EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS LETTERS, VOL. 17, PP. 151-155, 2006
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Maximum a posteriori decorrelating receiver for MC-CDMA systems Hatem Boujemˆaa and Mohamed Siala Higher School of Communications of Tunis, Route de Raoued Km 3.5, 2083 El Ghazala, Ariana, Tunisia
[email protected],
[email protected] Abstract— In this letter, we propose a maximum a posteriori decorrelating receiver for multicarrier code division multiple access systems. The decorrelating receiver consists in a maximum a posteriori combining of an uncorrelated version of the signal demodulated over the different subcarriers. The decorrelating receiver performance is compared to that of the conventional receiver which is based on a maximum ratio combining strategy.
I. Introduction The Conventional Receiver (CR) for MultiCarrier Code Division Multiple Access (MC-CDMA) systems is based on a Maximum Ratio Combining (MRC) of the demodulated signal over the different subcarriers [1-2]. The MRC rule is optimum when the channel is known. However, this rule is no longer optimum when the channel is estimated and the fading characteristics on the different subcarriers are correlated. The correlation of the observed channel at the different subcarriers depends on both the multipath delay profile of the channel and the subcarriers separation. In order to improve the receiver performance, we extend in this letter the structure of the Decorrelating Receiver (DR), proposed in [3] for Direct Sequence (DS) CDMA systems, to MC-CDMA systems. The proposed receiver consists in a Maximum A Posteriori (MAP) combining of an uncorrelated version of the signal demodulated over the different subcarriers. This uncorrelated version is provided by means of the Karhunen Lo`eve (KL) orthogonal expansion using the channel statistics. For a precise characterization and evaluation of the enhancement in performance provided by this new structure, the DR performance is investigated for perfect and Maximum Likelihood (ML) estimated channel statistics and compared to that of the CR. Finally, note that the proposed receiver can be used for both uplink and downlink. II. System model An MC-CDMA transmitter spreads the original signal using a spreading code in the frequency domain [1-2]. In the following, the expressions of the transmitted and received signals are given in the presence of a single user, with the abstraction of serial-to-parallel conversion of the transmitted symbols [1]. Without loss of generality, the number of subcarriers, N , is assumed to be equal to the spreading factor. The equivalent base-band transmitted
signal can be written as e(t) =
k
sk g(t − kTs )
N −1
ckN +m ej2πfm (t−kTs ) ,
(1)
m=0
where sk is the k-th transmitted symbol, Ts is the symbol N −1 period, {ckN +m }m=0 is a unit modulus spreading sequence, g(t) is a rectangular pulse response with unit useful energy and duration Ts = Tsu + ∆ where Tsu is the useful symbol period and ∆ is the guard interval, fm = f0 + m∆f is the m-th subcarrier frequency, f0 is the frequency of the first subcarrier and ∆f = 1/Tsu is the subcarrier separation. The symbol stream is assumed to be organized in time slots containing respectively Nd and Np data and pilot symbols. We denote by Es and Ep respectively the useful transmitted energy per data and pilot symbols. If the channel delay spread is lower than the guard interval ∆, then the restriction of the received signal to the interval [kTs + ∆, (k + 1)Ts ] can be rewritten as N −1 sk ckN +m ej2πfm (t−kTs ) H(fm ; t) + n(t), r(t) = √ Ts − ∆ m=0 (2) where H(fm ; t) = h(τ ; t)e−j2πfm τ dτ, (3)
h(τ ; t) is the impulse response of the multipath fading channel at time instant t and n(t) is an additive white complex gaussian noise. An MC-CDMA receiver uses a Discrete Fourier Transformation (DFT) to recover the transmitted signal over the different subcarriers [1-2]. In the following, a perfect synchronization on the different subcarrier frequencies is assumed. After removing the received signal during the guard interval and compensating the modulation due to the spreading sequence, the DFT outputs for symbol sk can be written as T
zk = (zk,0 , · · · zk,N −1 ) = sk Hk + nk ,
(4) T
where Hk = (H(f0 ; kTs ), · · · , H(fN −1 , kTs )) , nk = T (nk,0 , · · · , nk,N −1 ) is a zero mean complex Gaussian noise with covariance matrix N0 IN and (.)T is the transpose operator.
EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS LETTERS, VOL. 17, PP. 151-155, 2006
III. Conventional and Decorrelating Receivers A. The Conventional Receiver When the channel is known, the optimal soft output corresponds to that of the CR given by Λk =
H†k zk , N0
(5)
where (.)† denotes the Hermetian transpose operator. When the channel is estimated, the CR soft output is given by † zk H , (6) Λk = N0 is an estimate of H updated slot by slot using the where H set of pilot symbols. The conventional channel estimate used by the CR is deduced from the ML criterion. Assuming that the channel is constant during the pilot symbols of each slot, we have ML H
arg max p {zk }k∈Ap /H H arg min zk − sk H2 H
= =
(7) (8)
k∈Ap
where Ap is the set of pilot symbols indices in the current time slot. We then deduce the expression of the ML channel estimate ML = H
k∈Ap
s∗k zk
Np Ep
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B. The Decorrelating Receiver The expression of the subcarrier demodulator outputs in (4) is similar in form to that obtained in DS-CDMA systems [3]. Therefore, the Decorrelating Receiver (DR) structure, originally proposed in [3] for DS-CDMA systems, can be transposed to MC-CDMA systems. The DR is based on an uncorrelated version of the signal demodulated over the different subcarriers by means of the KL orthogonal expansion of the channel transfert vector H. More precisely, we have (12) yk = U† zk = esk + vk , N −1
where U = (u0 , · · · , uN −1 ), {ul }l=0 are the normalized eigenvectors of F, N −1 e = U† H = (e0 , · · · , eN −1 )T , {el }l=0 are uncorrelated zero-mean random Gaussian variables with decreasing variN −1 ances given by {Γl }l=0 the eigenvalues of F and vk = U† nk is a zero mean complex Gaussian noise with the same statistical properties as nk . Assuming an uncorrelated scattering [4], the (q,l)-th entry of matrix F is given by Fql = φH (fq − fl ) =
(9)
e† Wyk , N0
(14)
where e is an estimate of e obtained from the set of pilot symbols Ap of the current time slot as ∗ k∈Ap yk sk ML , e= = U† H (15) Np Ep
We can also use a MAP channel estimate to take into account the correlation of the components of H : MAP = arg max p {zk } H /H p (H) (10) k∈Ap W is a diagonal matrix with l-th entry H 1 1 , = arg min zk − sk H2 + H† F−1 H , wl = N0 2 H 1 + σ Es + 1 k∈Ap
N0
where F = E HH† . After some developements, we obtain MAP = H
−1 N0 −1 ML H IN + F Np Ep
(11)
The estimation of the channel covariance matrix F and the noise Power Spectral Density (PSD) N0 , needed in the derivation of MAP channel estimates, is treated in section IV. The CR soft output using ML or MAP channel estimates is no longer optimum when the components of H are correlated. The optimum implementation of the receiver according to the MAP criterion is detailed in the next subsection.
(13)
where φh (τ ) is the multipath intensity profile of the channel [4] and φH (∆f ) is the spaced frequency correlation function of the multipath channel. The optimum implementation of the receiver according to the MAP criterion is given by [3] Πk =
.
φh (τ )e−j2π(fq −fl )τ dτ,
(16)
Γl
σ 2 = N0 /(Np Ep ) is the channel estimation variance. If we examine the expression of the optimal decision variable (14), we learn two things. First of all, if the channel is perfectly estimated by increasing towards infinity the number Np and/or the energy Ep of the pilot symbols, all weighting coefficients wl converge to unity. Hence, we verify that the CR and DR are equivalent when the channel is perfectly estimated. The MRC rule is therefore optimum when the channel is perfectly estimated. Secondly, if the components of H are decorrelated, all weighting coefficients wl are equal so that the CR and the DR become equivalent. Hence, if the fading characteristics among the different subcarriers are decorrelated, the MRC rule is optimum even if the channel is estimated. Moreover, using (13), (15) and (16), the DR soft output can be written as
EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS LETTERS, VOL. 17, PP. 151-155, 2006
† zk H Πk = , N0
(17)
= UWU† H ML . H
(18)
where
Therefore, the optimum soft output can be obtained using instead of a MAP or the CR and the channel estimate H a ML channel estimate. IV. ML Noise and channel statistics estimation In a practical implementation of the DR and the CR with MAP channel estimates, the noise PSD N0 and the channel covariance matrix F must be estimated. In order to reduce the effect of noise on the quality of the statistics estimates, successive time slot estimates must be averaged. In this letter, we propose to use an exponential averaging window which allows a simple reduced-noise tracking of these statistics. According to this technique, the estimate of the channel covariance matrix at time slot m is given by 0m N m m−1 m m† F = (1 − ε) F +ε H H − IM , (19) Np Ep m is the estimate where 0< ε ≤ 1 is the forgetting factor, H of H at time slot m, and the noise PSD estimate at time slot m is expressed as
m N 0
=
m† 1 m−1 + ε (1 − ε) N zk zm k 0 (Np − 1)N k∈Ap mH m† , −Np Ep H (20)
where zm k is the demodulated signal over the different subcarriers for symbol sk at time slot m. Note that the terms multiplied by ε in (19) and (20) are respectively the Maximum Likelihood estimates of F and N0 over the m-th time slot [3]. The optimal choice of ε depends on the rapidity of the channel statistics variations. If the statistics vary slowly, they can be tracked using a small value of ε which allows a large averaging window. If the statistics vary rapidly, a large value of ε must be used in order to track the statistics evolution.
transmitted energy per data bit. The simulated channel is Rayleigh fading with classical Doppler power spectrum and normalized average energy. The maximum Doppler frequency is set to fD = 20 Hz. In figures 1 and 2, we compare the performance of the DR with estimated channel statistics for ε = 0.01 and ε = 0.03 to that of the DR with Perfect Channel Statistics (PCS), the DR with Perfect Channel State Information (PCSI) and the CR, for the ITU Indoor A, for a number of subcarriers N equal to 16 and 32, respectively. We see that the DR with estimated channel statistics for ε = 0.01 offers 2 dB gain compared to the CR with ML channel estimates. Moreover, the DR performance for ε = 0.01 is very close to that of the DR with PCS. This is explained by the fact that the channel statistics are constant so that a quasiperfect estimation can be obtained by using a small value of ε. The DR performance for ε = 0.03 is worse than that obtained for ε = 0.01 since the estimated channel and noise statistics are not sufficiently averaged for ε = 0.03. Finally, figure 1 shows that the CR with MAP channel estimates for ε = 0.01 offers only 0.3 dB performance improvement at low Eb /N0 compared to the CR with ML channel estimates. In fact, equation (12) shows that MAP and ML channel estimates become equivalent at high Eb /N0 . Figure 3 shows the different receivers performance for the ITU Vehicular A channel for N =16. We notice that the DR with estimated channel statistics for ε = 0.01 offers 1 dB gain compared to the CR with ML channel estimates. The more pronounced gain for the Indoor A channel, observed in figures 1 and 2, is explained by the stronger correlation between subcarriers due to the reduced Indoor A channel delay spread. VI. Conclusion In this letter, we have extended the decorrelating receiver structure, originally proposed for the optimum reception for DS-CDMA systems [3], to MC-CDMA systems. The decorrelating receiver with estimated channel statistics was shown to offer respectively 2 dB and 1 dB gain compared to the conventional receiver with ML channel estimates for the ITU Indoor A and the Vehicular A channels. The same order of gain can be observed for other practical channels. References [1]
V. Simulation results In this section, we compare the uncoded Bit Error Rate (BER) performance of the CR and the DR in the presence of a single user. The subcarrier separation is equal to ∆f = 50 kHz corresponding to Ts = 25 µs and ∆ = 5 µs. We recall that the spreading factor is assumed to be equal to the number of subcarriers. The used spreading sequence is a Gold sequence. The data stream is decomposed in slots containing Np = 4 pilot symbols and Nd = 6 data symbols. The transmitted symbols are dropped from a QPSK constellation. The transmitted energy per data and pilot symbols are equal. We denote by Eb the useful
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[2] [3] [4]
S. Shinsuke and R. Prasad, “Design and performance of Multicarrier CDMA systems in frequency selective Rayleigh fading channels”, IEEE Trans. on Vehicular Technol., Vol. 48, No. 5, pp. 1584-1595, Sept. 1999. S. Hara and R. Prasad, “Overview of multicarrier CDMA”, IEEE Commun. Magazine, pp. 126-133, 1997. M. Siala, “Maximum a posteriori decorrelating discrete-time Rake receiver”, Annals of Telecommunications, Vol. 59, No. 3-4, pp. 374-411, March/April 2004. J.G. Proakis, Digital communications, New York, M.C. GrawHill, Fourth edition, 2001.
EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS LETTERS, VOL. 17, PP. 151-155, 2006
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Biography
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Hatem Boujemˆ aa was born in 1974 in Tunis, Tunisia. He received the engineer Diploma from “Ecole Polytechnique”, Tunis, Tunisia, in 1997, the DEA of digital telecommunication systems from “Ecole Nationale Sup´erieure des T´el´ ecommunications”, Paris, France, in 1998 and the Ph. D. degree in electronic and communications from the same university in 2001. From October 1998 to September 2001, he prepared his Ph. D. degree at France T´el´ ecom R&D, Issy-lesMoulineaux, France. During this period, he participated in the RNRT project AUBE. From October 2001 to January 2002, he joined Ecole Sup´erieure d’Electricit´e, Gif-sur-Yvette, France, and worked on mobile localization for RNRT project LUTECE. In February 2002, he joined “Ecole Sup´erieure des Communications de Tunis”, Tunis, Tunisia, where he is an Assistant Professor. His research work are in the field of digital communications with special emphasis in DS-SSS, multicarrier systems, information theory, channel estimation, equalization, multi-user detection and antenna processing. Mohamed Siala was born in 1965 in Tunisia. He received the engineer Diploma from “Ecole Polytechnique”, Palaiseau, France, in 1988, the engineer Diploma from “Ecole Nationale Sup´erieure des T´ el´ ecommunications”, Paris, France, in 1990 and the Ph. D. degree from the same university in 1995. From 1990 to 1992, he was with Alcatel Radio-Telephones, Colombes, France, working on the GSM physical layer. In 1995, he joined Wavecom, Issy-lesMoulineaux, France, where he worked on IOTA multicarrier modulations for terrestrial digital TV and channel estimation for the ICO project. From 1997 to 2001, he worked at France T´el´ ecom R&D, Issy-les-Moulineaux, France, on the physical layer of the UMTS system. In 2001, he joined “Ecole Sup´erieure des Communications de Tunis”, Tunis, Tunisia, where he is a Professor. His research interests are in the area of digital communications with special emphasis on multicarrier systems, channel estimation, modulation and coding for mobile communications.
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Fig. 2. BER performance comparison of the CR and the DR for the ITU Indoor A channel (N =32).
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Fig. 1. BER performance comparison of the CR and the DR for the ITU Indoor A channel (N =16).
Fig. 3. BER performance comparison of the CR and the DR for the ITU Vehicular A channel (N =16).
