an extrinsic kalman filter for iterative multiuser decoding

642 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 4, APRIL 2004

Now, from the generating function definitions (11) and (22) and fromthe marginal definitions (24), it is clear that

J(��)(�; �) =

1

n=0

J(��)n (�)�n

and

I(��)(�; �) =

1

n=0

I(��)n (�)

�n

n!: (36)

Plugging these results into the LHS of (27), we obtain for the LHS

1

2�

1

�1

d�1

�1

d� e�i��

�

1

m=0

J(��)m (�)(i�)m

1

n=0

I(��)n (�)

�n

n!

=

1

m;n=0

J(��)m (�)I(��

�)n (�)

�

1

2�

1

�1

d�1

�1

d� e�i��(i�)m

�n

n!

(a)=

1

m;n=0

J(��)m (�)I(��

�)n (�)�mn

(b)= e

�� (37)

where (35) was used in step (a), and (26) allowed step (b). Thus, re-lation (27) is established.

ACKNOWLEDGMENT

The authors wish to thank Dr. Chris Lloyd for a reprint of his papercited here.

REFERENCES

[1] H. L. Van Trees, Detection, Estimation, and Modulation Theory. NewYork: Wiley, 1968, pt. I.

[2] T. M. Cover and J. A. Thomas, Elements of Information Theory. NewYork: Wiley, 1991, sec. 12.11.

[3] C. J. Lloyd, “Asymptotic expansions of the Fisher information in asample mean,” Statist. Probab. Lett., vol. 11, pp. 133–137, 1991.

[4] E. W. Weisstein. (1999) Cumulant. [Online]. Available: http://math-world.wolfram.com/Cumulant.html

[5] J. F. Kenney and E. S. Keeping, “Cumulants and the cumulant-gener-ating function, additive property of cumulants, and Sheppard’s correc-tion,” in Mathematics of Statistics, 2nd ed. Princeton, NJ: Van Nos-trand, 1951, pt. 2, sec. 4.10–4.12, pp. 77–82.

[6] E. W. Weisstein. -Statistic. [Online]. Available: http://mathworld.wol-fram.com/k-Statistic.html

[7] J. F. Kenney and E. S. Keeping, “The -statistics,” in Mathematics ofStatistics, 3rd ed. Princeton, NJ: Van Nostrand, 1962, pt. 1, sec. 7.9,pp. 92–100.

[8] P. R. Halmos, “The theory of unbiased estimation,” Ann. Math. Statist.,vol. 17, pp. 34–43, 1946.

[9] H. H. Barrett, J. L. Denny, R. F. Wagner, and K. J. Myers, “Objectiveassessment of image quality. II. Fisher information, Fourier crosstalk,and figures of merit for task performance,” J. Opt. Soc. Amer. A, vol. 12,pp. 834–852, 1995.

[10] R. Bracewell, The Fourier Transforms and Its Applications, 3rded. New York: McGraw-Hill, 1999, pp. 257–262.

[11] J. D. Gorman and A. O. Hero III, “Lower bounds for parameer esti-mation with constraints,” IEEE Trans. Inform. Theory, vol. IT-26, pp.1285–1301, Nov. 1990.

[12] R. Narayan and R. Nityananda, “Maximum entropy restoration in as-tronomy,” Ann. Rev. Astron. Astrophys., vol. 24, pp. 127–170, 1986.

[13] T. J. Schulz, “Multi-frame blind deconvolution of astronomical images,”J. Opt. Soc. Amer. A, vol. 10, pp. 1064–1073, 1993.

[14] P. McCullagh, Tensor Methods in Statistics. New York: Chapman andHall, 1987.

An Extrinsic Kalman Filter for IterativeMultiuser Decoding

Lars K. Rasmussen, Senior Member, IEEE,Alex J. Grant, Senior Member, IEEE, and Paul D. Alexander

Abstract—One powerful approach for multiuser decoding is to iteratebetween a linear multiuser filter (which ignores coding constraints) and in-dividual decoders (which ignore multiple-access interference). Subject toclearly formulated statistical assumptions and the history of input signalsprovided by the outer decoders over all previous iterations, an extrinsicKalman filter is suggested. This approach is motivated by the recent obser-vation that decoder outputs are loosely correlated during initial iterations.Numerical results show that iterative decoding using this filter providesbetter performance in terms of the supportable load and convergence speedas compared to previously suggested linear-filter-based iterative decoders.

Index Terms—Iterative decoding, Kalman filtering, multiple access, mul-tiuser detection, recursive filters.

I. INTRODUCTION

A multiple-access channel is a communications channel in whichseveral independent users transmit information to a common receiver.In certain cases it is not possible, or undesirable to provide orthog-onality between the users. This causes multiple-access interference(MAI), which limits the performance of any receiver which doesnot take it into account. The optimal (error probability minimizing)multiple-access receiver must make joint decisions, rather than treatingthe multiple-access interference as uncontrollable noise.The multiple-user detection problem is difficult. In general, optimal

detection [1] is NP-complete [2]. To make things worse, informationtheory tells us that multiple-user coding strategies should in fact be used[3, Ch. 14]. Once again, complexity is the limiting factor. Brute-forcejoint decoding of a random K user code is exponentially complex inboth the codeword length and the number of users. Optimal decodingfor convolutionally encoded users was considered in [4], resulting in atrellis with complexity growing exponentially with both the number ofusers and the constraint length of the code.One approach to the problem of coding for the multiple-access

channel is the use of single-user codes together with suboptimal,reduced complexity joint decoding methods which attempt to ap-proximate the effect of optimal joint decoding. Of particular interestis iterative multiuser decoding. The basic principle behind iterativedecoding is to decode independently with respect to the variousconstraints imposed on the received signal, rather than consideringthem jointly. The overall constraint is accommodated by iterativelypassing extrinsic information between the individual decoders. For

Manuscript received January 1, 2003; revised September 13, 2003. This workwas supported in part by the Swedish Research Council for Engineering Sci-ences under Grants 217-1997-538, 621-2001-2976, 621-2002-4533 and by theAustralian Government under ARCGrant DP0344856. The material in this cor-respondence was presented in part at the IEEE International Symposium onInformation Theory, Lausanne, Switzerland June/July 2002 and at the Interna-tional Symposium onTurbo Codes andRelated Topics, Brest, France, July 2003.

L. K. Rasmussen is with the Institute for Telecommunications Research,University of South Australia, Mawson Lakes, SA 5095, Australia (e-mail:[email protected]) and with the Department of Computer Engi-neering, Chalmers University of Technology, SE-412 96 Göteborg, Sweden(e-mail: [email protected]).

A. J. Grant and P. D. Alexander are with the Institute for TelecommunicationsResearch, University of South Australia, Mawson Lakes, SA 5095, Australia(e-mail: [email protected]; [email protected]).

Communicated by A. Kavcic, Associate Editor for Detection and Estimation.Digital Object Identifier 10.1109/TIT.2004.825031

0018-9448/04$20.00 © 2004 IEEE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 4, APRIL 2004 643

multiuser decoding, there are constraints due to the multiple-accesschannel and due to the individual users’ encoders.

Iterative multiuser decoding, therefore, separates multiuser channeldecoding and error control code decoding. Multiuser channel decodingrefers to an estimator that performs inference on the users’ signalswithout regard to the time-domain structure imposed by forward errorcorrection coding. The term “multiuser detection” is avoided in thiscontext, since that term is more properly associated with hypothesistesting, in which one of a number of decisions must be made. The in-formation usually exchanged is a posteriori probability (APP) on bitsor symbols, determined by separate APP decoders. Computationallyefficient algorithms exist for APP decoding of good codes with suit-able trellis structure, but the complexity of the multiuser channel APPdecoder remains exponential in the number of users. From a practicalpoint of view, decoders, or estimators with linear (or at worst polyno-mial) growth in complexity are preferred.

A variety of reduced-complexity iterative decoders exist in the liter-ature, mostly based on different types of multiuser channel decoding.Of particular interest is the use of linear filters and cancellation strate-gies. The linear soft parallel interference cancellation (PIC) strategysuggested in [5] was further developed in [6]. Cancellation followed byinstantaneous linear minimum mean-squared error (LMMSE) filteringwas proposed in [7]. This structure is identical to the feedforward feed-back LMMSE filter in [8]. An attempt to create a unifying frameworkfor describing these approaches has been presented in [9]. An inter-esting improvement to the PIC is given in [10], [11] where partial can-cellation as proposed in [12] for uncoded transmission is introduced.

The main result of this correspondence is an extrinsic Kalman filterwith potentially superior characteristics. This recursive filter yields es-timates based on the received signal and all the successive outputs pro-vided by the error control decoders over all previous iterations. Theapproach is motivated by the recent observation that these estimatesare loosely correlated during initial iterations [10], [11].

Notations: xxx 2 Sn shall be a column n-vector with elements xi =(xxx)i, chosen from the set S , commonly the reals . The space of prob-ability n-vectors (length n nonnegative vectors that sum to 1) is de-noted n. Similarly, XXX 2 Sm�n is an m by n matrix with elementsxij 2 S . The superscript t denotes the transpose operator. For randomvectors xxx and yyy, [xxx] is the expectation, xxx = [xxxtxxx] the variance,and xxx = hxxx; xxxi = [xxxxxxt] the covariance, respectively. Likewise,

(xxx; yyy) = hxxx; yyyi = [xxxyyyt].

II. SYSTEM MODEL AND ITERATIVE DECODER

Consider theK-user linearmultiple-access system of Fig. 1. Let userk = 1; 2; . . . ; K encode their binary information sequence bk[l] usinga rateR code C, to produce the coded binary sequence dk[i]. Transmis-sion occurs in blocks of L code bits per user, corresponding to KLR

information bits. Each user independently permutes their encoded se-quence with an interleaver �k . The sequence output from the inter-leaver of user k is uk[i], where i = 1; 2; . . . ; L is the symbol timeindex. The interleaved code bits uk[i] are then memorylessly mappedonto a binary phase-shift keyed (BPSK) constellation B = f�1; 1g,giving sequences ofmodulated code symbolsxk[i]. Extension to higherorder signaling constellations is straightforward, BPSK is chosen onlyfor simplicity.

At symbol time i, each user transmits sssk[i]xk[i], the multiplicationof xk[i] with the realN -chip spreading sequence,1 sssk[i] 2 f�1; 1gN .

1Although this model is cast in terms of direct-sequence code-division mul-tiple-access, the sequences [ ] may be used to represent any type of complexvectormodulation, thus admitting a general linearmultiple-access systemmodel[13].

Fig. 1. The transmission system model.

Spreading sequences with period much longer than the data symbol du-ration are modeled by letting each element of sssk[i] be independent andidentically distributed (i.i.d.) over users and time. For conceptual easeonly, users are symbol synchronized, transmit over an additive whiteGaussian noise (AWGN) channel, and are received at the same powerlevel. These assumptions, however, are not required for the followingdevelopment.Given symbol-synchronicity, the chip-match filtered received vector

rrr[i] 2 N at symbol time i = 1; 2; . . . ; L is

rrr[i] = SSS[i]xxx[i] + nnn[i] (1)

where SSS[i] = (sss1[i]; sss2[i]; . . . ; sssK [i]) is an N � K matrix with thespreading sequence for user k as column k. The vector xxx[i] 2 BK haselements xk[i] and the vector nnn[i] 2 N is a sampled i.i.d. Gaussiannoise process, with nnn[i] = �2III .It will not be required to identify specific symbol intervals, so symbol

indexes will be omitted. For later use, define the following notation:

xxx�k = (x1; x2; . . . xk�1; xk+1; . . . ; xK)t

to indicate deletion of user k fromxxx. Likewise, III�k denotes aK�1�K

matrix formed by deleting row k from theK �K identity matrix.Application of the “turbo-principle” to the coded multiple-access

system just described, where for each user, the error control code is con-sidered as one constraint and the multiuser channel (1) as the other con-straint, results in the canonical iterative multiuser of Fig. 2 [14]–[17].This is a user-by-user structure where each user has a separate decoderbranch. The outputs following each full iteration are, however, sharedamong all the decoder branches.At each iteration, the multiuser APP decoder takes as input the re-

ceived signal rrr and the set of extrinsic probabilities qqq(n�1)k from user

k = 1; 2; . . . ; K calculated in the previous iteration. The extrinsicprobability distribution qqq(n�1)

k [i] 2 jBj is on the transmitted sym-bols xk[i] 2 B of user k. With a small abuse of the vector subscriptingnotation, (qqq(n)k [i])x is the element of the vector qqq(n)k [i] correspondingto user k transmitting the symbol xk 2 B at time i (technically, thesesubscripts should be integers corresponding to an index mapping be-tween B and 0,1).The multiuser APP calculates the updated extrinsic probability

vector ppp(n)k [i] for user k and after appropriate de-interleaving, theextrinsics ppp(n)k are used as priors for independent APP decoding of thecode C by each user. These decoders are standard implementations ofthe forward–backward algorithm [18] on the trellis associated with C.These decoders produce (after interleaving) the extrinsics qqq(n)k , whichserve as priors for the subsequent iteration.Calculation of ppp(n)k [i] requires summation over jBjK�1 terms, which

limits the practical application of this receiver. Because of this pro-hibitive complexity, many lower complexity alternatives have been pro-posed while retaining the same basic architecture. Of particular interestare structures that replace the multiuser APP block with a bank of linearfilters as shown in Fig. 3. In this structure, there is a bank of filters


Fig. 2. Canonical iterative multiuser decoder.

Fig. 3. Iterative multiuser decoder with linear multiuser estimation.

��(n)k

, one for each user. The coefficients of these filters may be re-computed every iteration. For the first iteration, n = 1, the input toeach filter ��(1)

kis just the received signal rrr. For subsequent iterations

n = 2; 3; . . . ; the input to the filter for user k is the received signal rrrand a set of signal estimates for all the other users from previous itera-tions fx(m)

k: k0 6= k; m 2Mg, whereM� f1; 2; . . . ; n�1g is a set

defining the memory order of the iteration. Typically in the literature,M = fn� 1g, although recentlyM = fn� 1; n� 2g has been con-sidered [10], [11]. These input signals can in some sense be thought ofas extrinsic information since the estimates x(m)

kfor user k have been

excluded.The output of the filter��(n)

kis an updated sequence of estimates ~x(n)

k

of the corresponding code symbol for user k. These estimates are thenmapped from the signal space onto the probability vector space usinga symbol-wise mapping T : 7! jBj. The resulting sequence ofprobability vectors ppp(n)

kare used as priors for individual APP decoding

of the code C. These decoders can output either posterior or extrinsicprobabilities qqq(n)

k. The sequence of probability vectors qqq(n)

kis, in turn,

mapped back onto the signal space by a symbol-wise function U :jBj 7! .Typically, T calculates the vectors ppp

(n)k

assuming that ~x(n)k

isGaussian distributed with known mean and variance

~x(n)k� N ~�

(n)k

; ~�(n)k

:

Likewise, a common choice for U is the conditional mean. Design ofimproved maps T andU is an interesting topic, but is outside the scopeof this correspondence, where the focus is on the exploitation of all thesuccessive outputs from the decoders, rather than just the most recent.

The use of posterior and extrinsic probabilities qqq(n)k

as feedback in-formation has been investigated in the literature, e.g., [9], [11]. Whensuboptimal multiuser detection strategies are used in place of mul-tiuser APP detection, it is however still an open problem whether pos-terior or extrinsic probabilities should be used. For the numerical ex-amples presented here, posterior probabilities have been observed to

Fig. 4. The extrinsic Kalman filter �� . For = 1 the input signal iswhile for 2 the input signal is ^ .

provide better performance in conjunction with linear filters. The con-clusive solution to this problem is, however, outside the scope of thiscorrespondence.

III. EXTRINSIC KALMAN FILTER FOR MULTIUSER ESTIMATION

In the literature, work on linear filters for iterative decoding has fo-cussed on LMMSE filtering [7], [8], [19], [20], linear interference can-cellation [5], [6], [9], [21]–[23], and linear weighted cancellation [10],[11]. Furthermore, these filters have been designed based on the re-ceived signal rrr and the most current code symbol estimates of the in-terfering users xxx(n)�k

. After n iterations, there is, however, a sequence ofsuch estimates available, namely fxxx(1)�k

; xxx(2)�k; . . . ; xxx

(n)�kg together with

rrr. It has been observed that the estimates are not strongly correlatedduring the initial iterations [10], [11], so in principle, using all the pre-vious estimates in the filter design could improve performance. Themain result of this correspondence is the derivation of an extrinsicKalman filter��(n)

kfor use in the iterative decoder of Fig. 3 that outputs

the LMMSE estimate for xk , given rrr and the output of all the APPdecoders for every previous iteration. We call this filter an extrinsicKalman filter, since it does not use previous estimates of xk in its up-date procedure.


Fig. 5. BER performance versus number of users at = 5 dB with = 8.

Consider the following recursively defined vector of observations asinput to the filter ��(n)

k :

ccc(n)k =

rrr; n = 1

ccc(n�1)k

xxx(n�1)�k

; n = 2; 3; . . . :(2)

Direct application of the LMMSE design criterion results in

��(n)k = xk; ccc

(n)k ccc

(n)k ; ccc

nk

�1

:

It is clear, however, that��(n)k grows in dimension with n, which is im-

practical. Application of the principles of Kalman filtering [24], [25],or recursive Bayesian LMMSE estimation [26], solves this dimension-ality problem by giving a recursive form for ��(n)

k , subject to the fol-lowing assumptions.

A1 The received signal is rrr = SSSxxx+ nnn, according to (1) where nnn isGaussian with nnn = �2III , and �2 and SSS are known.

A2 The interleaved code symbol estimates of the interfering usersxxx(n)�k

coming out of the single-user APP decoders can be written

x(n)k = xk + v

(n)k (3)

where it is assumed that v(n)k is uncorrelated with xxx and alsouncorrelated over time and iterations, but not over users at agiven iteration

xxx; v(n)k =0 (4)

v(n)k ; v

(m)j =

0; n 6= m

qkj ; n = m.(5)

A3 The matrixQQQ(n)k defined asQQQ(n)

k = hvvv(n)�k

; vvv(n)�ki with elements

determined by (5), is known.

Let ccc(n)k be according to (2). Then the system has trivial state evolu-tion xxx(n) = xxx and observations

yyy(n)

=HHH(n)k xxx+ zzz

(n)

=SSSxxx+ nnn; n = 1

III�kxxx+ III�kvvv(n�1)

; n � 2.

The usual Kalman equations, underA1–A3, yield the LMMSE estimate~x(n)k of xk given ccc(n)k as the kth element of

~xxx(n)

= ~xxx(n�1)

+MMM(n)k yyy

(n) �HHH(n)k

~xxx(n�1)


Fig. 6. BER performance versus number of iterations at = 5 dB with = 8.

where

MMM(n)k

=WWW(n�1)k

HHH(n)k

t

HHH(n)kWWW

(n�1)k

HHH(n)k

t

+RRR(n)k

�1

WWW(n)k

= III �MMM(n)kHHH

(n)k

WWW(n�1)k

RRR(n)k

= zzz(n) =

�2III; n = 1

QQQ(n)k; n > 1

with initial conditionsWWW (0)k

= III and ~xxx(0) = 0.Fig. 4 shows how the filter exploits feedback to avoid growing di-

mensionality. Observe that the filter for each user is required to estimatethe code symbols for the interfering users based on the user-specificinput signals to the user-specific filter. These estimates are for internalfiltering use only and are not available in the external iterative structure.

Assumptions A1–A3 yield a tractable filter design problem and arethe key to obtaining the convenient iterative structure. In practice, theseassumptions hold only approximately, and some performance loss canbe expected due to this inaccuracy in modeling. The filter is referredto as the extrinsic Kalman filter since the estimate for user k fromthe error control decoders is excluded by choice from the observationvector input to the filter. This is not a requirement of the Kalman filterderivation, rather it is neccessary for good performance of the iterative

decoder. Similar deletion of user k has been widely used in the liter-tature with cancellation/filtering based receivers.It is envisaged that many other simpler filter structures exploiting

all or part of the iteration history of the decoder could be investigated.The Kalman filter just described provides however a useful benchmark,just as the postcancellation LMMSE-based iterative decoder providesa reference for first-order iterative decoders.

IV. NUMERICAL RESULTS

In this section, numerical results are used to illustrate the perfor-mance benefits obtained by the proposed filter. For the purposes ofsimulation, each user applies the maximum free distance four-stateconvolutional code (with generators (5; 7)) mapped onto BPSK. Bi-nary spreading sequences with length N = 8 were generated i.i.d. ateach symbol for each user. Transmission is symbol synchronous andall users are received at the same power level. The matrix QQQ(n)

kis de-

termined based on decision-directed estimation in the simulations. Theperformance of the new recursive filter (RF) is compared to the parallelinterference canceller (PIC) [6], the improved parallel interference can-celler (IPIC) [10], [11], and the LMMSE filtered parallel interferencecanceller (FPIC) [7], [8].


Fig. 7. BER performance versus decibels, =1–5, 7, 10, 20.

In Fig. 5(a), the bit-error rate (BER) performances for the four casesare shown as functions of the load in terms of the number of activeusers. Here, the number of iterations is limited to five and the per-formance is captured for Eb=N0 = 5 dB. Observe that the PIC cansupport up to eight users and still provide close to single-user perfor-mance. The IPIC can support nine to ten users, while the FPIC and theRF can support 12 to 13 users. As predicted, the RF provides betterperformance than other linear estimators, although it is only margin-ally better than the FPIC after five iterations.

In Fig. 5(b), the BER performances are shown after ten iterations asfunctions of the load. The PIC can now support nine users, the IPIC12 to 13 users, while the FPIC and the RF both can support up to 14users. After ten iterations, the advantage of the RF over the FPIC is stillmarginal, although becoming more significant.

The BER performances after twenty iterations are shown in Fig. 5(c)as functions of the load. The PIC can still only support 9 users, how-ever, for more iterations, 10 users may be supported. The IPIC cancomfortably support 13 users, while the FPIC has not improved withiterations and thus can still support 14 users. In contrast, the RF hasimproved with iterations and can now support 14–15 users with closeto single-user performance.

In order to investigate the convergence behavior for each system, theBER performance is now given as a function of the number of iterations

atEb=N0 = 5 dB for a load close to the maximum for the estimator inquestion.In Fig. 6(a), the BER performance for the PIC as a function of the

number of iterations is shown. As was the case in the previous figures,the PIC can support eight users after five iterations, nine users afterseven to eight iterations, and possibly ten users for a large number ofiterations. However, 11 users cannot be supported for any number ofiterations.In Fig. 6(b), the corresponding plot for the IPIC is shown. Again,

the supported loads observed in previous figures are confirmed. TheIPIC can support nine to ten users after five iterations, 11 users afterseven iterations, 12 users after nine iterations, and 13 users after 11iterations. It is also possible that 14 users can be supported for a largenumber of iterations, however, 15 users cannot be supported even for alarge number of iterations.The convergence behavior for the FPIC is shown in Fig. 6(c). Here,

it is observed that 11 users can be supported after only four iterations,12 users after five iterations, 13 users after six iterations, and 14 usersafter nine iterations. It is, however, not likely that the FPIC can support15 users even for a large number of iterations.Finally, the convergence behavior for the RF is demonstrated in

Fig. 6(d). As for the FPIC, 12 users can be supported after five


iterations, 13 users after six iterations, and 14 users after nine itera-tions. Note that 15 users can be supported at only a small number ofiterations above 20, providing an extra user at a processing gain ofN = 8. It can, however, also be observed that 16 users can most likelynot be supported even for a large number of iterations.

All of the above results have been forEb=N0 = 5 dB. The followingresults consider the convergence behavior for a range of Eb=N0 and anumber of iterations.

In Fig. 7(a), the performance of the PIC as a function of the Eb=N0

with K = 9 users is shown for m = 1–5, 7, 10, 20 iterations, re-spectively. The BER performance improves with the number of itera-tions up to seven iterations, where close to single-user performance isachieved at Eb=N0 = 5 dB. For an increasing number of iterations,single-user performance is achieved for lower Eb=N0. After ten iter-ations, single-user performance is achieved at Eb=N0 = 4.5 dB, andafter 20 iterations at Eb=N0 = 4.0 dB. This behavior can potentiallycontinue for an increasing number of iterations down to Eb=N0 =

3.0–3.5 dB where the threshold for the waterfall region appears to be.Similar behavior is observed for the IPIC in Fig. 7(b) for K = 12

users. Here, single-user performance is achieved at Eb=N0 = 6.0 dBafter seven iterations, at Eb=N0 = 5.0 dB after ten iterations and atEb=N0 = 4.0 dB after 20 iterations. Again, it is likely that single-userperformance can be achieved down to Eb=N0 = 3.0–3.5 dB as thenumber of iterations increase.

In Fig. 7(c) and (d), respectively, the signal-to-noise (SNR) con-vergence behavior for the FPIC and the RF are shown for K = 14

users. As previously noted, similar behavior is observed for 10 and20 iterations where single-user performance is achieved at Eb=N0 =

5.0 dB and Eb=N0 = 4.5 dB, respectively. It is, however, interestingto see that after seven iterations, the RF is at single-user performance atEb=N0 = 5.5 dB and very close at Eb=N0 = 5.0 dB, where the FPICis not at single-user performance before Eb=N0 = 6.0 dB. The advan-tage of the RF at a low number of iterations increases as the Eb=N0

increases as can be seen by the steeper BER slop at Eb=N0 = 6.0 dBafter five iterations.

V. CONCLUDING REMARKS

A Kalman filter has been described for use in iterative multiuserdecoding. Subject to certain statistical assumptions, it delivers theLMMSE estimate of each user based on the received signal andthe entire history of outputs from the single-user decoders. Sinceestimates of the code symbols pertaining to user k are excluded fromthe observation vector input to the filter for user k, the resulting filteris called an extrinsic Kalman filter. This filter was shown to performbetter than other suggested structures in all cases investigated. Ata processing gain of N = 8, an extra user can be supported for alarge number of iterations, as compared to the LMMSE filtered PIC.This is interesting as such benefits are expected to increase for largerprocessing gains. For a small number of iterations, the new filter alsoprovides better BER performance. This advantage becomes significantfor increasingEb=N0 as the slope of the BER curves for a low numberof iterations is steeper than for the other linear estimators. This isexpected since it has previously been observed that the estimatesprovided by the single-user decoders are loosely correlated for the firstfew iterations, thus, the output of each iteration can potentially provideadditional information, which otherwise would have been ignored.

The principles described here are also applicable to other systemcomponents and technologies such as space–time coding, equalization,coded modulation, and joint channel estimation/decoding. Another in-teresting avenue of further investigation would be the development ofother, simpler structures which approximate the extrinsic Kalman filter,yet retain the benefit due to the use of all, or part of the iteration history.

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[2] , “Computational complexity of optimum multiuser detection,” Al-gorithmica, pp. 303–312, May 1989.

[3] T. M. Cover and J. A. Thomas, Elements of Information Theory. NewYork: Wiley, 1991.

[4] T. R. Giallorenzi and S. G. Wilson, “Multiuser ML sequence estimatorfor convolutionally coded asynchronous DS-CDMA systems,” IEEETrans. Commun., vol. 44, pp. 997–1008, Aug. 1996.

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an extrinsic kalman filter for iterative multiuser decoding

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