ISIT2007, Nice, France, June 24 - June 29, 2007

Evolution Analysis of Iterative LMMSE-APP Detectionfor Coded Linear System with Cyclic Prefixes

Xiaojun Yuan, Qinghua Guo, and Li Ping, Member IEEEDepartment of Electronic Engineering, City University of Hong Kong, Hong Kong

Email: xjyuan@cityu.edu.hk, qh.guo@ student.cityu.edu.hk, and eeliping@cityu.edu.hk

Abstract-This paper is concerned with the iterative detec-tion principles for coded linear systems with cyclic prefixes.We derive a matrix-form low-cost fast Fourier transform(FFT) based iterative LMMSE-APP detector and propose anevolution technique for the performance evaluation of theproposed detector. Numerical results show a good matchbetween simulation and evolution prediction.Keywords-LMMSE-APP detection, Turbo equalization,

iterative detection, SNR-variance evolution.

I. INTRODUCTION

Iterative minimum-mean-square-error (MMSE) detec-tion has been studied in the context of a variety of applica-tions involving inter-symbol interference (ISI), multiple-input-multiple-output (MIMO) transmission, and multiple-access (MAC) [1-4]. It can provide performance compara-ble to maximum a posteriori (MAP) detection [5, 6] atconsiderably reduced cost.

In this paper, we study the linear MMSE (LMMSE)approach to systems with circulant channel matrices. Suchmatrices arise when cyclic prefixes are added to thetramnsmitted signals, which is a technique considered, e.g.,in 3GPP long term evolution systems [7]. We first derive amatrix form FFT-based iterative LMMSE-APP (APP for"a posteriori probability") detector, and then propose anSNR-variance evolution technique for its performanceevaluation. The extrinsic information transfer (EXIT) charttechnique [2, 8] is a useful tool for the analysis of iterativedetectors in fixed channels. However, its applicationbecomes difficult when channels are not fixed, such as inthe case of MIMO ISI quasi-static fading channels. In thiscase, the average performance of a system can becomputed by collecting statistics for a sufficiently largenumber of channel realizations. For each realization, adifferent pre-simulated transfer function of the detector isrequired for the EXIT chart analysis. It is generallyimpractical to store the pre-simulated EXIT transferfunctions for all possible channel realizations. In this paper,we therefore develop an alternative solution in which thetransfer function is generated analytically (rather than pre-simulated) at very low cost for each channel realizationduring the evolution process. The overall systemperformance can be evaluated by averaging over differentchannel realizations.

List of Notations: Vectors are expressed in bold lettersand are column vectors by default. Matrices are specifiedby bold capital letters. The linear equation in (1) (see be-low) can be expressed in a block form, where each blockmay represent a scalar, sub-vector or sub-matrix. Thelength of x, in blocks, is called the "frame length", and isdenoted by J. The transpose and conjugate transpose aredenoted by "T' and " respectively. An identity matrix

This work was fully supported by a grant from the Research Grant Coun-cil of the Hong Kong SAR, China [Project No. CityU 1314/04E].

with dimension n is denoted by In and sometimes the sub-script is omitted for simplicity. [.]Ij denotes the (1, j)-thentry of the matrix in the bracket. The operator 0 denotesthe Kronecker product. The operation diag{ } returns adiagonal matrix with the elements in the brace ordered onthe main diagonal, whilst ()diag returns a diagonal matrixthat only contains the diagonal part of the matrix in theparentheses. The operator E(t) is the expectation with re-spect to the joint pdf ofx and 77.

II. CHANNEL MODELS

A. System ModelConsider a coded system, as shown in Fig.1, character-

ized by the following linear matrix equationr=Hx+q7, (1)

where r is an observation vector, x a transmitted signalvector, H a channel matrix and 77 an additive white Gaus-sian noise (AWGN) vector with zero mean and covariancec2I . We assume that x is generated by an encoding device(denoted by ENC in Fig.1) using forward-error-control(FEC) codes and permuted by an interleaver (marked by Hin Fig. 1). Typical examples of (1) include ISI, MIMO andMAC channels.

Transmitter

DataOpENC

IlL----------

DataEstimation:

*

Iterative Receiver

Fig. 1. The general transmitter and (iterative) receiver structures for acoded linear system with channel input x, nl and El denoting the inter-leaver and the corresponding de-interleaver, respectively.

B. Circulant Systems

The channel matrix H in (1) is a JxJ circulant matrix if[H]ij = [H]i,1 for anyj - I = (j'- 1') modulo J. A circulantH can be represented as

ho h2

H h h h2 (2)

hj-l .. hoLet rj and xj (the jth entry in r and x, respectively) be thereceived and transmitted signal at time instant j, respec-tively. The matrix H in (2) is realizable in an ISI channel(with channel coefficients [ho, ..., hL_j] where L denotes

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ISIT2007, Nice, France, June 24 - June 29, 2007

the channel memory length) by padding x with a cyclicprefix (CP) [9] that consists of the last L-1 entries of x.

Let F be the normalized discrete Fourier transform (DFT)matrix with the (j, I)-th entry given byJ- 12exp(-i2zTjl/J),where i fT . Let {gj} be the DFT of {hj} (see (2)), i.e.,

g J-1/2J-1hk exp(-i2fjlI/ J) j =O .

The following properties are well known for a circulant H.

G FHFH is diagonal;G =J12diag{go, g1, , gj-i};H=FHGF.

C. Block Circulant SystemsA block circulant system, as a natural extension of the

scalar circulant one in II.B, can be represented byro] H° H2 H1 x0 1 'l 1

rI Hi H 2 xi Ili,(3a)

rJi-i. J-1 H H° x-1 J-iwhere H is block circulant (i.e., each block-column is acyclic shift of its previous block-column by one block),each H, is an MxN sub-matrix, rj (or 77j) an Mxl sub-vector,and xj an Nxl sub-vector. A typical example of such a sys-tem is a MIMO ISI channel with CP padding to eachtransmit antenna. Let L be the channel memory length, andN (resp., M) be the number of transmit (resp., receive) an-tennas. Thus, when = L, ... J-1, H, = 0; and when 1 =0,..., L-1,

[HI ]mn = h1 mrvfor m = 1, ..., M, n = 1, ..., N, (3b)where hl,m,n denotes the lth tap coefficient between the nthtransmit antenna and the mth receive antenna. The sub-vectors rj, xj and 7qj are the corresponding sub-blocks in r,x, and q at time instantj, respectively.

Let {GO, G1, ..., GJ-l} be blocks of the same size as HI,Vl, as given in (3). Define the block-wise DFT of {HI} as

Gj= -1/21 J-1 H, exp(-i2ffjlIJ) for j = 0, ..., J-1.

Denote FM =_F0PIM, and FN - F0IN. Similarly to the casefor a scalar circulant matrix in II.B, the followingproperties hold for a block circulant H:

G _ FMHFN is block diagonal;G = J"2diag{Go, G1, ..., G-1 };H = FMHGFN.

D. Real Representation of a Complex SystemThe discussion in II.B and C applies to both real and

complex channel models, since a complex linear systemcan be equivalently transformed into a real system (withdoubled dimensions) by equating the real and imaginaryparts separately. Hence, we only discuss real systems inthis paper.For simplicity, we assume that each element in x is

modulated by binary phase shift keying (BPSK) over {+1,-1}. Note that BPSK modulation for the converted realsystem model is equivalent to quadrature phase shift key-ing (QPSK) with Gray mapping for the original system.

III. THE ITERATIVE LMMSE-APP DETECTION PRINCIPLES

The lower part of Fig.1 shows the iterative LMMSE-APP receiver. The ESE (for "elementary signal estimator")

module is based on the LMMSE principle and the DECbased on the APP decoding principle, hence the name"LMMSE-APP". These two modules are connected by theinterleaver fl and the corresponding de-interleaver fl-H,and work iteratively. Note that H is assumed to be knownat the receiver.

A. The LMMSE Approach to the ESEThe ESE computes the extrinsic log-likelihood ratio

(LLR) for each xj as1 p(rIxj =+1) j= 1, 2, ... (4)

p(r xi = -1)with the FEC coding constraint ignored, i.e., the ESE oper-ates as if x contains un-coded bits. The exact evaluation of(4) can be realized by the MAP algorithm, but is usuallyprohibitively complicated. Let the diagonal matrix V be thecovariance of x. It is shown in [4] that the LLR in (4) canbe approximated by the following LMMSE estimator

j= 2hj'Rj-- (r - HE(x)+hiE(xj )), j = 1, 2,... (5)IvhhT -VT+02where R1 R vhh>, R=HVHT+&I, hj is the jth

column of H, and vj is the variance of xj. We adopt initialconditions E(x) = 0 and V = I (meaning no a priori infor-mation). During the iterative process, E(x) and V are up-dated by using the feedback information from the DECafterwards, as shown below.

B. The DEC OperationsThe DEC performs APP decoding using i- [AO, 2A,

lj, ...ITas its inputs. The DEC outputs are the extrinsicLLRs given by

nP( Xi = +1)p(2 xi = -1)' j = 1, 2, ....

C. The Overall Iterative ProcessAfter the DEC operations, the ESE operations can be

executed again to refine the estimates in (4) using thefeedback y- [Yo, Y1, , Y, ] from the DEC. Since xj istaken over {+1, -1}, we have

exP(71) -12E(xj) = = tanh(71 / 2) and vj = 1- (E(x1 ))2.exp(7) +1I

This process continues iteratively until the algorithm con-verges. Refer to [1-3] for details.

IV. A FAST IMPLEMENTATION TECHNIQUE FOR THE ESE

Since the APP decoding in the DEC has been well stud-ied, we focus on the implementation of the ESE here.

A. ESEfor CirculantH

Applying the matrix inversion lemma to Rj-1, we can re-write (5) as

h'TR-1 (r - HE(x)) + hjTR-lhE(xj)2 1- v1hi R-lh

or in a vector form as2(I V(H'HTh), 1-[HTR'(r-HE(x))-(HTR'H) E(x)]. (6)

Let v be the average of {v,}, i.e., v =J-1 vj [2].

We update V as follows.V ;vI

Recall that V is the variance of x. Eqn. (7) implies:(7)

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ISIT2007, Nice, France, June 24 - June 29, 2007

(i) xl and xj are uncorrelated if 1.], which is approxi-mately ensured by the interleaver H; and

(ii) All {xj} have the same variance v .

Note that (ii) is not an optimal treatment since we actuallyknow the individual variances {x1}. However, this sub-optimal treatment leads to greatly reduced complexity.Based on the properties given in II.B, we have

HTR-1H = FHGHF(j7FHGHFFHGF + &I)-lFHGF

= FHGH(iTGGH + &I)-IGF.Since GH(vJGGH +&2I)-'G is diagonal, HTR 'H is circu-lant. Further, we have

(HTR-'H)diag =ul (8)

wJ-1 1g 12 (vJIg. 12 + 2)-l Then we rewrite (6) as

= 2(1 _VU)-1 (FHGH (i7GGH + &72I)-F(r -HE(x)) + uE(x)) (9)The ESE can be implemented based on (9) as follows:

Step 1: Take the FFT of r - HE(x).Step 2: Multiply the results of Step 1 by a diagonal ma-

trix GH(v-GGH+&21)- 1

Step 3: Take the IFFT of the results in Step 2, then addthe weighted a priori mean uE(x), and finallyscale the result by 2(1- vu)-1 .

The overall complexity of the above method is roughlyO(log2J) per entry of x. Note that multiplyingGH (vGGH + U21)-1 can be regarded as symbol-by-symbolequalization in the frequency domain. Therefore the abovemethod is actually equivalent to the FDE method [1 1]. Werefer to this method as frequency-domain-equalizationMMSE (FDE-MMSE). For convenience of comparison,we refer to the implementation technique in [1, 2 and 4] astime-domain-equalization MMSE (TDE-MMSE). Onemajor advantage of FDE-MMSE is that its complexity isindependent of the channel memory length L, whilst that ofTDE-MMSE is proportional to L2 in general.

B. ESEfor Block CirculantH

Consider the block circulant H in (3). Denote by Vj thecovariance matrix of xj (which is an NxN diagonal matrix).Similarly to the scalar case, we approximate V by

V IJ OV (10)where V is the average of Vj for j = 0,..., J- 1. Based onthe properties given in II.C, it can be shown that

i 2(1 -VU)-lFNHGH (GVGH + &2I)-lFM (r - HE(x))

+2(I -VU)-'UE(x) (11)

where U _ (IJ®Usub)df1ag ZJlGH(JGVG +&I)1G(S)Um)dJa II,=The operations in (11) can be implemented similarly tothose in IV.A except that block-wise FFT should be used.

V. EVOLUTION ANALYSIS

In this section, we outline an evolution-based techniqueto characterize the behavior of the iterative LMMSE-APPdetector discussed previously.

A. Evolution Analysis for CirculantHThe evolution technique tracks a few parameters in the

iterative process using pre-calculated transfer functions.We require that the parameters involved be as few as pos-sible so as to reduce the complexity involved. At the same

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time we require that these parameters accurately character-ize the statistical behavior of the iterative process, so as topredict its performance. For a system with a circulantchannel matrix as in (2), we find that it is most convenientto use the input variance (denoted by v- ) and output sig-nal-to-noise ratio (denoted by p) to characterize the ESEand the input SNR p and output v to characterize theDEC. Thus, the density evolution of the iterative decodingprocess reduces to a simple recursion between p and v,and the transfer functions of the ESE and the DEC, respec-tively, can be denoted by

p= (v) and v (p).

p=b(v)

DEC ESEv =w(p)

Fig. 2. An illustration of the evolution process.

Similarly to the generation of EXIT functions [8], bothb(*) and V)(-) can be obtained by simulation. Provided thatb( ) and V'(-) are available, we can track the evolution of pand v during the iterative process, as shown in Fig. 2. Forexample, we can start with an initial value of v = 1. (ForBPSK modulation over {+1, -1 1, v = 1 implies no a pri-ori information.) Then the evolution process is as followsAfter the first iteration: p = OM1)After the second iteration: p = 0( 0(1)))

The SNR value resulting from a specified number of itera-tions can be used to predict the system performance.The above principle is similar to the EXIT chart method

in which mutual information is used. The EXIT chartapproach [2] is for fixed channels in which the transferfunctions can be pre-calculated by simulation and stored.This strategy, however, is not applicable if we want toevaluate the statistical system behavior in quasi-static fad-ing channels, since the transfer functions involved (in par-ticular, 0(*)) can be different for different channel realiza-tions. It is not practical to pre-calculate and store 0( ) andV'(-) for all possible channel realizations.In the following, we present a solution to this difficulty

that is obtained by making certain approximations. It isshown via numerical results that the proposed method canaccurately characterize the behavior of the iterative re-ceiver in Fig. 1.

First, we derive an analytical form for 0( ). This is basi-cally to find the output SNR p for the ESE with a given v .

We rewrite lj in (5) in a signal-plus-distortion form as(12)Aij = pxj + ;

where ,u = 2h'R1 'hj,Sj = 2h TRj l (r - HE(x) - hj (xj - E(xj ))) ,

R HH -ihh'+&I1T T+2IINote that ; is not a function of xj (since r - hjxj is not afunction of xj). Also, ,u is invariant to index j since

,u=2h'Rlhj =2h'R `h = 2ui J I IhRl I_ _-u1vhRh 1iu

ISIT2007, Nice, France, June 24 - June 29, 2007

where the second equality follows from the matrix inver-sion lemma, and the last equality follows from (8).Furthermore, it can be verified that the mean of ; is zeroand its variance given by E(,2) 4h R,'R,R,'hj The

average variance of j, denoted by 0T2 is given byJ-1 J-1

j2 j-1 EE(2) = 44J-l TRhRj'RR'lhji=o j=o

J-1 _ _ J-1

=4J-1 , v hjTR h,h,h R71h + cr2h.R 1R-'hj, =0jK' j=O'J-1 _ J-1

=4-1 Vl (hTj lhl 2 + E 072 T --Ih4J- VI RR1h1=0 j#I j=o

4J_1E Vjh. - vH- vhhT + U21 )Rj

=4J 1JhRHTh Rl

4u1 vu

It is known that the residue distortion at the output of theLMMSE estimator is approximately Gaussian distributed[10]. Thus, we introduce the following assumption.

Assumption I: ; is a Gaussian random variable with zeromean and variance o2.Under the above assumption, the ESE output in (6) can

be modeled as an observation of xj scaled by ,u (a constant)and corrupted by AWGN samples I 4}. A single SNR,given by

p = A22/ = u(1 - 1VU)- (13)is sufficient to characterize this model. From (8), u is afunction of v Thus (13) provides an analytical form forp = 5(v) . With (13), we can instantaneously evaluate 0( )at each step instead of pre-calculating and storing 0() forall possible channel realizations.The generation of A(-) is straightforward. Recall that I A)

are interleaved and delivered to DEC as its inputs. Basedon (12), lIAl can be modeled as the observation of {xj}through an AWGN channel. (The correlation among lAj)can be ignored due to interleaving.) Thus, we can generateA(-) by simulating the DEC in memoryless AWGN chan-

nels with various SNRs.

B. Evolution Analysis for Block CirculantHThe above evolution analysis can be generalized to

block systems. However, we found that the method be-comes inaccurate if the ESE outputs are still characterizedby a single SNR for MIMO systems. This is caused by thefact that the SNR values of the ESE outputs in a MIMOsystem can vary significantly because the signals transmit-ted via different MIMO inputs may experience differentchannel fading. It therefore becomes too inaccurate to usea single parameter for all of the ESE outputs. However, weobserved that the following approaches are valid.

The ESE outputs corresponding to different time in-stants but the same MIMO input can be characterizedby a single SNR value. (Note: A MIMO input is in-dexed by n. See (3b).)If there is only one decoder in the system, then theoutputs of this decoder can be characterized by a sin-gle variance value. However, if there are multiple de-coders, such as for a multiple access system, multiplevariance values should be used.

We will use a vector form SNR denoted byp. pn, denotingthe nth entry in p, represents the SNR value at the ESEoutputs corresponding to a specific index n of the MIMOinputs. For simplicity, we assume that there is only onedecoder and so one variance value is used, i.e., V = vIThus we need to find the following two functions, namely

p = O(v) and v= (p) .

For H given in (3), it can be shown that

p = (Usub)da (I-V(Usub)da ) 1 =_(iV), (14)where 1 is an all-one vector with length N. The derivationfor (14) is similar to (13). We omit details here due to thelack of space.Next we consider the DEC. We divide the entries of A

into N groups, each characterized by an entry in p. We cansimilarly model each 2i as a scaled observation of Xj cor-rupted by an additive Gaussian noise, i.e.,

2i = ,Ujxj + j . (15)

However, the scaling factor guj now has N different values,as specified by p. We refer to (15) as an N-state fadingchannel. Thus the inputs to the DEC can be treated as anLLR sequence coming from a memoryless N-state fadingchannel, and the corresponding transfer function can beobtained by simulating the DEC over different memorylessN-state fading channels. Note that the set of N values for{jt} are different for different channel realizations. Weneed to simulate and store all possible combinations, andthus an N-dimensional table is required. For convenience,we refer to this method as the full-table (FT) method.The FT method is impractical even for a moderate N.

The following low-complexity approach is inspired by therationale behind the EXIT chart analysis [8], i.e., given anLLR sequence A as the input of a turbo component (theESE or the DEC), its output behavior will not be affectedby replacing A with another sequence 2' that contains thesame amount of mutual information (with respect to thetransmitted sequence x) as A does. We write this as anassumption.

Assumption II: The decoder behavior over a channelmodeled in (15) remains the same if the amount of mutualinformation with respect to {xj} contained in lAj} is a con-stant.

According to the analysis in [8], the mutual informationcontained in an LLR A from an AWGN channel with SNRequal top (where x E 1+1, -1I} is transmitted) is given by

I 1 1 28)2;x)=+l8z log X+- e 8pO dA=-f(p). (16)

Let v = JAWGN(P) be the transfer function of the DEC in anAWGN channel. This function can easily be generated viasimulation and stored as a one-dimensional table. Now weadopt the following simple strategy to generate (). Weselect an effective SNR value Peff so that the observationson the AWGN channel with SNR = Peff contain the sameamount of mutual information with respect to {xj} as IAjIdoes, i.e.,

Peff f (N n= f (Pn))We then let (p) l/-AWGN(Peff), based on Assumption II.Note that both f(-) and JAWGN(-) can be pre-calculated

and stored using one-dimensional tables. We refer to this

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ISIT2007, Nice, France, June 24 - June 29, 2007

method as the equivalent-mutual-information (EMI) tech-nique.

VI. NUMERICAL RESULTS

In this section, numerical results are given to verify theproposed implementation method and the evolution analy-sis. Although the generic system model in Fig. 1 includes a

variety of communication channels, we restrict attention toMIMO ISI channels due to space limitation.

Consider quasi-static Rayleigh fading MxN MIMO ISIchannels. The ENC contains only one encoder employing a

rate-1/2 (7, 5)8 convolutional code with information length1024. The coded bits are BPSK modulated, and then are

randomly interleaved and separated into two streams toform the real and imaginary part of the transmit symbolvector. Then, this vector is separated into N streams forsimultaneous transmissions over N transmit antennas. Atypical implementation of this MIMO system can also befound in [4].The quasi-static Rayleigh fading channels are modeled

as follows: the fading coefficients are independently iden-tically distributed complex Gaussian random variableswith zero-mean, and the average total fading gain normal-ized to 1, i.e.,

[3] L. Liu, Li Ping, and W. K. Leung, "Iterative detection for inter-leaver division multiple access in channels with intersymbol inter-ference", IEICE Trans. Commun., vol. E87-1, no. 11, pp. 3274-3280, Nov. 2004.

[4] X. Yuan, K. Wu, and Li Ping, "The Jointly Gaussian Approach toIterative Detection in MIMO Systems," in Proc. IEEE Int. Conf: on

Commun., ICC'06, Istanbul, Turkey, 11-15 June 2006.[5] C. Douillard et al., "Iterative correction of intersymbol interference:

turbo equalization," Eur. Trans. Telecommun., vol. 6, pp. 507-511,Sept.-Oct., 1995.

[6] M. C. Reed, C. B. Schlegel, P. D. Alexander, and J. A. Asenstorfer,"Iterative multiuser detection for CDMA with FEC: Near-single-user performance," IEEE Trans. Commun., vol. 46, pp. 1693-1699,Dec. 1998.

[7] 3rd Generation Partnership project, Technical Specification GroupRadio Access Network, Physical Layer Aspects for Evolved UTRA(Release 7), 3GPP TR25.814 Vl.0.1 (2005- 11).

[8] S. ten Brink, "Convergence behavior of iteratively decoded parallelconcatenated codes", IEEE Trans. Commun., vol. 49, no. 10, Oct.2001.

[9] D. Falconer, S. L. Ariyavisitakul, A. Benyamin-Seeyar, and B.

Eldson. "Frequency Domain Equalization for Single-Carrier Broad-band Wireless Systems". IEEE Commun. Mag., pages 58-66, April2002.

[10] V. Poor and S. Verdu, "Probability of error in MMSE Multiuserdetection," IEEE Trans. Inform. Theory, vol. IT-43, pp. 835-847,May 1997.

[11] M. Tuichler, and J. Hagenauer, "Turbo equalization using frequencydomain equalizers," in Proc. Of Allerton Conference, Monticello,IL, USA, Oct. 2000.

Eh =E[Ih 12]=1, for V j,

where the expectation is taken over the channel fadingdistribution. Thus the bit-energy-to-noise-density ratio iscalculated as

Eb E( xj 12)EhE 1

No RCNO 2RcU2where Eb denotes the bit energy, No denotes the single-sideband noise power spectral density, and RC denotes therate of xj.

In simulation, at least 10 iterations are taken in thedetection to guarantee convergence. For each point on theperformance curve, at least 1000 errors are collected.

Fig. 3 shows the BER/FER performance of the system inquasi-static Rayleigh-fading 2x2 MIMO ISI channels withL = 4. Both FDE-MMSE and TDE-MMSE are considered.It can be seen that the performance loss with FDE-MMSE(due to the approximation in (10)) is not significant. Fig. 3also includes the performance curves predicted by evolu-tion. In Fig. 3, it can be seen that the performance of FDE-MMSE is accurately predicted by the full-table method(denoted by "FT-Evolution"). Moreover, the predictionmade by the EMI method (denoted by "EMI-Evolution")deviates only about 0.1-0.2dB away from the actual per-formance.

In Fig. 4, we consider a quasi-static Rayleigh-fading 4x4MIMO ISI channels with L = 2 and 4, respectively. Theevolution results are based on the EMI method since theFT method becomes too complicated here. We can see thatthe EMI method provides better predictions as the MIMOdimension and/or channel memory length are increased.

REFERENCES

[1] X. Wang and H. V. Poor, "Iterative (turbo) soft interference cancel-lation and decoding for coded CDMA," IEEE Trans. Commun., vol.47, pp. 1046-1061, July 1999.

[2] M. Tuchler, R. Kowtter, A. C. Singer, "Turbo equalization: Princi-ples and new results", IEEE Trans. Commun., vol. 50, pp. 754-767,May 2002.

1.

0.0 1.0 2.0 3.0Eb/No (dB)

4.0 5.0 6.0

Fig. 3. BER/FER performances and evolution predictions on quasi-staticRayleigh fading 2x2 MIMO ISI channels with L = 4.

1.E+00

1.E-01

=-1.E-02

0 1.E-03

° 1.E-04

1.E-05

1.E-06-3.0 -2.0 -1.0 0.0

Eb/No (dB)1.0 2.0

Fig. 4. BER/FER performances and evolution predictions on quasi-staticRayleigh fading 4x4 MIMO ISI channels with different L.

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