performs an iterative estimation of the channel accor- ding to the maximum a posteriori criterion, using the expectation-maximization algorithm. This estimation.
pp. 243-254
243
Iterative rake receiver with MAP channel estimation for DS-CDMA systems* Mohamed S I A L A * * Daniel D U P O N T E I L * *
Abstract We propose an iterative rake receiver structure using an optimum semi-blind channel estimation algorithm for DS-CDMA mobile communication systems. This receiver performs an iterative estimation of the channel according to the maximum a posteriori criterion, using the expectation-maximization algorithm. This estimation process requires a convenient representation of the discrete multipath fading channel based on the KarhunenLobve orthogonal expansion theorem. The rake receiver uses pilot as well as unknown control and data symbols optimally for improving channel estimation quality. Moreover, it can take into account the coded structure of all unknown transmitted symbols when channel estimation quality is poor or unsatisfactory. The validity of the proposed method is highlighted by simulation results obtained for the FDD mode of the UMTSinterface. Key words: Optimum receiver, Multichannel detection, Iteration. Statistical estimation, A posteriori probability, Code division multiple access. Direct sequence spread spectrum, Multipath propagation, Mobile radiocommunication, Doppler effect, Numerical simulation, Karhunen Lo?ve transformation.
RI~CEPTEUR EN RATEAU ITI~RATIF AVEC ESTIMATION DE CANAL AU MAXIMUM A POSTERIORI POUR SYSTEMES AMRC
contr6le et d'information pour amdliorer la qualit6 de l'estimation de canal. De phts, il peut prendre en compte la structure cod6e de tousles symboles inconnus lransmis quand la qualitd de l'estimation de canal n 'est pas satisfaisante. L'int6r6t de l'approche proposd est illustrd par des r~sultats de simulation obtenus dans le cadre applicatif du mode FDD de la norme UMTS.
Mots cl~s : R6cepteuroptimal, D6tection multicanal, It6ration, Estimation statistique, Probabilit6 a posteriori, Acc6s multiple code, Spectre6ta16 s6quence directe, Propagation trajet multiple, Radiocommunication service mobile, Effet Doppler, Simulation numdrique, Transformation Karhunen Lo6ve. Contents I.
Introduction Transmitted signal characteristics Multipath fading channel characteristics Signal model at the output of rake fingers V. Convenient representation of the discrete multipath fading channel VI. Semi-blind maxinTum a posteriori discrete channel estimation VII. Decoding information symbols VIII. Simulation results IX. Conclusion Appendix References (7 ref ) II. III. IV.
R6sum6 Une structure de rdcepteur en rCtteaufonctionnant de manibre itdrative est propos~e, elle utilise un algorithme optimal d'estimation semi-aveugle de canal, pour les syst&nes de radiocommunication avec les mobiles de type AMRC avec sdquence directe. Ce dispositif effectue une estimation itdrative du canal selon le crit~re du maximum a posteriori, en se servant de l' algorithme EM. Cette estimation ndcessite une representation convenable du canal multitrajet avec dvanouissements discrets, bas6e sur le thdor&ne de d~veloppement orthogonale de Karhunen-Lobve. Ce dispositif utilise, de manikre optimale, les symboles pilotes et les symboles inconmts de
I. INTRODUCTION
We propose in this paper an iterative rake receiver using an optimum b l o c k - b y - b l o c k semi-blind channel estimation algorithm (CEA) for the DS-CDMA mobile communication systems. This receiver performs an iterative channel estimation (CE) according to the m a x i m u m a posteriori (MAP) criterion, using the expectation-maximization (EM) algorithm [1-3]. It uses optimally pilot as well as information-carrying symbols in the optimization of the multipath Doppler CE. It can take into account the
*The work presented in this paper is partly financed by the ACTSACTS090FRAMESproject which is funded by the European comnmnity. **France Tdldcom CNET, 38-40, rue du G6n6ral-Leclerc -F 92794 Issy Moulineaux Codex 9, France. E-mail :{mohamed.siala, daniel.duponteil }@cnet.flancetelecom.fr 1/12
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M. SIALA - ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS
coded structure of the transmitted information-carrying symbols in order to improve its performance. It requires a convenient representation of the multipath Doppler channel, based on a Karhunen-Lo~ve (KL) orthogonal expansion [4] of each path of the discrete multipath Doppler channel seen by the rake receiver. The performance of this algorithm is evaluated by simulation in terms of raw bit error rate (BER) for the uplink speech service of the FDD component of the UMTS [5], assuming a slowly power-controlled vehicular environment. This evaluation is carried out as a function of the average received energy per information carrying symbol to noise plus (multiple access and inter-symbol) interference ratio. The document is organized as follows. Sections II and III are respectively aimed at characterizing the transmitted signal and multipath fading channel. Section IV is dedicated to modeling the discrete signal at the output of the rake receiver fingers. In Section V, a convenient representation of the discrete multipath Doppler channel seen by the rake receiver is derived. Section VI describes the semi-blind MAP iterative estinaation of this discrete channel. The decoding process following CE is treated in Section VII. Numerical and simulation results are presented in Section VIII. Appendix A provides the proof of the main proposition of Section VI. Finally, Appendix B deals with the minimum mean square error (MMSE) estimation of the multipath fading channel when all transmitted symbols are known by the receiver.
III. M U L T I P A T H F A D I N G C H A N N E L CHARACTERISTICS
The multipath fading channel seen by the transmitted signal is composed of several paths presenting timevariations due to the Doppler effect. Each path is characterized by its average power as well as its Doppler power spectrum (DPS) which depends on both environment and mobile station speed. Moreover, each path can present either Rayleigh or Ricean fading. In general, the shape of the ops is either classical or flat. The classic and flat DPS are met in outdoor and indoor environments, respectively. Next, we denote by B D the Doppler spread of the multipath fading chalmel and by J0(') the 0th-order Bessel function of the first kind. The autocorrelation function of one path with average power r is therefore given by
(1)
O(z) = 0(o) Jo(ZBDO
for the classic DPS and by (2)
~ r ) = 0(0) sin(xBDr) 7$BD r
II. T R A N S M I T T E D S I G N A L CHARACTERISTICS
The capacity of a CDMAsystem is naturally limited by multiple access interference (MAI). This MAI is minimized by controlling the transmitted power and reducing it periodically to a minimum while guaranteeing an acceptable reception quality. Typically, each power control period (PEr') is composed o f N D PSK modulated data symbols, a o, a l , . . . , aND -1, and N c PSK modulated control symbols, aND, aND + 1' .... aN - l, where N = N D + N c. Both symbol categories can be spread with different spreading factors (sF) and multiplexed together either in time or separately on the inphase and quadrature-phase components of the transmitted signal. Generally, the control symbols are composed of N p pilot symbols aND, aND ~ 1. . . . . aN o + Ne -1 known by the receiver and N c - N p symbols aNo + Np, aNn + Np + 1. . . . .
aN
for the fiat DPS. The average power O (0) varies from one path to the other and characterizes the multipath intensity profile.
IV. S I G N A L M O D E L AT T H E O U T P U T O F THE RAKE FINGERS
As shown in Figure 1, the rake receiver is composed of L fingers, each tracking one of the L most powerful paths of the channel. It also guarantees the constructive combination of the contributions of these paths. The number of fingers in the rake receiver is generally lower than the number of effectively received paths. This number depends strongly on the environment (outdoor or indoor) and the spreading chip rate. Usually used typical values range from 2 to 3 for indoor environments and 4 to 8 for outdoor environments. The /th finger output signal corresponding to the k th symbol a k can be written as
1 dedicated to physical layer
signaling.
(3)
For each transmitted PeP, we denote respectively by Pk and E k the time position and transmitted energy associated to the k th symbol within the PCP. The transmitted energy is usually common to all symbols within a given category but can differ from one category to the other.
where clk is the gain factor of the lth path seen by the symbol a t and Ntt is an additive complex noise including thermal noise as well as MAI and inter-symbol interference. To simplify both the analysis and the design of the
ANN. TI~LI2COMMUN.,54, n ~ 3-4, 1999
Rit = clt a t + NIlc
2/12
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M. SIALA -- ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS
enhanced rake receiver, this noise is assumed to be decorrelated and Gaussian with variance I 0.
'~. . . . . . .
A ....
Received signal
A = (A o, A I..... AN_l) r
(6)
transmitted during each PCP, with A~ = ak/] a l. Based on these notations, we can rewrite the kth component of the lth path received vector R~ as Rlk = Clk Ak + Nlk
(7)
where we have denoted by Clk the k th component of the vector (d)
(8) q=
( laolC,o,la, Ic,,..... laN_,lc,,N__,
of normalized gain factors corresponding to the I th path. FIG. 1. - - Iterative rake receiver using semi-blind discrete channel estimation
Rdcepteur en rfiteau itdratif utilisant une estimation semi-aveugle du canal discret
V. CONVENIENT REPRESENTATION OF T H E
Assuming independent scattering, the gain factors corresponding to a given path are also considered to be independent from those of other paths. The rationale behind this is that paths reaching the receiver with different delays have a great chance to follow different ways and meet different obstacles. However, the gain factors within one path are generally correlated. If E[.] and (.) denote respectively the expectation operator and the autocorrelation function of the lth path, then the discrete autocorrelation function corresponding to this path is given by
For semi-blind MAP CE, we need a convenient representation of the discrete multipath fading channel seen at the output of the rake fingers during each PcP. This representation is based on a discrete version of the KL orthogonal expansion theorem. For simplicity sake, we assume that all channel paths are Rayleigh faded.
DISCRETE MULTIPATH FADING CHANNEL
Proposition 1: The/th normalized discrete channel vector C l can be expressed in the form N--1
(9) (4)
E[Q,,, c],,] = Or(Pro- pn).
During each PCP, the rake receiver needs good estimates of the gain factors ct~ corresponding to all unknown data and control symbols. Taking into account the time-correlation of the gain factors of each of these L paths, we can enhance CE for an arbitrary PEP by using all its symbols (data and control) as well as those of several previous and subsequent PCPS. For the sake of simplicity, we assume that the estimation of all gain factors is carried out using exclusively one PCP. We also assume that the semi-blind estimation process is based on known as well as unknown symbols within this PfP. We denote by (.)r the transpose operator and introduce the vector (5)
R / = (Rl. o, Rj, 1..... RI,N_I )T
of samples corresponding to the lth path finger outputs during the considered PCP. We denote by [. I the absolute value operator and recall that the amplitude l ak [ = V ~ of the PSK modulated symbol a k depends only on its index k within the pce. To get rid of this amplitude dependence within each pcP, we introduce the normalized vector
3/12
C1 =
~_, G#nBtm m=O
]N-I are the normalized eigenvectors of the where [B ] tk~k=O I N - I are covariance matrix F t = E[CtC[ 7] of C, and /[ G 'kJk=0
independent complex zero-mean Gaussian coefficients. The variances of these coefficients, assumed next to be arranged in decreasing order, are equal to the eigenvalues _S-I o f the Hermltian ' {F 'k}k=0 matrix F,. The systems~[B ]X I
lklk= O'
l = 0,1,..., L - 1, form L orthonormal bases of the canonical complex space of N dimensions. When the L considered paths have identical DPS forms and Doppler spreads, the latter L orthonormal bases are equally identical.
Proof. This proposition is a special case of the more general continuous KL orthogonal expansion theorem [4]. ]L-1 where G t = (Gl. 0, Gt, 1. . . . . The vectors, [G /. tkll=0'
GI,N_I )T, are referred to as the convenient representation of the discrete channel seen at the output of the rake fingers during the received pcP. The probability density function (PDF) of the vector G t is given by (10)
N- I
[ Olk ] 2/rl k
k=0
7tFlk
&(Gl) = l-I e-
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M. SIALA - ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS
When the rake receiver has exact knowledge of the characteristics of the multipath fading channel, the (m, n) th entry of the /th Hermitian matrix F t is given explicitly by Flm n =- (~l(Pm - pn ) ~//--EmEn.
(11)
a posteriori conditional PDF
and/ \ t h_e~r e f ~ )maximizes r e. the
p({Gl}?-o~[{Rl}?_fol). Directly solving this equation is a n intractable problem. However, the solution can be reached easily by means of the iterative EM algorithm. This ~ % algorithm inductively reestimates the L vectors f}~ /t~/=o , that a monotonic increase in the
In practice, neither the forms of the DPSs and the corresponding Doppler spreads, nor the exact powers with which the PCPS are transmitted or known precisely by the receiver. In some cases, the rake receiver has only a rough estimate or an upper bound of the actual channel Doppler spread B D. As a consequence, it can adopt the least predictable (maxentropic from an innovation process viewpoint) multipath channel representation [6] with a flat PCp and a single Doppler spread value B D. In some other cases, the rake receiver has available a bank of eigenvectors and eigenvalues for different typical upper bounds of the actual Doppler spread to be able to adapt to all mobile stations speeds.
a posteriori conditional
s {RI}~fo1) is guaranteed. This monotonic PDFp({ G 1}/=0 increase is realized via the maximization of the auxiliary function L-1 IG '/L-I~ (14) Q({ G t}/:o, ['R'L-i A {G 1,z l~ , ,,,:o)= aZPt , ,/,=o, ,
l~
{'R'L-|A Ill=o, , {G)}~;= I )'
where the latter sum is operated over all possible transmitted normalized data vectors during the PCP. Given the L received vectors {R/}LL 1,the
EM
algorithm
starts with an initial guess {G(~l _}1/"~ = ~ of the L vectors
VI. SEMI-BLIND M A X I M U M A POSTERIORI DISCRETE CHANNEL ESTIMATION
{GI Jl=olC-lThe evolution from the estimate {G(~/0}IL=01to the new estimate {G(d+l)}L-I is performed via the auxiliary l
/=0
function by carrying implicitly the following expectation and maximization steps : The map criterion is the most adapted to the semiblind estimation of the discrete multipath fading channel seen at the output of the rake fingers since the expressions of the PDFS Pl (Gl) are known by the receiver. For error correction or/and channel characteristics estimation, some of the transmitted symbols are coded or fixed (pilot or reference symbols). The normalized vector A transmitted during a pcP is therefore characterized by an a priori probability distribution P(A). Given this transmitted vector and the convenient discrete channel
r r enta ,on I ,l :o ', and ta ,o
account t,e inde
pendence of the/I-~noise components, we can describe the vectors [fRIL ill=0,1 by the conditional PDF
L i) = (12) p { Rt}l= ~ A, {Gt} /=0
Maximization step: Find the reestimate {G (d+ J)1L 1 l
that maximizes Q {G(~ }it-01' t-
- Ak ,n=o ~'~
GtmBlmk '
where B~,,,k is the k th component of the mth orthonormal base v e c t o r Blm corresponding to the/th path.
f~lL-1 The MAP estimate } tit= ~ of the discrete multipath fading channel rGlz~ ] ill= ~ seen during the received pcp is
~G,~L_, l'l=o )~ as a function
of
/lr-~, I L - I
t ~ l ll= 0 9
In general, the auxiliary function has several global maxima. This leads to an ambiguity in the MAP estimation of the discrete multipath channel gain factors. This problem can be avoided by using the pilot (or reference) symbols which are known by the receiver. However, this often proves to be insufficient, because the auxiliary function has also several local maxima which can be reached by the EM algorithm instead of the global
Proposition 2" Let S~ denote the alphabet set taken by the k th n o r m a l i z e d symbol during an arbitrary pcp. The expression of the mth component of the lth path reestimate ~_ ( d t~ l ) i s given explicitly by N--I
(15)
lm
= =
\AUzS k
\
given by (13) t ,I,= o =
Jl=0
one. To solve this problem, the initial guess {GI~) }L1 /=(/ should be computed with great care using the pilot symbols.
L-1N-1
l=o k=o n'Io exp - ~0
Expectationstep:computeO({G(~l }lL=;1,{G~}lC=O1),
,,,,*' argmaxp({G,}~? ~-, = o' {Rt}t= 0 ,
{g;};=o \ ANN. TgLgCOMMUN., 54, n o 3-4, 1999
where (16)
1 Whn
1 + lo/Flm 4/12
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M. SIALA -- ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS
Proof. See Appendix A. The weighting coefficient Wtm depend on the mth eigenvalue of the covariance Hermitian matrix F t (which takes into account the average power 0l (0) of the l th path as well as the Doppler spread B D and the transmitted energies E~, k = 0,1 . . . . . N - 1 associated to data and control symbols) as well as the noise variance I 0` For a baseband DPS with even symmetry, such as the
]N-I ' flat and the classic DPSs, all orthonormal bases [fBlk[k=0 l = 0,1 ..... L - 1, can be chosen to be real. The conjugations appearing in the components of the orthonormal bases vectors are therefore unnecessary. Next, we denote by S the set of pilot symbols indexes within a pcp and by D t the value taken by pilot symbol A~, k e S. Since all pilot symbols corresponding to the pcp are known by the receiver, we have
P At
(17)
ill=0 ,
G(dhn~11c an be rewritten as = Whn
\ ~ , o , Rlk ,,A~s~AP ~A t - ~*B*lmk
+
2 kES
.
~" Rl~ D*kBt* k. k~S
When a part of the transmitted vector A is coded by a convolutional or block code, the conditional probability distribution
P(A k = A IIR/J/=o, IL-1 /r(d)~L-I~/ t o # / l = o / c a n b e calcu-
lated exactly 'using the trellis of this code, based for example on the Bahl algorithm [7]. Let ,~le{.} and ,~m{.} denote respectively the real part and imaginary part operators. If the coded data is interleaved over a large number of PCPS, the complexity of the estimation algorithm can be reduced by assuming that the unknown data and control symbols are uncoded and take equiprobably their values in the alphabet sets Sk, k ~ S. In agreement with the notations of Figure 1, using Bayes rule, we can transform the previous expression of G(d~ I) into 5/12
for BPSK modulation, and l)_ (21) ~(d+ ~ lm--Wl,n
~',
_~o
tanh [ V ~ !)~e{A(~ }j +
Rtk
~ 2 " tanh [g2"~m{A(~ }])" B*lmk -b 2
kES
Rlk D; B*t,,,k ) ,
for QPSK modulation, where
t
iol2oR,tf=
,
G(d+l) lm of G lm is biased not only because of the weighting coefficient Wlm but also because of the tanh[.] function. When a severe fade occurs at time k cr S, the term in "tanh" vanishes. The corresponding quantity Rlk B*/,,,k
* * )
symbols. Therefore, f;r ~ PSK modulated alphabet sets St, k ~ S, the mth component of the initial guess G(~) corresponding to the /th finger is given explicitly by Wlm
+ kES ~'~ Rlt Dt B;'k)'
Let sgn denotes the sign function. CE complexity can be further reduced by using either the soft-limiting function
P(A k {RI}~-1, {G(~ }L-1 = ) f o r eac h of these unknown
G C~ lm =
i ~eYs~
is unreliable and its contribution to G(ta,~1) is insignificant.
the maxentropic uniform conditional probabilities
(19)
Rlk, tanh 2Ne A(~2 B*hnk
= Wlm
~ r,r=0'
At the start of the EM algorithm, the receiver has no idea about the values of the data and unknown control symbols transmitted during the pcP. As a consequence, it can use for the determination of the initial guess {G(/]lC_O 1-'"
lm
We can see from this expression that the reestimate
for k c S. As a consequence, the previous expression of
Im
(20)
(23)
x qb0(x) = sgn x
- 1 -
