and frequency-nonselective (d/T
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2.2.5 The Tapped Delay Line Channel Model
The continuous time channel representation contains more information than is
required in a receiver, since the transmitted signal is bandlimited, and the channel
response outside the signal bandwidth is irrelevant. Moreover, a discretised or sampled
representation of the channel is more amenable to digital implementations. Therefore,
we consider a discretised, version of (2.3).
The transmitted signal can be represented as a weighted sum of its samples,
s s kT k s ( ) , as
s t st kT
Tks
sk
( ) =
sinc , (2.15)
where 1/Ts is chosen to be at least the Nyquist rate for ( )ts . Substituting this into (2.4),
we obtain
z snT
Tk a nT d s an k
r
s
rk
n k n k k
=
=
sinc
( , ) , , (2.16)
wherez z nTn s ( ) , the channel tap gains an k, are given by
( )n kr
sra
nT
T k a nT d , ,=
sinc
(2.17)
and the sampling rate 1/Tr is chosen to be faster than the Nyquist rate for the noiseless
received signal. Since the pulse shapes of interest have bandwidth of at least 1/2T, it is
incorrect4
to employ symbol-spaced taps as, noted in [142], when matched filtering is
not employed. There are an infinite number of non-zero channel taps in general, due to
the infinite duration of the sinc(.) functions in (2.16). The taps at large nT T k r s/
diminish, so the infinite summation in (2.16) can normally be truncated without
appreciable error.
It is simplest to use a common sampling period, Trthroughout. In addition, Tr, is
normally chosen for convenience such that Tis an integral multiple ofTr: i.e. T= rTr,
where usually r= 2 is sufficient. Thus we can define the vectors rrN rN r r r [ , , , ]1 2 ,z
rN rN z z z [ , , , ]1 2 , srN rN s s s [ , , , ]1 2 and n rN rN n n n [ , , , ]1 2 . The receivedsignal actually extends beyond t NT= [ , ]0 due to the tails of the transmitted pulse shapeand the delay spread, but we can ignore these edge effects forNlarge.
4
An equaliser with symbol-spaced delay taps suffers a performance penalty relative toone designed assuming with its taps at the Nyquist spacing.
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2.2.6 The Power Series Expansion Channel Model
The delay line model expands the channel impulse response as the weighted sum
of time-shifted sinc(.) functions as a function of delay, . Other choices of basis functionare available [134]. If the channel can be represented accurately using fewer basis
functions, fewer parameters need to be estimated, potentially leading to simpler
equalisers.
The optimal set of basis functions is obtained from the Karhunen-Love
expansion of the channel autocovariance [125]. However, this must be known or
estimated, to obtain the basis functions. Thus, there is more interest in choosing basis
functions known a priori to be good. A Taylors series expansion was proposed in
[134], so that the basis functions are polynomials. This is most appropriate for smoothly
changing functions, such as the variation of a t( , ) or A t f( , ) in torfand has recently
resulted in a Reduced Dimensionality Model [151,164,166] for doubly selectivechannels and a linearly time-selective distortion model for rapidly varying flat fading
channels [48,100,129,130].
2.3 The Receiver
The receiver must detect the information sequence while compensating the
channel distortions. At its core is the equaliser, which we discuss more thoroughly in
subsequent sections. Here, we briefly discuss some common receiver properties.
2.3.1 The Receiver Front End
An optimal receiver obtains a set of sufficient statistics [32] for recovering the
transmitted symbol sequence. When the channel is known, the output of a filter matched
to the received pulse can be sampled at the symbol rate to provide a set of sufficient
statistics [105,138], as
m r t p t kT a t d dt k =
( ) ( ) ( , )* * (2.18)
For this case, the channel may be modelled as a delay line with symbol-spaced
taps. However, this receiver usually cannot be implemented because the time-varying
channel is unknown a priori. Hence, a set of sufficient statistics is only easily obtained
by sampling the received signal at a rate, 1/Tr, that is at least its Nyquist rate. A low pass
filter to limit the noise bandwidth is needed before sampling.
There are other front-ends, which can make acceptable trade-offs between
performance and complexity. For slowly time-varying channels, the noise-limiting filter
is often chosen to be a filter matched to the transmitted pulse. Although near optimum
for the frequency non-selective channel [142], it is sub-optimum for the time-varying
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and/or frequency selective case. The received signal is Doppler-spread by the channel,
so such a filter has insufficient bandwidth. Another simplification is to sample the
filtered received signal below its Nyquist rate, typically at one sample per symbol. This
entails a power penalty, and in fast fading channels, leads to an error floor [71,160].
2.3.2 Classifying Equalisers
Many different equaliser structures have been investigated; however, most are
based on similar ideas or share similar properties. Hence, it is instructive to identify
equaliser groupings:
One grouping is according to the class of channel for which the equaliser isdesigned. This grouping is hierarchical, in that equalisers for the general time- and
frequency-selective channel have simplified counterparts in the time-invariant and
frequency-flat classes of channels. A second grouping is according to the statistical basis of their decision rules. In
particular, in designing an equaliser according to some optimality criterion [32],
additive noise may be considered or ignored and the multipath channel may be
treated either as known or as a stochastic process.
We consider two examples in the second group in order to provide a first insight
into the practical problems encountered in optimal design. For the first, the channel is
assumed known and Gaussian noise is included. Then the probability density function of
the vector rrN of received samples, conditioned on the symbol sequence, ~cN , is given
by [72]
( )( )
( ) ( ) ( )[ ]p rN N Nr N
rN rN N
H
r N rN rN Nr c
C cr c C c r c|~
~exp (~ ) ~ (~ )=
11
.
(2.19)
where indicates the determinant, and where ( ) { }rN N rN N N E~ | ~c r c c = and
( ) ( ) ( ){ }C c r c r c c cr N rN rN N H
rN rN N N N E~ (~ ) (~ ) | ~ = are the mean vector and
covariance matrix of rrN conditioned on c cN N= ~ . Since the channel is known, theexpected value of the received signal is its noiseless version, corresponding to
c cN N
= ~ . Hence, ( ) ( )rN N rN N ~ ~c z c= and the covariance matrix is the noiseautocovariance,C n n
r rN rN
HE= { }, so the determinant in (2.19) is independent of thehypothesised symbol sequence. Assuming that n rN to be white noise, the log-
likelihood function of the received vector rrN conditioned on c cN N= ~ is found from(2.19) as
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( ) ( ) r c crN N k k N k
rN
r z|~ ~= =
2
1(2.20)
The Maximum Likelihood (ML) estimate cN [32] of the transmitted symbol sequence is
then
( )~
argmax |~ c
r c cN
rN N N . (2.21)
In treating the channel as known, it is realised that in practice it must be
estimated a priori and the estimate used in computing the sequence metric (2.20). This
class includes adaptive MLSE receiver structures [135,142,144,182] and the differential
detectors used in time-varying, frequency-flat channels. In the latter case (at least
conceptually), the received signal during one symbol interval is divided by the detected
symbol and the result used as a channel estimate for the next symbol interval [19].
In the second example, both the Rayleigh fading channel and noise are treated as
stochastic processes. The probability of observing the vector rrN
conditioned on ~cN
, is
again given by (2.19). However, the channel is unknown and purely random, so the
expected value of the received signal vector is the null vector, ( )rN N rN ~c 0= and the
received signal autocovariance ( )C cr N~ depends on ~cN . Then the log-likelihood of thereceived signal vector r
rNconditioned on c c
N N= ~ is from (2.19),
( ) ( ) ( ) r c C c r C c rrN N r N rN H
r N rN|~ ln ~ ~= + 1 (2.22)
It consists of a bias term and a quadratic form. The ML detection strategy is again given
by eq. (2.21) with ( ) r crN N|~ given by (2.22). However, evaluation of the metric of(2.22) is much more complex than that of (2.20). This metric characterises the second
type of equaliser which is most suited to fast randomly time-varying channels.
Finally we note that Rician channels have both specular and diffuse components.
If we assume the specular component to be known, we may conclude that the optimal
receivers should combine the equalisers from both categories above.
3. Frequency Non-Selective Channel Equalisation
Here we consider equalisation for the time-selective or frequency-flat fading
channel. As noted in Section 2, a fading channel is frequency-flat when its delay spread
is so small that the multipath effect results in a complex time-varying multiplicative
distortion.
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The first relevant studies concern optimal diversity detection of digital signals on
time-selective fading channels and date back more than forty years [3-18]. In particular,
a clear understanding of the general problem of Maximum A Posteriori (MAP) detection
of digital signals transmitted through a Gaussian random channel was provided by
Kailath [15-18]. He found the MAP receiver for an M-ary set of digital signals{ }q t i M i ( ), , , ,= 1 2 transmitted on a purely random channel to consist ofMbranches,each an estimator-correlator structure
5. The optimal receiver operates by making in its i-
th ( i M= 1, , ) branch a Minimum Mean Square Error (MMSE) estimate of the fadeduseful signal. This is provided by a time-varying, unrealisable filter [32] designed under
the assumption that q ti ( ) is the transmitted signal. A decision metric is derived by
correlating this estimate with the received signal and adding a bias term6
to the
correlation result. If the l-th branch output produces the largest metric, the receiver
decides that the signal q tl ( ) has been transmitted. The estimator-correlator is illustrated
in Fig. 2 for a binary communication system. If the channel is not purely random and itsmean response is known a priori, the metric computed in the l-th branch of the receiver
is the sum of two terms: one evaluated by an estimator-correlator processing only the
random component of the received signal and the other evaluated by a correlator
extracting information from the deterministic component [15]. We also note the
following:
a) Implementation of the estimator-correlator detector, employing realisable time-
varying filters is discussed in [32] and [41] for discrete and continuous time
signals (see also [21-24]), respectively;
b) Since the optimal receiver computesMMMSE estimates of the fading distortion,
one for each hypothesis, we can interpret the MAP receiver as either: (1) a form of
equaliser because the time-varying channel gain is estimated and its effect is
compensated by correlating the fading estimate with the received signal; or (2) a
form of partially coherent detection since the receiver, in correlating the channel
estimate with the received signal, tries to compensate for the channel phase effects
[25,26]. Coherent detection is achieved in the limit of perfect estimation of the
fading process;
c) MMSE estimation of the fading requires the mean and autocorrelation function of
the random channel. In general, these are not known and must be estimated.d) The receiver can employed for both one-shot and sequence detection [32].
However, receiver complexity increases exponentially with sequence length and
leads to unacceptable complexity in practical applications.
5 This interpretation of the optimal receiver was first proposed by Price for a fading
channel with a single scatter path [4]. Kailath extended Price's ideas to an arbitrary
random channel.
6
In general the bias term depends on the fading and noise second order statistics andon the hypothized signalling waveform.
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Following the above, considerable research has been directed to the analysis of
simple noncoherent one-shot receiver structures (see, for instance, [28-31]) because of
their practical importance. Numerous solutions of this type have been proposed for the
detection of digital signals transmitted through frequency flat fading channels. They can
be roughly categorised as:
a) Noncoherent detectors;
b) Coherent detectors employing pilot tones or symbols as a phase reference.
c) Sequence detectors.
In the remainder of this section these will be examined. A final subsection is devoted to
receiver structures for fast fading channels.
3.1 One-Shot Noncoherent Detectors
Channel estimation is difficult in fading channels [42]. Noncoherent receiver
structures allow detection of a signal in the absence of an explicit channel estimate and
offer the advantage of simplicity. Strictly speaking these are not equalisers since they
make no attempt to estimate the channel. However, they are widely used in wireless
transmission and their analysis provides a basis for approaching that of more complex
equalisation structures. If the linearly modulated signals of (2.1) are considered, a
differential receiver can be employed, provided that differential encoding is
accomplished at the transmitter [19]. An analysis of the error performance of differential
PSK receivers on fading channels can be found in [14, 29] and [33-37].
Noncoherent detectors are also available for the CPM signals of (2.2) and
comprise differential detectors, discriminators and matched filter & envelope detectors
[4]. An analysis of their error performance is provided in [38-40] and [46, 94] for the
differential receivers, in [43-47] for the discriminators and in [3, 7, 13, 14, 48] for the
matched filter & envelope detectors. These detectors all suffer from two drawbacks: (1)
there is a Signal-to-Noise-Ratio (SNR) loss with respect to coherent detection; (2) if the
fading is fast (changes appreciably in a symbol interval), the detector error performance
will exhibit an error floor [14, 95]. This is largely due to the quick phase changes which
the signal experiences during deep fades [49].
3.2 Reference Based Techniques for Coherent Detection
Coherent detection is possible if a reference (or sounding) signal [50, 51] is
transmitted with the information bearing signal. In practice an accurate phase reference
cannot be generated by a Phase-Locked-Loop (PLL) because a PLL cannot track the
rapid phase changes of the channel fading [52]. A coherent reference can be made
available to the receiver by transmitting a time-continuous sounding signal (pilot tone)
[53-62] or by transmitting a sequence of known symbols (pilot symbols) interspersed
with the data symbols [62-67]. Several pilot tone techniques have been proposed asfollows:
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a) That described in [53] which consists of sending a continuous wave sounding
signal together with a data BPSK signal. The data and the sounding signals can be
separated since they are kept orthogonal in phase;
b) The Transparent-Tone-in-Band (TTIB) technique [57], where the baseband
spectrum is split into two segments. The segment in the upper frequency band istranslated up in frequency by an amount equal to the 'notch' width and a reference
pilot tone is added at the center of the resulting notch.
c) The Tone Calibration Technique (TCT) [55] creates a spectral null in the data
signal by means of a zero DC encoding technique (e.g. Manchester coding [19])
and inserts a pilot tone in the null. The TCT scheme is illustrated in Fig. 3 together
with the baseband spectrum of the transmitted signal.
d) The Dual-Pilot Tone Calibration (DPTC) Technique [54], where two pilots are
symmetrically located outside the data spectrum near the band edges. DPTC
provides better bandwidth efficiency than TCT at the price of increased sensitivity
of the pilots to frequency shifts [54].
Pilot-tone techniques lead to robust and simple receiver structures, as evidenced
by Fig. 3. The pilot tone can be separated by relatively simple circuitry from the received
signal. The use of coherent detection substantially lowers the error floor level of the
receiver. Their main disadvantage (and also of the pilot symbol techniques) is that a
fraction of the transmitted power is wasted in transmitting reference signals.
Simpler transmitter and receiver processing is achieved by pilot symbol assisted
modulation (PSAM) [62-67], although frame synchronisation is required at the receiver.In PSAM transmission the transmitter periodically sends known symbols, from which
the receiver derives its amplitude and phase reference. The PSAM transmitter and
receiver schemes are shown in Fig. 4 together with the transmitted data format. Here, the
data symbol rate is equal to ( ) /K KT 1 , 1 / ( )KT being the pilot symbol rate7. Likepilot tone modulation, PSAM suppresses the error floor and offers the further advantage
of enabling multilevel modulation without requiring a change of the transmitted pulse
shape or of the peak to average power ratio. A comparison of PSAM with TTIB [62] has
shown it offers substantially better energy efficiency for any practical power amplifier.
Finally, we note that reference-based techniques for coherent detection were first
proposed for linearly modulated signals. Recently, Ho et al. [68] have shown that a pilot
symbol assisted detection strategy can be implemented for CPM signals.
7
The pilot symbol rate should be at least 2( )BD MAX , ( )BD MAX being the largest valueof the Doppler bandwidth BD [63].
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3.3 Sequence Detectors
This subsection examines sequence detection techniques. In order to present a
unified framework, a recursive formulation [70] of the MAP detection strategy [15,69]for the linearly modulated signal of (2.1) on a slow fading channel is derived. It is then
shown how the different solutions can be related to the optimalstrategy.
In a MAP receiver for the linearly modulated signal of (2.1) transmitted over a
slow fading channel [70] the received signal is passed through a matched filter. The
symbol-rate samples at the filter output can be expressed as [70]
y a c wk k k k = + (3.1)
for k N= 1 2, , ,
, where ak represents the fading distortion, ckis the k-th transmittedsymbol and wk is AWGN
8. Note that yk is a sufficient statistic if it is assumed that
a t( ) is constant (and equal to ak) over the duration of the transmitted pulse p t kT( )(slow fading assumption). Then the MAP estimate cN of the symbol sequence is found
as cN Nc c c [ , , , ]1 2
arg max (~ | )~
c c yc
N N NN
p=
(3.2)
where [ ]yN Ny y y 1 2, , , , ~cN is a hypothesised data sequence and ( )p N N~ |c y is theprobability density function of ~cN conditioned on yN . The solution of (3.2) can be
found by exhaustive search over the set of hypothesised sequences{ }~cN . This entails acomputational burden increasing as MN . The evaluation of the conditional density
( )p N N~c y over the set { }~cN becomes numerically simpler if a recursive formula isderived. To begin, we observe that [70]
( )( )
( )( )p
p y
p yp ck k
k k k
k k
k k~ |
| , ~
|
~ |~c yy c
yc=
1 1
1
1 (3.3)
In (3.3) ( )p ck k~ |~c 1 represents a transition probability and equals 1 /M if the sequence
is uncoded (if ~ck
is coded ( )p ck k~ |~c 1 depends on the code structure). The MAP strategy
then becomes the ML strategy. Moreover, ( )p yk k|y 1 does not depend on the trialsequence ~c
N. Taking this and the recursive formula (3.3) into account, the MAP
strategy (3.2) can be rewritten as
8
A similar signal model exists for the samples of a CPM signal taken at the output of
the receiver front end filter (for instance [71]). Thus the developments of this sectioncan be easily extended to the problem of CPM detection.
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( )~ arg max ,~~
c y cc
N k kk
N
N
=
=
1
, (3.4)
where
( ) ( ){ } ( ){ } y c c y ck k k k k k k p c p y, ~ ln ~ |~ ln | ,~= + 1 1 1 . (3.5)
The conditional probability density function ( )p yk k k| , ~y c 1 1 is Gaussian [72] and from(2.19) and [70] may be expressed as
( )p yk k
y k k
k kk k k y
k y
y
| ,~( | )
exp( | )
( | )=
1 2
2
2
1
1
1
1y c
, (3.6)
where the mean y k k( | ) 1 and the variance y k k2 1( | ) are given by
{ }y k k kk k E y ( | ) | , ~ = 1 1y c , (3.7)
{ }y k y k kk k E y k k 22
11 1 ( | ) ( | ) | ,~ = y c . (3.8)
Moreover, the mean y k k( | ) 1 can be rewritten as [70]
( )y k k kk k c a k ( | )~
| ,
~
= 1 1 1y c , (3.9)
where ( ) | , ~a k k ky c 1 1 is the MMSE one-step prediction of the fading sample akassuming that the sequence ~ck1 has been transmitted. In general the evaluation of
( ) | ,~a k k ky c 1 1 and the variancey k k2
1( | ) can be accomplished by a time-varying
Wiener filter [72,73]. However, if the process { }ak can be characterised by a Gauss-Markov model [72,74], both quantities can be computed recursively by means of a
Kalman predictor9
[73] for a given sequence ~cN .MN Kalman filters are needed, one for
each possible symbol sequence. In general, the complexity of the MAP (or ML) receiveris large and increases exponentially with sequence length. Some studies [71,72,75,77]
have indicated that it can be simplified if one of the following two assumptions hold:
A.1) The sequence { }a wk k+ of fading plus noise samples is an Auto Regressive (AR)
process of finite order L [71,72,77].
9
The recursive evaluation of likelihood functions was first described by Schweppe[76].
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A.2) The autocovariance function of the fading process has finite support i.e.
C kTa ( ) = 0for k L> [75].
In both cases the channel is said to have finite memoryL [75,77]. However, it can be
shown that only (A.1) leads to a simplified optimum structure [71], whereas a reduced
complexity receiver based on (A.2) is not optimum. Under (A.1), evaluation of
y k k( | ) 1 and y k k2
1( | ) requires only the fixed-length vectors
[ ]y k k k k LL y y y 1 1 2( ) , , , and [ ]~ ( ) ~ ,~ , , ~ck k k k LL c c c 1 2 instead of the variablelength vectors y k1 and
~ck. This reduces the number of estimation filters from M
N
toML (the number of possible data vectors [ ]~ ( ) ~ , ~ , , ~ck k k k LL c c c =1 1 2 ), independentof the actual sequence length. Under this assumption, the Kalman filters can be replaced
by time invariant Wiener predictors [71, 75, 77, 86]. This leads to substantial complexity
reduction. Under (A.2) or other similar assumptions, optimality, strictly speaking,
requires infinite-length predictors and complexity reduction is not possible. Finally, we
note that:
a) The sequence { }e k k y k k y k y( | ) ( | ) 1 1 (see (3.6)) represents a discrete-time innovations process [76,77]. Thus the optimal sequence receivers are in fact
innovations-based receivers [77];
b) The optimal receiver minimising the symbol error probability is closely related to
the optimal sequence estimator. In fact, the optimal symbol decision ck on the k-th
symbol ck ( )1 k N , givenyN , is given by [72]
( )c pkc
N Nc
k N k
=
arg max ~ |~ ( )
c yc
(3.10)
where ( ) ck denotes the set of all trial sequences ~cN such that ~c ck k=
. The
conditional probabilities ( ){ }p N N~ |c y of (3.10) are also evaluated from the sequencedetector (see (3.2)).
c) y k k( | ) 1 in (3.7) or (3.9) is an MMSE estimate of the faded signal components a ck k k in (3.1). Thus, an optimal sequence detector implicitly evaluatesmultiple channel estimates, i.e. as many channel estimates as the hypotheses on
the transmitted sequence. However, an unambiguous phase reference can be
computed only if some symbols are known (as in PSAM transmission) or if the
transmitted sequence is coded [86].
3.3.1 Symbol-by-Symbol Receivers
Both optimal sequence and symbol-by-symbol receivers are complex structureseven when assumption (A.1) holds. An alternative is the class of sub-optimal decision
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feedback receivers [25-27], [70] and [78-83]10
. These are based on the idea that in order
to detect the k-th symbol ck coherently, an estimate of the fading distortion sample ak is
required. If the data decisions on previous symbols are reliable, they can be used to
remove the modulation from the corresponding received signal samples and to predict
ak. A general scheme for such a receiver is shown in Fig. 5. Decision feedback leads toa reduction in the number of channel predictors from MN (orML ) to one. The single
channel estimate can be computed by a Wiener predictor [25,26,81], by a Kalman filter
[70,78,79] or by an Extended Kalman filter11
[80]. A drawback of these receivers is that
a periodic refresh of their memory with a string of known symbols is required to prevent
loss of channel tracking (receiver runaway) [26] and to solve the phase ambiguity
problem.
The error performance of a decision feedback receiver can be improved (with
increased detection latency) if the two-stage architecture of [83] is used (see Fig. 5). In
this case, the first stage consists of a symbol-by-symbol detector with a MMSE channel
estimator. The data decisions of the first stage are delivered to the second stage which
generates an improved channel estimate by means of an optimal smoother. Finally, this
estimate is used to produce new (more reliable) data decisions. A similar architecture
has been proposed by Kam in [118].
K-lag symbol-by-symbol MAP estimation has been investigated by Seymour and
Fitz in [82] for PSK and QAM signals. They derive a suboptimal receiver by resorting to
decision feedback, to a thresholding technique for discarding unlikely hypotheses on the
past data decisions and to assumption (A.2) on the finite memory of the channel. A
feature of their solution is that it produces soft information for the data decisions i.e. an
estimate of the a posteriori probability ( )p ck K N~ | y for each possible symbol ~ck K . Thiscan be used as a decoding metric in interleaved coded modulation systems [82] and in
iterative decoding schemes.
3.3.2 Block Receivers
In detecting a long data sequence, the sequence of received samples can be
partitioned into blocks of lengthNand a block-based algorithm can be employed at thereceiver. Block detectors can be roughly divided into two classes:
1) multiple-symbol ML detectors [88-90] and [94,95];
2) ML detectors employing the Expectation-Maximisation (EM) algorithm [91-93].
10
Most decision feedback receivers are designed for linearly modulated signals and,
in particular, for PSK. However, they can be designed for CPM (see, for instance [84,
85, 87]).
11
An Extended Kalman filter allows estimation of both the fading distortion and otherrandomly-varying system parameters even if the joint estimation problem is nonlinear.
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rate value as the signal-to-noise ratio increases). If coding [107, 177] and/or explicit
diversity techniques [49] (e.g., space, time or frequency diversity) cannot be employed
in a digital communication system operating in fast fading, reduction of the error floor
can be pursued by exploiting the implicit time diversity of the channel [125]. This can
lead robust receiver structures for fast fading channels and two different approacheshave been proposed; namely,
1) the multisampling approach [71, 126, 127];
2) the double-filtering approach [100, 129].
The first stems from the fact that, since the channel is rapidly changing, accurate
estimates of the fading process can be obtained only if closely spaced samples of the
received signal are available. This requires processing more than one sample of the
received signal per signalling interval. Multisampling14
was employed in [78,79] and
later in [127]. These used decision feedback symbol-by-symbol receivers for PSKsignals assuming a rectangular signalling pulse p t( ) of duration equal to the signalling
interval. A multisampling receiver for CPM signals was developed in [71] and one for
bandlimited PSK signals in [126].
The second approach, developed in [100] for PSK and in [129] for CPM
assumes a linearly time-selective channel model [134]. This consists of approximating
the fading distortion by a straight line over the duration of the transmitter pulse p t( ) for
linearly modulated signals and over the signalling interval for CPM signals, thus
allowing for linear changes in the fading with time. The implicit time diversity can then
be extracted by two matched filters (for a linearly modulated signal) [100] or two filter
banks [129] (for CPM signals). In the PSK case [100], the symbol-rate samples at the
outputs of the two filters are expressed by
y a c wk k k k 0 0= + (3.13)
y a c wk k k k 1 1= + (3.14)
where ak0
denotes the value of the fading distortion at the center of the k-th signalling
interval and ak1
the slope of the straight line approximation in the same interval. Thesequences { }yk
0and { }yk
1can be processed using the same techniques as in the MAP
receivers processing one sample of the matched filter output per signalling interval. An
example is the VA-based receiver of [100].
A variation of this receiver results in a so-called blind detector. FIR channel
estimators are employed and their tap coefficients are evaluated based on geometrical
considerations, independent of the statistical properties of the fading and the received
14
Theoretical consideration of multisample processing in optimal detection can befound in [128].
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signal to noise ratio. The resulting receivers are known as blind detectors because they
ignore the statistics of the fading process. Blind VA-based algorithms have been derived
for CPM signals and PSK signals in [129] and [100]. They provide good error
performance and fast acquisition, usually with no training. When linearly modulated
signals are employed, the double filter receiver requires a transmitter pulse with aspectrum equal to the square root of a raised cosine function with 100% roll-off in order
to avoid ISI in the filter outputk
y1
[100]. This reduces the bandwidth efficiency with
respect to receivers operating at one sample per symbol, for which smaller values of
pulse roll-off are usually selected. Finally, we note that the double-filtering approach can
be used to design noncoherent detectors for PSK [130] and FSK [131] that provide a
low error floor in fast fading.
4. Equalising the Doubly Selective Channel
Although the frequency-flat channel model is simple and low-complexity
equalisers have been designed for it, it is often a poor model for actual wireless
channels. Under some circumstances, the delay spread can reach 20s [211]. Systemswith a channel symbol rate exceeding a few thousand symbols per second are adversely
affected unless this dispersion is equalised. Therefore, we now consider equalisers for
the time- and frequency-selective channel, also known as the doubly spread (delay and
Doppler spread) channel.
Equalisation for doubly spread channels is a challenging problem. Instead of onerandom process to estimate as in the flat fading case, there are many, one for each tap in
the tapped delay line model of (2.16). However, the problem of frequency-selectivity
arose first in the telephone channel [2], and the HF channel is doubly spread [49, 176],
so a considerable body of theory and practice is already available.
We categorise equalisers according to the statistical basis of their decision rules
(rather than the more superficial historical or structural classifications). The most
successful schemes account for both the channels double spreading and the additive
noise. These are described in later subsections, initially addressing equalisers which treat
the channel as a deterministic or known process, and subsequently describing equalisers
which treat the channel as a random process.
When the Doppler and delay spreads are small, they may be ignored and the
performance of low-complexity non-coherent, differentially coherent or coherent
receivers (without equalisation) is satisfactory [198-201]. Some of these are described in
Section 3.1. However, for larger values of delay spread, or at higher SNRs, ISI leads to
an error floor, where an increase in transmitter power does not improve the BER. If the
floor is too high, then a more complicated receiver incorporating an equaliser is
required. As a general rule, the delay spread becomes significant when its normalisedvalue,
d T/ , exceeds approximately 10% [198-200].
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4.1 Inverting Equalisers
The doubly spread channel may be considered as a linear time-varying filter and
linear equalisers are applicable (we discuss decision feedback equalisers (DFEs) in
section 4.2.8). Linear equalisers are usually transversal filters, where the tap gains are
either symbol spaced (and operate on the output of a symbol-rate sampled matched
filter) [1, 2] or fractionally spaced (so the equaliser incorporates the matched filter) [2,
178, 214]. The tap coefficients are calculated to invert (at least approximately) the
channels transfer function so as to eliminate or to reduce ISI, according to either the
zero forcing or minimum mean square error criterion [2]. A hard decision is made on the
output resulting in low equaliser complexity. At high SNR, under most criteria of
optimality, these equalisers have the form of a filter with the sampled impulse response
hi k, at the ith symbol interval designed to satisfy
r h cir k i k k
i , (4.1)
Even for a known channel, this is not straightforward [153]. When the channel is
unknown, the coefficients, hi k, , must be estimated, and continually adapted to track the
changing channel [e.g., 162, 194]. This requires knowing the transmitted symbols, and
so a training sequence is often transmitted. The tap weights are acquired during this.
They are then tracked in a decision-directed mode, as in Fig. 7 [189, 215]. The
receivers decisions should be highly reliable, and decision-directed adaptation is
normally successful. However, a string of decision errors (e.g., during a deep, widebandfade) can cause decision-directed estimators to fail [189]. Thus regular retraining is often
needed, at a frequency several times that of the Doppler spread [176]. An alternative
strategy, when the fading is sufficiently slow [191, 216], is to adapt the tap coefficients
weights during the training sequence only. Over long intervals or when the fading is
faster, training sequences must be regularly interleaved with segments of data [195,
196].
The common adaptation algorithms are the Recursive Least Squares (RLS) and
Least Mean Square (LMS) algorithms [2, 73]. RLS acquires the channel taps rapidly
[73], and is a good choice during the training sequence [217]. The LMS algorithm is
substantially less complicated than RLS or its variants [2, 73, 218, 219, 220], and
therefore, is often preferred as a decision-directed estimator. For uncorrelated inputs,
the tracking performance of the LMS [e.g., 73, 155] and RLS [e.g., 73, 179] algorithms
are comparable. Unfortunately, LMS is slow to converge and causes significant noise
enhancement when the eigenvalues of the input autocovariance are widely spread (i.e.
the input is correlated) [2, 73, 155, 156]. This is typical in linear equalisation due to the
ISI [221] from delay spread. Thus, computing the tap coefficients from an estimate of
the channel impulse response (equalisation through channel identification) offers faster
convergence.
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Both the RLS and LMS algorithms track the channel, in the sense that their
estimate of the tap coefficients is computed to equalise the channel as it was in the past.
This lag error may be only partially diminished by increasing the LMS step-size [155] or
by decreasing the RLS forget factor [179, 222], since then the estimation noise
dominates [179]. Thus linear equalisers are best suited to quite slowly varying channels[175]. The performance of linear equalisers in specific situations is addressed in [223,
224]. Novel approaches to linear equalisation are described in [225-227]. In [225], it is
the real bandpass signal that is equalised, rather than the complex baseband signal.
Linear equalisers can eliminate the phase distortion of the channel satisfactorily,
but amplitude distortion cannot be ameliorated without noise enhancement [2]. Since
deep frequency selective fades are characteristic of wireless channels, DFEs are
generally preferred to linear equalisers since their complexity is comparable and their
performance suffers less under amplitude distortion. Linear equalisers, DFEs, and
adaptive MLSDs are compared in [175, 194, 195, 196, 217, 228, 229].
4.2 Equalisers for the Deterministic Channel
Here, we consider equalisers derived under the assumption that the channel is
known a priori. All equalisers in this section share a common metric, the squared
Euclidean distance of (2.20), although they may compute it in different ways. We
include blind equalisers in this section, since they also try to estimate the channel.
As discussed previously, statistical detection theory offers two main criteria of
optimality: minimum probability of sequence error through maximum likelihood
sequence detection (MLSD or MLSE), and minimum probability of symbol error
through maximum a posteriori symbol detection (MAPSD). Both approaches are
discussed. Reduced complexity variants of each are also described: reduced state
sequence detection (RSSD or RRSE) and Bayesian decision feedback, respectively.
The assumption of a known channel is reasonable in the slowly time-varying case,
where an adaptive estimator can readily track the channel. However, channel estimation
is much harder in fast fading channels, and impossible in overspread channels.
4.2.1 MLSD with an Actually Known Channel
Here, the ideal case is discussed, where the channel is assumed known, (e.g., a
genie-aided equaliser). Although there is much earlier research [e.g.15,105], it was not
until the work of [105] that a finite complexity ML sequence detector for time-invariant
channels was synthesised. The receiver front-end has the form of a whitened matched
filter, sampled at the symbol rate. Its outputs, yn , are compared with noiseless
hypothesised signals using a squared Euclidean metric, thus creating a number of branch
metrics15
,
15
This metric is not well-defined as r .
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( )
i n k n n kr k L i
i
n ir
i r
y c h= = + +=
+
,1
21 1
(4.2)
where hn n kr , is the sampled received pulse shape defined by substituting (2.1) into (2.4),
L is the (assumed) finite duration of the overall channel response, and we assume the
receiver front-end to comprise a noise-limiting filter and a fractionally-spaced sampler.
There areML
branch metrics due to the ISI combinations. Sequence metrics similar to
(2.20) are computed by recursively summing the branch metrics of (4.2), and the search
for the maximum sequence metric is efficiently conducted by the Viterbi algorithm. In
[135], a structure is proposed that replaces the whitened matched filter with the
traditional matched filter, and [142, 157] describe two other variations.
MLSD performance for arbitrary channels is bounded in [105, 135, 230, 231].For fading channels, it is necessary to average over the pdf of the fading process [19].
For WSSUS channels, the averaging must be conducted over each value of delay,
although more convenient mathematical techniques are available [133]. In addition, it
has been found that the BER can be significantly improved by exploiting the implicit
delay diversity [19,232,233], as in CDMA systems, through RAKE detection [234].
It is instructive to consider MLSD for known, time-varying channels. Although a
channel estimator for this case is much more difficult to devise, [15] alludes to the
solution, and it is described in [136, 138]. The two path case is addressed in [235], andjoint equalisation and decoding are considered in [236]. With linearly modulated signals,
the receiver front-end is a filter matched to the received pulse (the transmitter pulse
convolved with the time-varying channel) [138]. The filter output is sampled at the
symbol rate, and processed as in the time-invariant case of [135]. Analysis shows two
forms of implicit diversity: due to delay and Doppler spreading [138].
4.2.2 Pilot Information
Pilot signals allow for channel estimation prior to data detection. In the time-
invariant channel only a single training sequence is required; in the frequency-flat fadingchannel a single pilot tone or a sequence of pilot symbols suffice, as in section 3.2. In the
doubly spread channel, a single training sequence cannot track the time-varying channel
[191, 208, 237], nor can a pilot symbol sequence efficiently measure frequency-
selectivity [198, 238], since the adjacent data symbols overlap the pilot symbols and
obscure the channel information in the known pilot symbol. More general methods are
needed, so as to maintain orthogonality between the pilot and data-bearing signals at the
receiver, and to allow the pilot information to be extracted before detection [51, 138]. In
general, a comb of pilot tones is required to characterise the channel in frequency as well
as in time [138].
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In many systems, the channel is slowly time-varying but highly frequency-
selective. Then, the method of [180, 197, 239] is superior, where training sequences of
pseudo-random symbols are transmitted periodically. The equaliser estimates the
channel response from each. By interpolating [180, 239] or Wiener filtering [197]
between training sequences, the channel may be estimated over the whole transmission.The sequences are usually several times the length of the received pulse, and must be
spaced at the Nyquist rate for the channels Doppler spread. Thus throughput is low
except for small Doppler spread. For faster fading, very short training sequences may be
used as when isolated pilot symbols are employed [138].
4.2.3 Adaptive MLSD
The MLSD of section 4.2.1 was derived assuming a known channel. In practice
channel estimation is required. Adaptive MLSD provides the simplest structure. Asingle channel estimator is employed. The transmitted symbol sequence is detected
using the estimated channel impulse response (i.e. the known channel impulse response,
hn n kr , , at time t = nTr, in (4.2) is replaced by its estimate,
,hn n kr ). This estimate is
updated according to the detected sequence, as shown in Fig. 8 [135, 144]. When a
matched filter is used as a front-end, it too must be updated [135, 240].
Transmission normally starts with a training sequence to initialise the channel
estimator. The receiver is then switched to a decision-directed mode, where tentative
decisions from the survivor sequence with best metric [241] are fed-back to theestimator from the Viterbi processor after some delay [182, 242, 243]. The Viterbi
decision delay is on the order of five times the channel memory [19], so the channel
estimator has only out-dated information available, and the estimate suffers a lag error
[140, 241]. For time-varying channels, this must be traded off against the accuracy of the
tentative decisions, so the tentative decision delay may be chosen to be less than the
Viterbi decision delay [144]. The lag error may also be diminished by predicting the
channel estimate [188, 203, 209].
As with linear equalisers, a fast acquiring algorithm is desired [188]. For
tracking, the low complexity LMS algorithm is invariably favoured in adaptive MLSD
and adaptive PSP MLSD (see subsection 4.2.4) [135, 140, 144, 146, 243]. The estimator
inputs are the MLSDs tentative decisions. When the input data correlation is low, the
tracking ability of the LMS [73, 155, 156, 244, 245] and RLS [73, 179, 246-250]
algorithms approximately match. The BER performance of the LMS-adaptive MLSD is
analysed in [203]. A floor occurs when the fading is too fast (e.g. a BER floor of 10-3
was observed for a channel with two taps, of equal mean power, spaced by one symbol
interval, for a normalised Doppler spread,fDT, of 10-3
). Thus, adaptive MLSD is suited
only to very slowly fading channels.
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Performance is simulated for GSM-like systems in [182, 242]. Trellis-coded-
modulation over doubly spread channels is addressed in [243, 251], where the need for
effective, low complexity equalisers compatible with interleaving is identified. Adaptive
MLSD is compared with other equalisation strategies in [194, 195, 252].
4.2.4 Adaptive PSP MLSD
The tentative decision delay of adaptive MLSD is unsatisfactory for time-
varying channels and to avoid this, Per Survivor Processing (PSP) [106] may be applied
[140, 141, 142, 146, 253, 254]. Each survivor in the trellis has an associated channel
estimator and an estimate, ( ) ~,hn n kr k c , which may be updated with no lag using the
survivors hypothesised symbols [106]. A generic block diagram is shown in Fig. 9.
Adaptive PSP MLSD is motivated by the inadequacy of adaptive MLSD in time-
varying channels, but it is truly optimal only in the time-invariant channel, using theRLS algorithm for channel estimation [141, 142]. When used in time varying channels,
the LMS algorithm is preferred [140, 141, 142, 146]. Error floors can still occur, even in
relatively slow fading. In [241], adaptive MLSD and adaptive PSP MLSD are studied.
For the non-fading but Doppler shifted channel, the additional complexity of adaptive
PSP MLSD is found to be unwarranted, although in [140], it outperformed adaptive
MLSD. As a performance-complexity trade-off, the number of channel estimators may
be reduced [149].
4.2.5 Reduced Complexity MLSD
Receiver complexity is governed by the number of branches multiplied by the
complexity of the branch metric. There are ML
branches, where Mis the constellation
size andL is the length of the received pulse. The trellis is usually infeasibly large, and
thus there is considerable interest reducing it. In many cases the energy in the tails of the
received pulse is small, and can be neglected without significant penalty. This leads to
truncated sequence detection [255, 256, 257]. Further simplification is obtained by
observing that reliable decisions can be made once most of the pulse energy (i.e., its
main lobe) has been received. The postcursors can then be dealt with by decision
feedback [104, 188, 258, 259]. Since insufficient energy is available to make reliabledecisions on the precursors, all combinations must be hypothesised through the MLSD.
This is called delayed decision feedback sequence detection (DDFSD or DDFSE) [104].
Reduced state sequence detection (RSSD or RSSE) refines this idea. Instead of a full
trellis for the precursor and a single decision history per survivor for the postcursor, set
partitioning principles [109] are applied to steadily reduce the number of hypothesised
symbols as more of the received pulse arrives [102, 103]. In [260], finer control of the
number of states is achieved. One analysis of RSSD is presented in [102] and tighter
performance bounds are provided in [261].
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The span of the received pulse may also be reduced by adaptive prefiltering to
obtain some desired impulse response [159, 262], using a linear equaliser [145, 159] or a
decision feedback equaliser [158, 161, 263]. MLSD is applied to this pre-filtered signal.
However, the pre-filter colours the additive noise, thereby reducing performance if it is
not taken into account [157]. Also a DFE pre-filter exhibits error propagation, so ahybrid structure that delivers the MLSDs soft outputs to the DFE is superior [161, 181,
188]. This structure is closely related to the DDFSD of [104].
Another approach is to retain the full trellis, but to search it more intelligently.
The Viterbi, single stack, Fano, 2-cycle, stack, merge, bucket and M- algorithms are
reviewed and compared in [264]. More recent algorithms are presented in [265, 266].
Equaliser performance using the M-algorithm [187, 267, 268] and the Fano algorithm
[269] has been studied. It is clear that exhaustive trellis search via the Viterbi algorithm
is wasteful, as the other algorithms attain excellent performance at reduced complexity.
The complexity of computing the branch metric may be reduced, by adopting a
reduced complexity channel model. Provided the model is sufficiently accurate, little
degradation results. For channels causing sparse ISI, complexity savings may be
obtained without loss of optimality [270]. In [151, 164, 166], the channel transfer
function is modelled as a power series in frequency with time varying coefficients [134],
instead of the usual tapped delay line. The latter requires a large number of taps if path
delays do not match the model delays (which happens invariably). The work on series
models has shown that only a modest number of terms (i.e. a low-order polynomial) is
required to obtain excellent performance for a large class of channels. Computing the
branch metric is simpler, and the number of channel parameters to be estimated is much
reduced. A dual approach is adopted in [48, 100, 129, 130, 131], where a polynomial
model of the channels time-variation is adopted.
4.2.6 Known Channel MAPSD
When the criterion for optimality is the maximum likelihood sequence, the
equaliser is efficiently implemented by the Viterbi algorithm [105]. However, BER is
normally used to compare equalisers, so a better criterion is to detect the symbols with
maximum a posteriori probability [271]. Little attention has been paid to MAP symboldetection (MAPSD), since its implementation requires a two-way recursion [272,
Appendix] unsuited to long transmissions. There is renewed interest in MAPSD, since
the soft symbol probabilities preferred by decoders are directly computed and its low
SNR performance is utilised in Turbo decoding [273]. In doubly spread known channels
with a noise-limiting receiver front-end followed by a fractionally-spaced sampler, an
inefficient implementation of the MAPSD branch metric is [274]
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( )
( )
i
r
r
n k n n kr k L i
i
rn ir
i r
N T
y c h
N T=
= + +
=
+ 1
0
1
2
0
1 1
exp,
(4.3)
4.2.7 Reduced Complexity MAPSD
A D-lag MAPSD or MAP symbol-by-symbol detector is proposed in [275]. It
makes a MAP decision on a symbol, given all past samples but only the nextDrsamples
beyond. Under this constraint, only a forward trellis pass is required [276]. The trellis
size is exponential inD, and for near-optimal performance,D must be at least as large as
the received pulse length, L, and possibly as large as 5L [277]. Problems, such as the
need for multiplication and exponentiation, and the potential for underflow, have been
largely overcome [278]. In the doubly spread channel, MAPSD is complicated by theneed to track the channel. Kalman filtering is used in [193, 279], and LMS adaptation is
considered in [279].
Bayesian decision feedback equalisers (BDFEs) are analogous to DDFSDs or
RSSDs, since both use decision feedback to reduce the complexity of the optimal
structure [280]. However, Bayesian DFEs adopt a structure that is quite different to the
recursive D-lag MAPSD; namely that of a DFE, but with the feedforward and feedback
tap coefficients replaced by a nonlinear structure [205].
The BDFE treats detection as a nonlinear classification problem [281]. In
decoding a symbol, a fixed-length vector of received samples in the vicinity of the
symbol identifies a point in a multidimensional space. The space is divided into M
nonlinear regions, one per possible decision. These regions are each the union of smaller
decision regions, one for each of the ML ISI combinations. The MAPSD makes a
decision by locating the decision region in which the received sample vector lies.
Computational savings are achieved in BDFE over the full MAPSD implementation,
since past decisions reduce the number of allowed ISI combinations, and thus the
number of smaller decision regions that exist [205]. A block approach is presented in
[204].
Neural networks are effective nonlinear identification algorithms. Several
models have been applied to BDFE [281, 282, 283], to achieve the equalisation of non-
linearly distorted signals [284] and to suppress adjacent channel interference (ACI)
[282] and co-channel interference (CCI) [283]. A promising technique for doubly spread
channels is pursued in [202, 205, 285, 286]. In [202], the LMS algorithm is used to
adapt a complex radial basis function network in a GSM-like channel. BDFE
outperforms the adaptive MLSD of section 4.2.3.
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4.2.8 Decision Feedback Equalisation
In the extreme case, the nonlinear boundaries of MAPSD become hyperplanes,
and it simplifies to the well known decision feedback equaliser (DFE) which comprises
feedforward and feedback transversal filters connected to a decision device. In the time-
varying channel, their impulse responses are time-varying. The feedforward filter,
{ },fi k , processes the input signal so as to minimise the channels phase and amplitude
distortion, and the feedback filter, { },gi l , processes the decision devices output so as to
subtract postcursor ISI from the decision devices input (assuming correct past
decisions). The input to the decision device equals
f y g c ci k ir k k
i l i ll
i, ,
>
0
(4.4)
where ci
is its output. A DFE performs better than a linear equaliser on channels with
severe amplitude distortion.
When an decision error is made, its effect lingers due to the feedback filter,
potentially causing error propagation. However, for most practical channels, the DFE
recovers quickly [175, 287-289]. Its BER performance is analysed in [290, 291].
Analysis is simplified when the effect of incorrect past decisions is neglected. Under this
assumption, the mean square error is calculated in [174], and for two path diversity
outage probabilities are calculated in [292] and the BER is bounded in [293, 294].
DFE filters are normally designed under the mean square error criterion [19,
174, 175], and may be separately optimised [295]. The filter coefficients may be adapted
directly from the error between the decision devices input and output, normally by the
LMS [174, 176, 190] or RLS [176, 190] algorithms. Other schemes include equalisation
by channel identification [191, 296], a more robust approach in time-varying fading.
Interpolating channel estimates or tap coefficients obtained from periodic training
sequences is one means of channel identification [180, 297]. Others are described in
subsections 4.2.2 and 4.2.3. A fractionally spaced adaptation scheme is described in
[168] and [259] presents a variant for channels with long delay spread. The consequence
of a poor channel estimate is studied in [191], where again an error floor is observed intime-varying channels.
The DFE offers a nice balance of complexity and performance. As with RSSD,
best performance is achieved when the channel is minimum phase, yet this is difficult to
assure in a time-varying channel. Reversing time may convert a channel with a
undesirably large precursor into one that can be equalised more easily [189, 298, 299].
When DFEs are used with interleaved coded modulations, more reliable
decisions are available from the decoding algorithm, and it is advantageous to use themin the DFE feedback loop [300]. A modified DFE structure is required [173], and [300]
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shows that this relatively simple equalisation scheme can approach channel capacity. In
[301], a similar approach is adopted, and the fading frequency-selective channel is
addressed.
DFEs are also used in equalising CPM signals, where ISI is due in part to thememory in the modulation scheme rather than a dispersive channel [44, 84, 97], and in
equalising differentially demodulated signals [302]. Block equalisers, for short
transmissions between training sequences, may also exploit past decisions [195,196].
4.2.9 Blind Equalisers for Frequency Selective Channels
Thus far, some form of reference signal has been explicitly or implicitly
assumed: training sequences, pilot tones or pilot symbols. These reference signals are
used to estimate the channel impulse response, so equalisation may commence. The
reference signals lead to reduced complexity equalisers and ensure fast acquisition, butthey are an unnecessary overhead as the blind equalisation literature shows. Blind
equalisation potentially has many benefits in wireless channels, which are characterised
by unpredictable deep fades. An equaliser which can recover automatically from these is
indeed worthwhile.
Most of the literature on blind equalisation is directed at wireline applications
and other very slowly varying channels. The early schemes of [303-307] converge
slowly. In [308], it is shown that second order cumulants of symbol-spaced sampled
signals are inadequate to identify non-minimum phase channels, and thus many blind
equalisers have been proposed based on higher order cumulants, as in [309-315]. These
require significant time for accurate computation, and are generally unsuited to the short
packet lengths and time-varying channels typical of wireless systems.
These deficiencies have been realised [312, 316-318], and recently, blind
equalisers with fast acquisition times have been developed [316, 317]. An effective
method is to employ sequence detection [141], fractional sampling and second order
cumulants together [142, 316, 318-320], although fractional sampling is not always
sufficient [321] (a deficiency avoided in [322]). Such blind equalisers acquire in
approximately a hundred symbol periods [141, 322, 323], so packets may be processedin two passes: one to estimate the channel, and a second to make decisions. However,
this does not attain the goal of swift, automatic recovery after a fade, when it is desirable
to re-acquire within a few tens of symbols. One approach is to account specifically for
the channels time-variation in the blind equalisers channel model [317]. Other blind
equalisers are presented in [279, 318, 324-326]. Blind equalisation in conjunction with
antenna arrays is investigated in [328, 329].
4.3 Equalisers for the Random Channel
It is surprising that little attention has been paid to this class of equaliser, sincethey are explicitly designed for doubly spread channels. Most others have been ad hoc
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solutions employing the LMS or RLS algorithms to provide adaptivity. Early work was
aimed at the ionospheric and tropospheric scattering channels [4, 5, 76, 147]. The
problem was identified as detecting Gaussian signals (i.e. Rayleigh or Rician faded
signals) in Gaussian noise, through their different autocovariances [5, 15, 22, 24, 76,
147]. Only the time-varying channel and noise autocovariances need to be calculated,and the time-varying channel impulse response is not required, although in some special
cases it is actually estimated [77, 86, 98, 171]. The detection of non-Gaussian signals
has also been investigated [21, 23, 143].
Kailaths estimator correlator structure is well known and theoretically elegant
[15-18]. In coloured noise, a non-causal filter is required [15], but this case is generally
unimportant. A disadvantage of these structures is that sequence detection is undertaken
using a brute force search [15, 32]. As discussed in section 3.3.3, the Viterbi algorithm
may be introduced under the assumption of finite memory [19, 20]: that is, the minimum
mean square estimate, y k k( | ) 1 , of yk, conditioned on a data sequence and the past
received samples, requires only a finite number of past samples16
. This is shown more
fully in [77] for realistic pulse shapes and Rayleigh fading channels. Using Bayes
theorem, coupled with the Innovations process or the Cholesky decomposition [71, 77],
the sequence metric of (2.10) can be decomposed into the conditionally independent
branch metrics (also derivable from (3.6)),
( )
ik y
yk ir
i r
y
y k k
k k
k k=
+ =
+ ( | )
( | )
ln ( | )1
1
1
2
2
1 12
(4.5)
where y k k2
1( | ) is the conditional variance of the prediction. This metric may be
interpreted as the squared Euclidean distance between the received signal and the
expected received signal, normalised by its variance. The bias term, ln ( | )y k k2
1 , may
be neglected for some channels and modulations [77, 86, 98, 171]. The conditional
mean, y k k( | ) 1 , is computed as the weighted sum of past received samples. The
metrics data dependence arises in the conditional mean and variance, and their
computation requires the channel and noise autocovariances. Several methods for
estimating these have been proposed [127, 139, 330, 331].
The error performance of the innovations-based MLSE scheme, described in
[77] for a Rayleigh fading doubly spread channel is analysed in [331]. Alternatives to
the MLSE solution are the ML-based sliding block detection scheme of [332] and the
Kalman filter based MLSE receiver of [148, 164], where the received pulses time
variation and the channel are modelled as ARMA and AR processes respectively. MLSE
16
Estimates based on a finite number of samples are, strictly speaking, optimal onlyunder assumption (A.1).
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employing Kalman filter channel estimation is analysed in [169]. Another sequence
detector for unknown channels is the MAP receiver of [193] which employs a bank of
extended Kalman filters to jointly estimate the channel, data and symbol timing. Finally,
it can be shown, given a high enough sampling rate (Nyquist rate for the received signal)
and sufficiently accurate predictors, the error floor in fast doubly selective fading can belowered arbitrarily [71, 160].
5. Combined Equalisation and Decoding
In order to achieve the required error performance, most digital wireless systems
require some form of forward error correction coding [177]. However, wireless systems
usually operate in a tightly band limited environment which must be used in a spectrally
efficient manner. Conventional error correction techniques require additional bandwidth
to maintain the required data rate and this is often unacceptable. Consequently, there ismuch interest in the use of spectrally efficient coding techniques such as trellis coded
modulation and multi-level coded signalling for digital wireless transmission.
The use of these bandwidth efficient techniques, where the coding is embedded
in an expanded signal set leads to a requirement for combined equalisation and decoding
techniques. In the time-invariant ISI channel, two major approaches, reduced state
sequence estimation (RSSE) [102] and [103] and delayed decision feedback [104], have
been investigated. To date, there have been a few attempts to develop similar techniques
for use in the digital wireless environment. In the case of trellis coded modulation,
researchers have considered receivers which perform joint MLSE and decoding on an
enlarged trellis structure often resulting in unacceptable system complexity [86, 103].
One of the problems with coded transmission is that in slow frequency selective
fading, interleaving is required in order to randomise the effects o