Decoding and EM Channel Estimation
Mari Kobayashi y
and Joseph Boutros y
and Giuseppe Caire
?
April 26, 2001
y Ecole Nationale Superieure des Telecommunicationsde Paris,
64 rue Barrault { 75634ParisCEDEX, France.
Email: [email protected], [email protected].
? Institut Eurecom,
2229 Route des Cretes, B.P. 193
06904Sophia-AntipolisCEDEX,France.
Email: [email protected]
Abstract
We derive a low-complexity receiver scheme for joint multiuser decoding and
parameterestimation of CDMA signals. The resulting receiver processes the users
seriallyanditeratively,andmakesuseofsoft-insoft-outsingle-userdecoders,ofsoft
interference cancellation and of expectation-maximization parameter estimation as
themainbuildingblocks.
Computersimulationsshowthattheproposedreceiverachievesnearsingle-user
performanceat very highchannelload(numberofusers perchip)and outperforms
conventionalschemes withsimilarcomplexity.
Keywords: Interference cancellation,joint data detection and parameter estimation.
1 Introduction
AmongtheseveralmultiuserdetectionschemesproposedforCDMA[1],SerialandParallel
Interference Cancellation(SIC and PIC)are particularlyattractive becausethey process
directly the output of a bank of single-user matched lters (SUMF). The receiver front-
end is identical to that of conventional detection. Therefore, these methods can be seen
asan\add-on"post-processingtoenhancetheperformance ofaconventionalbase-station
receiverwhenparticularlyhigh channelloadisneeded, andcan beappliedeasilytoeither
short and long spreading sequence formats [2, 3,4].
ThemainperformancelimitationofSIC/PICschemesare: 1)errorpropagationcaused
by feeding back erroneous symboldecisions; 2)imperfect interference cancellation due to
non-ideal knowledge of channel parameters (e.g., the complex amplitudes and delays of
theusers' multipathchannels). In thiswork, weproposeareceiverscheme whichhandles
successfully both problems.
SICisbothsimplerandmorerobustthanPICwithrespecttoerrorpropagation,since
users can beranked according to their signal-to-interference plus noise ratio (SINR) and
decodedinsequence[5,6,7,8]. Hence,wefocusonSICschemes. Inearlyworks[6,5],SIC
isappliedtouncodedtransmissionandharddecisionsareusedateachstagetoremovethe
already detected users from the received signals. In order to prevent error propagation,
the use of soft (or partial) interference cancellation and iterative SIC schemes has been
proposed in dierent forms and by dierent authors (see for example [9, 10, 8]). More
recently,the SICapproach has been combined with channel coding and Soft-InSoft-Out
(SISO) decoding [11]. The number of works in this direction is overwhelming. Without
theambitionofbeingexhaustive,wereferto[12,13,14,15,16,17,8,18,19,20,21,22,23]
and references therein. A common feature of these algorithms is that single-user SISO
decoders provide at each iteration an estimate of the a posteriori probabilities (APP)
for the user code symbols, which are used to form a soft estimate of interference to be
subtracted from the received signal. In this way, the contribution of a user is eectively
subtracted fromthe signal only if itssymbol decisionsare suÆciently reliable.
A unied framework toiterative multiuserjointdecoding based onfactor-graphs and
sum-product algorithm[24] is provided in [25]. In this framework, almost all algorithms
previously proposed (notably, those of [12] and of [23]) have been re-derived in a simple
direct way. Moreover, as a consequence of the sum-product approach, it is found that
extrinsic (EXT) probabilities [26] rather than APPs should be fed back to form the soft
interference estimate. As conrmed experimentally by [27], APP-based soft interference
signal,and the APP-basedalgorithmsof[12, 23] attainaworse overall spectraleÆciency
than their EXT-based counterparts derived and analyzed in[25].
Inordertoreduce parameterestimationerrors,iterativeSICschemescanbenaturally
coupledwithiterativeparameterestimationinorderto(hopefully)improvethe estimates
with the iterations, aslong as the signalis \cleaned-up"from interference (see for exam-
ple [28]). In [29] the trade-o between the number of users per chip (channel load) and
the amountof trainingsymbolsis investigated ina general iterative joint decoder which
re-estimates the channel parameters ateachiteration.
Wepropose alow-complexityiterativesoft-SIC algorithmfor jointdatadetection and
channel parameterestimation. The main buildingblocks ofour receiver are SISO single-
user decoders, soft interference cancellation stages and a channel parameter estimation
updating step which is formallyequivalent toone step of the Expectation-Maximization
(EM)algorithm[30,31]. Thekeyideatoachievepolynomialcomplexityinthenumberof
usersistoapplyEM\locally",i.e.,insteadofusingthetrueaposterioridistributionofthe
missingdatagiventheobservationandthecurrentparameterestimate,weusetheproduct
distributioninducedbytheaposteriorimarginal(symbol-by-symbol)probabilitiesoutput
by the SISO decoders ateach receiver iteration.
Werestrictour treatmenttosynchronous CDMA withfrequency non-selectivepropa-
gationchannels. Users are synchronous atthe chip, symboland framelevelandencoding
and decoding is performedframe by frame. We assumealsothat the channelparameters
remain constant over each frame. The reason for adoptingthis simple modelis twofold:
on one hand, this model allows the development of the algorithmin a simple and clear
way, on the other hand frame-synchronous transmission with piecewise constant channel
parameters is quite realistic in systems like UMTS-TDD [3], applied to indoor and pic-
ocells with slowly moving user terminals. Generalization to asynchronous transmission
andcontinuously time-varyingmultipathchannelsisleftasaninterestingtopic forfuture
work.
Related work can befound, for example, in[32] (see also[31] and references therein),
where EM channel estimation is appliedto SIC in an uncoded system. In [33], joint pa-
rameter estimation and data detection in a multiuser multipath environment is tackled
by using an alternating maximization strategy and EM is used to solve the parameter
estimates updating step. In [34, 35, 36], the EM approach is applied to the joint data
detection and parameter estimation in a single-user space-time coded system. In [37],
the SAGE algorithm [38] is applied to joint MAP symbol-by-symbol detection and pa-
rameter estimation in an asynchronous CDMA system. The algorithmsobtained in [37]
(as opposed to what we do here). Classical references on the application of EM in com-
munications problems are [39], where EM is applied to parameter estimation in digital
receivers, and [40], where several iterative multiuser schemes for uncoded CDMA (with
perfectly known parameters) are derived as applications of EM and SAGE.
Thepaperisorganizedasfollows. InSection2thesynchronous CDMAsignalmodelis
presented. In Section3we derivethe proposedreceiverstructure In Section4we present
some numericalresults and in Section5 we summarize our conclusions.
Notation conventions:
Let A be a matrix, then a
n
;a k
and a
k;n
(or equivalently [A]
k;n
) denote the n-th
column,the k-th row and the (k;n)-thelement of A.
z N
C
(;) indicates that the randomvector z is complex circularly-symmetric
jointly Gaussianwith mean E[z]= and covarianceE[(z )(z ) H
]=.
The superscript H
indicates Hermitiantranspose.
A/B indicates that A and B dier by a multiplicativeterm.
A :
=B indicates that A and B dierby anadditive term.
Probability density functions (pdf) are denoted by p() and probability mass func-
tions (pmf)are denoted by Pr ().
2 System Model
Weconsidertheuplinkofacodeddirect-sequence CDMAsystemwithsynchronoustrans-
mission overfrequency-non-selective channels and Nyquist chip-shaping pulses [41]. The
system is frame-oriented, i.e., encoding and decoding is performed frame-by-frame and
users are synchronous also at the frame level. In each frame, the complex baseband
equivalent discrete-time signal originated by sampling at the chip rate the output of a
chip-matched lter is given by [1]
(
Y =SWX+N Data transmission phase
Y (t)
=SWX (t)
+N (t)
Trainingphase
(1)
where:
Y 2 C LN
and Y (t)
2 C LT
are the arrays of received signal samples in the data
N2C LN
and N (t)
2C LT
arethe correspondingarraysofnoise samples,assumed
complexcircularly-symmetric Gaussiani.i.d. N
C (0;N
0 ).
S2C LK
contains the user spreading sequences by columns.
W=diag(w
1
;:::;w
K
) contains the user complex amplitudes w
k .
X2C KN
is the array of transmitted code symbols.
X
(t)
2C KT
is the array of transmitted trainingsymbols (known at the receiver).
N;T;L and K denote the code block length and the training sequence length (in
symbols), the spreading factor (number of chips per symbol) and the number of
users, respectively.
The total frame length in symbols is equal to N +T. Since the channel amplitudes
remain constant over the whole frame and the system is synchronous, the position of
training symbols in the frame is irrelevant and arbitrary. 1
With reference to the above
modeland toour notationconventions, s
k
;x k
;y
n
and x
n
denote the k-thuser spreading
sequence, the k-thuser code word, the received signal vector inthe n-thsymbol interval
and the transmitted symbol vector in the n-th symbol interval, respectively. The user
spreading sequences are normalized such that js
k j
2
= 1 for all k. Hence, the signal-to-
noise ratio (SNR) of user k is given by SNR
k
= jw
k j
2
=N
0
. The corresponding system
block-diagramis shown in Fig. 1.
At each frame, each user encodes a sequence of information bits into a code word
x k
2C
k
, whereC
k
isthe codebookof userk,dened overagivencomplexsignalset(e.g.,
a PSK or QAM constellation). In this paper we consider non-systematic non-recursive
convolutionalcodeswithtrellistermination,mappedontoBPSK,sothatx
k;n
2f 1;+1g.
Each code word is independently interleaved beforetransmission.
3 Iterative joint data detection and parameter esti-
mation
Without loss of generality, we assume that the user decoding order at each iteration is
k =1;2;:::;K. Decodingof userk atiterationm inthesoft-SIC receiverisbasedonthe
1
Inpractice,forslowly-varyingfrequency-selectivechannelsitisconvenienttoplacethetrainingphase
inthemiddleofeachframe[3].
observed signal sequence
z (m)
k;n
= 1
b w
(m)
k s
H
k y
n
| {z }
SUMF output k 1
X
j=1 s
H
k s
j b w
(m)
j
b w
(m)
k b x
(m)
j;n
| {z }
currentiteration K
X
j=k+1 s
H
k s
j b w
(m)
j
b w
(m)
k b x
(m 1)
j;n
| {z }
previous iteration
(2)
for n = 1;:::;N, where fwb (m)
j
: j = 1;:::;Kg are estimates of the user amplitudes at
iterationm, fbx (m)
j;m
:j =1;:::;k 1gare estimates of the user symbols already decoded
atiterationm andfbx (m 1)
j;m
:j =k+1;:::;Kg areestimates ofthe usersymbolsprovided
by the previousiteration,since these users are not yetdecoded at iterationm.
DecodingisperformedbyaSISOdecoder,whichinthecaseofconvolutionalcodescan
beimplementedeÆcientlybytheforward-backwardBCJRalgorithm[42]. Letp(z (m)
k;n jx
k;n
=
a) be the conditional pdf of z
k;n
given x
k;n
= a, with a 2 f 1;+1g. The SISO decoder
for user k produces a marginalEXT pmffor x
k;n
, given by
EXT (m)
k;n (a)/
X
c2C
k :cn=a
Y
`6=n p(z
(m)
k;`
jx
k;`
=c
`
) (3)
where the normalizationEXT (m)
k;n
(+1)+EXT (m)
k;n
( 1)=1is enforced. The corresponding
APP isgiven by
APP (m)
k;n
(a)/p(z (m)
k;n jx
k;n
=a)EXT (m)
k;n
(a) (4)
with again the normalizationAPP (m)
k;n
(+1)+APP (m)
k;n
( 1)=1.
Assuming that z
k;n
is conditionally(marginally) circularly-symmetriccomplex Gaus-
sian given x
k;n
,the pdf p(z (m)
k;n jx
k;n
=a) can be approximated as
p(z (m)
k;n jx
k;n
=a)/exp jz
k;n aj
2
(m)
k
!
(5)
where (m)
k
= E[jz (m)
k;n x
k;n j
2
] is the residual interference plus noise variance, which is
independent of n undermilduniformity conditions onthe user codes [25].
The SISO decoders outputalso APPsfor the informationbits, whichwillbeused for
nal symbol-by-symboldecisionsinthe lastiteration. For simplicity,weassume that the
total number of iterations M is xed for all users. In practice, M should be optimized
according to the SNR and channel load K=L. Also, some dynamic stopping criterion
might be used in order to minimize the number of iterations. We leave this interesting
topic for future work.
Next, we address the estimation of the residualinterference plus noise variance (m)
k ,
the estimation of the code symbols x and the estimation of the user amplitudes w
usedinthe soft-SIC(equation(2)). Wealsoaddressthe initializationof thereceiverwith
training-based parameter estimation and some methods to combine training-based and
EM-based estimation. Finally, we summarize the resulting soft-SIC receiver with joint
data detection and parameter estimation.
3.1 Estimation of the residual interference plus noise variance
The variance (m)
k
isunknown, and must beestimated on-line before each SISO decoding
step. Let (m)
k;n
= z (m)
k;n x
k;n
denote the residual interference plus noise term in (2). A
simple estimatorfor (m)
k
is given by 2
b
(m)
k
= 1
N N
X
n=1 jz
(m)
k;n j
2
1 (6)
Besideits simplicity,the motivations for using(6) toestimate (m)
k are:
1. If (m)
k;n
and x
k;n
are uncorrelated, then b (m)
k
is anunbiased estimator.
2. If x
k;n
is i.i.d., uniformly distributed on f 1;+1g (as in our case), (m)
k;m
is i.i.d.
N
C (0;
(m)
k
), and x
k;n
; (m)
k;n
are uncorrelated, then the error variance of b (m)
k is
given by
E
(m)
k b
(m)
k
2
= 1
N
4 (m)
k
+( (m)
k )
2
while the error variance of the Maximum-Likelihood (ML) estimator with known
x
k;n
is given by
E 2
4
(m)
k
1
N N
X
n=1 jz
(m)
k;n x
k;n j
2
2
3
5
= 1
N (
(m)
k )
2
Hence,if4
k
=N 1theproposedestimatorperformsveryclosetotheMLestimator
for known code symbols.
3. Ifthecomplexamplitudeisestimatedreliably,i.e., wb (m)
k
w
k
,andifx
k;n
isuncorre-
latedwith bx
j;n
for j 6=k, then (m)
k;n
and x
k;n
are practicallyuncorrelated. Moreover,
undermildconditions on the user amplitudes, for large K the residualinterference
term (m)
k;n
is asymptotically Gaussian [43, 25]. We conclude that for large N and
2
Itiseasilyshownthatb (m)
k ;n
istheMaximum-Likelihoodestimatorforthevarianceoftheprocess (m)
k ;n
from the observation z (m)
k ;n
= x
k ;n +
(m)
k ;n
when x
k ;n and
(m)
k ;n
are white, statistically independent, and
Gaussianwithx
k ;n
N (0;1)and (m)
k ;n
N (0;
(m)
k ).
K the estimator b (m)
k
performs very close to the ML estimator for known coded
symbols.
Inthe actualreceiverimplementation,theEXTand APPpmfs(3)and(4)arecalculated
by using (5)where (m)
k
isreplaced by its estimateb (m)
k
given by (6).
3.2 Soft estimation of the code symbols
The (non-linear)MMSE estimateof symbol x
k;n
given the observation Y is given by the
conditional mean[44]
x mmse
k;n
=E[x
k;n
jY]=+Pr(x
k;n
=+1jY) Pr(x
k;n
= 1jY)=2Pr(x
k;n
=+1jY) 1 (7)
where Pr (x
k;n
= ajY) is the a posteriori pmf of symbol x
k;n
given the observation Y.
We are tempted to replace Pr(x
k;n
= ajY) by APP (m)
k;n
(a) given by the SISO output at
iteration m and let bx (m)
k;n
= 2APP (m)
k;n
(+1) 1, and claim that this choice minimizes the
residualinterference varianceand itis thereforeoptimal. Unfortunately, this reasoningis
incorrect. Anintuitiveway ofseeingthis isby contradiction: ifthe truea posterioripmfs
Pr(x
k;n
= ajY) were available at some iteration, then optimal symbol-by-symbol MAP
decisionscouldbe madeandthere would benoneed forfurtherinterference cancellation.
Moreover, theexactcalculationofaposterioriprobabilitiesPr (x
k;n
=ajY)isingeneralan
NP-complete problem [1]. Therefore, if aftera nite number of iterations m an iterative
algorithm(withpolynomialcomplexityinK)obtainsexactvalues forPr (x
k;n
=ajY)the
NP-completeness would be violated. Hence, we conclude that APP (m)
k;n
(a) 6 Pr(x
k;n
=
ajY),for any nite numberof iterationsm.
Interestingly,theabove\non-linearMMSEargument"hasbeenusedinseveral papers
(e.g., [22, 23, 8, 18]), sometimes with claim of optimality. On the contrary, by using a
rigorous derivation based on factor-graphs and on the application of the sum-product
algorithm,itcan beshown that [25]:
1. Even forperfectlyknown amplitudesand SISOinput variances(i.e., wb (m)
k
=w
k and
b
(m)
k
= (m)
k
), the residual interference term (m)
k;n
= z (m)
k;n x
k;n
in (2) when using
b x
(m)
k;n
= 2APP (m)
k;n
(+1) 1 is conditionally biased and the bias tends to cancel the
useful signal, i.e.,
E[
(m)
k;n jx
k;n
=a]= (m)
k;n a
where (m)
k;n
isanon-negativequantitythat maydependonk;n andonthe iteration
2. Byusing EXT-based instead of APP-based symbolestimates, i.e., by using xb (m)
k;n
=
2EXT (m)
k;n
(+1) 1,the resultingresidualinterference termisconditionallyunbiased,
i.e., E[
(m)
k;n jx
k;n
]=0,and the overall soft-SIC algorithmattains better performance
thanits APP-basedversion. Remarkably, this eectis not visibleforsmall channel
loadbut,asK=Lincreases,the dierencebetween APP-basedand EXT-basedsoft-
SICschemes is more and more evident[27].
Inpassing,wenoticealsothatabiasedresidualinterferenceimpliesthatx
k;n and
(m)
k;n are
correlated (even for perfect amplitude estimation). Hence, the variance estimator (6) is
asymptoticallyoptimal for large N;K onlywhen the symbol soft estimates are obtained
fromEXT pmfs.
Drivenbytheresultsof[25]andbytheaboveconsiderations,weshallusethefollowing
soft symbol estimates
b x
(m)
k;n
=2EXT (m)
k;n
(+1) 1 (8)
whichcan beregarded asa\local"MMSEestimateof x
k;n
assumingthat theaposteriori
pmfof x
k;n
is EXT (m)
k;n
(a)(even if it isnot true!).
3
3.3 Estimation of the user complex amplitudes
Let w = (w
1
;:::;w
K )
T
denote the vector of complex amplitudes to be estimated. The
ML estimateof w given the observation Y isgiven by
w ML
=arg max
w
logp(Yjw) (9)
where p(Yjw)is the conditional pdf of the observed signal given w, given by
p(Yjw) / X
X
p(YjX;w)Pr(Xjw)
/ X
x 1
2C
1
X
x K
2C
K exp
1
N
0 N
X
n=1 jy
n SX
n wj
2
!
(10)
where we have dened the diagonalmatrix X
n
=diag(x
1;n
;:::;x
K ;n
) and where wehave
used the factthat the channelinput Xis independent of the channelamplitudes, sothat
3
In [25], expression (8) is derived as a direct consequence of the application of the sum-product
algorithm, without any heuristic motivation based on MMSE estimation. The fact that EXT-based
algorithmsperformbetterthanAPP-basedalgorithmsjust putsinevidencethepowerandgeneralityof
thesum-productapproachtostatisticalinferenceproblemsonBayesiannetworks(see[45]andreferences
Pr(Xjw)=Pr (X)= uniform on the Cartesian product of the code books C
1
C
K
and zero outside, since each user k selects its code word with uniform probability onits
code book C
k
and independently of the other users. From (10) itis clear that direct ML
estimation of w is infeasible in any practical case, as it has complexity proportional to
the total number of user code words Q
K
k=1 jC
k j.
Now, assume that the estimate wb (m)
and the a posteriori probability Pr(XjY;wb (m)
)
are available at iteration m. Then, we can produce an updated estimate wb (m+1)
for
next iterationby following the EM approach. In the languageof the EM algorithm[31],
Y;X and fY;Xg play the role of incomplete, missing and complete data. The EM
update consists of computing the expected log-likelihood function of the complete data
conditionally on the incomplete data and on the current parameter estimate (E-step),
and maximizing the result with respect to the parameter (M-step) [31]. In our case, the
complete data log-likelihoodfunction is given by
logp(Y;Xjw) :
= logp(YjX;w)
:
= 1
N
0 N
X
n=1 jy
n SX
n wj
2
:
= 2
N
0 Re
r H
w 1
N
0 w
H
Rw (11)
where wedene the vector
r= N
X
n=1 X
n S
H
y
n
= N
X
n=1 2
6
6
6
6
4 x
1;n s
H
1 y
n
x
2;n s
H
2 y
n
.
.
.
x
K ;n s
H
K y
n 3
7
7
7
7
5
(12)
and the KK matrix
R= N
X
n=1 X
n S
H
SX
n
(13)
with (i;j)-th element
[R]
i;j
= (
N for i=j
s H
i s
j P
N
n=1 x
i;n x
j;n
for i6=j
Byusing (11) we obtain the E-step in the form
Q(w;
b
w (m)
) = E[logp(Y;Xjw)jY;
b
w (m)
]
= X
X
Pr(XjY;wb (m)
)logp(Y;Xjw)
:
= 2
N Re
r H
w 1
N w
H
Rw (14)
where welet r=E[rjY;wb (m)
] and R=E[RjY;wb (m)
]. These are given by
r= N
X
n=1 X
n S
H
y
n
(15)
and by
[R]
i;j
= (
N for i=j
s H
i s
j P
N
n=1 x
i;n x
j;n
for i6=j
(16)
where X
n
= diag(x
1;n
;:::;x
K ;n
) and where x
k;n
and x
k;n x
j;`
denote the rst and second
momentsof the joint aposterioripmf Pr(XjY;wb (m)
),given by
x
k;n
= X
X x
k;n
Pr(XjY;wb (m)
)
x
k;n x
j;`
= X
X x
k;n x
j;`
Pr(XjY;wb (m)
) (17)
By noticing that (14) is a quadratic form in w and that R is non-negative denite, the
M-step is readily obtained as
b w
(m+1)
=arg max
w
Q(w;wb (m)
)=R 1
r (18)
The aboveprocedure has stillcomplexityexponentialinK,since the computationof the
moments(17) isequivalenttothe marginalizationof thejointpmfPr (XjY;wb (m)
), which
has complexityexponentialinK. Then,we shallapply the aboveEM step\locally",i.e.,
by replacing Pr(XjY;wb (m)
) by the productof the marginalAPPs producedby the SISO
decoders atthe end of iterationm. Namely, weuse the approximation
Pr (XjY;wb (m)
) K
Y
k=1 N
Y
n=1 APP
(m)
k;n (x
k;n
) (19)
As shown in the previous section, the APPs do not coincide in general with the true
marginalsof thejointpmf Pr(XjY;wb (m)
). However, the utilityofthe approximation(19)
is twofold: on one hand, the product pmf in the LHS is readily available from the SISO
outputs. On the other hand, thanks tothe product form, the exponential complexity of
the moment computation is reduced to linear. In fact, the moments of the product pmf
are given by
e x
k;n
= +APP (m)
k;n
(+1) APP (m)
k;n
( 1)=2APP (m)
k;n
(+1) 1
^ x
k;n x
j;`
= (
1 for (k;n)=(j;`)
e
x xe otherwise
(20)
Finally,the proposed approximated EM updating step consists of computing (18) where
R and r are given by (15) and by (16) when replacing the true moments (17) by their
approximations (20).
The complexity of (18) is then dominated by the matrix inverse R 1
, which must be
computed ateachiteration. Asuboptimal M-step that doesnot require a matrix inverse
canbeobtainedbynoticingthat,undermildconditionsonrandominterleavingandonthe
uniformity of user codes, the averaged symbolsxe
k;n
are symmetrically distributed (their
distributionisinducedby thenoiseand by therandomchoiceofthe usercode words over
the code books). Moreover, xe
k;n
and xe
j;n
are weakly correlated for k6=j. Then, 1
N
RI
for large block length N. Hence, under these conditions (18) can be approximatedby
b
w (m+1)
= 1
N
r (21)
Notice that both (18) and (21) are directly computed from the SUMF outputs, since r
dened in(15) depends onthe observed signalY onlythrough the SUMF outputss H
k y
n .
3.4 Initialization and combining with the training phase
Theoveralliterativesoft-SICalgorithmneedsasuÆcientlyreliableinitialestimatewb (0)
of
the complex user amplitudes. Otherwise, for completelyunknown w, the SISO decoders
at the rst iteration yield APPs very close to 1=2, i.e., xe
k;n
0 for all k and n. This
yields r 0 and R = NI, which in turns yields wb (1)
0, so that the receiver never
\bootstraps" and remainsstuck atthe \zero"xed point.
Forthesakeofinitialization,ajointMLestimateofthecomplexamplitudesisobtained
fromthe training phase. This isreadily given by [44]
b w
(t)
= R
(t)
1
r (t)
(22)
where r (t)
and R (t)
are given by (12) and by (13), respectively, when replacing N by T
and the code symbols x
k;n
by the known trainingsymbolsx (t)
k;n
. If the trainingsequences
are mutually orthogonal, i.e., such that (X (t)
) H
X (t)
= TI, we obtain R (t)
= TI and no
matrix inverse is needed in (22). It can be shown that this choice also minimizes the
estimation error variance [46]. Then, if a set of mutually orthogonal training sequences
exists, this choice shouldbepreferred.
4
The receiver isinitializedbyletting b
w (0)
= b
w (t)
. Then, atiterationsm=1;2;:::; the
receiverexploitstheupdatedestimatewb (m)
providedbytheEMstep(18)bycombiningit
4
Theexistenceofsuch setoftrainingsequencesdependsonthetrainingsymbolalphabet andonthe
insomewaywiththetraining-basedestimate.
5
Weinvestigatethefollowingtwomethods
for combining the training phasewith the EM update.
The mixing method. Form=1;2;:::,the \local" EMestimation described aboveis
appliedtothe incompletedatafY;Y (t)
gwith missingdataX, by treatingX (t)
asknown
parameters. The sameresult is obtained by includingthe known trainingsymbolsin the
missing data and by dening their marginal pmfs as APP (t)
k;n
(a) = 1 if a = x (t)
k;n
and 0
if a 6= x (t)
k;n
, so that for training symbols we have xe
k;n
= x (t)
k;n
[29]. After straightforward
algebra,completely analogous to the derivation of the previous section and not reported
here for the sake of space limitation,we obtainthe mixing estimatoras
b
b w
(m)
mix
=
R+R (t)
1
r+r (t)
(23)
The combining method. Assume for simplicity that the training sequences are mu-
tually orthogonal. Then, wb (t)
= w + (t)
with (t)
N
C (0;
N0
T
I). In particular, the
training-basedestimatorwb (t)
is unbiased.
Now, from(15) and (1)we obtain
r= N
X
n=1 e
X
n S
H
(SX
n w+n
n )=R
0
w+
where R 0
= P
N
n=1 e
X
n S
H
SX
n
and N
C (0;N
0 R
00
) with R 00
= P
N
n=1 e
X
n SS
H
e
X
n . By
using this into (18) we have
b w
(m)
=R 1
R 0
w+R 1
(24)
Since R 6= R 0
unless the code symbols are perfectly known, the result of EM is biased.
For the sake of simplicity, we assume that N is suÆciently large so that the following
approximations hold
R NI
R 0
diag N
X
n=1 x
1;n e x
1;n
;:::; N
X
n=1 x
K ;n e x
K ;n
!
R 00
diag N
X
n=1 jex
1;n j
2
;:::; N
X
n=1 jex
K ;n j
2
!
(25)
5
TheeÆcient useof the available trainingsymbolsin addition to someblind parameter estimation
(this follows by the fact that, under mildconditions, the out-of-diagonal terms are nor-
malized empirical correlations between uncorrelated zero-mean sequences, which vanish
forlargeN). Byusing(25) in(24)weobtainthe biasedEMestimateof userk amplitude
as
b w
(m)
k
=
k w
k +
0
k
(26)
where
k
= 1
N N
X
n=1 x
k;n e x
k;n
and where 0
k
N
C (0;
N
0
N
2
k
)is the k-thcomponent of R 1
,with
2
k
= 1
N N
X
n=1 jex
k;n j
2
Now, our goal is toobtain acombined estimatorinthe form
b
b w
(m)
k;comb
=a
k b w
(m)
k +b
k b w
(t)
k
(27)
wherethecoeÆcientsa
k andb
k
are choseninordertominimizetheerrorvariancesubject
tothe unbiased constraint, i.e., they are the solutionof
(
minimize E[ja
k
0
k +b
k
(t)
k j
2
]
subject to a
k
k +b
k
=1
Since 0
k
and (t)
k
are mutually independent (they depend on the mutually independent
noise samplesN and N (t)
inthe data and trainingphases), weobtain easilythe solution
of the above problemas
a
k
=
k
2
k +
T
N
2
k
b
k
= T
N
2
k
2
k +
T
N
2
k
(28)
Onelastproblemisrepresentedbythefactthat
k
dependsontheunknowncodesymbols
x
k;n
. Then, anestimate of
k
can be obtained asfollows. We notice that
x
k;n e x
k;n
= (
jex
k;n
j for sign(ex
k;n )=x
k;n
jex
k;n
j for sign(ex
k;n )6=x
k;n
Sincesign(ex
k;n
)isthe maximumaposteriorisymbol-by-symboldecisiononthe codesym-
bolx
k;n
basedontheaposterioripmfAPP (m)
(a)outputbythe SISOdecoderatiteration
m, for large N the following approximation holds
k
(1 2
k )
1
N N
X
n=1 jex
k;n
j (29)
where
k
is the symbol error probability (on the coded symbols, not on the information
bits!) at the output of the SISO decoder for user k at iteration m. If the residual
interference plus noise process (m)
k;n
is Gaussian with variance (m)
k
, the error probability
k
isaknownfunctionof (m)
k
,determinedbytheusercodeC
k
. Thiscanbepre-computed
and stored in a look-up table, and an estimate b
k of
k
can be easily obtained from the
estimateb (m)
k
given by(6). Finally,
k
canbeapproximatedby replacing
k byb
k
in(29).
Remark. Weprovideaqualitativeandintuitivediscussiononthebehaviorofthemixing
and combiningmethods.
The mixingmethodsuers frombiasinthecase oflarge K=LandT=N 1(whichis
clearly the most interesting case, as it is usually desirableto maximize the channel load
and minimizethe lengthof the trainingphase). In fact, suppose that at iterationm=0
the signal at the input of each SISO decoder is \very noisy", since the interference has
not been removed yetand K=L is large. Then, the averaged symbols ex
k;n
output by the
SISO decoders are all close to zero. Assuming orthogonal training sequences (the best
case), the mixingmethodyields R+R (t)
(N+T)I,r 0and r (t)
=Tw+noise. The
resultingestimatoris
b
b w
mix
T
N +T
w+noise
which is clearly biased. In particular, if T=N 1, the bias might prevent the whole
receiverto bootstrap.
6
On the contrary, the combining method (assuming
k
known) provides an unbiased
estimateateachiteration. Attherst iterations,when
k
1=2, thena
k
0,b
k
1and
b
b w
(m)
comb wb
(t)
, i.e., only the result of training-based estimation is used. As the soft-SIC
cleans-upthe signalfrominterference and
k
becomessmall(convergingtothesingle-user
performance), then jex
k;n
j 1,
k
2
k
1 and a
k
N
N+T , b
k
T
N+T
. These limiting
valuesarepreciselythemaximal-ratiocombiningcoeÆcient[41]forestimatingwfromthe
6
Interestingly,in [29] trainingsymbolsareused in aniterativejointdecoder andchannel estimation
scheme according to the mixing method. The analysis in [29] is uniquely based on propagating the
variancesof residual interference and of channel estimation errors from one iteration to the next, and
doesnottakeintoaccountthebias. Unfortunately,theinterferencecancellationalgorithmof[29]isbased
onAPPsandhenceitisplaguedbybiasedresidualinterference[25],andthemixingmethodyieldsbiased
unbiased noisyobservations w+ (t)
and w+, with (t)
and independent, Gaussian,
with covariances N
0
T
I and N
0
N
I, respectively. Comparisons between the mixing and the
combiningmethodsare provided in Section4.
3.5 Algorithm summary
Fig.2showsthe block diagramofthe proposedreceiver. Theusersare ranked indecreas-
ingorder of their estimated signal-to-interference ratio, given by
jwb (t)
k j
2
P
j6=k js
H
k s
j j
2
jwb (t)
j j
2
Without loss of generality, we assume that the decoding order is k = 1;2;:::;K. The
algorithmisinitializedby letting wb (0)
=wb (t)
, xb ( 1)
k;n
=0for allk and n and m=0. Then
we have:
Userloop: Fork =1;:::;K, do
Symbol loop: Forn =1;:::;N, do
Compute the soft-SIC signal samples according to (2) and the estimated residual
interference plus noise varianceb (m)
k
according to(6).
Compute the k-th SISO decoder EXT and APP outputs and compute the soft
interference estimatexb (m)
k;n
according to (8) and the average symbols xe
k;n
according
to(20).
End symbol loop.
End user loop.
Parameter estimation update: If the mixingmethodis used, compute the updated
amplitude estimate according to (23). If the combining method is used, compute
the EM amplitude estimate according to (18) and the updated estimate according
to(27).
Ifm =M, makesymbol-by-symbol decisions on the informationbits APP outputs
of the SISO decoders, otherwise letm :=m+1 and goback tothe user loop.
4 Results
Inordertodemonstratetheperformance ofthe proposed soft-SICreceiver, weconsidered
the followingsimulation setting,looselyinspiredby the UMTS-TDD system [3]:
Spreadingfactor L=16,QPSK chips with\short"randomspreading sequences. A
newsetofKsequencesisgeneratedrandomlyandindependentlywithi.i.d. elements
ateachframe. Obviously,theBERisaveragedoverseveralframes sothatthe eect
of the randomsequences is smoothed.
The user code is the same for all users. For the sake of simplicity, we chose the 4-
staterate-1/2convolutional code(CC) with generators(5;7)
8
(octalnotation[41]).
CodeblocklengthN =2000codedsymbols,correspondingto1000informationbits
perframe.
K =32and 40users, corresponding tochannel loads of 2:0and 2.5 users perchip,
respectively.
Trainingsequence lengths T =4 and 32symbols.
Users havethe same received power. The channelcomplexamplitudes are given by
w
k
= p
R E
b e
j
k
where R is the user coding rate (R = 1=2 in our case), E
b
is the
energy per information bit and
k
is a uniformlydistributed random variable over
[ ;],independently generated for eachuser.
We considered a xed maximum number of SICiterations M =10,in allcases.
In these examples we considered only the equal-rate equal-power users for the sake of
space limitation and since this is a worst-case for iterative soft-SIC decoders (see the
discussion in [14, 15]). In [25], by using the technique of density evolution, whichis now
a standard tool for the analysis of iterative \message passing" algorithms (see [45] and
referencestherein),itisshownthatthesoftinterferencecancellationalgorithmconsidered
here at target BER = 10 5
, with perfect channel parameter knowledge, CC (5;7)
8 user
codes and equal power users attains channel load of 3 users/chip. The required E
b
=N
0
is 6 dB. Fig. 3 shows the BER curves for K = 40 users and perfect channel knowledge
(all BER curves show the worst user performance, which in the equal power case is
usually, but not necessarily, obtained by the user decoded rst). The load inthis case is
40=16 = 2:5, below the limit of 3 predicted by the analysis of [25]. For E
b
=N
0
5 dB
for smaller K the convergence to the single-user BER occurs with less iterations and at
lower E
b
=N
0
threshold.
Fig. 4, 5 and 6 show the BER of the system with K = 32users and T =32 training
symbolsperframe,with trainingestimationonly,andEM+trainingestimationwithmix-
ingand combining methods, respectively. Training-onlyestimation prevents the receiver
toachieve the single-userBER, since interference cannot becanceled completelybecause
of the estimation errors which do not vanish with iterations. The combining method
shows faster convergence than the mixing method. This conrms the qualitative bias
analysismadeintheremarkof Section3.3. However, forsuch\light"load 7
thedierence
between the two methods is not very signicant.
Fig. 7, 8 and 9 show the BER of the system with K = 32 users and T = 4 training
symbolsperframe,withtraining-estimationonly,andEM+trainingestimationwithmix-
ingand combiningmethods, respectively. Withonly4 trainingsymbols,the degradation
ofsystemwithtraining-onlyestimationisveryevident(noticethat forT =4and K =32
itis obviouslynot possible tomake the trainingsequences mutuallyorthogonal, and this
contributes to poor channel estimation). Also, the better convergence properties of the
combining method versus the mixing method are more evident: the combining method
attains the single-user BER at E
b
=N
0
= 4 dB, while the mixing method attains it at
E
b
=N
0
=6dB.
In order to put in evidence that the bias in the mixing method might prevent the
receiver to converge to the single-user BER while the combining method still works, we
consider the case K = 40 and T = 32 (again, orthogonal training sequences are not
possible here). Fig. 10 and 11 show the BER of this system. The mixing method does
not converge for the range of E
b
=N
0
considered in our simulations, since the estimated
amplitudes after the rst iteration are biased by a factor 32=(2032) = 0:0157, which
preventscancellation,and the received doesnot bootstrap. On the contrary,the combin-
ingmethodis stillable to converge for E
b
=N
0
7dB. By comparing Fig. 3 with Fig. 11
we canquantify the degradationdue tounknown channel amplitudes: with M =10iter-
ations this is about 1.6 dB at BER = 10 4
, 0.8 dB at BER = 10 5
and 0.0 dB at BER
410 7
, since inthis BER rangeboth systems achieve the single-user performance.
7
ItisworthwhiletopointoutherethatK =32userswithspreadingfactorL=16isaloadalready
farbeyondanyconventionalpracticalCDMAsystem[2,3]. Wecallthisload\light"sinceitisfarfrom
thethresholdloadpredictedbytheanalysisof[25].
5 Conclusions
We proposed a low-complexity iterative soft-SIC algorithm for joint data detection and
channel parameter estimation, based on SISO single-user decoders and soft interference
cancellation. The channel parameters estimates are updated along with the receiver
iterations. The updating operation has the form of a likelihood function expectation
of followed by maximization, i.e., itis formallyequivalent tothe basic EM step.
Even thoughsimilar algorithmscan befound (with minorvariations)in several other
works(seethe discussioninSection1),hereweinvestigated inthe detailsseveral newim-
portant aspects, namely: a simple and eÆcient way to estimatethe residualinterference
plus noise variance atthe SISO inputs; the issue of soft interference estimation based on
EXT pmfs versus the conventional approach of using APPs; the correct formulation of
EMestimationwithchannelcoding, andthekeyapproximationtobringcomplexityfrom
exponentialdown topolynomialinthe numberofusers; the useof training-basedestima-
tiontogether with EMupdating. In particular,we provided anew methodfor combining
theunbiasedchannelestimatesprovidedbyMLtraining-basedestimationwiththebiased
estimates provided by EM. The newmethod(referred toas\combining")providesmuch
betterconvergenceoftheoverallreceiverthanthemoreconventionalmethodconsistingof
treatingtrainingsymbolsand unknown code symbolstogether (referred toas \mixing").
The fullinvestigationof the optimaltrade-obetween trainingsymbols fractionT=N
and channel load K=L is out of the scope of this paper. However, from the simulation
results shown here we can get some conclusions on the overall benet of the proposed
approach. With our receiver, we can t = 40=16 = 2:5 users/chip with coding rate
R = 1=2 bit/symbol at BER = 10 5
, with actual channel estimation (T = 32 training
symbolsoutofN =2000codedsymbolperframe)andnon-recursive4-stateconvolutional
codes. The required E
b
=N
0
is about 6.7 dB, i.e., user SNR 3:7 dB, with 10 iterations
(10SISO decodingperuser perframe). InUMTS [3,2],conventional SUMFreceivers are
envisaged,butverycomplexandpowerfuluserchannelcodesareconsidered(eitherturbo-
codesor256-stateconvolutionalcodes). Considerforexampleaconventionalsystemwith
turbo-codes of rate R = 1=2, optimized interleavers of size 1024 [26] (corresponding to
coded block lengthN =2048, similartoour case), recursive systematic4-stateCCs with
generators(1;5=7)
8
and 8fulliterations, correspondingto16SISO decodingperuserper
frame.
8
8
Weallow morecomplexity in SISO decoding forthe conventional system (16SISO decoding steps
instead of 10) since our system requires also interference cancellation, which involves someadditional
In the conventional system we assume perfect channel estimation, since channel esti-
mation is much less critical than in the soft-SIC system. The turbo-code achieves BER
=10 5
at SINR = 1 dB. The SINR at the output of the SUMF for equal-power users
and randomspreading sequences, inthe limitfor K;L!1 with K=L=[48], isgiven
by SINR =
SNR
1+SNR
. Then, the limit load of the conventional turbo-encoded system is
= 1
SINR 1
SNR
. By letting SINR = 1 dB (as required by the target BER perfor-
mance) and SNR = 3:7 dB (as in the soft-SIC system), we obtain = 0:83. Even by
letting SNR !1, the maximum possible channel load is not larger than = 1:26. We
conclude that the proposed receiver with actual channel estimation is able to (at least)
doublethecellcapacityatroughlythesamecomplexityoftheconventionalturbo-encoded
system.
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C 1
C 2
C K
Info frame user K
X
Ν Ν Ν Ν
Y Π 1
Π 2
Π K
Info frame user 2 Info frame
user 1
x 1
x 2
x K
s 1 w 1
s 2 w 2
s K w K
Figure 1: Coded synchronous DS/CDMA system (
k
denotes interleaving, dierent for
eachuser).
SISO IC
SISO IC
SISO IC
Training estimation
Y Y (t)
APP
EXT
EXT
EXT
EM update
SISO IC
SISO IC
SISO IC
APP
EXT
EXT
EXT
EM update
Figure 2: Block diagram of the proposed soft-SIC receiver with iterative EM channel
estimation (only two iteration stages are shown for simplicity). APP and EXT denote
softcode symbolestimates obtainedfromAPPand EXTSISO outputs. The \IC"blocks
denoteinterference cancellation and matched ltering.
10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0
-2 0 2 4 6 8
BER
E b /N 0 (dB)
Perfect knowledge, K=40, L=16, CC(5,7)
Single-user Iter.#1 Iter.#2 Iter.#3 Iter.#4 Iter.#5 Iter.#6 Iter.#7 Iter.#8 Iter.#9 Iter.#10
Figure3: K =40;L=16,N =2000,CC (5;7)
8
, perfect channel knowledge.
10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0
-2 0 2 4 6 8
BER
E b /N 0 (dB)
T=32 (Training only), K=32, L=16, CC(5,7)
Single-user Iter.#1 Iter.#2 Iter.#3 Iter.#4 Iter.#5 Iter.#6 Iter.#7 Iter.#8 Iter.#9 Iter.#10
Figure 4: K =32;L=16,N =2000, CC (5;7) , training-onlyestimation with T =32.
10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0
-2 0 2 4 6 8
BER
E b /N 0 (dB)
T=32 (mixing), K=32, L=16, CC(5,7)
Single-user Iter.#1 Iter.#2 Iter.#3 Iter.#4 Iter.#5 Iter.#6 Iter.#7 Iter.#8 Iter.#9 Iter.#10
Figure 5: K = 32;L = 16, N = 2000, CC (5;7)
8
, EM+training estimation with T = 32
and the mixingmethod.
10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0
-2 0 2 4 6 8
BER
E b /N 0 (dB)
T=32 (combining), K=32, L=16, CC(5,7)
Single-user Iter.#1 Iter.#2 Iter.#3 Iter.#4 Iter.#5 Iter.#6 Iter.#7 Iter.#8 Iter.#9 Iter.#10
Figure 6: K = 32;L = 16, N = 2000, CC (5;7)
8
, EM+training estimation with T = 32
and the combining method.
10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0
-2 0 2 4 6 8
BER
E b /N 0 (dB)
T=4 (Training only), K=32, L=16, CC(5,7)
Single-user Iter.#1 Iter.#2 Iter.#3 Iter.#4 Iter.#5 Iter.#6 Iter.#7 Iter.#8 Iter.#9 Iter.#10
Figure7: K =32;L=16,N =2000, CC (5;7) ,training-onlyestimation with T =4.
10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0
-2 0 2 4 6 8
BER
E b /N 0 (dB)
T=4 (mixing), K=32, L=16, CC(5,7)
Single-user Iter.#1 Iter.#2 Iter.#3 Iter.#4 Iter.#5 Iter.#6 Iter.#7 Iter.#8 Iter.#9 Iter.#10
Figure 8: K = 32;L = 16, N = 2000, CC (5;7)
8
, EM+training estimation with T = 4
and the mixingmethod.
10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0
-2 0 2 4 6 8
BER
E b /N 0 (dB)
T=4 (combining), K=32, L=16, CC(5,7)
Single-user Iter.#1 Iter.#2 Iter.#3 Iter.#4 Iter.#5 Iter.#6 Iter.#7 Iter.#8 Iter.#9 Iter.#10
Figure 9: K = 32;L = 16, N = 2000, CC (5;7)
8
, EM+training estimation with T = 4
and the combining method.
10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0
-2 0 2 4 6 8
BER
E b /N 0 (dB)
T=32 (mixing), K=40, L=16, CC(5,7)
Single-user Iter.#1 Iter.#2 Iter.#3 Iter.#4 Iter.#5 Iter.#6 Iter.#7 Iter.#8 Iter.#9 Iter.#10
Figure10: K =40;L=16, N =2000, CC (5;7)
8
, EM+training estimation with T =32
and the mixingmethod.
10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0
-2 0 2 4 6 8
BER
E b /N 0 (dB)
T=32 (combining), K=40, L=16, CC(5,7)
Single-user Iter.#1 Iter.#2 Iter.#3 Iter.#4 Iter.#5 Iter.#6 Iter.#7 Iter.#8 Iter.#9 Iter.#10
Figure11: K =40;L=16, N =2000, CC (5;7)
8
, EM+training estimation with T =32
and the combining method.
List of Figures
1 Coded synchronous DS/CDMA system (
k
denotes interleaving, dierent
foreach user). . . 26
2 Blockdiagramof theproposedsoft-SIC receiverwith iterativeEMchannel
estimation (only two iteration stages are shown for simplicity). APP and
EXTdenotesoftcodesymbolestimatesobtainedfromAPPandEXTSISO
outputs. The \IC" blocks denote interference cancellation and matched
ltering. . . 27
3 K =40;L=16, N =2000, CC (5;7)
8
,perfect channel knowledge. . . 28
4 K =32;L=16,N =2000,CC(5;7)
8
,training-onlyestimationwithT =32. 28
5 K = 32;L = 16, N = 2000, CC (5;7)
8
, EM+training estimation with
T =32and the mixingmethod. . . 29
6 K = 32;L = 16, N = 2000, CC (5;7)
8
, EM+training estimation with
T =32and the combiningmethod. . . 30
7 K =32;L=16, N =2000, CC (5;7)
8
,training-onlyestimation with T =4. 30
8 K = 32;L = 16, N = 2000, CC (5;7)
8
, EM+training estimation with
T =4 and the mixingmethod. . . 31
9 K = 32;L = 16, N = 2000, CC (5;7)
8
, EM+training estimation with
T =4 and the combiningmethod. . . 32
10 K = 40;L = 16, N = 2000, CC (5;7)
8
, EM+training estimation with
T =32and the mixingmethod. . . 33
11 K = 40;L = 16, N = 2000, CC (5;7)
8
, EM+training estimation with
T =32and the combiningmethod. . . 34
