Adaptive Bayesian decision feedback equaliser for alpha stable ...

Adaptive Bayesian decision feedback equaliser for alpha-stable noise environments Apostolos T.. We 1rst derivethe optimum Bayesian decision feedback equaliser and present a novel analyti

Adaptive Bayesian decision feedback equaliser for alpha-stable

noise environments Apostolos T Georgiadis∗, Bernard MulgrewDepartment of Electronics & Electrical Engineering, University of Edinburgh, Mayeld Road, Kings Building,

EH93JL Edinburgh, UK Received 22 March 2000; received in revised form 21 November 2000Abstract

In some communication systems the channel noise is known to be non-Gaussian due, largely, to impulsivephenomena The performance of signal processing algorithms designed under the Gaussian assumption maydegradeseriouslyin such environments In this paper we investigate the problem of adaptive channel equalisation in animpulsive noise environment The impulsive interfering noise is modelled as an
-stable process We 1rst derivethe optimum Bayesian decision feedback equaliser and present a novel analytical framework for the evaluation ofsystems in in1nite variance environments A family of generalised adaptive channel identi1cation algorithms forthis in1nite variance noise environment is also presented The combination of a Bayesian equaliser and a channelestimator operating as an adaptive channel equaliser is experimentallystudied and its performance is comparedwith that of a traditional system designed under the Gaussian assumption The experimental data suggest that theproposed combination of equaliser and channel estimator outperforms the traditionallydesigned adaptive equaliser interms of error probability We 1nally provide some useful approximations concerning the practical implementation

of an
-stable adaptive equaliser.? 2001 Elsevier Science B.V All rights reserved

Keywords: Adaptive equalisation; Impulsive noise; Alpha-stable noise; Non-Gaussian Bayes decision theory; Adaptive channel estimation

1 Introduction

High speed data transmission over communication channels is subject to intersymbol interference andnoise The intersymbol interference is usually the result of the restricted bandwidth allocated to thechannel and=or the presence of multipath distortion in the medium through which the information istransmitted Equalisation is the process which reconstructs the transmitted data combating the distortionand interference of the communication link The most simple architecture in the class of equalisers makingdecisions in a symbol-by-symbol basis is the linear transversal 1lter The optimal solution, however, isthe Bayesian approach which is also known as the maximum a posteriori (MAP) symbol-by-symboldecision equaliser [1]

∗Corresponding author Tel.: +44-131-650-5580; fax: +44-131-650-6554.

E-mail addresses: atg@ee.ed.ac.uk (A.T Georgiadis), bernie@ee.ed.ac.uk (B Mulgrew).

0165-1684/01/$ - see front matter ? 2001 Elsevier Science B.V All rights reserved.

PII: S0165-1684(01)00075-5

(a; b) open interval in R; the set {x∈ R: a ¡ x ¡ b}

[a; b] closed interval in R; the set{x∈ R: a6x6b}

[ ] matrix or vector

[ ]T transpose of a matrix or vector

Aci scalar centre of the channel (06i ¡ 2N)

ci vector centre of the channel (06i ¡ 2K)

D feedback order of the equaliser

f
(s) univariate
-stable pdf, de1ned in s∈ R

K length of the input vector (= N + M−1)

L length of the residual input vector (= K−D)

M order of the equaliser

N length of discrete-time channel impulse response

Nc number of vector centres (= 2K)

NDFc number of DFE vector centres (= 2L)

Nsc number of scalar centres (= 2N)

n(k) noise sequence added at the output of the channel

r(k) received (observation) signal sequence

r(k) received (observation) vector

x(k) transmitted data sequence

ˆx(k) estimate of transmitted symbol x(k)

x(k) transmitted symbols vector

xi all possible discrete states of vector x(k) (06i ¡ Nc)

xch(k) channel input vector

xch i all possible discrete states of vector xch(k) (06i ¡ Nsc)

y(k) noise-free output sequence of the channel

y(k) noise-free channel output vector

Although the Bayesian equaliser and its adaptive implementation has been thoroughly studied in theliterature (for example see [19] and the references therein), byand large, the results are related to theassumption that the interference noise is Gaussian However, in manyphysical channels, such as urban,indoor radio and underwater acoustic channels [18,27,29], the ambient noise is known through experimen-tal measurements to be non-Gaussian, mainlydue to the impulsive nature of man-made electromagneticinterference It is well known that non-Gaussian noise can cause signi1cant performance degradation inconventional systems based on the Gaussian assumption [27]

A number of models have been proposed for impulsive phenomena in communication systems, either

by 1tting experimental data or based on physical grounds Recently, it has been suggested [27] thatthe familyof
-stable random variables provides an appropriate model for manyimpulsive phenomena,including interference in communication channels Stable distributions share de1ning characteristics withthe Gaussian distribution, such as the stabilitypropertyand central limit theorems

In the following, after a quick overview of stable processes (Section 2), we derive in Section 3 theoptimum Bayesian decision feedback equaliser (DFE) for
-stable noise environments The problem ofevaluating communication systems in in1nite variance environments is addressed in Section 4 and a newanalytical framework in this direction is presented Some preliminary experimental results are given inSection 4.1, showing a promising performance bene1t compared with a Bayesian DFE designed under the

Fig 1 System model for FIR channel and 1nite memory equaliser.

Gaussian assumption.1 Section 5 discusses the problem of estimating the channel and noise characteristics

in an
-stable noise environment A familyof recursive algorithms for channel identi1cation in suchenvironments is presented and studied The adaptive Bayesian DFE is then experimentally studied inSection 6 Some useful approximations concerning the simulation and implementation of such an equaliserare 1nallydiscussed in Section 7

2 The class of stable random variables

The familyof stable random variables (RV) is de1ned as a direct generalisation of the Gaussian law.The main characteristic of a non-Gaussian stable probabilitydensityfunction (pdf) is that its tails areheavier than those of the normal density This is one of the main reasons why the stable law is regardedsuitable for modelling signals and noise of impulsive nature

The symmetric
-stable (S
S) pdf f
(s) is de1ned bymeans of its characteristic function2 F(!) = exp(i!−|!|
) The parameters
;  and  describe completelya S
S distribution The characteristicexponent
(0 ¡
62) controls the heaviness of the tails of the stable density; a smaller value impliesheavier tails, while
= 2 is the Gaussian case The dispersion parameter  ( ¿ 0) plays an analogousrole to the variance and refers to the spread of the distribution Finally, the location parameter  iscomparable with the mean of the distribution In fact theyare identical for 1 ¡
62

Theoretical justi1cations for using the stable distribution as a basic statistical modelling tool comefrom the generalised central limit theorem [8] Unfortunately, no closed-form expressions exist for thestable density, except the Gaussian (
= 2) and Cauchy(
= 1) distributions An important propertyofall stable distributions is that onlythe lower order moments are 1nite That is, if x is a stable RV, then

Ex{|x|p}¡∞ iH p ¡
A well known consequence of this propertyis that all stable RVs with
¡ 2have in1nite variance For a more detailed discussion of
-stable processes refer to [26] Moreover, [27]presents a signal processing framework for
-stable processes

3 Bayesian equaliser

The model of the system considered is depicted in Fig 1 We assume that the data sequence {x(k) =+1;−1}, consisting of independent and equiprobable binarysymbols, is passed through a noiseless linear

1 Hereinafter, a Bayesian equaliser designed under the Gaussian assumption will be referred to as traditional Bayesian equaliser.

2 The characteristic function F(!) of a RV is the Fourier transform of its probabilitydensityfunction f(s).

dispersive channel with 1nite impulse response (FIR) which spans over N symbols:

H(z) =N−1

i=0

hiz−i; h = [h0; h1; : : : ; hN−1]T: (1)

If xch(k) = [x(k) x(k−1) · · · x(k−N + 1)]T is the channel input vector, then the observation sequence

{r(k)} is formed byadding the
-stable random noise n(k) to the output of the channel y(k) = hTxch(k),i.e., r(k) = y(k) + n(k) In 1nite memoryequalisers, the M most recent samples of the observationsequence{r(k)} are stored in the observation vector

r(k) = [r(k) r(k−1) · · · r(k−M + 1)]T: (2)

A decision function fd(·) is then evaluated on r(k) and passed through a quantiser to provide an estimate

of the transmitted symbol x(k−d) Here, d is the decision lag of the equaliser

3.1 Feed-forward equaliser

Let x(k) be the vector with all the transmitted symbols that inJuence r(k), i.e.,

x(k) = [x(k) x(k−1) · · · x(k−K + 1)]T; (3)where K = N + M−1 The state equation that relates the received vector r(k) to x(k) is

Note that Eq (7) reduces to the traditional MAP equaliser for
= 2 The actual estimate is given bythesign of fd, i.e.,

Eq (8) partitions the M-dimensional observation space spanned bythe received signal vector r(k) intwo sub-spaces Therefore, the solution of equation fd(r(k)) = 0 de1nes the optimum decision boundary.Since fd(r(k)) is related to the pdf of the noise, the corresponding Bayesian decision boundaries will beinherentlydiHerent for Gaussian and non-Gaussian distributions In [19] a radial basis functions networkimplementation of Eq (7) is suggested However, it has been demonstrated [9] that in non-Gaussiannoise environments the basis functions are not radially symmetric The actual radial asymmetry of theM-dimensional stable noise pdf is responsible for the radical discrepancies between the Gaussian andnon-Gaussian decision boundaries

3.2 Decision feedback equaliser

Without loss of generality, we can assume that the D decisions ˆx(k−L); ˆx(k−L−1); : : : ; ˆx(k−K + 1)are correct (here L = K−D) Replacing these decisions [4,5,30] on the trailing part of vector x(k) wehave

The sub-matrices HR; HD; xR, and xD are de1ned in an obvious manner.4

The eHect of the decisions contained in xD(k) can then be removed from the observation vector r(k)

to produce a residual observation vector, de1ned as

rR(k),r(k)−HDxD(k) = HRxR(k) + n(k) = yR(k) + n(k): (11)

We can now applya Bayesian decision function to rR(k) rather than r(k)

A decision feedback equaliser implementing this scheme is depicted in Fig 2 In Fig 3 we can seethe optimum boundaries for the Bayesian DFE with feedback order D = 2 and a varietyof values forthe characteristic exponent The features of the optimum decision boundaries are signi1cantlydiHer-ent compared to the boundaries of a traditional MAP equaliser Therefore, it is reasonable to expect

a considerable performance degradation of the traditional Bayesian equaliser in a non-Gaussian noiseenvironment

4 The subscript R stands for residual while D stands for feedback.

Fig 2 Decision feedback equaliser Fig 3 Observation space and decision boundaries

of Eq (7) for the Bayesian DFE The channel is H(z) = 0:3482 + 0:8704z−1+ 0:3482z−2 (the stars∗indicate

a centre in the S− subset and the circles a centre in the

S + subset).

4 Evaluating systems in in"nite power noise environment

The traditional performance measures are usuallyplots of the bit-error ratio (BER) against the noise ratio (SNR) In non-Gaussian stable noise environment (
-stable noise with
¡ 2), however, thevariance of the noise is in1nite [27], making the use of SNR meaningless Nevertheless, all receivers inpractice have a 1nite input dynamic range Let us consider the generic receiver depicted in Fig 4 Thelimiter at the front end of the receiver is assumed to be an ideal saturation device, with transfer functiong(s; G) =

The distribution of the received signal r(k) is

of the received signal and its tails are concentrated at the points +G;−G where theyappear as Dirac

5 Scalar centres are all the discrete noise-free channel outputs.

Fig 4 Generic adaptive equaliser with saturation device at the front end.

Fig 5 The pdf of (a) r L (k), and (b) ˆn(k) for Gaussian (
= 2) and
-stable noise (
= 1) The channel is H(z) = 0:3482 + 0:8704z−1+ 0:3482z−2(the circles◦denote the corresponding scalar centers): (a) For Gaussian case  = 0:135, and for
-stable case  = 0:1 G = 2:2; (b) For Gaussian case  = 1:67, and for
-stable case  = 0:72 G = 4.

impulses (s) (Fig 5(a)) The pdf of the limited received sequence rL(k) is therefore

The receiver removes the channel output estimate ˆy(k−d) from the limited received signal rL(k−d)

to form an estimate of the noise samples ˆn(k−d) (Fig 4) We can assume, without loss of generality,

that the samples ˆy(k) are correct The pdf of the noise estimate ˆn(k) will then be

In Fig 5(b) we can see an example for the pdf of the noise estimate sequence ˆn(k) Due to the symmetry

of scalar centres, fˆn(s) is symmetric Therefore, the mean of ˆn(k) is zero, while its variance can bewritten as

G2exp

−G2 2

4  + G1exp

−G2 1

where vy is the variance of the noise-free channel output

In practice for a given SNRrcv, characteristic exponent
and dynamic range G, it is possible tonumericallysolve Eq (22) for the noise dispersion  For the values of
that it is not possible toanalytically compute Eq (22) the variance of the noise estimate ˆn(k) maybe experimentallymeasured

in order to compute the working SNR However, in Section 7 we suggest an approximate method tocompute the variance vˆn(
; ; G) for a given dispersion  Accordingly, using an analogous approximation

we can obtain  for a given SNRrcv

Fig 6 Performance of the optimum (solid lines) and traditional (dashed lines) feed-forward Bayesian equalisers for a channel with 3 taps and a varietyof values for
.

Fig 7 Performance of the optimum (solid lines) and traditional (dashed lines) Bayesian DFE for
= 1 (M = 2, D = 2, d = 1,

G = 4): (a) Correct data for the feed back; (b) Detected data for the feed back.

4.1 Experiments

In order to assess the Bayesian equaliser in an
-stable noise environment, the experimental mance of a number of feed-forward and DF equalisers was recorded The simulations were performedfor a channel with transfer function

perfor-H(z) = 0:3482 + 0:8704z−1+ 0:3482z−2: (23)For the moment we assume that the equaliser has perfect knowledge of the channel model and the noisecharacteristics The dynamic range of the receiver is G = 4

For the 1rst set of experiments we simulated the feed-forward MAP equaliser in varying noise ronments (
= 1; 1:5; 2) The length of the observation vector was M = 2 and the equalisers operated with

envi-a decision lenvi-ag d = 1 The performenvi-ance of the optimenvi-allydesigned MAP equenvi-aliser wenvi-as recorded, envi-alongwith that of the traditional Bayesian equaliser The BER performance of both equalisers is plotted in Fig

6 It can be clearlyseen that the optimum MAP equaliser outperforms the traditional Bayesian equaliserwhen the noise is non-Gaussian In fact, the further the noise deviates from the Gaussian distribution,the more signi1cant the performance degradation of the traditional Bayesian equaliser is

For the simulations concerning the Bayesian DFE,
was set to 1 Again, both optimum and traditionalequalisers were studied The equaliser forward order was M = 2 and the decision lag d = 1, while the

Fig 8 Probabilityof exceedence of
-stable distribution for a varietyof values for
(* = 0:6, G = 4).

feedback order was D = 2 Fig 7(a) shows the performance of the equalisers in this highlyimpulsive

-stable noise environment For comparison, the BER graphs of the feed-forward and DF equalisers inGaussian noise environment are given as well In this experiment the correct transmitted data were fed

in the feedback vector xD (for the DF equalisers) The results show that for 0:001 BER the mance bene1t from the feed-forward optimum equaliser compared to the traditional one is 4:18 dB Thecorresponding gain for the DF equaliser is 8:88 dB

perfor-For the next experiment (Fig 7(b)) the actual decisions of the DF equaliser were fed into the feedbackvector xD As expected, the performance gain is slightlyinferior (due to error propagation), but stillconsiderable For the DF equalisers this gain is 8:08 dB at 0:001 BER That is, the use of the actualdecision data results in a gain loss of 0:8 dB

It is interesting to note that the actual shape of the BER graphs for non-Gaussian stable noise is entlydiHerent from the traditional graphs in Gaussian noise: the probabilityof error in a communicationsystem is highlyrelated to the probabilityof exceedence6 Px¿* of the underlying noise distribution Forthe Gaussian case (
= 2) the probabilityof exceedence Px¿*(
; ) is

In Fig 8 we plot this probabilityas a function of the SNR at the receiver Eq (22) was used in order

to map the values of  to the corresponding values of SNR The similarityof Fig 8 with Figs 6, 7(a)and (b) is clear

5 Training the equaliser

The optimum Bayesian equaliser derived in Section 3 is fullyde1ned bytwo sets of parameters: (a) thevector centres ci and their associated signs si, and (b) the parameters of the probabilitydensityfunction

of the S
S noise, namelythe characteristic exponent
and dispersion  (see Eq (7)) This sectionaddresses the problem of determining these parameters for the Bayesian equaliser in a non-Gaussian

-stable noise environment For the estimation of the equaliser centres, the most popular approach 1rst

6 The probabilitythat the RV x exceeds (is greater than) a given *.

Fig Performance... to assess the Bayesian equaliser in an
-stable noise environment, the experimental mance of a number of feed-forward and DF equalisers was recorded The simulations were performedfor a channel...
-stable distribution for a varietyof values for
(* = 0:6, G = 4).

feedback order was D = Fig 7(a) shows the performance of the equalisers in this highlyimpulsive

-stable