[PROAUDIO]Ecualizacion Formulas Matematicas

7/27/2019 [PROAUDIO]Ecualizacion Formulas Matematicas

1/64

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.


2/64

2

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].


3/64

3

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


4/64

4

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.


5/64

5

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].


6/64

6

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


7/64

7

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


8/64

8

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.


9/64

9

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


10/64

10

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.


11/64

11

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


12/64

12

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


13/64

13

( ) ( ) 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.


14/64

14

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.


15/64

15

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:


16/64

16

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].


17/64

17

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.


18/64

18

( )~ 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].


19/64

19

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


20/64

20

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.


21/64


22/64


23/64

23

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].


24/64

24

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].


25/64

25

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.


26/64

26

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 .


27/64

27

( )

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].


28/64

28

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.


29/64

29

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].


30/64

30

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]


31/64

31

( )

( )

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.


32/64

32

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]


33/64

33

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


34/64

34

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).


35/64

35

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

[PROAUDIO]Ecualizacion Formulas Matematicas

Documents

Transcript of [PROAUDIO]Ecualizacion Formulas Matematicas