Novel Exact Low ComplexityMMSE Turbo Equalization

Vladimir D. TrajkovicSchool of Electrical Engineering and Computer Science

University of NewcastleCallaghan NSW 2308, AUSTRALIA

Email: Vladimir.Trajkovic@newcastle.edu.au

Abstract-In this paper we analyze low complexity turboequalization that combines interference canceling and Soft-InputSoft-Output (SISO) decoding. We derive a new, exact MMSEsolution for the equalization part taking into account the cross­correlation between the linear equalizer output and the softdecisions in the turbo equalization feedback loop as well asthe auto-correlation function of the decoder output. Also, thissolution implicitly takes into account the error propagation in thefeedback loop of the turbo equalizer. We also introduce a non­linear element in the feedback loop that suppresses less reliablesoft decisions. The level of suppression depends on the reliabilityof a soft decision, which is directly proportional to its absolutevalue. The simulation results for a common scenario show thatthe proposed solution outperforms all previously known turboequalizers of similar computational complexity.

I. INTRODUCTION

Turbo equalization is a technique that combines channelequalization and coding in an iterative detection scheme. It wasoriginally proposed in [1] and named turbo equalization dueto its similarity to previously proposed turbo codes [2]. As thediscrete-coefficient channel model is a finite-state machine thatcan be described by a state diagram and a corresponding trellisdiagram, the turbo equalizer in [1] combines Maximum Likeli­hood (ML) or trellis-based equalization and ML channel codedecoding in a serial concatenated scheme in order to improvethe Bit Error Rate (BER) performance at the receiver side.The scheme proposed in [1], although very powerful, suffersfrom high computational complexity since the computationalcomplexity of the ML based equalizer grows exponentiallywith the number of discrete channel coefficients and becomesvery high for long channels.

Alternative low complexity schemes employ low complexityequalizers (filters) instead of the ML equalization. The firstlow-complexity turbo equalizer was proposed in [3], whereML equalization has been replaced by a low complexityadaptive least Means Square (LMS) Interference Canceler. AMinimum Mean Square Error (MMSE) Linear Equalizer (LE)for turbo equalization has been introduce in [4], [5]. Althoughthis scheme has been shown to achieve excellent BER per­formance, the computational complexity is still dramaticallyhigh, requiring one matrix inversion per each transmitted bitper one turbo iteration. However, the problem of the highcomplexity has been solved in the same work [5] by employingan approximate solution with a certain degradation in the

978-1-4244-2644-7/08/$25.00 ©2008 IEEE

SNR-BER performance. A low-complexity turbo equalizationscheme that takes into account the feedback error propagationhas been analyzed in [6]. All low-complexity turbo equaliza­tion schemes presented in [3]-[7] are structurally equivalentwith the only difference in a way how the filter coefficientsare determined.

In this paper we analyze the low-complexity turbo equaliza­tion and investigate the ways to further improve its SNR-BERperformance.

The first novelty of this paper is that we propose a newset of MMSE coefficients for the turbo equalizer. This newset of coefficients takes into account the cross-correlationbetween the linear equalizer output and the decoder output(soft decisions) as well as the auto-correlation function ofthe soft decision sequences. These correlation functions areusually considered to be equivalent to the Dirac function,because it is assumed that any correlation between the symbolscan be destroyed by an interleaver. This is, however, generallytrue only at the decoder input. After decoding, the softdecisions are deinterleaved (put back in the original order).Hance, the correlation between the neighboring bits is beingre-established.

The second novelty of the paper is a non-linear element thatwe introduce in the feedback loop of the turbo equalizer. Thisnon-linear element serves as a gate for the soft decisions, andits purpose is to suppress unreliable soft decisions.

In order to avoid any confusion, at this point we wouldlike to point out that throughout the text we use the term'equalizer' for the equalization part of the turbo equalizeronly (i.e, interference canceler), and when we use the term'turbo equalizer', we refer to the detector as a whole. Wewould also like to point out that in this paper, we refer to thelow-complexity turbo equalizers as those detectors where theequalizer coefficients are recalculated not more than once perone turbo iteration. In the other words, once determined, thecoefficients remain fixed during the turbo iteration.

This paper is organized as following: In Section II wepresent the systems model, In Section III we derive theexact MMSE solution for the low-complexity turbo equalizer.In Section III we also introduce the non-linear feedback­suppression element. In Section IV we present some numericalresults while Section V concludes the paper.

n

Fig. 1. System Model

II. SYSTEM MODEL

(5)

(6)

(4)

f = RU1hk

b=H!5r

equalizer produces the following output

Xk = fTrk - bTXD.

where f and b are feed-forward and feedback coefficients,respectively, and XD is a vector containing previous decisionsthat are assumed to be available to the detector (e.g. inthe turbo equalization case, the vector XD contains the softdecisions from the previous iteration). This turbo equalizationstructure was originally proposed in [3] as an adaptive LMSturbo equalizer, and later investigated in [4]-[7], [10]. Themost remarkable results are obtained in [5] where it wasshown that if the extrinsic information from the decoderoutput is used to determine the MMSE coefficients of theequalizer, remarkable BER performance can be achieved. Theidea behind the work in [5] is very similar to the iterativeMMSE multiuser detection proposed in [11]. However, as alsopointed out in [5], the computational complexity of such ascheme is very high, considered that the MMSE coefficientsneed to be recalculated with each transmitted bit at each turboiteration. Although the scheme is referred to as a linear MMSEturbo equalization, it is designed as an interference cancelerbut the canceling filter is implicitly presumed by using theextrinsic information to remove the lSI. In previous work onthe turbo equalization (e.g. [10]), the MMSE criterion hasbeen considered as the optimization criterion in determiningthe equalizer coefficients. Based on the conventional perfectdecision feedback assumption, a general expression for MMSEcoefficients can be written as

where hk is the k-th column of the channel matrix H related tothe symbol being currently detected. The matrix H D relates toalready decided symbols and it is very similar to the channelmatrix H, with the only difference that the k-th column ofthe H D contains all zeros. The matrix Ru = HuH~ + a 21N ,

where H» = H - H D , IN is an N x N identity matrix and Nis the forward filter length. As it can be seen from (5) and (6),with the perfect feedback assumption, the coefficients r and bare fixed at each iteration. However, it was shown in [6] thatthe perfect feedback assumption is not always valid especiallyat low SNRs, where sometimes it is a better choice to limit oreven not to use the feedback at all due to very severe feedbackerror propagation.

A. The Exact MMSE Solution

(1)

The block diagram of the system model is shown in Fig. 1.The information bits are encoded with a convolution encoderwhich can generally be of any rate p/q. The coded bits areinterleaved using an interleaver and the interleaved bits aresymbol mapped and the symbols are transmitted through thelSI channel. In this work we limit ourselves to the BPSK mod­ulation. At the receiver, the received signal can be expressedby

where H is the Toeplitz channel impulse response matrix [8,eq. (4)], x~ is the input vector of transmitted symbols at timeinstant k, and denotes interleaving. Ok is the vector of AdditiveWhite Gaussian Noise (AWGN) samples with covariancematrix a~I and k denotes a time instant. As the interleaverdoes not affect the rest of our analysis here, the interleaver sign( , ) will not be used in the rest of the paper due to simplicityof notation. The received sequence is equalized, deinterleaved,and decoded using the SISO decoder. The decoder finds softoutputs as Log-Likelihood Ratios (LLR) defined as [9]

( - II-(n))L(n)(Xk) = log p Xk - X ( ) . (2)

p(Xk = -Iii n )

where i is the sequence at the SISO decoder input delivered bythe equalizer at the iteration n. Once the LLRs are calculated,the soft decisions at the iteration n used at the followingiteration are obtained as [3], [7]

A (n) L(n) (Xk) (3)x k = tanh 2 .

The equalizer block differs at the first from those at higherturbo iterations. At the first iteration MMSE LE is employed,while at the higher iterations the equalizer is designed toperform as an interference canceler [3], [7].

III. THE PROPOSED DETECTOR

As mentioned in the Introduction, all so-called low­complexity turbo equalization schemes are structurally equiv­alent, i.e. they consist of a forward filter and a feedback filter.The feedback filter is placed in the turbo equalization feedbackloop and processes soft decisions from the decoder, so that the

In order to derive an exact expression for the low­complexity turbo equalization, we start with the expression forthe equalizer (interference canceler) output at the n-th turboiteration, i.e,

xin) = (r<n))Trk - (b(n))Txin) =

= (r<n))T(Hxk + Ok) - (b(n))Txin) (7)

(13)

(10)

(18)

where vectors rn ) and b(n) contain feed-forward and feedbackfilter coefficients at the n-th iteration, respectively, while xin )

is the vector containing the soft decisions from the previousiteration n - 1, but only those decisions relating to pre­and post-cursor interference, and it does not contain the softdecisions about the currently detected symbol. The error at theequalizer output at iteration n is then

ein) = ((rn))Trk - (b(n))Txin)) - Xk (8)

The MSE then can be found as

€(n) = E[lein)12 ] = a; + (rn))Tm<n) + (b(n))TR~~b(n)_

-2(r<n))THR~~b(n) + 2(b(n)) Tbkn)- 2(r<n))Thk (9)

where R = HHT + a~IL, R~~ = E[xin)(xin))T], R~~ =E[Xk(xin))T], and bkn) = E[XkXin)]. I L is an L x L identitymatrix and a; is the average power of the received symbolsand it can be normalized to 1 without the loss of generality. Inour previous work on decision feedback turbo equalization in[6], the analysis of the low-complexity turbo equalization wassimplified by modeling the feedback signal as xin ) + (nb)k,

where (nb ) k is considered to be the k-th sample of thethe feedback error or feedback noise, and (nb ) k was alsoassumed to be an i.i.d. random variable uncorrelated to xin ) .

With this simplification, it is straightforward to show that thepreviously introduced quantities become R1~ = (1 + a~)I,

(n) A(n) .Rx x = H D , and hk = O. The model In [6], although anapproximation, offered significant improvement relative to theconventional interference canceler that assumes the perfectdecision feedback. However, in what follows we will showthat a more accurate solution can be obtained if no assumptionabout the feedback signal (i.e. feedback error) model has takenplace. Starting from (9), we find the MMSE solution for thecoefficients r n) and b (n) as

r<n) = [R - HR(~)(R~~))-l xxx xx

X (R~~)THT]-l[hk - HR~~)(R~~)-lbkn)]

andb(n) = (R~~)-l[-bk + (R~~)THTrn)]. (11)

The MMSE becomes

€(n) = 1 + bk(R~~)-lbk - [h, - HR~~ (R~~)-lbkn)]T

[R - HR~~(R~~)-l(R~~)THT]-l[hk - HR~~(R~~)-lbkn)](12)

and the mean of the symbol estimate becomes

j.t(n) = E[XkXin)] =

[h - HR(~)(R~~))-lb(n)]T[R - HR(~)(R~~))-l(R(~))THT]-lk xx xx k xx xx xx

[h - HR(~)(R~~))-lb(n)] - b (R~~))-lbk xx xx k k xx k·

In order to determine Signal to Interference plus Noise Ration(SINR) at the output of the equalizer at iteration n, we use

an assumption as in [12] which says that the interference plusnoise at the output of the canceler is Gaussian. In that casewe write the estimates (7) as

xin) = j.t(n)Xk + ein) (14)

where ein ) is a Gaussian random variable with variance a 2( n )

~k

and ( )SINR(n) = (M 2

n)2 (15)

a (n)~k

After deinterleaving, SISO decoder delivers Log LikelihoodRatios (LLRs), that are interleaved again. Interleaved LLRregarding bit Xk then can be expressed as

L (n) 2 -(n) (L )(n)k = (a(n»)2xk + e k " (16)

where (a(n))2 = l/SINR(n) and (Le)in) is the extrinsicinformation obtained as in [2]. Soft decisions used in the turboequalization feedback loop are [7]

A(n+l) =t h(Lin))

=t h(_l__ (n) + (Le)in))

xk an 2 an (a(n))2 Xk 2·(17)

In order to calculate feed-forward and feed-back coefficientsof the equalizer at iteration n, we need to estimate the matrices

(n) (n) A (n) . .Rx x ' Rx x ' and vector hk , I.e we need to estimate thefollowing

(L )(n-l)E[ A(n) A(n) ] = E[t h ( 1 -(n-l) + e k )

xk xk-m an (a(n-l))2 xk 2 x

(L )(n-l)( 1 -(n-l) e k-m )]

x tanh (a(n~1»)2xk-m + 2

and

(L )(n-l)[ A(n)] E[ h ( 1 -(n-l) e k-m )]

EXkXk_m = xk"tan (a(n-l»)2xk-m + 2 "

(19)By substituting (7) in (18) and (19), we get

E[ A(n) A(n) l >x k x k - m -

E[tanh( 1 ((r<n-l))T(Hx +0 ) _ (b(n-l))Tx(n-l))+(a(n-l))2 k k k

(Le)in-1

) ) h ( 1 ((..t'n-l))T(Hx )+ 2 x tan (a(n-l))2 I' k-m + 0k-m -

(L )(n-l)_(b(n-l»)Ti~~~»)+ e ~-m )] (20)

and

E[XkXi~m] =

E[Xk . tanh ((a(n~1»)2((t'n-l»)T (Hxk-m +Dk-m)-

(L )(n-l)_(b(n-l»)Ti~'-"-~»)+ e ~-m )]. (21)

0.3 .

0.2 . . .

0.3 . - .

-0.1 .. .. , - ..

-0.2-15 -10 -5 0 5 10 15

-0.1 .. .. .. . .. ...

-0.2-15 -10 -5 0 5 10 15

Fig. 3. The auto-correlation function the soft decisions signal x' after thefirst turbo iteration for Channel 1

0.1 ".' .

0.4 _.

0.5 .

0.6....------.-----....------,-----r------,-----,

Fig. 2. The cross-correlation function between the transmitted sequence x'and the soft decisions x' after the first turbo iteration for Channell

0.6....------r-----.,-------,-----.,------,-------,

0.5 .

0.1

0.4 .

0.2 .

The value in the exponent in (23) is empirically selected fromthe range of values varying between 0 and 4 as the value thatminimizes the SNR turbo threshold. The work on finding moreanalytic explanation of the choice for the exponent in (23) ispart of an ongoing effort.

IV. SIMULATION RESULTS

In this section we show the simulation results that comparethe proposed method with some of already existing algorithms.Channel 1 already mentioned in the previous section is used asthe lSI channel. This channel is commonly used in the turboequalization literature and is considered to be a severe lSIchannel [4], [5]. For the consistency, the other parameters arealso chosen as in [5], i.e. the channel code is an 1/2 code rateRSC code with the generator polynomials [5 7] in octal form.Modulation is BPSK, the frame size is 216 bits while the linearfilter length is N == 15. The maximum preset number of turbo

The expectations in (20) and (21) can be either numer­ically calculated (provided that the channel impulse re­sponse is known) or estimated during the turbo detectionprocess. Fig. 2 shows the measured cross-correlation func­tion between the transmitted sequence and the sequenceof soft decisions after the first turbo iteration over adiscrete-coefficient channel whose impulse response is hI ==[0.227 0.46 0.688 0.46 0.227]T from [13]. We will furtherrefer to this channel as Channell. Fig. 3 shows a measuredsoft-decision auto-correlation function after the first turboiteration for the same Channell. What can be observed fromboth figures is that the current symbol Xk is highly correlatedwith the soft decisions relating to the neighboring symbols,and that this correlation has to be taken into account duringthe design of the MMSE filters (interference canceler), i.e.

when calculating R~~), R~~, and bin). The result in Fig. 3 issomehow surprising because it is expected that any correlationthat exists between the estimates at the filter output wouldbe destroyed due to high non-linearity of the SISO decoder.However, this results may be explained by analyzing (21)and (22) where it can be observed that the soft decisionsabout the symbol Xk depend on the term (Hx k - m + Dk-m),

which at early iterations (especially after the first iteration)and for a small m (e.g. 1 or 2 for the considered case, butthis generally depends on the channel impulse response) stillcontains significant interference not removed by the feedbackfilter. It is also worthwhile pointing that this correlation cannotbe destroyed by interleaving because the soft decisions are fedback into the canceler after the deinterleaver in the feedbackloop, so that any effect of interleaving between the filter andSISO decoder is reversed. Clearly, the purpose of interleaveris to destroy possible correlation prior to the SISO decoderonly.

B. Feedback with non-linear suppression

Here we present an idea of introducing a non-linear feed­back suppression element. The element is placed in the feed­back loop immediately after the soft decider. The level ofsuppression of soft-decisions depends on their reliability andas a measure of this reliability we use the probability of errorof the hard decisions for corresponding soft-decisions. e.g.

Lk . LkPe == 1 - p(tanh(2)) == 1 - P(slgn(tanh(2)) == Xk).

(22)

An example of how the probability Pe in (22) looks like afteriterations between 1 to 8 is shown in Fig. 4 for previouslyintroduced Channell. As it can be observed, the probabilitydiffers at different iterations and the soft decisions havingsmaller values become less reliable with the increasing numberof iterations. Therefore, we introduce a reliability parameterbased on the probability in (22) and the corresponding softdecision is then multiplied by the parameter, i.e.

i; 2 t.;Xk == p(tanh(2)) . tanh( 2)· (23)

10-4

0 ••• 0.0 •

10-3

00000000

: ::: -- lSI-free (AWGN):: :: -e- exact MMSE (Tuch)

~ ~ ~ ~ ------ proposed: : ::~ approx. MMSE (Tuch):::: ~exactLCMMSE

10-1

00 ••••• 0 ........

10-2

..............

10° fT.T.."T.IT. .ToomTTTT'T'T'1~""TT"T~T'TT~~n;III:II:r:r::D:::o::II:::IJ::I:::OD:IIII::rn

........

10-5

.000000.

0::wm

::::::::::::::-::::::::::::::-:::::::::::::::::::::::::::::-::::::::::::::

10° .....----~---___,;__---___y_---___r---__,

............. -: :- : < .

· . . ............................................. : .••••••••••••••• •••••••••••••• ••••••••••••••• 1 •••••••••••••••••· .

••••••••••••• ••••••••••••• :- ••••••••••••• : ••••••••••••••••••••••••••• 0

• 0............................................................................................ : .· . . .

10985 6 7SNR [dB]

4

::::::::::::::::::::::::::::::::::.:::::::::::::::::::::::::::::::::::1::::::::1::::::::(::::::::1::::::::1::::::::1::::::::: : : : : : : : ~ : : : : : : : : ! : : : : : : : : : : : : : : : : : ! : : : : : : : : : : : : : : : : : ! : : : : : : : :

........ : : : ; : ; .

. ; ; ; ; ; ; .. .

3

10-6

........

10-7 L....-_--L.__...........__....L--__"--_---'__---L..__--J.-_----'

20.80.4 0.6tanh(U2)

0.210-4 L....- ......L.- ---' --L- --L. ---J

o

Fig. 4. Probability of error of the hard decisions versus corresponding softdecision magnitude after iterations 1-8

Fig. 5. BER results comparing the performances of various turbo equalizationalgorithms

iterations is n m a x == 15 (in [5] the iterations are denoted fromo to 14, while here we denote them from 1 to 15). The resultsare shown in the Fig. 5 and compare the proposed methodwith the approximate MMSE LE [5], the turbo DFE that takesinto account feedback error propagation [6] and the exact LEMMSE. As expected, the best results are achieved by the exactLE MMSE. However, the exact MMSE LE also requires thecomputational complexity which is much higher than any otherscheme considered here because it performs an N x N matrixinversion per each transmitted bit per iteration, where N is thelinear equalizer length. On the other hand, when comparedwith the approximate hybrid LE MMSE and the imperfectdecision feedback turbo equalizer from [6], the proposedmethod exhibits about 0.5 dB SNR gain at the BER of 10-4 .

The proposed method also reaches the lSI-free performance ofthe considered convolutional code on AWGN channel muchearlier than the other two low-complexity methods. Anotheradvantage of this scheme compared to the approximate schemein [5] is that it does not require switching from the extrinsicinformation to the hard decision mode. This switching involvesthe use of an EXIT chart in order to determine after how manyiteration one mode performed better than the other one.

V. CONCLUSION

In this paper we have shown that the SNR-BER performanceof the low-complexity turbo equalization can be improved bya careful choice of the filter coefficients without increasingits computational complexity. This improvement is achievedby taking into account the correlation between the equalizeroutput and the soft decisions, and without assuming that thesoft decision auto-correlation function (or the feedback signal)is a Dirac function. Additional improvement is achieved bya non-linear feedback suppression element, which attenuatesunreliable soft decisions. Our simulations show that for theconsidered common turbo equalization scenario, the proposedmethod closes the gap to the exact LE MMSE turbo equalizer

and achieves a gain of 0.5 dB comparing to the approximateLE MMSE and the imperfect decision feedback turbo equal­izer.

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