channel estimation algorithm (CEA) for DS-CDMA systems. This algorithm performs an iterative channel estimation (CE) according to the maximum a posteriori.
MAXIMUM A POSTERIORI MULTIPATH FADING CHANNEL ESTIMATION FOR CDMA SYSTEMS Mohamed Siala, Daniel Duponteil France T616com -BD - CNET -DMR -1IM 38-40, rue du Gdn6ral-Leclerc - 92794 Issy Moulineaux Cedex 9 - France Phone: +33 1 45 29 61 96 Fax: +33 1 45 29 41 94 E-mail: { mohamedsiala, daniel.dupontei1} @cnet.francetelecom.fr Abstract - We propose an optimum channel estimation algorithm for DS-CDMA systems. This algorithm performs an iterative estimation of the channel according to the maximum a posteriori criterion, using the Expectation-Maximization algorithm. It requires a convenient representation of the discrete multipath fading channel based on the Karhunen-Lo&e orthogonal expansion theorem.
I. INTRODUCTION We propose in this paper an optimum block-by-block channel estimation algorithm (CEA) for DS-CDMA systems. This algorithm performs an iterative channel estimation (CE) according to the maximum a posteriori (MAP) criterion, using the Expectation-Maximization (EM) algorithm [l-31. It uses profitably not only pilot symbols but also information-carrying symbols in the optimization of the multipath Doppler CE. It also takes into account the 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 KarhunenLobve (KL) orthogonal expansion [4] of each path of the discrete multipath Doppler channel seen by the RAKE receiver. The evaluation of the performance of this algorithm is based on the UMTS UTRA/FDD uplink 8 kb/s speech service [5]. The paper is organized as follows. Sections I1 and 111 are respectively aimed at characterizing the transmitted signal and multipath fading channel. Section IV is dedicated for modeling the discrete signal at the output of the correlators of the RAKE receiver. In Section V, a convenient representation of the discrete multipath Doppler channel seen by the RAKE receiver is derived. Section VI is aimed at describing the MAP iterative estimation of this discrete channel. The decoding process following CE is treated in Section VII.
0-7803-5565-2/99/$10.00 0 1999 IEEE 1121
Simulation results are presented in Section VIII.
11. TRANSMITTED SIGNAL CHARACTERISTICS The capacity of a CDMA system 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 (PCP) is composed of N, PSK modulated data symbols, a,,a ,,...,aN,-,, and N, PSK modulated control ,aND+ ,,...,aN-,, where N = N, + N, . symbols, aND Both symbol categories are generally spread with different spreading factors (SF) and can either be multiplexed in time or in the inphase and quadrature phase components of the transmitted signal. Generally, the control symbols are composed of N , pilot symbols aND ,..., known by the receiver and N, - N , symbols dedicated to physical layer signaling.
In what follows, we denote respectively by p k and Ek the time position and transmitted energy associated to the k symbol within the PCP. The transmitted energy is usually common to all symbols of a given category but can vary from one category to the other.
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111. MULTIPATH FADING CHANNEL CHARACTERISTICS The multipath fading channel seen by the transmitted signal is composed of several Rayleigh fading paths.
Each path is characterized by its average power as well as it Doppler power spectrum (DPS). In general, the shape of the DPS is either classic or flat. The classic
DPS is met in outdoor environments. The corresponding autocorrelation function, for one path with average power #(O) ,is given by #(z) = $(o)JO(nBDz)
V. CONVENIENT REPRESENTATION OF THE DISCRETE MULTIPATH FADING CHANNEL
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where BD is the Doppler spread of the channel and Jo (-) is the Oth-orderBessel function of the first kind.
For MAP CE, we need a convenient representation of the discrete multipath fading channel seen at the output of the RAKE correlators during one PCP. This representation is based on a discrete version of the KL orthogonal expansion theorem [4].
IV. SIGNAL MODEL AT THE OUTPUT OF THE RAKE CORRELATORS
Proposition 1: The 1 th normalized discrete channel vectors C , can be expressed as
The RAKE receiver tracking one of the channel. The 1 corresponding to the
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is composed of L fingers, each L most powerful paths of the path correlator output signal k th symbol ak can be written as
R, = CQak + NI,
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The gain factors are independent from one path to the other but time-correlated within the same path. Due to this time-correlation, successive PCPs can be used jointly to enhance CE. For the sake of presentation simplicity, CE is carried out using exclusively the symbols of one PCP. Let (-)' denote the transpose operator. We introduce for each PCP the vector
R, = (RiotRll,... ,Rl,N-I)'
' path
correlator output samples. Let also
6
A = (&,A,,...,AN-I)T with A, = a k I lakl. Based on this, we rewrite the
R, = C,Ak
R, as
+ N,
where C, the k th component of the normalized vector
of 1
th path
where {Blk}k=oare the normalized eigenvectors of the
are independent complex zero-mean Gaussian coefficients. The variances of these coefficients, arranged in decreasing order, are equal to the eigenvalues {rlk}:: of the Hermitian matrix F, . The system {B
}:io1constitutes an orthonormal base of
the complex space of The
vectors
N dimensions. {G,
where
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G , =(C,,,G,, ,...,G,,N-,)T,are referred to as the convenient representation of the discrete channel seen at the output of the RAKE correlators during the received PCP. The probability density function (PDF) of G , isgivenby
I{
denote the absolute value operator. To get rid of the of the PSK dependence of the amplitude modulated symbol ak on the index k , we introduce the normalized transmitted vector
components of the received vector
k=O N-l
covariance matrix F, = E[C,C;'] of C , and {Glk}r:
9
where c, is the gain factor of the 1 path seen by the symbol a, and Nlk is complex AWGN with variance
of 1
N-I
gain factors.
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For a RAKE receiver with perfect local oscillator frequency stability (LOFS) and exact knowledge of the th entry of the 1 channel characteristics, the (m,n) Hermitian matrix F, is given by
'
~
n
=n fi (Pm - pn )dEmEn
a
In practice, the receiver has neither a perfect LOFS nor a precise knowledge of the channel characteristics. As a consequence, it adopts the least predictable multipath channel representation with a flat DPS and a virtual Doppler spread B , summing the additive effects of Rayleigh fading and oscillator frequency stability.
algorithm starts with an initial guess {G:o'}:
VI. MAXIMUM A POSTERIORI DISCRETE MULTIPATH CHANNEL ESTIMATION
{G,},::.
of
L-I
The evolution from the estimate {G~d')I=O to
the new estimate {G"+''}z is performed via the
The MAP criterion is used in the estimation of the discrete multipath fading channel seen at the output of the RAKE correlators since the expressions of the PDFs p,(G,) are known by the receiver. For error correction or/and channel characteristics estimation, some of the transmitted symbols are coded or fixed. The normalized transmitted vector A is therefore characterized by an U priori probability distribution P(A). Given this transmitted vector and the
auxiliary function by carrying implicitly an expectation step followed by a maximization step.
Proposition 2: Let S, denote the alphabet set taken by the k th normalized symbol in the PCP. The expression of the m th component of the 1 th path reestimate Cy'" is explicitly given by GF1) =
Wl,ll
convenient discrete channel representation {Gl}/:: , and taking into account the independence of the noise components, we can describe the received vectors
where
{R,}ra-l by the conditional PDF 1=0
N-l
fiE-!-exp[ /=o k=O d o
- d ( R l k - Ak~Gl,,,B,, IF0
where B,, is the k * component of The MAP estimate {GI}:
B, .
of the discrete multipath
Next, we denote by S the set of pilot symbols indices within a PCP and by Dk the value taken by pilot symbol A,, k E S . At the start of the EM algorithm, the receiver has no idea about the values of the transmitted data and physical layer control symbols. Therefore, assuming centro-symmetric PSK modulated alphabet sets S, , k E S , we can use as m th component of the initial guess GT'
fading channel {Cl},:: is given by
Directly solving this equation is an intractable problem. However, the solution can be reached easily by means of the iterative EM algorithm. This algorithm inductively reestimates the L vectors
L-1
so that a
monotonic increase in the a posreriori conditional PDF in (1) is guaranteed. This monotonic increase is realized via the maximization of the auxiliary function
where the latter sum is operated over all possible transmitted normalized data vectors.
R I } : , Given the L received vectors {
When the transmitted vector A is coded, the conditional probabilities in (2) can be computed using for instance the Bahl algorithm [ 6 ] .
Let %e{.} denote the real part operator. For uncoded data and information control BPSK modulated symbols, using Bayes rule, the previous expression of GE+') can be simplified into
where
the EM For a good fulfillment of the EM algorithm steps, the
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RAKE! receiver should have available not only an upperbound of the virtual Doppler spread B, but also an estimate of both the noise variance I, and the individual average powers $l (0) of all considered paths.
VII. INFORMATION SYMBOLS DECODING The
iterative
estimation
representation {G,
1:,
of
the
convenient
channel is performed a fixed number of times D so guarantees an
unnoticeable degradation with respect to the optimum estimate
{ G , } : .
In Figure 1, we represent the raw BER versus E / I,. The curves associated to the MAP CEA are obtained after D = 5 iterations using exclusively the 6 most important vectors of the orthonormal bases { B B } ~ ~ ~ .
of the discrete multipath fading
that the reached estimate
For the characterization of the enhancement in performance provided by our MAP CEA, we consider constant minimum mean square error (CMMSE) CEA, linear MMSE (LMMSE) CEA and perfect channel state information (PCSI) as benchmarks.
Based on this estimate, the
maximum-ratio combining RAKE receiver provides the soft outputs AY' used for recovering the transmitted data and control sequences. For uncoded BPSK modulated symbols, the decision on symbol Ak is given simply by
For PCSI, we have shown both simulated and theoretical curves [4]. Figure 1 shows that MAP CE with SPS outperforms all other benchmarks. It also shows that MAP CE with GPS has always poorer performance than with SPS. For high raw BER, the poor performance presented by GPS is mainly due to the incapacity of the receiver to follow accurately fast channel variations. For low raw BER, this poor performance is mainly due to the inability of the gradient-like MAP CEA to converge to the global maximum. 1o'
................................. . . . . . . . . . . . . . . . . . . . . . . . .
VIII. SIMULATION RESULTS
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For illustration purpose, we consider exclusively the UMTS UTRA/FDD uplink 8 kb/s speech service [SI. A PCP of length 0.625 ms is composed of N, = 20 data symbols and N, = 10 control symbols, multiplexed in the inphase and quadrature phase components of the transmitted signal, respectively. All symbols have common transmitted energy E and average received energy We use N = 6 pilot symbols for CE. We treat the case of grouped pilot symbols (GPS) at the beginning of each PCP as well as the less conventional case of spread pilot symbols (SPS) with three groups of two pilot symbols placed at the beginning, the middle and the end of each PCP.
E.
For the sake of simplicity, we assume a perfect knowledge of channel and local oscillator characteristics at the receiver. We consider a classic DPS Rayleigh-faded channel with L = 3 equal average power paths, a vehicular speed v =SO0 km/h and a carrier frequency f, = 1.92 GHz. We also consider a local oscillator with frequency stability E (expressed in ppm) and a worst case residual frequency shift E f, .
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*--a
+ - -+
CMMSESPS LMMSESPS MAPGPS CMMSE GPS LMMSE GPS PCSl Simulation PCSl Theory
I
I
Figure 1: Raw BER and BEP versus and v=500 kmk.
E / I,, for
L =3
For a further characterization of the MAP CEA and its ability to follow fast channel variations caused by the Doppler effect, we show in Figure 2 the raw BER versus data symbol indices in the PCP for E / I, = 10 dB. This figure shows that the flatness of the raw BER curves is achieved exclusively when MAP CE is combined with pilot symbols spreading. For a final characterization of the MAP CEA and its ability to follow fast discrete channel variations caused by the additional residual frequency shifts generated by an imperfectly locked local oscillator, we depict in Figure 3 the raw BER versus the LOFS E , for
-
E / I, = 10 dB. This figure shows that pilot symbols spreading provides better robustness against local oscillator imperfections, whatever the used CEA. It also shows that the performance of the MAP CEA with SPS is almost insensitive to local oscillator residual frequency shifts as significant 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. 100
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Our algorithm provides a dramatic enhancement in RAKE receiver performance, especially for high-speed mobile stations and large carrier frequencies. It can also counter efficiently significant residual frequency shifts generated by uncontrolled or imperfect low-cost local oscillators. Finally, it alleviates the important overhead due to pilot symbols, especially for low data rate services.
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ACKNOWLEDGEMENT
U
I
c
The work presented in this paper is partly financed by the ACTS AC090 FRAMES project which is funded by the European community.
10
10
simulation results, in comparison to conventional CEAs, we have observed that the degradation in performance presented by our algorithm with respect to perfect CSI is very small. We have equally noticed that the spreading of pilot symbols within each PCP flattens the curves representing the raw BER as a function of symbols indices, even for high mobile station speeds.
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REFERENCES 2
4
6
8 10 12 Data symbol posnmn k
14
16
18
20
Figure 2: Raw BER versus data symbol position, for E / I , =10 dB, L = 3 and v = 5 0 0 W h .
(1) Dempster, A.P., Laird, N.M., Rubin, D.B., "Maximum Likelihood from Incomplete Data via the EM Algorithm," Journal of the Royal Statistical Society, Ser. 39, 1977.
(2) Kaleh, G.K., "Joint Carrier Phase Estimation and Symbol Decoding of Trellis Codes," European Transactions on Telecommunications and Related Technologies (ETT), Focus on Channel Coding and Information Theory, No. 2, March-April 1993. (3) Georghiades, C. N., Han, J.C., "Sequence Estimation in the Presence of Random Parameters Via the EM Algorithm," IEEE Transactions on Communications, Vol. 45, No. 3, March 1997.
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(4) Proakis, J.G., Digital Communications,McGrawHill, New York, 1989. loo
LOFS [ppm]
Figure 3: Raw BER versus LOFS, for L = 3 and v = 5 0 0 W h .
E/I, = 10 dB,
(5) Dahlman, E., Gudmundson, B., Nilsson, M., Skold, J., "UMTS/IMT-2000 Based on Wideband CDMA," IEEE Communications Magazine, September 1998.
(6) Bahl, L.R., Cocke, J., Jelinek, F., Raviv, J., "Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate," IEEE Transactions on Information
IX. CONCLUSION
Theory, Vol. IT-20, March 1974.
We have proposed an iterative algorithm for maximum a posteriori estimation of the multipath fading channel met in CDMA systems. Based on numerical and
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