CHANNEL EQUALIZATION USING ADAPTIVE FILTERING WITH AVERAGING

Georgi Iliev and Nikola KasabovDepartment of Information Science, University of Otago

P.O. Box 56, Dunedin, New Zealandgiliev@infoscience.otago.ac.nz, nkasabov@otago.ac.nz

Abstract – The recent digital transmission systemsimpose the application of channel equalizers withshor t training time and high tracking rate. Theserequirements turn our attention to adaptivealgor ithms, which converge rapidly. From thispoint of view the best choice is the recursive leastsquares (RLS) equalizer . Unfor tunately thisequalizer has high computational complexity andstability problems. Therefore it is wor thinvestigating a good alternative of the classic RLSequalizer . In this contr ibution we present such achannel equalizer based on adaptive filter ing withaveraging (AFA). The main advantages of AFAequalizer could be summar ized as follows. I t hashigh convergence rate comparable to that of theRLS equalizer and at the same time lowcomputational complexity and possible robustnessin fixed-point implementations.

I. INTRODUCTION One of the most important advantages of the digitaltransmission systems for voice, data and videocommunications is their higher reliability in noiseenvironment in comparison with that of their analogcounterparts. Unfortunately most often the digitaltransmission of information is accompanied with aphenomenon known as intersymbol interference (ISI).Briefly this means that the transmitted pulses aresmeared out so that pulses that correspond to differentsymbols are not separable. Depending on thetransmission media the main causes for ISI are:• cable lines – the fact that they are band limited;• cellular communications – multipath propagation. Obviously for a reliable digital transmission systemit is crucial to reduce the effects of ISI and it is wherethe adaptive equalizers come on the scene [1], [2].Two of the most intensively developing areas ofdigital transmission, namely digital subscriber lines(DSL, HDSL, ADSL and VDSL) and cellularcommunications (GSM and IS-54) are stronglydependent on the realization of reliable channelequalizers [3], [4], [5]. One of the possible solutions isthe implementation of equalizer based on filter withfinite impulse response (FIR) [6], [7] employing thewell known least mean squares (LMS) algorithm foradjusting its coefficients [8], [9]. An enhancement ofthis equalizer is the so-called decision feedbackequalizer, which is a combination of two adaptivefilters. These two variants are most often encounteredin practice [10] and they work as a rule in two modes:

• training – the adaptive equalizer identifies thechannel characteristics and works with known(training) sequence;

• tracking – the equalizer operates in so-calleddecision direction fashion.

Recently blind equalizers have received greatresearch interest [11], [12] since they do not requiretraining sequence and extra bandwidth. The mainweaknesses of these approaches are their highcomputational complexity and slow adaptation [12]. Namely the requirement for fast adaptation turns outthe most important factor in applications where thechanges in channel characteristics could be quiterapid. This is the basic stimulus for the utilization ofadaptive algorithms, which have a good rate ofadaptation, of course, at the price of morecomputations. An example is Kalman/Godardalgorithm [13] also known as recursive least squaresalgorithm (RLS) [14]. Kalman filter maintains therequirement for fast convergence but it has twodisadvantages:• high computational complexity;• stability issues [15], which are mainly due to the

method for updating the Kalman gain vector. To summarize the requirement for fast adaptationrate, which is vital especially in applications ascellular communications where the changes in channelcharacteristics may be quite rapid, forces theinvestigation of adaptive algorithms that could be agood alternative of the classic Kalman/Godardalgorithm. They are expected to avoid the problemsrelated to high computational complexity and lowstability in fixed-point implementations. The need ofsuch algorithms is the main motivation for theinvestigations conducted in this contribution. Here wepresent the results of the realization of an adaptiveequalizer using adaptive filtering with averaging. Theresults show that this channel equalizer could be agood alternative of Kalman equalizer maintaining itsfast adaptation rate and at the same time eluding itsbasic weaknesses.

II. CHANNEL EQUALIZATION As mentioned in the introduction the intersymbolinterference imposes the main obstacles to achievingincreased digital transmission rates with the requiredaccuracy. Traditionally, ISI problem is resolved bychannel equalization in which the aim is to constructan equalizer such that the impulse response of thechannel/equalizer combination is as close to z-d aspossible, where d is a delay. Frequently the channel

parameters are not known in advance and moreoverthey may vary with time, in some applicationssignificantly. Hence, it is necessary to use the adaptiveequalizers, which provide the means of tracking thechannel characteristics. In Fig. 1 a digital transmission system using channelequalization is depicted. Here, as most often inpractice, the analog signal is sampled at Nyquist rateand then the different samples are coded with binarysequences taking the two possible values –1 and 1. InFig. 1 s(n) denotes the transmitted signal where thesampling period T=1. Actually s(n) is a randomsequence of –1s and 1s. Here the channel is dispersiveand can be modeled by a FIR filter. Thus the channeloutput is written as

x(n) = ∑ −=

N

0ii )in(sa (1)

where N is filter order and ai, 0≤i≤N, are filtercoefficients. In addition

y(n) = x(n) + n(n) (2)

is the adaptive equalizer input, where n(n) is theinevitably present additive noise. Obviously the problem to be considered is that ofusing the information represented in the observedequalizer inputs y(n) to produce an estimate oftransmitted symbols in the sequence s(n). There is adelay through the adaptive equalizer, which meansthat the estimate is delayed by d symbols. In Fig.1 thisis denoted as d(n-d) that is an estimate of s(n-d). In thescheme shown in Fig. 1 a decision device applies a setof thresholds to recover the original data symbolsselecting the symbol which is closest to the estimated(n-d). For example, if the transmitted signal takesvalues ±1, the decision device would simply be

)]dn(d[sng)dn(d −=− (3)

where sng [ ] is the sign function. Here the object is

that )dn(d − = s(n-d).

In such a digital transmission system a transversalfilter can be used as an adaptive equalizer. Thisarrangement is shown in Fig. 2 as the equalizer formsthe linear combination of input samples as follows

d(n-d) = ∑ −−−=

d

djj )jdn(y)n(w (4)

where wj, -d≤j≤d, are the filter coefficients. The objectof the adaptive algorithm is to adjust the filtercoefficients in such a manner that d(n-d) ≈ s(n-d). Acriterion that can be applied to adapt the coefficients isthe minimization of the output mean square errorE[e2(n-d)] where

e(n-d) = d(n-d) – s(n-d). (5)

In (5) the estimation error is determined as thedifference between the estimate and the originalsignal, which implies that the transmitted sequence isknown in advance. In this case the equalizer operatesin so-called training mode. The second possible fashion of working is decisiondirection mode where

e(n-d) = d(n-d) - )dn(d − . (6)

Notice that here the estimation error is defined as thedifference between the estimate and the detected datasymbols at the decision device output. This manner ofadaptation is acceptable for tracking a slowly varyingchannel, but can not be used for initial channelidentification. In this case the approach is to utilize adeterministic data sequence and a training period. Thetime it takes for training of the equalizer is stronglydependent on the choice of adaptive algorithm. Themain concern is to reduce the training time as much aspossible and this suggests the usage of adaptivealgorithms, which converge rapidly. Two of the most widely spread algorithms inadaptive equalization practice are LMS and RLSalgorithms. For the adaptive channel equalizerdepicted in Fig. 2 the needed steps for thesealgorithms are presented in Table 1 and Table 2.Comparing the two algorithms, it is clear that thestrongest point of LMS algorithm is its lowcomputational complexity. Unfortunately thisalgorithm has low adaptation rate. In contrast, RLSalgorithm provides a tracking rate sufficient for a fast-fading channel, but the price to be paid is an order-of-magnitude increase in complexity. Moreover RLSalgorithm is known to have stability issues [16] due tothe covariance update formula P(n) (see Table 2). Thisfact creates a lot of problems especially in fixed-pointimplementations. All mentioned above turns ourattention to search for adaptive algorithms, whichhave fast convergence that means short training timeand good tracking rate and at the same time are free ofthe basic RLS’ shortcomings, namely high complexityand stability problems. In the next section we presentsuch a candidate algorithm to be used in channelequalization.

n(n) s(n) x(n) Channel

)dn(d − d(n-d) y(n)

Decision Adaptive Detected Device Equalizer Data

Fig. 1. Digital transmission system using channelequalization.

y(n) z-1 z-1 z-1

w-d w-d+1 w-d+2 wd-1 wd

d(n-d)

s(n-d) e(n-d)

Fig. 2. Transversal channel equalizer.

Table 1. LMS equalizer.

Signal estimation:

d(n-d) = ∑ −−−=

d

djj )jdn(y)n(w

Signal detection:

)]dn(d[sng)dn(d −=−Estimation error:

e(n-d) = d(n-d) – s(n-d) - training

e(n-d) = d(n-d) - )dn(d − - tracking

Coefficients update:wj(n+1) = wj(n) - µe(n-d)y(n-d-j)

for -d≤j≤d

Table 2. RLS equalizer.

Signal estimation:

d(n-d) = ∑ −−−=

d

djj )jdn(y)n(w

Signal detection:

)]dn(d[sng)dn(d −=−Estimation error:

e(n-d) = d(n-d) – s(n-d) - training

e(n-d) = d(n-d) - )dn(d − - tracking

Covariance update:

P(n) = P(n-1) - )n(Y)1n(P)n(Y/1

)1n(P)n(Y)n(Y)1n(PT

T

−+δ−−

Gain update:

K(n) = )n(Y)1n(P)n(Y/1

)n(Y)1n(PT −+δ

−

Coefficients update:wj(n) = wj(n-1) – kj(n)e(n-d)

for -d≤j≤d0<δ<1 P(0) = αI α large

III. A CHANNEL EQUALIZER USING ADAPTIVEFILTERING WITH AVERAGING

In this section the application of an algorithm baseon adaptive filtering with averaging in channelequalization is investigated. For simplicity the channelequalizer implemented by this adaptive algorithm isdenoted as AFA (adaptive filtering with averaging)equalizer all through the rest of this paper. The mainmotivation for the utilization of such an algorithm arethe results reported in [17] that show the goodconvergence properties of this kind of algorithms,which are vital for the requirements stated in theprevious section. Recalling the problem outlined in section II andusing the same notations, we can re-define theproblem in the following manner. To recursivelyupdate the vector of the filter coefficients as depictedin Fig 2, so that the estimation error given by (5) and(6) is minimized. A standard algorithm is of the form:

W(n+1) = W(n) – a(n)Y(n)e(n) (7)

where

W(n) = [w-d(n), w-d+1(n),…, wd-1(n), wd(n)]T is thecoefficients vector,

Y(n) = [y(n), y(n-1),…,y(n-d),…,y(n-2d)]T is the inputvector and a(n) is a sequence of positive scalars asa(n)→0 for n→∞. In (7) the estimation error can be written as

e(n) = YT(n)W(n) – s(n). (8)

The equation (7) could be transformed in two steps.First, we take the averages of W and in this way cometo the following algorithm

)n(e)n(Yn

1)n(W)1n(W

γ−=+ (9)

∑==

n

1k)k(W

n

1)n(W

1/2<γ<1

The investigations presented in [17] show that suchan algorithm could be unstable in the initial period. Inorder to improve the stability we undergo the secondstep, namely to average not only trough W but alsothrough the product of Y and e. This leads us to thealgorithm:

∑==

n

1k)k(W

n

1)n(W

∑−=+=γ

n

1i)i(e)i(Y

n

1)n(W)1n(W (10)

1/2<γ<1

We term this algorithm AFA algorithm and theneeded steps for the implementation of an AFAequalizer are presented in Table 3. Going through Table 3 it could be concluded thatfirst, the averaging here does not create additional

burden since the terms )n(w j and )n(yej can be

recursively computed from their past values. Second,the algorithm does not use the covariance matrix, sothere is no need of covariance estimate. This implieslow computational complexity and escape fromstability issues related to P(n).

Table 3. AFA equalizer.

Signal estimation:

d(n-d) = ∑ −−−=

d

djj )jdn(y)n(w

Signal detection:

)]dn(d[sng)dn(d −=−Estimation error:

e(n-d) = d(n-d) – s(n-d) - training

e(n-d) = d(n-d) - )dn(d − - tracking

Coefficients update:

∑==

n

1kjj )k(w

n

1)n(w

∑ −−−==

n

1ij )di(e)jdi(y)n(ey

wj(n+1) = )n(eyn

1)n(w jj γ

−

for -d≤j≤d and 1/2<γ<1

IV. EXPERIMENTAL RESULTS In this section we evaluate the performance of theproposed AFA equalizer. Since the main parameter ofconcern is the rate of adaptation, this parameter istested and a comparison with the LMS and RLSequalizers is implemented. In the conductedsimulations the transmitted signal s(n) is a randomsequence of –1s and 1s, SNR=20 dB, the channel ismodeled with a FIR filter of second order and theequalizer is realized as a FIR adaptive filter of 6th

order. The LMS, RLS and AFA equalizers areimplemented according to the steps presented inTables 1-3 as for the LMS algorithm - µ= 0.02, for theRLS algorithm - δ= 0.98 and for the AFA algorithm -γ = 0.5. Under these conditions the trajectories of theequalizer coefficients are shown in Figures 3-5. Fig. 6provides the results for MSE obtained over 100independent trials. Obviously the AFA equalizeroutperforms the LMS equalizer and its convergencerate is comparable to that of the RLS equalizer.

Fig. 3. Trajectories of equalizer taps – LMS.

Fig. 4. Trajectories of equalizer taps – RLS.

Fig. 5. Trajectories of equalizer taps – AFA.

0 50 100 150 200 250 300 350 400 450-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Number of Iterations

Taps

0 50 100 150 200 250 300 350 400 450-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Number of Iterations

Taps

0 50 100 150 200 250 300 350 400 450-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Number of Iterations

Taps

Fig. 6. Learning curves of LMS, AFA and RSLequalizers.

V. CONCLUSIONS The basic goal of this paper is to investigate theapplication of an algorithm based on adaptive filteringwith averaging in channel equalization. Here the mainconcern is to achieve a high convergence rate in orderto meet the requirements for short training time andgood tracking properties. In this light the obtainedresults show that the AFA equalizer is very promising.Its main advantages could be summarized as follows:• high adaptation rate, comparable to that of the

RLS equalizer;• low computational complexity and possible

robustness in fixed-point implementations. A realization based on DSP to test the AFAequalizer performance in real environment is subjectof future investigations. A demonstration program is available on the WWWfrom:http://divcom.otago.ac.nz/infosci/KEL/CBIIS.html.

ACKNOWLEDGEMENT Partial support from grant UOO 808, University ofOtago, funded by the Foundation for Research,Science and Technology, New Zealand.

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