A Tree Search Method for Iterative Decoding of Underdetermined Multiuser Systems

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by  Adriel Kind

A Tree Search Method for Iterative Decoding of
Underdetermined Multiuser Systems
Adriel P. Kind Alex Grant
Agere Systems Australia Institute for Telecommunications Research
University of South Australia
Abstract—Application of the turbo principle to multiuser using various different list detectors [12–19]. Most of these
decoding results in an exchange of probability distributions list-detection based methods rely on Cholesky decomposition
5
between two sets of constraints. Firstly, constraints imposed by
0 of the correlation matrix as a first step. In underdetermined
themultiple-access channel,and secondly, individualconstraints
0 systems (more users than signaling dimensions), this decom-
imposed by each users’ error control code. A-posteriori proba-
2 bility computation for the first set of constraints is prohibitively position cannot be performed. First steps towards avoiding

n complex for all but a small number of users. Several lower this problem have been made in [20,21], based on filtering,
a complexity approaches have been proposed in the literature. followed by lattice reduction.
J One class of methods is based on linear filtering (e.g. LMMSE).
The main contributionof this paper is a simple transforma-
8 A more recent approach is to compute approximations to
tion which creates a virtual full rank system, which permits
2 the posterior probabilities by marginalising over a subset of
sequences (list detection). Most of the list detection methods are Cholesky decomposition and the straight-forward application
] restricted to non-singularsystems. In this paper, we introducea oftree-searchmethodsin underdeterminedmultiusersystems.
T transformation that permits application of standard tree-search The method requires less computational complexity that the
I methodsto underdeterminedsystems. Wefindthat theresulting one described in [20,21]. Numerical results demonstrate the
. tree-search based receiver outperforms existing methods.
s superior performance of the approach, which is compared to
c
[ I. INTRODUCTION other techniques from the literature.

1 It is well known that joint decoding can improve perfor-
v II. SYSTEMMODELAND CANONICAL DECODER
mance in multiple-access systems. Joint maximum likelihood
1
(ML) decoding, which minimizes the overall probability of Consider a multiple-access system with K radio terminals,
8
0 error is however prohibitively complex [1]. Brute force com- or users, simultaneously transmitting forward error correction
1 putationofthejointlyMLcodewordsequencesforK usersis coded digital data across an additive white Gaussian noise
0 O(QKκ) forQ-arymodulationand constraintlength κ codes. (AWGN) channel. The encoder for user k = 1,2,...,K
5
The good performance and low complexity of the turbo operates as follows. A length I frame of independent equi-
0
decoder [2] led to application of the turbo principle to joint probableinformationbitsu isencodedbyarateR code .
/ k C k
s multiuserdecoding.Figure1showsaschematicrepresentation TheI/R codedbitsc arethenpermutedwiththeinterleaCver
c C k
: of the “canonical” iterative multiuser decoder [3–6]. This Πk,andparsedintolengthlogQsegments.Thesesegmentsare
v
decoder treats the users’ forward error correction codes as mappedontoastreamd ofI/R logQconstellationsymbols
i k C
X an “outer code” and the interdependency introduced by the accordingtosomememorylessmapping,andthenmultiplexed
r multiple access channel as an “inner code”. The decoder onto the symbol sequences of length I/RClogQ. Each user
a iterates between a-posteriori probability (APP) computation transmits at a rate of R logQ bits per channel use.
C
fortheinnercodeandindividualAPPdecodingofeachuser’s A data vector d = (d1, ,dK)T K represents all
FEC code. The multiuser APP computation is O(QK), an users’ symbols in a given sy·m··bol interv∈alD(assuming symbol
improvementover joint ML decoding, but still prohibitive. synchronous transmission for simplicity of explanation). The
One low complexity alternative is to replace the inner APP complex constellation C has = Q unique elements,
decoderwithalinearfilter.Examplesincludesoftinterference with moment constrainDts⊂E[d]=0|,Dan|d E[dd∗]=PI , and
K
cancellation [7,8] and linear minimum mean-squared error symbolsareequiprobable.Theaveragetransmitpowerperuser
filtering[9].Theseapproachescanworkquitewell,butthereis is P.
stillroomforimprovementcomparedtotheexactcomputation Each symbolis multiplied by a lengthL modulationvector
of the multiuser APP. s , which has real random elements chosen uniformly from
k
A more powerfulapproachis to computean approximation 1/√L. A vector z CL with independent white zero-
of the multiuser APP by marginalizing over a subset of se- ±meanGaussianelement∈representsthermalnoisewithvariance
quences(inmanycasesonlyasmallsubsetisrequired),found σ2 per real dimension. In a coded system where each user
employs a rate R code and transmits with power P, the
ThisworkwasundertakenwhileA.KindwasattheInstituteforTelecom-
munications Research. A. Grant is supported in part by the Australian appropriate signal-to-noise measure we will use is Eb/N0 =
GovernmentundergrantDP0344856. P/2σ2RlogQ.
Weassumethateachuser’ssignalsarereceivedwithidenti- III. APPROXIMATION OF THEMULTIUSERAPP
calpower,phaseanddelay,althoughthesearenotfundamental
Theposteriorlogjoint-probabilityofaparticularhypothesis
restrictions imposed by the proposed receiver. After standard
sequence d′ is equal to
manipulationsthemultiple-accesschannelmayberepresented
by logp(r,d′ S,N0)=c 1 r Sd′ 2+logp(d′) (2)
r=Sd+z (1) | − N0 k − k
where c is a constant and p(d′) is the prior probability of the
where S=(s1, ,sK) 1/√L L×K. We only consider sequence. Expand the squared-distance term as
··· ∈{± }
the case that K > L, ie. the number of users exceeds the
number of independent observations. r Sd′ 2 =r∗r 2 e r∗Sd′ +d′∗ Sd′ 2
k − k − ℜ { } k k (3)
ThecanonicaliterativedecoderisshowninFig.1.Thegoal =c+ e yd′ + Sd′ 2
ℜ { } k k
istoinferthevalueofu ,k=1, ,K,basedonr,Sandthe
k ··· where y = 2r∗S. In order to simplify the search for
constraints . The module labelled Multiuser APP computes
the marginaClkposterior probability matrix ω(d) PQ×K, sequences d′ −that minimise (3), a recursive expression in
∈ d′, ,d′ may be obtained if G=S∗S is positive-definite.
which has as columns the probability mass functions for the 1 ··· K
This cannot be the case when K >L. In our model S is not
correspondingsymbols,basedonalltheavailableinformation
even guaranteedto have rank L. In order to obtain an equiva-
using the constraint (1), as well as the prior probability
lent full-ranksystem, we exploitthe following representation.
matrices ω (d). This inner decoder is the focus of this work.
a
The other constraint, separated from the first by an inter- K K
leaver, is the single-user decoders, which calculate extrinsic Sd 2 =d∗Gd= g d 2+d∗ g d
k k X kk| k| kX jk j
probabilities ωe(ck) based on the codes and ωa(ck) for all k=1 j=1
k. The process repeats by iteratively exchanging information j6=k
in the form of extrinsic probability matrices between the two
K  K 
modules. The individual APP decoders also compute, on the = ρ d 2+d∗ g d +(g ρ )d 2
final iteration, the data sequence probabilities ω(u ). X k| k| kX jk j kk− k | k|
k k=1 j=1 
 j6=k 
whereρ Risafreeparameter.Thetermsinbracketsdefine
ωa(dK) (cid:27)ωe(cK) the colukm∈ns of a new matrix, and a sufficient condition for
ΠK
that matrix to be positive-definite is that each term is positive
irrespectiveofd.Thefollowingprocedureisusedtotransform
ωa(d1) Π1 (cid:27)ωe(c1) the log-likelihood into an additive recursive metric with K
terms. This technique, which we believe to be new, is the
? ?
r - ωe(d1-) Π−11 ωa(c1-) A1PP ω(-u1) ma1in) cCohnotroisbeutaiopnoosiftitvheiscopnapstearn.t ρ R+ satisfying
∈
MuAltPiuPser ... ... ... ρ>(K 1)max−ℜe{Di∗Dj} (4)
− i,j i 2
S - ωe(dK-) Π−K1 ωa(cK-) AKPP ω(-uK) where 1, , Q are the elemen|tDs o|f .
2) ConstruDct t·h·e·veDctor u = diag(G) ρ1D RK, where
− ∈
1 is the all-ones vector.
Fig.1. Iterative joint-APP multi-user receiver. 3) Construct a new matrix G˜ by setting all diagonal ele-
ments of G equal to ρ. The matrix G˜ is guaranteed to
The computation of the marginal symbol posteriors in be positive-definite.
the APP detector entails marginalising the joint probability 4) Compute the factorisation T∗T = G˜, where T
p(d,rr,S,N0) over each possible sequence d. Brute force RK×K is lower-triangular. ∈
|
marginalisation for each user is therefore a summation with
QK−1 terms, clearly impractical for all but small numbers of By setting the prior term L(d′)=−N0logp(d′), and assum-
ing statistical independence of the prior symbol probabilities
users. In practice however, nearly all of the probability for
due to the interleaver, (2) may be written as
systems of interest is contained in a relatively small subset
of of those terms [10], and provided those terms can K
be iPsolated, the marginalisation can become computationally −N0logp(r,d′|S,N0)=c+Xℜe{ykd′k}+
tractable. This is the idea goes back to [11] (in an uncoded k=1
2 (5)
context) and has seen a recent revival in the framework of k
iterative processing [12–19]. The goal of the next section is (cid:12)(cid:12)(cid:12)Xtkjd′j(cid:12)(cid:12)(cid:12) +L(d′k)+uk|d′k|2
to approximate this sum with greatly reduced complexity. (cid:12)j=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
ForconstantenergysymbolconstellationssuchasQ-aryPSK, marginalisation.
the last term in (5) is absorbed into c, and we only require At low SNR in the early iterations, or with few receiver
ρ>(K 1). observations compared to the number of transmitters, large
−
Quadraticformssuchas(5)admitatreerepresentation[11]. numbers of paths will exist with similar path weight. Due to
The equation represents a Q-ary tree of depth K, where each complexityconstraintsin this scenario the numberof retained
of the Qk nodes at depth k represent a partial sequence pathsateachdepthmustbelimitedto ,andthealgorithm
max
P
with an associated positive path weight, and the leaf nodes essentially becomes the -algorithm with = . In
max
M M P
represent sequences d′ with total path weight equal to c more favourable circumstances however, very few paths are
−
N0logp(d′,rS,N0). Hence, the problem reduces to finding required and the T-algorithm adapts automatically to take
|
the leafnodesinthetreewithminimumweight,whichmay advantage of the conditions with greatly reduced complexity.
P
be approximated using tree search techniques. When no prior information is available the T-algorithm
The simple manipulations applied to G described above adapts very well to the channel, automatically finding a good
artificially create a virtual full rank channel from a rank- performance/complexity trade-off through the parameter T.
deficient one, assigning a positive path weight to every node As a general rule, the T-algorithm only tends to approach
in the tree and allowing sequential search to be applied to the bound in the early iterations, since the search is
max
P
(5). This is notto say that extrainformationis obtainedabout greatly facilitated by the prior probabilitiesonce they become
the symbols via the transformations described; only that the available. When very strong prior information is available
information about the interfering signals is spread out onto a however, the only sequences retained will be those dictated
greaternumberofeffectiveobservations,sothatanysequential by the priors, since other paths will be discarded in the early
search techniques developed for a full rank channel may also depths. In this case the detector will glean little new infor-
be applied in the overloaded or singular case. mation, and the information about the symbols will quickly
A transformation for overloaded linear systems was pre- become correlated over iterations. Hence, another parameter
sented in [20,21], which similarly creates a virtual full-rank must also be set, forcing the algorithm to consider a
min
P
system to which the full Q-ary tree may be assigned. The certain minimum number of sequences at each depth. The
approach is based on a minimum mean square error gener- effectisonlysignificantinhighlyloadedsystemswheremany
alised decision feedback equaliser filter, followed by lattice iterations are required for convergence.
reduction,columnre-ordering,andthentriangularfactorisation TheT-algorithmfindsfull-depthpathsthroughthetree,and
(if tree/sphere decoding is used). These transformations are the priorprobabilitiesareincorporatedin a naturalfashion.In
significantly more complex than our procedure, and may contrast, the depth-first strategy of [14,15] required special
be unsuitable for time-varying channels. The approach also handling of the prior probabilities, while the method of [18]
colours the noise, so that the system no longer lends itself required an initial breadth-first search in order to exploit
naturally to the iterative APP framework. the priors during the main search. The -algorithm based
M
A depth-first tree-search was used in [14,15,18], which approach used in [12,13] did not directly incorporate priors
necessitatedspecialtreatmentofthepriorprobabilityonthose (thiswasdoneinaseparatecombiningstep).Otherapproaches
paths that did notreach full depth.We propose a breadth-first incorporatethe prior probabilityinto spherical or branch-and-
search using the T-algorithm. The T-algorithm was used in bound decoders in various ways [16,17,19], but tend to be
[22,23] for near-optimal hard-decision decoding of channel quite complex and unsuitable for large-dimensionalsystems.
codes up to a pre-determined minimum-distance with signifi- The receiver complexity is dominated by Cholesky factori-
cantcomplexitysavingsovertheViterbialgorithm.Therelated sation of the K K matrix G˜, and then by the tree search
×
-algorithmretainsexactly pathsateachdepth,regardless during the iterations. The complexity of the T-algorithm is
M M
of the actual weights of each partial path (this approach was upper bounded by K node computations per iteration,
max
P
usedin[12]).Inpracticethestatisticalnatureofthenoiseand but this bound only ever tends to be reached in the early
the spreading sequences for each transmission may require iterations, as discussed above. Contrast this for example with
a different number of sequences to approximate the APP. It the LMMSE filter [9] as the inner detector, which requires an
should also be noted that any other tree search algorithm initial matrix inverse, and then a matrix inverse per user on
could be used, with slightly varying levels of performance each subsequent iteration.
and complexity. The key step is (5), which admits such tree
representations for overloaded systems. IV. NUMERICAL RESULTS
We exploit the heuristic observation that paths with very Inthissectionweconsiderabenchmarkmodel,withlength
large partial path weight are unlikely to be components of L = 8 random PN spreading sequences and no fading. The
low weight paths. Rather than retaining a fixed number of model is difficult to work with, since a significant probability
paths at each depth, the T-algorithm attempts to adapt to existsthatthespreadingmatrixSwillhavelinearlydependent
the channel conditions by only retaining paths at each depth rows.
with weight not exceeding the best weight by more than T, The individual users transmit BPSK symbols, which are
where T is a parameter of the algorithm. When the algorithm encoded with a nonsystematic 4-state rate 1/2 convolutional
terminates at the leaves, the best sequences are used in the code, described by the feed forward generator polynomials
P
(05,07).A length 2I =1000interleaverbetween the encoder computations, assuming the best case that S has rank L.
and the transmitter is generated randomly for each user. An extrinsic-information transfer chart [24] shows that the
We consider joint iterative decoding of the system using T-algorithm detector is very well matched in shape to the
the T-algorithm, where is set to 512 and the threshold particular code, which helps to explain the good performance
max
P
T is set to 16N0. The bound max is deliberately set large after many iterations, even at very high loads. The charts,
P
in order to demonstrate the performance advantage of closely which we do not include here, also predict very accurately
approximating the APP. The T-algorithm for the overloaded the convergence characteristics shown in Figures 2 and 3.
case is furnished by the matrix manipulations proposed in
Section III for computing the log-likelihood. 100
20users
100 10−1
2 5 0 i ti et e r ra a t it oi o n ns s 15users 17 19
10−2 16 18
10−1
R
PIC wiltihstTd-eatl.g BE10−3 single-userbound
10−2
R
BE MMSE 10−4
10−3
10−5
10−4 10−6
2 2.5 3 3.5 4 4.5 5 5.5 6
Eb/N0
0 5 10 15 20 Fig. 3. CDMA system BER performance after 20 iterations as a function
number ofusers K ofSNR.Spreading gainL=8.
Fig. 2. Comparative CDMA system BER performance as a function of
numberofusersK.SpreadinggainL=8,Eb/N0=5dB. Figure 4 shows the spectral efficiency of the receiver for
the system described above, as a function of Eb/N0, mea-
The performance of the T-algorithm in the above model sured as the maximum number of users for which single-
is shown as a function of the number of users in Figure 2, user performance is reached. Also shown is the maximum
at Eb/N0 = 5 dB. Also shown is the performance of two spectral efficiency C achievable by using both an optimal
linearfilterscommonlyusedasthemulti-userdetectorinsuch jointreceiver,and an MMSE detector followedby single-user
systems,thePIC[8]andtheLMMSE[9]filters.Theparameter decoding. These curves were approximated using the large-
for the T-algorithm is set to 32 for K 16, =64 systemsexpressionsforrandomspreadinggivenin[25]under
min min
P ≤ P
for K =17,18, and =128 for K =19,20 users. These the constraint C =KR/L with R=1/2.
min
P
values were found by experiment to be sufficiently large for Thereceivereasilyapproachesoptimaljoint-processingdata
the loads considered. rates at low SNR, but cannot maintain this slope with in-
While very computationally efficient, the PIC can only creasingSNR.Nevertheless,theiterativeT-algorithmreceiver
support 9 users after 20 iterations, and is clearly not suitable outperforms any other practical algorithm we are aware of in
for highly loaded systems. The MMSE filter performs better, terms of system load for the given channel model. Various
supporting 14 users after 20 iterations, but requires a matrix other results for randomly spread CDMA in AWGN using
inversion per user per iteration. a rate 1/2 code are available in the literature, utilising the
Estimating the detector APP is a highly non-linear cal- same canonical receiver structure but differing in the multi-
culation, and the linearised models and assumptions used user detector implementation. These are shown in Figure 4,
by the above filters are not necessarily valid in the model along with references to the relevant papers.
under consideration. The performance of the T-algorithm in
Figure 2 clearly demonstrates the performance advantage of V. CONCLUSION
approximatingthe APP directly using the rulesof probability. We have shown that near-optimal performance may be
List-detection using the T-algorithm supports 16 users with achieved with low complexity in a randomly spread CDMA
only 5 iterations and 19 users after 20 iterations at 5 dB. channel by employing the turbo principle in an iterative
To our knowledge we have not seen loads in such a system receiver. This is not a new observation; our contribution is
approaching those achieved here. to show that by attempting to calculate the true symbol-APP
In Figure 3 is shown the performance of list detection distributions in the inner detector, the performance is signif-
as a function of Eb/N0, for various number of users, after icantly improved over detectors that employ linear filters, or
20 receiver iterations. Note that without the log-likelihood otherstructuresderivedusingalternativeconsiderations.Close
transformation of Section III, the tree search with 20 users approximation of the desired APP distributions in overloaded
would require at least one stage with 2K−L = 4096 node or singular channels is practically facilitated by the simple
3
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