Adaptive MMSE Equalization

7/28/2019 Adaptive MMSE Equalization

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Adaptive MMSE Equalizer with Optimum

Tap-length and Decision Delay

Y G X A-S

School of Systems EngineeringThe University of Reading, Reading

RG6 6AY, UK

Email: {goghogkus}@rdgcukTel: +44 (0) 118 378 8581, Fax: +44 (0) 118 378 8583

Absact-In li dpive MMSE equlizer, he ruure since the optimum is different for different N. n general,(or he numer of he free oeien) nd he deiion dely the best SE is achieved when N re oen xed eiher o ome ompromie vlue or imply from experiene, mking i hrd o hieve he e poenilperformne. In hi pper, we propoe novel mehod ojoinly dp he plengh nd deiion dely o h hee performne n e hieved wih minimum plenghnd deiion dely. The propoed pproh need lile exromplexiy. I derie n innovive wy o implemen he

dpive equlizer where no only he ruure i opimized ulo he deiion dely i elf uned.

. NTODUTION

The adaptive minimum mean square error (SE) equal

izer as a classic approach has been widely used in commu

nications. n a traditional design, the structure (or the tap

length) and decision delay of the equalizer are normally xed,

making it suboptimum in most cases. While the taplength

refers to the number of free tap coecients, the decision delay

determines which symbol is detected at the current time.

Both the taplength and decision delay can signicantly

affect the performance of the equalizer. On the one hand, if the decision delay is xed, the SE of the equalizer

is a monotonic decreasing function of the taplength, but the

SE improvement due to length increase always becomes

insignicant when the taplength is large enough. There exists

an optimum taplength that best balances the performance and

complexity. Length adaptation algorithms can be used to nd

the optimum taplength of the equalizer with the decision delay

being xed. Among varies length adaptation algorithms (e. g.

[1]-[4]), of particular interest is the fractional taplength (FT)

algorithm due to its robustness and simplicity [4], in which the

length adaptation is based on a pseudo fractional taplength

whose integer part gives the true taplength. The performance

of the FT algorithm can be further improved with convex

combining of two FT algorithms [5].

On the other hand, for a given taplength, there exists an

optimum decision delay that minimizes the SE. The

optimum depends on specic channels. As will be shown

later, should be set as 0 and N -1 for minimum and max

imum phase channels respectively, where N is the taplength.

n general, even with accurate channel state information, it

is still not straightforward to obtain the optimum which

normally lies between 0 zero and N -1. This can be even

more complicated if the taplength N is also to be determined

n classic applications, the decision delay is oen chosen

based on either the preassumption of the channel or simply

from experience (e. g. [6, Section 9.7]). This normally requires

the taplength to be long enough such that there exist a large

number of "appropriate which have the similar SE

to that for the best

.Then the chance that the selected

happens to be one of those "appropriate can be high.

Obviously such approaches have no guarantee to have the

optimum . oreover, too large the taplength contradicts

the requirement that the taplength should remain as small as

possible without sacricing the performance.

Delay selection is an important issue in varies equalization

approaches including blind equalization [7], decision feedback

equalization (DFE) [8] and multichannel equalization [9]-[ 11].

The rst decision delay adaptation algorithm for a xed length

linear equalizer was proposed in [12]. This algorithm may lose

convergence in some cases because it is based on tracking the

mean squared error (SE) for dierent decision delays which

are not always reliable. A joint length and delay optimization algorithm was proposed in [13] for the DFE equalizer but

cannot be applied in the linear adaptive equalizer. Up to now,

the only joint taplength N and decision delay optimization

for the linear adaptive equalizer was derived in [14], in which

a linear prewhitener is applied prior to the equalizer. The pre

whitener makes the equivalent channel be maximumphase so

that the two dimensional search for the optimum N and reduces to one dimensional search by letting = N -1.This approach is far from ideal: First, since the decision delay

is xed as large as N -1, it brings too much delay in the

equalization in many cases. For instance, for the minimum

phase channel, the optimum should be Secondly, the equalizer has to be longer as now both the channel and pre

whitener need to be "equalized. Finally, the length of the

prewhitener needs to be adapted as well, which may diverge

the overall system if it is not well handled.

Therefore, it is of great interest to have a robust and simple

way to joint adapt the taplength and decision delay so that

the best SE performance of the equalizer can be achieved

with minimum taplength and decision delay. As generally a

two dimensional search problem, such joint approach is hard

to realize adaptively.

n this paper, we rst investigate how the taplength and



decision delay aect the MMSE performance based on anequivalent model of the equalizer. We then propose to useforward and backward length approaches to jointly adapt thetaplength and decision delay. The proposed algorithm is therst robust yet simple approach in literature for joint taplengthand decision delay optimization. It describes an innovative way to implement the adaptive equalizer.

II. YSTEM ODEL

The model of the adaptive equalization is illustrated in Fig.1, where x( n) is the transmitted information signal, H(z) isthe channel transfer function, (n) is the channel noise, y(n)is the received signal, z- is the decision delay, z (n) is theequalizer output, e(n) is the error signal and W(z) is thetransfer function of the equalizer.

:c(n)- Ie(n

Fig. 1. daptive channel equalization

The MMSE equalizer [6] is obtained by minimizing theMMSE cost function

(1)

with respect to the tapvector w(n) where (n)[y(n),··· , y(n N + W which is the tapinput vector, Nis the taplength and d(n) = x(n . The decision delay

determines which symbol is being detected at the currenttime n or the current equalization output z(n) is an estimateof x(n .

The maximum potential performance of an MMSE equalizeris achieved by the ideal equalizer with N extendingfrom to as (see [15]):

If the decision delay is the ideal equalizer is also delayedby and is denoted as w(n). Then (1) can be rewrittenas:

� = Ele(n) ((n) w T(n)(n))12, (3)

where e(n) d(n) w (n)(n) and (n)w (n)(n). Sn Ele(n)'12 i a constant andEle(n)(n)1 = due to orthogonality principle [6],

minimizing (3) is equivalent to minimizing

(4)

Therefore, the equalization can be regarded as the systemmodelling shown in Fig. 2, where W (z) is the transferfunction ofw(n) and (n) = (n)w T(n)(n). The modelin Fig. 2 is very useful in analyzing the taplength and decisiondelay.

y

e

Fig. 2. The equivalent model for the MMSE equalizer.

III. PTIMUM PLENGTH ND EISION ELY

Fr an equalizer with xed ecision delay if the talength N is too large, some last tap coecients will be closeto zero. Then N can be decreased by remving these nearzerotaps with little performance lost. If N is not large enough,all tap coecients have signicant values. Then it is highly possible that the MMSE performance can be improved withone or more taps so that N should be increased. Thus anoptu taplength that best balances the MMSE performance

and complexity can be dened as the minimum N satisfying(5)

where � N is the mean square error (MSE) with taplengthNand decision delay �} K is the forward trunked MSEwith the last K tap coefcents being removed from theequalizer and E is a small positive constant.

On another hand, for a given taplength N, there exists aminimum that minimizes the MMSE. As is shown in Fig.2, the equalization is equivalent to using an FIR lter to modelthe unconstraint optimum equalizer W(z) which is shiedby the decision delay.

as The transfer function of the ideal equalizer can be expressed =

W(z) = L (n)z- + L (n)z- , (6)

=-

where the rst and second summation are the noncasual andcasual terms respectively. In fact, when the noise is small, wehave W(z) R 1/H(z) so that the noncasual and casualterms of W(z) are obtained by inverting the maximum andminimum parts of the channel, which contain the zeros outsideand inside the unit circle respectively.

Since the noncausal part cannot be modelled by the causal

FIR equalizer no matter how long the taplength is, thedecision delay is introduced to shi noncausal terms intothe causal range. In practice, while W(z) has innite numberof noncausal and causal terms, we always have (i) 0when i ± Therefore, for an equalizer with taplength Nand decision delay the MMSE (or the equalization error)is contributed by, beside the noise, the noncausal terms of z - W (z) and the causal terms with time index larger thanz- N.

The optimum that minimizes the MMSE depends onspecic channels. For the minimum phase channel that allzeros of H(z) are within the unit circle, W (z) only contains



causal terms so that we should have � = O. For the maximumphase channel that all zeros of H z are outside the unit circle,W z only contains noncausal terms so that we shouldhave � = N - 1. In general, unfortunately, there is no xedrelationship between N and �. If the decision delay � is toolarge, too many noncausal terms of W z are moved intothe causal range so that some initial taps of w(n) are close to

zero. Then the decision delay can be decreased to move thesenearzero taps back to the noncausal range with little affecton the MMSE performance. On the other hand, if � is notlarge enough, some noncausal terms with signicant valuesare still in the noncausal range so that the MMSE is too large.Then � should be increased to move these "signicant noncausal terms into the causal range. Based on these observation,the optimum decision delay is dened as the minimum � thatsatises b K , - , � E (7)

where jK is the bckwrd trunked MSE with the rst Ktapcoecie�ts being removed from the tapvector.

Overall, the joint optimum taplength and decision delay isdened as the the minimum N and � that satisfy (5) and (7)

simultaneously.

IV. OINT PLENGTH ND EISION ELY

ADPTTION

While the taplength and decision delay are jointly optimized by satisfying both (5) and (7), accordingly, they can beadapted by the forward and backward length adaptation as isshown below.

A Forwrd legth dptto

At time n we assume the decision delay is xed at �which is obtained from the previous backward adaptation attime (n - 1. The forward length adaptation is then appliedwith the fractional length algorithm to adapt the taplength [4].To be specic, letting nf be the pseudo fractional taplengthwhich can take real positive values, the length adaptation ruleis constructed as:

nf(n + 1 = (nf(n) - a) - '. [leN(n)12 -H�_K(n ) 12 l ,

t8where a and are small positive numbers, e (n) is the er

ror signal of the equalizer with taplength N(n) e] _K(n)is the forward trunked error signal with the last K tap coefcients being removed from the equalizer. a is the leaky factor

and satises a« T Initially we can have nf(O) = N(O).The "tue taplngth N(n) adusted only the change

of nf(n + 1 is big enough as:

N(m + 1) = { lnf(n + l)J, IN(�) - nf(n + 1) � 6

N(m) otherwIse(9)

where 6 is a positive threshold value and l.J rounds theembraced value to the nearest integer. ote that we use adifferent time index m for N because the length adjustmentmay happen in both forward and backward approaches at timen.

In the forward approach, the taplength adjustment is applied at the "end of the tap vector. Specically, if N (n + 1is increased by K K zero taps are appended aer the last tap,and otherwise the last K tapcoecients are removed from thetapvector. Or we have

(10)

where 6(m) = N(m + 1 - N(m).

B Bckwrd legth dptto

While the forward length adaptation adjust the taplength atthe "end of the tapvector, based on the symmetry betweenthe cost functions of (5) and (7), a backward approach can beapplied to search for the optimum decision delay dened (7).

To be specic, if the initial tapcoecients are large, thedecision delay should be increased, for instance by 6 to

shi more noncausal terms into to the casual range. Atthe same time, the taplength should also be increased by padding 6 zero taps before the rst tap to prevent the last6 tapcoefcients from moving out of the equalization range.Similarly, if the decision delay is decreased by 6 the taplength should also be decreased by removing the rst 6 fromthe equalizer. Therefore, the decision delay adjustment shouldbe synchronized with the taplength adjustment applying atthe "beginning of the tapvector.

Similar to the forward approach, we dene nb as thefractional backward taplength with the adaptation rule as:

nb(n + 1 =

(nb(n)- a

)- '.

[leN(n)12 -le _K(n

)1 2 ,

dl)where e _K(n) is the backward trunked error signal of

the equalizer with the rst K tapcoefcients being removed.Initially we set nb(O) = nf(O).

If the change of nf(n + 1 is big enough, the taplength Nis adjusted as

N(m + 1) = { lnb(n + l)J, IN(�) - nb(n + 1) � 6

N(m) otherwIse(12)

Unlike the forward approach, the taplength adjustment hereis applied at the "beginning of the equalizer as

(13)

where 6(m) = N(m + 1 -N(m).The decision delay adjustment is synchronized with (12) as

�(n + 1 = �(n) + 6(m) (14)

We must ensure �(n) 0 at all time to maintain causality asotherwise the equalizer is detecting future data.



C The lgorth

The forward and backward lengh adapaion are appliedalernaively unil hey converge o he opimum aplengh anddecision delay. A any ime he forward lengh adapaionis rs applied o adap he aplengh a he "end of heapvecor wih he decision delay being xed a wha hebackward approach obained a ime 1. Aer ha, he

backward lengh algorihm is applied o adap he decisiondelay and adjus he aplengh a he "beginning of he apvecor. Since he aplengh adjusmen can happen a bohforward and backward approaches a ime we mus apply f+1 = f+5 and +1 = +5 before(8) and (11) respecively.

For sabiliy, he aplengh and decision delay should belimied as Nmin Nmax and 0 respecively,where Nmin and Nmax are se based on sysem requiremens.I is clear ha, during he adapaion, is always smallerhan so ha is auomaically upper limied.Accordingly, f and adapaion mus be limied o

saisfy he he consrains on he aplengh and decision delay.To be specic, since f only affecs he aplengh, isconsrain is se as same as ha for N :

(15)

I is known ha adapaion may change boh he aplengh and decision delay. On he one hand, we mus have Nmax N (n) o ensure < Nmax. On he oherhand, should also be larger han Nmin and o ensure : Nmin and 0 respecively.Therefor, we hae

maxNi, Nn - �n ' nbn ' N - Nn (16)

Because he forward and backward approaches are highly ineracive, of paricular imporance for convergence is o makesure ha (15) and (16) are checked aer he lengh adapion(8) and (11) respecively a every ime .

The convergence analysis of he proposed algorihm canfollow ha for he original FT algorihm [4 bu is morecomplicaed as now here are wo FT running in parallel. Thedeails are no shown in his paper due o he space consrain.

V. UMEIL IMULTIONS

Compuer simulaions are given o verify he proposedalgorihm. Three channels are considered:

I = [1 0.4 0.3 O.l]T2 = [0.1 0.3 0.4l]T

minimum phase

maximum phase

mixed phase 3 = [0.1 0.3 0.41 0.40.3 O.l]T(17)

For comparison, we deliberaely le I and 2 be nearly symmeric, and 3 be he combinaion of I and 2.

Fig. 3 plos he MMSE wih respec o he decision delay for he mixed phase channel 3 , where he aplengh N is xed a 10 1 30 and respecively. I is clearly shown ha for a xed he MMSE is a nonincreasingfuncion of N. For a xed N, on he oher hand, here exiss

an opimum ha minimizes he MMSE. Overall, he joinopimum aplengh and decision delay are he minimum N and corresponding o he similar MMSE as he bes MMSE,where he bes MMSE is achieved when N I is shownin Fig. 3 ha he opimum N and for 3 are around 1 and10 respecively.

For he minimum and maximum channels I and 2, we

can also plo he MMSE vs. N and curves which are noshown here due o he space consrain.

0 ,�-�----.• * 0 "l •

N=

(�

N= N=

:$ 0 O

o

·�H ..$$$$$$ ·��++:+ :�-:'o 5 5 3 4

Decision Delay t

Fig. 3. MMSE YS Decision Delay for the channel h.

Fi. 4 shows he learn urs of he eng adapan. Iis clear ha he aplenghs converges o around he opimumvalues. I is ineresing o observe ha, he lengh learning

curves for heI

and2

converge o similar value becauseI and 2 are nearly symmeric. On he oher hand, he mixedchannel requires longer aplengh as i is he combinaion of he oher wo.

C<:! 9IC

2

6

· r -) I : r . . .

. . . . : r " : : '

" - r ' � l ' -:.

! I

- . - Mnimum phaseMaximum ase

Mxed phase

40L-�00--1000--100--20--250--3000

No of sybols

Fig. 4. Tap-length leaing curves.

Fig. 5 plos he learning curves of he decision delay



adaptation, where it is shown that the decision delay for the

minimum phase channel I is 3 and those for the maximum

and mixed phase channels 2 and 3 are both around 10.This well matches our previous analysis: First, the decision

delays for minimum and maximum phase channels should be

around 0 and N - 1 respectively; Secondly, the decision delay

is determined by the maximum phase part of the channel, if

we note that the maximum phase part of 3 is just as same as 2.

9

8

> 7u

5 6'uTlQo

3

00

�I1

000

1

Miniu ase Maxiu ase

Mixed hase

- i 1r , I J

00 2000 200No o syols

Fig. 5. Decision delay leaing curves.

3000

Finy, Fig. 6 shows h MSE aning cuvs fo the thr

channels to indicate the converence of the equalizer.

100��===Minimum phase

+ .. Maxiu ase : Mixed hase

,

; " ....

10-2L_____________- 00 000 00 2000 200 3000N f symbs

Fig. 6. MSE learning curves.

V. CONLUSION

This paper proposes a novel approach to jointly adapt the

tap-length and decision delay of the adaptive equalizer so that

the best MMSE performance can be achieved with minimum

tap-lenth and decision delay. This is achieved by running

the forward and backward length adaption toether at every

iteration. Numerical results have been iven to verify the

alorithm. This is rst joint tap-length and decision delay

adaptive lorithm in literature with little extra complexity.

t describes an innovative way to implement the adaptive

equalizer with wide applications.

AKNOWLEDGEMENT

This research was sponsored by the UK Engineering and Phys-ical Sciences Research Council and DSTL under grant number EP/H0125 6/1.

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Adaptive MMSE Equalization