ISPACS{98, Melbourne, Australia (November 1998), pp. 432-436
1
JOINT MAP DETECTION AND CHANNEL ESTIMATION FOR CDMA OVER FREQUENCY-SELECTIVE FADING CHANNELS Linda Davis
Iain Collings
Dept. Electrical & Electronic Engineering University of Melbourne Parkville, VIC 3052 Australia fldavis,
[email protected]
ABSTRACT This paper presents a centralized multi-user maximum a posteriori (MAP) detector for code division multiple access (CDMA) signals on frequency-selective fast-fading channels. Importantly, the algorithm does not assume perfect channel state information at the receiver. The key idea is to expand the hypothesis trellis for the purpose of joint channel estimation and equalization for all users. Using the additional hypotheses, minimum mean square error (MMSE) channel estimation techniques are coupled with the MAP algorithm. This algorithm is applicable to synchronous systems, as well as asynchronous systems provided that the user timing o sets are known. 1. INTRODUCTION Over recent years, there has been much interest in developing multi-user receivers for code division multiple access (CDMA) communication systems. For the case of Gaussian channels, the maximum likelihood (ML) optimal receiver was presented by Verdu [1]. Since then, many lower-complexity linear receivers have been presented [2, 3]. These are generally based on nding a lter which provides the minimum mean square error (MMSE) estimate of the data. Again, the channel is assumed to be Gaussian (i.e. known channel with additive white Gaussian noise introduced at the receiver). Generally, mobile radio communication channels are fading (or time-varying) channels. Recently, MMSE linear receivers have been extended to include adaptation. They may be designed to adapt to the unknown channel [4, 5], as well as the spreading codes of other users when these are unknown (down-link transmission) [6]. However it turns out that the rate of adaptation for these linear techniques is not sucient to track fast-fading chan This author was partially supported by the Australian Telecommunications and Electronics Research Board.
nels. Thus a more sophisticated approach is required. In this paper we focus on the up-link in this fading multi-user environment, where the centralized receiver knows the spreading codes of each of the users. We propose a maximum a posteriori (MAP) detector to jointly estimate the channels and the incoming data for each of the users. This approach is more computationally intensive than previously proposed linear receivers. However, we are better able to track the fast-fading nature of the environment. This algorithm is applicable to synchronous systems, as well as asynchronous systems provided that the user timing o sets are known. The development of the multi-user MAP detector in this paper follows on from previous work in single user systems [7, 8]. The MAP algorithm is a symbol-by-symbol estimator which accepts a priori symbol probabilities (soft-inputs) and produces a posteriori symbol probabilities (soft-outputs). Its operation on a channel with memory may be represented using a state trellis [9]. The key idea is to expand the trellis for the purpose of joint channel estimation and equalization [8]. Expansion of the state-space is made possible by the fact that the lowpass nature of the fading e ectively introduces correlation into the received signal. Using an MMSE predictor, the expanded state can then be used to form channel estimates based only on the state, as required by the MAP algorithm. Our MMSE predictor for the channel estimates should not be confused with the MMSE symbol predictors of linear multi-user receivers. Our use of the MAP algorithm is motivated by the bene ts of soft-input soft-output algorithms at the receiver [10], a property not a orded by the maximum likelihood sequence estimation (MLSE) approach based on the Viterbi algorithm (VA) which is commonly used in single user systems.
2. TRANSMISSION SYSTEM MODEL Consider an asynchronous CDMA system with a
ISPACS{98, Melbourne, Australia (November 1998), pp. 432-436 {u1,n }
s1
Oversample R = Tc / Ts
{x 1,k }
2
Equivalent Channel f 1,k Multi-user
{u^1,n }
MAP
+
+z
k
Detector
wk AWGN {uM,n }
s
M
Oversample R = Tc / Ts
{x M,k}
{u^M,n } Equivalent Channel f M,k
Figure 1: Asynchronous DS-CDMA system single-antenna at the centralized receiver. The system has M users, each transmitting symbols using a known direct-sequence (DS) spreading code with processing gain G (i.e. G chips per symbol). For user m, the data sequence of symbols of period T is designated fum;ng, n = 0; : : : ; N ? 1. The G 1 spreading code is sm = [sm;0 ; : : : ; sm;G ]T , where ()T denotes transpose. The information is transmitted over a frequencyselective fading channel using Q-ary chips of period Tc , with bandlimited pulse shape gm ( ). The physical frequency-selective fading channel for the m-th user has impulse response cm (t; ). The physical channel is usually assumed to have Gaussian fading statistics, thus resulting in a Rayleigh or Rician channel. When the channel statistics are wide-sense-stationary and uncorrelated scatterers (WSSUS), the scattering function (or delay-Doppler pro le) completely describes the channel statistics. The signals from each of the users are superimposed at the receiver where additive white Gaussian noise (AWGN) is introduced. In order to accommodate the asynchronous nature of the incoming signals from each of the users, the received signal is oversampled by an integer factor R = Tc=Ts where Tc is the chip period (Tc = T=G), and Ts is the sampling period. Figure 1 depicts the discrete equivalent system. The sampled received signal at time k is:
zk =
M `X m ?1 X m=1 =0
xm;k? fm;k; + wk
(1)
where fm;k; is a discrete equivalent channel (of length `m) for the m-th user, representing the combined e ect of the transmitted pulse shape, the physical channel and the receiver front-end lter. The complex Gaussian noise samples, wk , are zero mean i.i.d. with variance w2 = No =(2Ts). No =2 is the two-sided spectral noise density [9]. The sequence fxm;k g represents the oversampled chips for
user m, related to the spreading code sm , and the data sequence fum;n g. To be precise: (
k?m xm;k = sm;mod( k?Rm ;G) um;b k?RGm c R integer
0
otherwise
(2) where user m has a relative delay of m sample periods. The timing o sets m need only be known as a fraction of a symbol interval at the receiver, since whole symbol intervals are transparent to the symbol hypotheses of the MAP algorithm as will be seen in the following section. Note that we use k to index quantities sampled at the higher rate, 1=Ts = (RG)=T , i.e. k = 0; : : : ; RGN ? 1. At the symbol rate 1=T , quantities are indexed using n. It is apparent from (1) that each sample, zk depends on fxm;k?`m +1 : : : xm;k g, m = 1; : : : ; M . i.e. for user m, `m ? 1 values previous to xm;k interfere with the contribution to the output zk . Therefore, considering the worst case, there are L ? 1 = d(` ? 1)=(RG)e additional bits of intersymbol interference (ISI) at the symbol rate, where ` = max m `m . Thus the trellis for MLSE has QM (L?1) states (L > 1).
3. MULTI-USER MAP DETECTION Here we review the MAP algorithm for symbol-bysymbol detection of the user transmitted sequences fum;ng, m = 1; : : : M . In this section we assume that the channel samples, fm;k;, are known at the receiver. In Section 4, we will incorporate estimates of the channels by expanding the trellis. The MAP algorithm calculates the a posteriori probability of each M 1 set of transmitted symbols, i.e. Pr(un = q j Z0RGN ?1; ), for each possible M 1 set of Q-ary symbols, q, where un = [u1;n; : : : ; uM;n]T , Zkk12 is the set of observations fzk1 : : : zk2 g and represents the channel model. The decoded symbol set is declared to be that with
ISPACS{98, Melbourne, Australia (November 1998), pp. 432-436
~ u1,n-1 ~ u2,n-1 j=1
00
j=2
01
j=3
10
j=4
11
an-1,11 bn-1,11
~ u1,n ~ u2,n
3
The a priori state transition probabilities are derived from the a priori information provided to the demodulator (soft-inputs).
an?1;ij = Pr(Sn;j j Sn?1;i ; ) = Pr(~un?L+1;j : : : u~ n;j j u~ n?1?L+1;i : : : u~ n?1;i ; ) Since the observation noise is Gaussian, the observation probabilities bn?1;ij are given by: an-1,44 bn-1,44 α n-1,j
β n-1,j
α n,j β n,j
n-1
n
where
the maximum a posteriori probability. As with the MLSE Viterbi algorithm, the MAP algorithm uses a trellis. Figure 2 shows the trellis for binary symbols with M = 2 users, assuming known fading channels with ISI such that L = 2. The j th state of the trellis at time n is Sn;j . This state represents one of the QM (L?1) possible values for fun?L+1; : : : ; un g. We denote the particular value by fu~ n?L+1 ;: : : ; u~ n g where \tilde" indicates a hypothesized value. Note that the multi-user trellis has QM transitions exiting (and entering) each state. The a posteriori state probabilities, n;j = Pr(Sn;j j Z0RGN ?1 ; ), can be calculated using the forward{ backward procedure [11]. The forward variable is n;j = Pr(Sn;j ; Z0RG(n+1)?1 j ). The backward RGN ?1 ; ). We devariable is n;j = Pr(Sn;j j ZRG (n+1) note the a priori transition probability from state Sn?1;i to state Sn;j by an?1;ij , and for that transition, we denote the probability of observations RG(n+1)?1 by b ZRGn n?1;ij . The recursions for the forward and backward variables are:
n;j =
?
i=1
bn?1;ij an?1;ij n?1;i
(3)
?
Q X
bn;ij an;ij n+1;j (4) j =1 where Q? = QM (L?1) is the number of states in n;i =
the trellis. The a posteriori state probabilities are then given by: (5)
k;j = PQ ?k;j k;j i=1 k;i k;i
n
RG(nY +1)?1
o
1 e ? 21w2 (zk ?z^k;ij )2 (6) p = 2w2 k=RGn
Figure 2: Trellis for Q = 2, M = 2, L = 2
Q X
RG(n+1)?1 j S bn?1;ij = Pr(ZRGn n?1;i ; Sn;j ; )
z^k;ij =
M `X m ?1 X m=1 =0
In vector notation:
z^k;ij =
M X m=1
x~m;k?;ij fm;k;
x~m;k;ij fm;k
(7)
where x~m;k;ij = [~xm;k;ij ; : : : ; x~m;k?`m +1;ij ], and fm;k = [fm;k;0; : : : ; fm;k;`m?1]T .
Our general multi-user MAP detection algorithm is appropriate for asynchronous systems, provided the relative timing o sets are known. This is because o sets are incorporated into x~m;k;ij using (2), and the observation probabilities (6) are written in terms of x~m;k;ij . In the next section we describe how the channels fm;k are estimated by expanding the trellis statespace. In fact we will have a di erent channel estimate for each transition. We will de ne ^fm;k;ij to be the estimate of the channel for the m-th user, under the assumption of a transition i to j . When the channel is being estimated, the variance w2 , in 2 which accounts for the (6), will be replaced by ^k;ij extra variance due to channel estimation errors. Of course, the MAP algorithm is often implemented in the log domain, to minimize the required computations. In this case, log-probabilities are referred to as metrics. Pilot symbols can be used to assist with channel estimation and resolution of phase-ambiguity [7]. In our multi-user MAP detector, these known symbols are taken into account by adjusting the a priori trellis transition probabilities, an;ij . For this reason, it is important to treat the transition probability metric and observation probability metric separately. The ratio of pilot symbols to data symbols can be adjusted to re ect the channel conditions.
ISPACS{98, Melbourne, Australia (November 1998), pp. 432-436
4. MMSE CHANNEL ESTIMATION In the previous section, we reviewed the MAP algorithm for symbol-by-symbol detection in a multiuser environment where perfect channel knowledge was available at the receiver. Now we consider the fading case when channel estimates are required. We generate these estimates using MMSE techniques. The key idea is to expand the state-space of the trellis. The extra states arise because the low-pass nature of the fading e ectively introduces correlation into the received signal. The states in the expanded trellis contain the extra information needed in order to form channel estimates based only on the state using an MMSE predictor. Each state has a di erent set of channel estimates based on the hypotheses. This approach is related to persurvivor processing (PSP) and therefore has signi cant advantages in terms of channel tracking compared to the standard approach of separately (from the detector) forming channel estimates using past decisions. When p additional samples of zk and hypotheses x~m;k (at the sample rate) are used in the estimation, the expanded trellis has QM (P +L?1) states, where P = dp=(RG)e. The maximum a posteriori symbol probability for a particular value of un is given by summing the probabilities of states corresponding to that value. Thus: Pr(un = q j Z0RGN ?1 ; ) =
X
j for which u~ n =q
Pr(Sn;j j Z0RGN ?1; )
So far, we have discussed forming estimates of the channel, ^fm;k;ij , in order to calculate z^k;ij in (7) for the branch metrics for the MAP detector. However, in this section we directly evaluate the MMSE estimates, z^k;ij , without explicitly estimating ^fm;k;ij . Using our expanded trellis, the estimates z^k;ij (one for each state transition i to j ), are calculated using the observations fzk : : : zk?p : : : zk?p?`+1 g and the hypotheses fx~m;k : : : x~m;k?p : : : x~m;k?p?`+1 g, for m = 1; : : : ; M . This is achieved with a MMSE linear predictor. First, we write the observation equation (from (1)) in vector-matrix notation:
zk?1 =
M X m=1
Xm;k?1Fm;k?1 + wk?1
where zk?1 = [zk?p ; : : : ; zk?1 ]T ,
(8)
4
wk?1 = [wk?p ; : : : ; wk?1 ]T , and: 2 xm;k?p 0 6 ... 6 0 6 Xm;k?1 = .. .
6 4
2
Fm;k?1 =
6 6 6 6 4
...
0
fm;k?p .. . .. .
fm;k?1
3
3
0 .. .
0 0 xm;k?1
7 7 7 7 5
7 7 7 7 5
where xm;k = [xm;k : : : xm;k?`+1 ] and fPm;k = [fm;k;0 : : : fm;k;`?1]T . In this notation, zk = M x f + w . The estimate, z^ , is obk k;ij m=1 m;k m;k tained by prediction, and is of the form:
z^k;ij = hk;ij zk?1
(9)
where of course hk;ij is di erent for each state in the trellis. Clearly there will be a di erent estimate, z^k;ij , for each transition in the trellis. Conceptually at least, this is similar to MLSE PSP. Minimizing the mean square error, 2 = E (zk ? z^k;ij )2 conditioned on the ij hyk;ij pothesis gives [12]:
hk;ij = E zk;ij zHk?1;ij E zk?1;ij zHk?1;ij ?1 (10) where the ()H denotes Hermitian transpose and the conditioning on the hypothesis is implied. For what follows, we drop the subscripts fgk;ij and
?
use the \tilde" to indicate matrix values under the hypothesis of an ij transition. Assuming that the channels for the users are independent, the result of evaluating expectations in (10) is: M X
M X
m=1
m=1
hk;ij =( x~mrmX~ Hm ) = ??1
!?1
X~ mRmX~ Hm + NoI (11)
where the covariance matrices rm = E [fm;k FHm;k?1 ] and Rm = E [Fm;k?1 FHm;k?1 ] are assumed to be known (from the fading statistics or scattering function of the channels) for each user, m = 1; : : : ; M . We can now substitute (11) into (9) so that the branch metrics (6) for the MAP algorithm can be calculated. The prediction error variance is given by: 2 = 2 ? ??1 H ^k;ij z 2 where z = E [zk zk ] ( represents the complex conjugate). Since the hypotheses, X~ m and x~ m , depend only on the index of the preceding state, i, and the transition de ned by ij , we note that the
ISPACS{98, Melbourne, Australia (November 1998), pp. 432-436 MMSE coecients hk;ij , and the prediction error 2 are independent of k , and can be variance ^k;ij pre-calculated. It is interesting to note that the explicit MMSE estimates ^fm;k;ij used in (7), can be seen to be: ^fm;k;ij = rm X~ Hm
M X m=1
!?1
X~ m RmX~ Hm + NoI
zk?1 (12)
5. JOINT MAP DETECTIONESTIMATION PROPERTIES In this section we consider the properties of the proposed MAP detector with joint channel and data estimation. Near-far resistance Unfortunately, the role of the spreading codes in forming the MMSE estimates for the multi-user MAP detector is unclear in (11). In the simpli ed case of a synchronous multi-user system operating in a slow at-fading environment, the spreading codes can be explicitly incorporated using (2) [13]. As might be expected, it can be shown that if the spreading codes are orthogonal, the channel estimate for each user is independent of the other users, and in particular, their power. Therefore, since the MAP algorithm with perfect channel information and high SNR is near-far resistant, so too is our expanded MAP detector (which does not assume perfect channel information). Performance in severe fading Per-survivor techniques have been demonstrated to provide better channel estimation and detector performance in fading conditions [14]. Our MAP detector mimics the per-survivor approach by expanding the state-space of the trellis, and therefore has similar advantages in this multi-user environment. There is a trade-o between the amount of expansion of the state-space and the accuracy of the channel estimates. In severe fading or low SNR conditions, it is often necessary to have a large expansion in order to have accurate channel estimates. However the expansion should not exceed the coherence time of the channel. Naturally, trellis expansion carries with it an increase in computational intensity. Soft-output The MAP algorithm is known to provide optimal soft-decisions on a symbol-by-symbol basis. Thus our algorithm is well-suited to receiver structures in which subsequent stages (e.g. outer decoding) utilize soft-decisions. In particular, soft-input soft-
5
output processing enables iterative (turbo) processing at the receiver. Turbo processing has been shown to approach the Shannon limit. 6.
REFERENCES
[1] S. Verdu, \Minimum probabilty of error for asynchronous Gaussian multiple access channels," IEEE Trans. on Information Theory, vol. 32, no. 1, pp. 85{96, 1986. [2] R. Lupas and S. Verdu, \Linear multiuser detectors for synchronous code-division multiple access," IEEE Trans. on Information Theory, vol. IT-35, pp. 123{136, January 1989. [3] Z. Xie, R. Short, and C. Rushforth, \A family of sub optimal detectors for coherent multi-user communications," IEEE Journal on Selected Areas in Communications, pp. 683{690, May 1990. [4] P. B. Rapajic and B. Vucetic, \Adaptive receiver structures for asynchronous CDMA systems," IEEE Journal on Selected Areas in Communications, vol. 12, pp. 685{697, May 1994. [5] L. J. Zhu and U. Madhow, \Adaptive interference suppression for direct sequence CDMA over multipath fading channels," Preprint, 1997. [6] M. L. Honig, U. Madhow, and S. Verdu, \Blind adaptive multiuser detection," IEEE Trans. on Information Theory, vol. 41, pp. 944{960, July 1995. [7] M. J. Gertsman and J. H. Lodge, \Symbol-bysymbol MAP demodulation of CPM and PSK signals on Rayleigh at-fading channels," IEEE Trans. on Communications, vol. 45, pp. 788{799, July 1997. [8] L. Davis, I. Collings, and P. Hoeher, \Joint MAP equalization and channel estimation for frequencyselective fast-fading channels," in Proc. IEEE GLOBECOM-98 (accepted), 1998. [9] J. Proakis, Digital Communications. McGraw Hill, 3rd ed., 1995. [10] P. Hoeher, \Advances in soft-output decoding," in Proc. IEEE GLOBECOM-93, pp. 793{797, Nov 1993. [11] L. R. Rabiner, \A tutorial on Hidden Markov Models and selected applications in speech recognition," Proceedings of IEEE, vol. 77, pp. 257{285, Feb 1989. [12] S. J. Orfanidis, Optimal Signal Processing: An Introduction. Macmillan, 1985. [13] L. Davis and I. Collings, \MAP decoding for at{ fading multiple access CDMA channels," in Proc. Int. Symposium on DSP for Communication Systems (DSPCS) (submitted), 1999. [14] R. Raheli, A. Polydoros, and C.-K. Tzou, \Persurvivor processing: A general approach to MLSE in uncertain environments," IEEE Trans. on Communications, vol. 43, pp. 354{364, Feb 1995.
