ISlT 2002, Lausanne, Switzerland, June 30 -July 5,2002
Efficient Implementation of Iterative Multiuser Decoding
Ralf R. Muller
DoCoMo Communications Lab Europe GmbH' Landsberger Straae 308-3 12 80687 Munich, GERMANY e-mail: mueller@docomolab-euro. com
I. INTRODUCTION
Iterative multiuser decoding based on the sum-product algorithm can give excellent performance [I]. To reduce complexity, inherent maximum-aposteriori detection is often replaced by linear MMSE de- tection combined with parallel successive cancellation.
The powers of the users are often unbalanced, either by design to improve performance as proposed in [2] or due to independent fading in the reverse link. During iterations, the power distribution of the residual interference evolves. This can lead to a severely mismatched MMSE detector after few iterations unless it is re-calculated for every iteration as in [3]. Here, a computationally less costly approach is presented.
11. PROPOSED SCHEME
Let the synchronous Gaussian CDMA channel in vector notation [4] be given as y p = S m x p
+
np where np is unit variance, zero-mean AWGN, x p is the vector of transmitted symbols at time instant p , g p is the vector of chip samples at time instant p, the L x K matrix S denotes the spreading sequences SI, . . .,
S K , and the diagonal matrix W is composed of the users' powers w1,. . .,
W K .W.l.o.g., let EezH = I.
Each user's signal is encoded and interleaved in discrete time p.
Consider now a parallelly operating iterative multiuser decoder based on interference cancellation as considered in [2]. Then, the signal of user k fed into the decoding unit at iteration number m is given by
here f
r
should be chosen appropriately, e.g. according to the uncon- ditional MMSE criterion, cf. [2].The unconditional MMSE detector leads to
(fr)H
=& B ~ ( S W V " S ~
+
I - W k V r 8 k S : ) - ' with the error covari- ance matrix V" = E(z, -* T ) ( z p
-ST)".
We assume that the estimates2p
result from extrinsic information only and the girth of the code's graph is greater than m. Thus,V m
is diagonal, its entries are denoted by v r .Let the users' powers be quantized in J
<
K levels and K j denote the set of users belonging to class j. W.l.o.g., we assume that the powers of users in class j are greater than the powers of users in class j+
1, b'j. Now, we propose a suboptimum approach that calculates only J different detectors. Detector #j assumes the powers in classes i<
j to be zero, and the powers of in classes a3
j to be W k V r for IC EIC;.
That means that only users that have not (almost) been canceled out are suppressed by linear detection.Define the diagonal matrix Z j such that z j , k is zero if k belongs to a class smaller than j and one otherwise. Decompose the spread- ing matrix into S = S Z j
+
S (I - Zj). Note that the exact MMSE detector, depends on the iteration index m via the matrixV".
This is overcome by the approximationV m
E p m V j Z j withVj
de- noting the matrix V m 0 frozen at iteration mo when the last switch of detectors took place and pm being an arbitrary real-valued scalar depending on the iteration index m.'This work was performed while R. Muller was with Forschungszentrum Telekommunikation Wien (ftw.), Vienna, Austria.
Giuseppe Caire
Institut EURECOM 2229 Route des Crktes, B.P. 193 06904 Sophia Antipolis CEDEX, FRANCE
e-mail: caire@eurec!om. f r
We introduce the singular value decomposition (SVD) S J m = T j D j U y , such that Tj and Uj are unitary and Dj is diagonal up to some additional columns or rows which are all zero. Define Qj = T r S . With (l), we find
+I)-'
( ~ j " ~ ,
- Q ~m * p )
dcp = q '
(
+&2zkq ; k ( p n D j D , r + I ) - ' q j , k
Given Qj and the SVD, this involves only O ( K ) multiplica- tiondadditions per user, symbol, and iteration, as the calculation of T ~- Q j ~
mai.F
, is common to all users and the other operations are inner products of vectors with diagonal kernels. Only when a switch from detector j to detector j+
1 takes place, a SVD and a matrix multiplication are needed.111. SWITCHING CRITERION
Assume i.i.d. random spreading and K
>>
1 enabling use of large system results. To decide if time has come to switch, the multiuser efficiencies at the outputs of the actual and the altemative detector are calculated. Then, that detector promising better performance is chosen for the next iteration.Assume the residual interference power of user k at iteration m is
pr.
Making use of some results in [ 5 , 61, it can be shown that the multiuser efficiency converges almost surely to the large-system limitwhere 77" is the unique positive solution to the fixed point equation
(vm)-'
= 1+ E:='=,
p;Ez(l+vT"pT)-l and the interference power is approximated byJ wkv;Opm k E
U
Kii = j
io
otherwiseWkV? k E
U
K i ,pT
=Pr
={
otherwise i=;+lin case the detector is switched, not switched, respectively.
Certainly, one would like to chose pm in such a way as to maxi- mize multiuser efficiency, i.e. p m = argmax6". A suboptimum, but computationally less costly altemative achieving satisfying results is P" =
E;='=,
vr/c;=1 U.":REFEFLENCES
[ I ] J. Boutros, G . Caire. Iterative multiuser joint decoding: Unified frame- work and asymptotic analysis. IEEE Trun.F. Inf: Th., Submitted.
[2] G. Caire, R. Muller. The optimal received power distribution for IC- based iterative multiuser joint decoders. In Proc. of 39th An. Allerfon Con$ on Commun., Contr: & Comp., Monticello, IL, Oct. 2001.
[3] X. Wang, V. Poor. Iterative (turbo) soft interference cancellation and de- coding for coded CDMA. ZEEE :Truns. Comm., 47(7):1046-1061, 1999.
[4] S. Verdli. Multiuser Detection. Cambridge University Press, NYC, 1998.
[5] D. Tse, S. Hanly. Linear multiuser receivers: Effective interference, ef- fective bandwidth and user capacity. lEEE Trans. In$ Th., 45(2):641- 657,1999.
Output MA1 distributions of linear MMSE multiuser receivers in DS-CDMA systems. ZEEE Trans. Znf: Th., 47(3):1128-1144,2001.
[6] J. Zhang, E. Chong, D. Tse.
O-78O3-75Ol-7lO2l$l7.OO@2OO2 IEEE. 446
