pp. 243-254 243

Mohamed SIALA**

Daniel D U P O N T E I L * *

Iterative rake receiver with MAP channel estimation for DS-CDMA systems*

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 accor- ding to the maximum a posteriori criterion, using the expectation-maximization algorithm. This estimation process requires a convenient representation of the dis- crete multipath fading channel based on the Karhunen- Lobve 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 esti- mation quality is poor or unsatisfactory. The validity of the proposed method is highlighted by simulation results obtained for the FDD mode of the UMTS interface.

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

R6sum6

Une structure de rdcepteur en rCtteau fonctionnant 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 opti- male, les symboles pilotes et les symboles inconmts de

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 lrans- mis 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 : R6cepteur optimal, D6tection multicanal, It6ration, Estimation statistique, Probabilit6 a posteriori, Acc6s multiple code, Spectre 6ta16 s6quence directe, Propagation trajet multiple, Radiocom- munication service mobile, Effet Doppler, Simulation numdrique, Transformation Karhunen Lo6ve.

Contents

I. Introduction II. Transmitted signal characteristics

III. Multipath fading channel characteristics IV. 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 )

I. INTRODUCTION

We propose in this paper an iterative rake receiver

using an opt imum block-by-block semi-bl ind channel

estimation algorithm (CEA) for the DS-CDMA mobile com-

munication systems. This receiver performs an iterative

channel est imation (CE) according to the maximum a

posteriori (MAP) criterion, using the expectation-maximi-

zation (EM) algori thm [1-3]. It uses opt imal ly 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 ACTS ACTS090 FRAMES project 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

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244 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 Dop- pler 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 trans- mitted 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 pre- sented 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) esti- mation of the multipath fading channel when all trans- mitted symbols are known by the receiver.

I I I . M U L T I P A T H FADING C H A N N E L C H A R A C T E R I S T I C S

The multipath fading channel seen by the transmitted signal is composed of several paths presenting time- variations due to the Doppler effect. Each path is charac- terized 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$B D r

II. T R A N S M I T T E D S I G N A L C H A R A C T E R I S T I C S

for the fiat DPS. The average power O (0) varies from one path to the other and characterizes the multipath intensity profile.

The capacity of a CDMA system is naturally limited by multiple access interference (MAI). This MAI is minimi- zed by controlling the transmitted power and reducing it periodically to a minimum while guaranteeing an accep- table reception quality.

Typically, each power control period (PEr') is compo-

sed o fN 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 trans-

mitted signal. Generally, the control symbols are compo-

sed of Np pilot symbols aND, aND ~ 1 . . . . . aN o + N e -1

known by the receiver and N c - N p symbols aNo + Np,

aNn + Np + 1 . . . . . aN 1 dedicated to physical layer

signaling.

For each transmitted PeP, we denote respectively by Pk and E k the time position and transmitted energy associa- ted 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.

IV. S I G N A L M O D E L AT T H E O U T P U T OF T H E R A K E F I N G E R S

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 num- ber 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

(3) Rit = clt a t + NIl c

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 interfe- rence. To simplify both the analysis and the design of the

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M. SIALA -- ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION

enhanced rake receiver, this noise is assumed to be decorrelated and Gaussian with variance I 0.

Received signal

'~ . . . . . . . A ....

(d)

FIG. 1. - - Iterative rake receiver using semi-bl ind discrete channel es t imat ion

Rdcepteur en rfiteau itdratif utilisant une estimation semi-aveugle du canal discret

FOR DS-CDMA SYSTEMS 245

(6) A = (A o, A I ..... AN_l) r

transmitted during each PCP, with A~ = ak/] a l. Based on these notations, we can rewrite the k th component of the lth path received vector R~ as

(7) Rlk = Clk Ak + Nlk

where we have denoted by Clk the k th component of the vector

( (8) q = laolC,o, la, Ic,, ..... laN_,lc,,N__,

of normalized gain factors corresponding to the I th path.

V. CONVENIENT REPRESENTATION OF T H E D I S C R E T E M U L T I P A T H FADING C H A N N E L

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 diffe- rent 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

(4) E[Q,,, c],,] = Or(Pro- pn).

During each PCP, the rake receiver needs good esti- mates of the gain factors ct~ corresponding to all un- known 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 esti- mation 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 l th 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 modu- lated 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

For semi-blind MAP CE, we need a convenient repre- sentation of the discrete multipath fading channel seen at the output of the rake fingers during each PcP. This repre- sentation is based on a discrete version of the KL ortho- gonal expansion theorem. For simplicity sake, we assume that all channel paths are Rayleigh faded.

Proposition 1: The/th normalized discrete channel vector C l can be

expressed in the form

N--1

(9) C 1 = ~_, G#nBtm m=O

[B ]N-I where ] tk~k=O are the normalized eigenvectors of the [ G I N - I covariance matrix F t = E[CtC[ 7] of C, and / 'kJk=0 are

independent complex zero-mean Gaussian coefficients. The variances of these coefficients, assumed next to be arranged in decreasing order, are equal to the eigenvalues

F _S-I f ' systems~[B ]X I { 'k}k=0 o the Hermltian matrix F,. The lklk= O'

l = 0,1,..., L - 1, form L orthonormal bases of the cano- nical complex space of N dimensions. When the L consi- dered 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]. [G ]L-1 The vectors, /. tkll=0' where G t = (Gl. 0, Gt, 1 . . . . .

GI, N_I )T, are referred to as the convenient representation of the discrete channel seen at the output of the rake fin- gers during the received pcP. The probability density function (PDF) of the vector G t is given by

N - I [ Ol k ] 2 / r l k (10) &(Gl) = l-I e-

k=0 7tFlk

3/12 ANN. TELECOMMUN., 54, n ~ 3-4, 1999

246

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

(11) Flm n =- (~l(Pm - pn ) ~//--EmEn.

In practice, neither the forms of the DPSs and the cor- responding 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 pro- cess 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.

VI. SEMI-BLIND MAXIMUM A POSTERIORI D I S C R E T E C H A N N E L E S T I M A T I O N

The map criterion is the most adapted to the semi- blind estimation of the discrete multipath fading channel seen at the output of the rake fingers since the expres- sions of the PDFS Pl (Gl) are known by the receiver. For error correction or/and channel characteristics esti- mation, 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

fRIL 1 vectors [ ill=0, by the conditional PDF

{ L i) (12) p Rt}l= ~ A, {Gt} = /=0

L - 1 N - 1

l=o k=o n'Io exp - ~0 - 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 ~ l L - 1 The MAP estimate } tit= ~ of the discrete multipath

rGlz~ fading channel ] ill= ~ seen during the received pcp is

given by

(13) t ,I,= o argmaxp({G,}~? o' = ~-, = {Rt}t= 0 ,

{g;};=o \

M. SIALA - ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS

and/ _ ~ . \ t h e r e f ~ ) r e maximizes the a posteriori conditional PDF

p({Gl}?-o~[{Rl}?_fol). Directly solving this equation is an

intractable problem. However, the solution can be rea- ched easily by means of the iterative EM algorithm. This

f ~ / ~ % algorithm inductively reestimates the L vectors } t~/=o ,

that a monotonic increase in the a posteriori conditional

G s PDFp({ 1}/=0 {RI}~fo 1) is guaranteed. This monotonic

increase is realized via the maximization of the auxiliary function

G L-1 (14) Q({ t}/:o, IG '/L-I~ [ 'R 'L - i A {G 1,z l~ , , , , : o ) = aZPt , ,/,=o, ,

{'R'L-|A {G)}~; I l ~ Ill= o, , = )'

where the latter sum is operated over all possible trans- mitted normalized data vectors during the PCP.

Given the L received vectors {R/}LL 1, the EM algorithm

starts with an initial guess {G(~ }1/"~ l _ = ~ of the L vectors

{GI Jl=olC-l The evolution from the estimate { G(~/0 }IL= 01 to 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 :

Expectationstep:computeO({G(~l }lL=;1, {G~}lC=O1),

Maximization step: Find the reestimate {G (d+ J)1L 1 l Jl=0

~G,~L_, ~ that maximizes Q {G(~ }it-01' t - 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 esti- mation 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

one. To solve this problem, the initial guess {GI~ ) }L- 1 /=(/

should be computed with great care using the pilot symbols.

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 ~ ( d ~ l ) i s the mth component of the lth path reestimate _ t

given explicitly by N - - I

(15) lm = = \ A U z S k \

,,,,*' where

(16) 1

Whn 1 + l o / F l m

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M. SIALA -- ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS

Proof. See Appendix A. (20) 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 fB ]N-I flat and the classic DPSs, all orthonormal bases [ lk[k=0 '

l = 0,1 . . . . . L - 1, can be chosen to be real. The conju- gations 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

(17) P A t ill= 0 ,

for k c S. As a consequence, the previous expression of G(d ~ 11 c hn an be rewritten as

Im = Whn , Rlk AP A t \ ~ , o ,,A~s~ ~ ~ r,r=0'

- ~*B*lmk kES * * ) + 2 .

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 - ' " the maxentropic uniform conditional probabilities

L-1 h P(A k {RI}~ -1, {G(~ } = ) f o r eac of these unknown

symbols. Therefore, f ; r ~ PSK modula- ted alphabet sets St, k ~ S, the mth component of the ini- tial guess G(~ ) corresponding to the /th finger is given explicitly by

(19) G C~ ~" Rl~ D*kBt* k. lm = Wlm 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 inter- leaved 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 S k, k ~ S. In agreement with the notations of Figure 1, using Bayes rule, we can transform the previous expression of

G(d~ I) into

247

lm = Wlm Rlk, tanh 2Ne A(~2 B* hnk i ~eYs ~

+ kES ~'~ Rlt Dt B;'k)'

for BPSK modulation, and

~(d+ l)_ [ V ~ !)~e{A(~ (21) ~ lm--Wl,n ~', Rtk tanh }j _ ~ o

~ 2 " tanh [g2"~m{A(~ }])"

+

B* B* ) lmk -b 2 Rlk D; t,,,k , kES

for QPSK modulation, where

t iol2oR,tf= ,

We can see from this expression that the reestimate

G(d+l) of is biased not only because of the weigh- lm G lm ting coefficient Wlm but also because of the tanh[.] func- tion. When a severe fade occurs at time k cr S, the term

B* in "tanh" vanishes. The corresponding quantity Rlk /,,,k

is unreliable and its contribution to G(ta,~ 1) is insignificant.

Let sgn denotes the sign function. CE complexity can be further reduced by using either the soft-limiting function

x - 1 - < x - - 1 (23) qb0(x) = sgn x elsewhere

or the hard-limiting function

(24) ~l(x) = sgn x

instead of the tanh[.] function. For a good fulfillment of the EM algorithm steps, the

rake receiver should have available not only an upper bound of the Doppler spread B D but also an estimate of both the noise variance I o and the individual average powers Ot(0) of all considered paths. However, when the hard-limiting function is used instead of the tanh[.] func- tion, we only need to know the ratios Ot(O)/lo in addition to the Doppler spread estimate B o. Moreover, when the average powers of all paths are unknown, the receiver can adopt a maxentropic uniform multipath intensity pro- file where all Ol(0), l = 0,1 . . . . . L - 1 are taken equal.

VII . D E C O D I N G I N F O R M A T I O N S Y M B O L S

As illustrated in Figure 1, the iterative semi-blind

IG ~L-I estimation of the convenient representation t ul=0 of

the discrete multipath fading channel during a PCP is car- ried out a fixed number of times D. This number is

chosen so that the reached estimate/G!O)IL--lguarantees L ~ / g ~ 0

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248 M. SIALA -- ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS

an unnoticeable degradation in performance with respect / ~ / L l

to the optimum estimate t D l = O �9

Based on this estimate, the maximum-ratio combi-

ning rake receiver provides the soft outputs A(~ ) which can be used by an error-correction Viterbi decoding algo- rithm for recovering the transmitted information sequence. For BPSK modulated data and control symbols, only the real part of the previous soft-outputs are needed in the decoding process. Moreover, for uncoded BPSK modulated information symbols taking equiprobable vahles in the alphabet set { - I, + 1 }, the decision on symbol A k is given simply by

(25) A k = sgn ,qD{A(~) }.

VI I I . S I M U L A T I O N RESULTS

m

We denote by Elk the average received energy asso- ciated to the/th path and generated by the k th transmitted symbol in the PcP. We also denote by E k the correspon- ding average total received energy associated to all L considered paths. We have explicitly

Elk = r Ek (26)

a n d

(27) L - I

l--0

For the sake of illustration, we restrict our treatment to the case of the UMTS FDD uplink speech service on the vehicular environment [5]. All other uplink and down- link services of similar DS CDMA systems can be treated in the same way. We recall that the data and control sym- bols of the uplink speech service are transmitted respecti- vely in the inphase and quadrature phase components of the modulated signal. We also recall that the SF used in the inphase component is half that of the quadrature phase component. We finally recall that the average transmit power associated to the inphase component is twice that of the quadrature phase component. Conse- quently, for each PCP, all data and control symbols are transmitted with the same energy. This common trans- mitted symbol energy is denoted by E, its corresponding total average received energy is denoted by_E and its associated lth path contribution is denoted by E l.

Similarly, the control symbol rate is half that of the data symbol rate. Hence, the data symbol duration T D is half the control symbol duration T c. More specifically, we have T c = 2T D = 62.5 ~s, N D = 2N c = 20 and therefore N = 30 data and control symbols are transmitted per PCP. The time position of these symbols with respect to the beginning of their corresponding pcp is given by

(28) P k = ( k + 1/2) T D , k = O , 1 . . . . . N D - 1

for data symbols and by

(29) P k = ( k - N D + I/2) T c , k = N D,N D + I . . . . . N - I

for control symbols. Finally, for presentation simplicity, we assume that

the DPS and the Doppler spread B D of the multipath fading channel and the local oscillator characteristics are perfectly known by the receiver. We recall that the Dop- pler spread B D, also given next in the normalized form B D T D, is proportional to the mobile speed v and the car- rier frequency f0'

For our evaluation, we consider the three typical vehicular speeds v = 120, 250 and 500 km/h and the lowest carrier frequency f0 = 1.92 GHz in the uplink spectrum allocated to the UMTS FDD mode. The resulting normalized Doppler spreads are then B D T D = 1/75, 1/36 and 1/18. We also consider a vehicular environment mul- tipath channel with a Rayleigh fading classic Doppler spectrum. We finally consider a local oscillator with fre- quency stability (LOFS) e (generally expressed in ppm) and a worst case residual frequency shift c f0.

VIII.1. Numerical results

For the sake of illustration, we have shown in Figure 2 three curves representing the largest normalized eigenva- lues Ftm/~, for arbitrary path index l, as a function of the second index m. These curves correspond to the three previously considered Doppler spreads B D T o = 1/75, 1/36 and 1/18. Based on this figure, we can see that only a small number of eigenvectors contribute significantly to the convenient representation of the discrete channel. This number is approximately proportional to the norma- lized Doppler spread B D T D.

FIG. 2 . - - Main normalized eigenvalues

Principales valeurs propres normdes

Moreover, we have represented in Figure 3 the Im'gest weighting coefficients Wtm as a function of the second index m for all three Doppler spread values and

ANN. TgL[COMMUN., 54, n ~ 3-4, 1999 6/12

M. SIALA - ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION

if, l~1 o = - 3 and 0 dB. These weighting coefficients are close to one only for significant normalized eigenvalues. Hence, important weighting factors are always associa- ted with those good quality channel representation coeffi- cients, providing more information on the channel than noise. Stated otherwise, channel representation coeffi- cients with negligible normalized eigenvalues make very noisy contributions to channel reconstruction and are therefore weighted by negligible factors. For complexity reduction, all these negligible coefficients can be remo- ved from the expressions of the estimation algorithm with an unnoticeable degradation in performance.

FIG. 3 . - - Main weighting coefficients

Principaux coefficients de ponddration

FOR DS-CDMA SYSTEMS 249

stations since it guarantees a better observation of fast multipath channel variations during an arbitrary time period. Heuristically speaking, the spreading of pilot symbols guarantees for noisy multipath channel recons- truction the equivalent of the Shannon sampling theorem for noiseless band-limited signals. From a practical view- point, the spreading of pilot symbols prevents our gra- dient-like semi-blind MAr, CEA from converging to a local maxima of the a posteriori conditional probability.

Also for a good characterization of the enhancement in performance provided by our MAP CEA, we consider three rake receivers with conventional constant mini- mum mean square error (CMMSE) CE, linear minimum mean square error (LMMSE) CE and perfect channel state information (PCSI) as benchmarks. The CMMSE CEA com- pensates the modulation of all received pilot symbol samples and computes L partial averages as an esti- mation of the L paths of multipath fading channel. The LMMSE CEA equally compensates the modulation of recei- ved pilot symbol samples and carries a lineal" interpola- tion and/or extrapolation following the MMSE criterion.

For the characterization of the MAP CEA and the three considered benchmarks, we have shown in Figures 4 through 8 the behaviour of the raw BER as a function of the average received energy per symbol to noise ratio E/I o. This characterization is provided for all previously considered values of the paths number L and the mobile station speed v. The raw BER corresponding to the MaP CEA is obtained after D = 5 iterations using exclusi- vely and respectively the 4, 5 and 6 most important

fB I N - 1 vectors of the orthonormal base [ ~Jt=0 for the mobile

station speeds v = 120, 250 and 500 km/h.

VIII.2. Simulat ion results

We assume that all considered paths have the same Doppler spread B D and a classic DPS form. We consider exclusively the case of the max_entropic uniform multi- path intensity profile where all E t, l = 0,1 . . . . , L - 1 are equal. The general case where the channel paths have different average powers can be treated optimally if these powers are known by the receiver. The other case where this knowledge is not available can be treated using the least predictable uniform multipath intensity profile. Finally, we restrict our treatment to the three typical cases where the multipath channel has L = 1,3 and 5 paths.

We restrict out" investigation to the case where N# = 6 pilot symbols pet" PCe are used for semi-blind cE. We treat the case where the pilot symbols are grouped at the beginning of each pcp as well as the less conventional case where all these symbols are spread over each Pcr,. For the latter case, we assume that three groups of two pilot symbols are placed at the beginning, the middle and the end of each pcP. From a theoretical viewpoint, pilot symbols spreading is beneficial for high-speed mobile

FIG. 4. - - Raw BER and raw BEP as a function of ~,/1o, for

v = 1 2 0 k m / h a n d L = 1

Taux d'erreur binaire (TEa) bj'ut et probabilit6 d'erreur binaire (PEB) brute enJbnction de E/I O, pour v = 120 knv'h et L = 1

For the rake receiver with PCSI, we have represented both simulated and theoretical curves. The theoretical curves represent the bit error probability (BEP) P of uncoded Br'SK over a Rayleigh multipath fading channel

7/I 2 ANN. TI~Lr 54, n ~ 3-4, 1999

250 M. S |ALA -- ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS

with uniform multipath intensity profile. We recall that for a channel with L paths, this BEe is given explicitly by [4]

L--I l (30) P =(1---~-2 )LL (L-l+1) (1+-~2) e m l

where

(31)

FIG. 7. - - R a w BER and r a w ~EP as a f u n c t i o n o f t : I / o , for

v = 1 2 0 k n g h a n d L = 5

rEtJ brut et t'EB brute en Jonction

de E/l(y pour v = 120 km/h et L = 5

FIG. 5. - - R a w BER and r a w BEP as a f u n c t i o n o f E/1 o, for

v = 5 0 0 k m / h a n d L =- 1

TEB brut et I'Et~ brute en fonc t ion

de El@ p o u r v = 5 0 0 km/h et L = I

These figures show that the enhanced rake receiver with spread pilot symbols (ses) and MAP CE outperforms all other practical benchmark receivers. They equally show that the enhanced rake receiver with grouped pilot symbols (GPS) has poorer performance than the one with sPs, especially for high-speed mobile stations. For high raw BER, the poor performance presented by GPS is mainly due to the channel estimator incapacity to follow accurately fast multipath channel variations. For low raw BER, this poor performance is mainly due to the inability of the gradient-like MAP estimation algorithm to converge to the true global maximum.

By way of example, for a raw BER of 10 -2, Figure 8 shows that the enhanced rake receiver with sps guaran-

FIG. 6. - - R a w BER and r a w BEe as a f u n c t i o n of E / l o, for

v = 2 5 0 k m / h and L = 3

TEB brut et PEB brute en fonct ion

de Eli o, p o u r v = 2 5 0 krn/h et L = 3

FIG. 8. - - R a w BER and r a w aEP as a f u n c t i o n o f Eli O, for

v = 5 0 0 k m / h and L = 5

rEt~ brut et l'Ell brute en Jbnction

de E/1 o, pour v --- 5 0 0 km/h et L = 5

ANN. TI~L~COMMUN., 54, n ~ 3-4, 1999 8/12

M. SIALA - ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS 251

tees a gain of around 1.5 dB with respect to the most favorable of the CMMSE and LMMSE CEA receivers. Moreover, for the same raw BER, this figure shows that the enhanced receiver presents a degradation of only 0.9 dB with respect to the pcsI rake receiver.

For a further characterization of the MAP CEA and its ability to follow channel variations for large mobile sta- tion speeds, we have equally shown in Figures 9 and 10 the behavior of the raw BER as a function of data symbols indices within a pcP, for a fixed value E/I o = 10 dB. These figures show that the flatness of the raw BER curves is achieved exclusively when the MAP CEA i s com- bined with spread pilot symbols.

For an additional characterization of the MaP CEA, we have depicted in Figure 11 the behavior of the raw BER as a function of the number of accomplished iterations D, for different values of the number of orthonormal base vectors (NOBV) used in the representation of the discrete multipath fading channel. This figure shows that the best achievable raw BER for v = 500 km/h and L = 3 is almost reached after 5 iterations using a channel repre- sentation with only 5 base vectors.

Fro. 9. - - R a w BER as a f u n c t i o n o f d a t a s y m b o l p o s i t i o n k, f o r

E/I o = 10 dB , v = 500 k m / h and L = 1

7~~ brut en fonction de la position des symboles de donndes k, pour E/l o = lO dB. v = 5 0 0 km/h et L = 1

FIG. 11. - - B e h a v i o r o f r a w BER as a f u n c t i o n o f the n u m b e r o f

c a r r i ed i t e ra t ions D , f o r Eli o = 10 dB, v = 500 k ln /h and L = 3

yet* brat en fo_nction du nombre d'itdrations effbctuges D. pour E/1 o -- 10 dB, v = 500 klrffh et L -- 3

For a complementary characterization of the MAP CEA, we have shown in Figure 12 the behavior of the raw BER when the optimum tanh(.) is replaced by either of the soft

FIG. 10. - - R a w BER as a f u n c t i o n o f d a t a s y m b o l p o s i t i o n k, f o r

E/I o = 10 dB, v = 5 0 0 k m / h and L = 5

rl:'~ brnt en Jbnction de la position des symboles de donndes k, pour E/10 = 10 dB, v = 5 0 0 kndh et L = 5

FIG. 12. - - R a w BER as a f u n c t i o n El1 o u s i n g the MAP CEA,

f o r v = 5 0 0 k rn /h

YES brat en Jbnction de ~;/1 o utilisant l'algorithme d'estimation de canal au MAC, pour v = 500 km/h

9/12 ANN. TELI~COMMUN., 54, n ~ 3-4, 1999

252

and hard limiter functions O0(.) and O1(.). This figure shows that the replacement of the optimum tanh(.) func- tion in the MAP CEA by the soft limiter function O0(. ) leads to an unnoticeable degradation in performance. However, when the hard limiter function ~1( ') is used instead, an extra degradation in E / I o of around 0.5 dB is observed.

For a final characterization of the MAP CEA and its ability to follow fast discrete channel variations caused by a residual frequency shift e f0 generated by an imper- fectly locked local oscillator, we have depicted in Figure 13 the raw BER as a function of the LOFS E, for E/1 o -- 10 dB. For this case only, the rake receiver adopts a maxentropic flat DPS representation with a total bandwidth B o + 2e f0, including the combined effects of Doppler and imperfect local oscillator frequency stabi- lity. This figure shows that the spreading of pilot symbols provides better robustness against local oscillator imper- fections, whatever the used CEA. It also shows that the performance of the MAP CEA with SPS is almost insensi- tive to local oscillator residual frequency shifts as signifi- cant as 1 ppm. This behavior is not presented by the CMMSE and LMMSE CEAS for which the raw BER increases rapidly towards 0 .5 starting from residual frequency shifts as low as 0 .2 ppm.

M. SIALA -- ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS

significant degradation of more than 2 dB, especially for large values of the mobile station speed.

FI6. 14. - - Channel estimation SNR as a function of E/I o

Rapport signal gl bruit de I'estimation de canal en fonction de Eli o

IX. C O N C L U S I O N

m

FIG. 13. --- Raw BER as a function of LOFS E, for E/1 o = 10 dB, v -- 500 km/h and L = 3

TEB brut en Jbnction_de la stabilitd fr~quencielle de l' oscillateur local ~ pour E/I o = 10 dB, v = 500 krrdh and L = 3

For an evaluation of the CE quality of the MAP CEA, we have__represented in Figure 14 the CE SNR as a func- tion of E / I o for different values of the mobile station speed and paths number. As a benchmark, we have inclu- ded in this figure the optimum SNR obtained in Appen- dix B for the MMSE CE receiver, based on a full knowledge of all transmitted symbols. This figure shows that, for large values of E / I o, the MAP CEA SNR presents almost no degradation with respect to that of the MMSE. However, for small values of E / I o, this SNR presents a

We have proposed an iterative rake receiver using a semi-blind maximum a p o s t e r i o r i estimation of the mul- tipath fading channel met in DS-CDMA mobile communi- cation systems. Based on numerical and simulation results, we have noticed that the degradation in perfor- mance presented by this algorithm with respect to perfect channel state information is very small compared to other more intuitive algorithms. We have also noticed that, even for high mobile station speeds, the spreading of pilot symbols within each power control period flat- tens the curves representing the raw BER as a function of the symbols indices.

Our algorithm can be used in the DS-CDMA systems to enhance the rake receiver performance, especially for high-speed mobile stations and high carrier frequencies. It can also be used to counter residual frequency errors generated by uncontrolled or low cost local oscillators. Finally, it can be used to alleviate the important overhead represented by pilot symbols, especially for low data rate services.

Appendix A: proof of proposition 2

Following the approach of [2], we calculate the next

reestinaate {G(d/1) }L~01 which maximizes Q({G (~ }L-,,

of {GI} l o " First, we nonce that {G'I}L--01 ) as afunction ,L=-I . t .1 ,=o

ANN. TgLgCOMMUN., 54, n ~ 3-4, 1999 10/12

M. SIALA - ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS

the logarithmic term in the expression of Q(.,.) can be expressed as

(A. 1) logp {R;}_ ,,a, o;/t= ~ ) =

'ogP(A) q- logp G}t ~ {.l}lt;l A I.Gtl.L-I~ t DI=O )"

The first term in this equation doesn't intervene in the ~ t , L - I

maximization of Q(.,.) with respect to {t/t)t= o . The second term can be written explicitly as

' d ff/k are given respectively by to ,Olk a n

(A. 6) cTp;~'--~Q({G(~}Lol,{G}',~;') =

(A. 2) logp { G~ }f~-01 (=/=

l=0, k=O

Finally, the third term can be rewritten as

(A. 3) logp({R/}L201 A , {G}}L291) = - LNIog(~[O)

L--1,N--I Rlk N--I 2.

I 0 l=O,k=O m = 0

253

2 / , R 1 L 1 iG(d)~L-I~ ~o P~ &-o"- I ~t=o )

F t (1 + I0/ ,m)P,m

Z P(A I ""-' / I lit o' )~e{RlkA*e JG,,B;,,~} ~ k = O A E S k \ = { G(/d) } l t ; I

\and

3 ( c-l {~,,L-I\ (A. 7) "~tm Q {G(~}/= O, o11/= 0 ) =

/ I 0 P {RI}/L=; I' ' ,=0

I ) k=O2 A~skP(A 'Rt 1)1=0 "L-1 {G(/d)}/L;I ~.)~e{RlkA,je_jff#,,B;,,I.. }

Substituting these expressions in that of Q(.,.) and reordering the sums, we obtain

L--I ,N--I / 'R 'L-I c-1 ~ _ _ c - ,,=o,

G,zk 2

Flk

1 I 0

L-,,N--, {IR/s A = {G(d)} L-l) E E Pit /'1=0' k A, / l=0]

l=0, k=O AES k

N--I 12 R/k - A E G' lm Blmk

m = 0

The facts that the norms of the normalized symbols

A k are constant (equal to 1) and that the L bases { =Bl,l~ o , _ l = 0,1 . . . . . L - 1, are orthonormal have been used to derive and simplify these equalities. Equating these deri- vatives to zero yields

(A. 8) (l + I~ ) G (d+') F h n hn =

z . ,,,=o k = 0 A~S k

Letting

1 (A. 9) Wlm= 1 + lo/Flm and reordering the previous expression, we obtain finally

where C is an additive factor independent of { G}}tt~ I.

For the derivation of Q(.,.) with respect to I~'IL-1 t~Nl= 0 , ,I ~ ' ~ L - I , N - 1

we represent the complex coefficients t Cilk)l=O,k=O in

polar form. We write the coefficient Gl' k in the following form

N- I G(d ~ I) (A. 10) l;,, = Wtm ~,

k = O \ A E S k

I)" {6(~~ } L-~ B* l l=0 hn/."

t t (A. 5) GI~ =PI~ eJCf;~,

where P~k and 0'l~ arerespect ively the amplitude and

the phase and j = ~ v / - 1. For l = 0,1 . . . . . L - 1, and k = 0,1 . . . . . N - 1, the derivatives of Q(.,.) with respect

Appendix B: MMSE estimation of the discrete multipath fading channel

We assume that the normalized vector A transmitted during a ecP is entirely known by the receiver. Without

11/12 ANN. T~LdCOMMUN., 54, n ~ 3-4, 1999

254

any loss of generality, all the components of this vector can be taken equal to one. The received vector corres- ponding to the l th finger is therefore given by

(A. 11) Rl= C: + N:,

where N / =- (Nz0, NZl . . . . . Nj, N -IY denotes the corres- ponding noise vector.

We denote respectively by QI = (Ql0, Qll . . . . . Qt, u -tY and M I = (Mz0, Mr1 . . . . . Mj, N-IY the expansions of the I th path received vector R t and noise vector N t on the

[B ]N-I orthonormal basis l tk.f ~=0' The equivalent noise vector

Mj is also decorrelated and Gaussian with zero mean and

variance I 0. For each of the L considered paths, we can write

(A. 12) Qlk = Glk + Mlk' k = 0,1 . . . . . N - 1.

For the sake of MMSE CE, we express the lth path esti- A

mate C~ as N--I

(A. 13) CI = 2 Wtk Qlk Blk' k=0

where wlk, k = 0,1, . . . , N - 1, are arbitrary complex weighting coefficients to be determined by means of the MMSE criterion. This criterion consists in minimizing the performance index

(A, 14) J = E C l - C l kl=O

with respect to these weighting coefficients�9 Taking into account the orthonormality of each of the L basis B ]N-1 zkl. 0, I = 0,1 . . . . , L - 1, and the decorrelation of the

,f ~ ~L-1, K-I channel coefficients t Chk ~ t=0.k=0 and the noise coeffi-

�9 L I N - I czents {Mzk}l=0'k= 0 , we can show that the optimum weighting coefficients minimizing the performance index J are real and given by

M. S1ALA - ITERATIVE RAKE RECEIVER WITH MAP CHANNEL ESTIMATION FOR DS-CDMA SYSTEMS

(A. 15) wlm = 1 + [O[Flm

The corresponding minimum value of the perfor- mance index J is given explicitly by

L - I , N - - I

(A�9 16) G i n = [0 N" Wlk, / , /=0, k=0

and its associated CE signal to noise ratio (SNR), iS also given explicitly by

L - I , N ~ I

(A. 17) kl=O = l=O, k=O L--I,N 1

Jmin [0 ~" Wlk l=O, k=O

Manuscrit requ le 27 fdvrier 1999

R E F E R E N C E S

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12] KALEH (G. K,). Joint carrier phase estimation and symbol deco- ding of trellis codes�9 European Transactions on Telecommunica- tions and Related Technologies, San Diego, Ca, (January 1990),

[3j GEOaOHIADES (C. N.), HAN (J. C.). Sequence estimation in the presence of random parameters via the EM algorithm. IEEE Tram sactions on Communications, 4K n ~ 3, (March 1997),

[4] PROAVaS (J, G.). Digital communications. McGraw-Hill, New York (1989).

15] DAHLMAN (E.), GUDMUNDSON (B.), NILSSON (M.), SKOLD (J.). UMTS/IMT-2000 based on wideband CDMA. IEEE Communications Magazine, (September 1998).

[6] SCHARF (L.L.). Statistical signal processing: detection, esti- mation and time-series analysis. Addison-Wesley Publishing Company, New York, (1991),

[71 BAHL (L. R.), COCKE (J.), JELINEK (E), RAVIV (J.). Optimal dec(J- ding of linear codes for minimizing symbol error rate. IEEE Transactions on Ind'brmation Theoly, 20, (March 1974).

ANN. T~.LI~COMMUN., 54. n ~ 3-4, ]999 12/12