MULTISTAGE MULTIUSER DETECTION FOR CDMA WITH SPACE-TIME CODING Yumin Zhang and Rick S. Blum
EECS Department, Lehigh University Bethlehem, PA 18015
[email protected]
ABSTRACT
The combination of Turbo codes and space-time block codes is studied for use in CDMA systems. Each user's data are rst encoded by a Turbo code. The Turbo coded data are next sent to a space-time block encoder which employs a BPSK constellation. The space-time encoder output symbols are transmitted through the fading channel using multiple antennas. A multistage receiver is proposed using non-linear MMSE estimation and a parallel interference cancellation scheme. Simulations show that with reasonable levels of multiple access interference ( 0:3), near single user performance is achieved. The receiver structure is generalized to decode CDMA signals with space-time convolutional coding and similar performance is observed.
1. INTRODUCTION
Space-time codes [1]-[4] use multiple transmit and receive antennas to achieve diversity and coding gain for communication over fading channels. High bandwidth eciency is achieved, with performance close to the theoretical outage capacity [1]. Turbo codes [5] are a family of powerful channel codes, which have been shown to achieve near Shannon capacity over additive white Gaussian noise channels. Since their introduction, both space-time codes and Turbo codes have received considerable attention. In the CDMA2000 Radio Transmission Technology (RTT) proposed for the third generation systems, both space-time codes and Turbo codes have been adopted [6]. Although papers treating either just space-time codes or Turbo codes abound, jointly considering space-time codes and Turbo codes in CDMA systems is a relatively new topic. In this paper, we initiate a study on this topic where we focus on space-time block codes [3][4]. Our research develops suboptimum low-complexity receivers, which will be needed. This paper is organized as follows. Section 2 rst sets up the system con guration and develops the received signal model. A brief review of space-time block
codes is given in Section 3. The structure of our multistage receiver is discussed in Section 4. Section 5 presents simulation results. Conclusions are given in Section 6.
2. SYSTEM CONFIGURATION AND RECEIVED SIGNAL MODEL
Fig. 2 depicts a K user synchronous CDMA system with combined Turbo coding and space-time block coding. There are N transmit antennas and M receive antennas in the system. Suppose user k, k = 1; :::; K , has a block of binary information bits fdk (i); i = 1; :::; L1g to transmit. These bits are rst encoded by a Turbo code with rate R1 = LL . The bits which are produced by the Turbo encoder, denoted by fd~k (i); i = 1; :::; L2g, are passed to a space-time block encoder. This spacetime block code uses a transmission matrix GN [3] with a BPSK constellation, generates N output bits during each time slot, and has rate R2 = LL . During time slot l, N bits are transmitted, which are denoted by fbnk (l); n = 1; :::; N g, for l = 1; :::; L. The bit bnk (l) 2 f?1; +1g is spread using a unique spreading waveform sk (t) and transmitted using antenna n. For convenience we denote the vector of nth output bits from all K users as bn (l) = [bn1 (l); :::; bnK (l)]T , and we note that all of these bits are transmitted by antenna n during time slot l. We de ne the set of bits fbn (l); l = 0; :::; L ? 1g as one frame of data. The fading coecient for the path between transmit antenna n and receive antenna m is denoted by nm . In our research, we assume a at quasi-static fading environment [3], where the fading coecients are constant during a frame and are independent from one frame to another. Further we assume for simplicity that perfect estimates of all fading coecients are available at the receiver. The received signal at antenna m is 1 2
2
rm (t) =
N X K LX ?1 X
n=1 k=1 l=0
nm Ak bnk (l)sk (t?lT )+m(t) (1)
where T is the bit period, Ak is the transmitted signal
amplitude for user k, and m (t) is the complex channel noise at receive antenna m. The received signal rm (t) is next passed through a matched lter bank, with each lter matched to one user's spreading waveform. Denote the matched lter outputs at receive antenna m for the time slot j by ym (j ) = [ym1 (j ); :::; ymK (j )]T . The equation describing ym (j ) can be represented in vector form as
ym(j )
=
RA
N X n=1
nm bn (j ) + nm(j )
m = 1; :::; M; j = 0; :::; L ? 1: (2) where R is the K K cross-correlation matrix of the spreading codes, A = diag(A1 ; :::; AK ), and nm (j ) is the K 1 complex noise vector after matched ltering. Assuming the channel noise is Gaussian with zero mean and autocorrelation function 2 ( ), nm (j ) has a multidimensional Gaussian distribution N (0; 2 R).
3. SPACE-TIME BLOCK CODES
An extensive discussion of space-time block codes is given in [3][4]. Here we consider only N = 2 antenna cases. Extension to N > 2 cases is straightforward. A BPSK space-time block code with two transmit antennas is described by the transmission matrix
s s 1 2 G2 = ?s2 s1 :
(3)
The encoder works as follows. The block of L2 Turbo coded bits enter the encoder and are grouped into units of two bits. Each group of two bits are mapped to a pair of BPSK symbols s1 and s2 . These symbols are transmitted during two consecutive time slots. During the rst time slot, s1 and s2 are transmitted simultaneously from antenna one and two respectively. During the second time slot, ?s2 and s1 are transmitted simultaneously from antenna one and two, respectively. The code rate of G2 is 1. In [3][4], the transmission matrix is designed so that the columns are orthogonal to each other. This allows a simple receiver structure using only linear processing. We illustrate this using the code described in (3) as an example. Extension to N > 2 cases is straightforward. Assuming there are M receive antennas, the received signal at antenna m during the rst and second time slots, denoted by ym (1) and ym (2), are
ym(1) = 1m s1 + 2m s2 + nm (1) ym(2) = ? 1ms2 + 2m s1 + nm (2)
(4)
where nm (1) and nm (2) are two iid complex Gaussian noise samples with variance n2 . The observations in
(4) can be combined to yield the improved quantities
s~1 and s~2 using s~1 = 1m ym (1) + 2m ym (2) = (j 1m j2 + j 2m j2 )s1 + 1m nm (1) + 2m nm (2) s~2 = 2m ym (1) ? 1m ym (2) = (j 1m j2 + j 2m j2 )s2 + 2m nm (1) ? 1m nm (2)
Combining quantities obtained at each receive antenna yields ~s1 = ~s2 =
M X ( 1m ym (1) + 2m ym (2)) Cs1 + n1
m=1 M X m=1
( 2m ym (1) ? 1m ym (2)) Cs2 + n2 (5)
where
C=
M X m=1
(j 1m j2 + j 2m j2 ):
(6)
The Gaussian noise variables n1 and n2 have variance
b2 = n2
M X m=1
(j 1m j2 + j 2m j2 )
(7)
It is easily seen from (5), (6) and (7) that after this simple linear combining, the resulting signals are equivalent to those obtained from using maximal ratio combining [7] techniques for systems with 1 transmit antenna and 2M receive antennas. This combining technique will be used in two places in our low-complexity receiver as discussed in the next section.
4. LOW-COMPLEXITY MULTISTAGE RECEIVER
The optimum receiver that minimizes the frame error rate should construct a \super-trellis" for decoding. The super-trellis combines the trellis of Turbo codes and the structure of the multiuser channel and spacetime block codes. Due to the interleavers used in the Turbo codes, it is very hard to construct such a supertrellis. In fact, \optimum decoding" for Turbo codes alone is impossible in practice. This is why suboptimum iterative decoding schemes are used to decode Turbo codes [5]. Thus instead of trying to nd an optimum receiver, which would obviously have a prohibitively high complexity, our goal in this section is to develop a low-complexity suboptimum receiver. We suggest the multistage receiver structure depicted in Fig. 2. The output of the matched lter bank is rst passed to a decorrelating detector [8], which attempts to eliminate the multiple access interference (MAI) completely with perfect estimation. The output
When perfect estimate of bn (j ) is available, ym (k) (j ) o ers K di erent observations of the signal from user k, contaminated only by channel noise. For simplicity, we N X use the kth element of ym (k) (j ) for processing, which y~m(j ) = (RA)?1ym(j ) = nm bn(j ) + n~m(j ) (8) gives the highest SNR for user k. The kth elements n=1 of ym (k) (j ), m = 1; :::; M , at all receive antennas are where we de ned the noise vector n~m (j ) = (RA)?1 nm (j ), combined using the techniques discussed in Section 3. The improved observations are passed to another set of which has a Gaussian distribution with covariance maTurbo decoders to perform the second stage of decodtrix ing. These Turbo decoders produce the nal \hard" ~R = 2 (ARA)?1 : (9) decisions on each user's transmitted bits. The elements from y~1 (j ), ..., y~M (j ) corresponding to 5. SIMULATION RESULTS the kth user, denoted by y~1k (j ),...,~yMk (j ), are combined using the technique discussed in Section 3 to proMonte Carlo simulations are carried out to study the vide improved observations for user k. These improved performance of the proposed multistage receiver. Conobservations are sent to a single user Turbo decoder to sider a 4 user synchronous CDMA system with 2 transperform the rst stage of decoding. The Turbo decoder mit antennas and 2 receive antennas. Each user's bits produces posterior probabilities for user k's transmitare rst encoded by a rate 1/3 Turbo code with conted bits. These posterior probabilities, together with straint length = 5 and generator 23, 35 (octal form). the diversity combined observations, are used by a soft The random interleaver chosen for the Turbo code has estimator to form soft estimates of user k's transmitted length 128. The block of Turbo coded data is encoded bits. using a space-time block code with the code matrix The soft estimator uses non-linear minimum mean G2 from (3) and a BPSK constellation. Next the outsquare error (MMSE) estimation [9] to form the soft put bits are spread using each user's spreading waveestimates. From (5), it is seen that the diversity comform and the results are transmitted using 2 antennas bined observations for user k can always be represented over the fading channel. The path gains are modeled in the form of y = Cb + n, where y is the noisy obseras samples of independent complex Gaussian random vation, b is the transmitted bit, C is a known constant variables with variance 0.5 per dimension (real or imagand n is a complex Gaussian noise sample with variinary). Quasi-static fading is assumed. For the CDMA ance denoted by b2 . The soft estimate of b is obtained channel, we use the symmetric channel model where by the cross-correlation between all pairs of two users is the common value . The SNR for user k is de ned as Re Cy ? Re Cy
of the decorrelating detector at receive antenna m and time slot j is
2
(
)
2
(
)
Pr(b=+1) e b2 ? e b2 Pr (b=?1) E fbjyg = 2Re(Cy ) ; 2Re(Cy ) Pr(b=+1) e b2 + e? b2 Pr(b=?1)
(10)
where the prior probabilities Pr(b = 1) can be updated using the posterior probabilities obtained by the Turbo decoders. The transmitted signals are reconstructed using the soft estimates as if they were binary digits. Denote the reconstructed encoder output for antenna n and user k during time slot j as ^bnk (j ) and de ne b^n (j ) = [^bn1 (j ); :::; ^bnK (j )]T . The reconstructed signals fb^n (j ), n = 1; :::; N , j = 0; :::; L ? 1g are used in soft MAI cancellation to produce \cleaner" received signals for each user. To cancel MAI for user k, we rst de ne a vector b^(nk) (j ) equal to b^n (j ) except that its kth element is zero. The MAI-reduced observation for user k at receive antenna m is obtained using
ym k (j ) = ym(j ) ? RA ( )
N X n=1
nm b^(nk) (j )
(11)
k SNRk = 2NA R1 R2
(12)
Fig. 3 gives the BER performance of the proposed multistage receiver in Gaussian noise when all users have the same power (A = I). The BER performance for the rst stage and second stage decoding are both plotted, which we denote by \S 1" and \S 2" on the graph. For comparison, we also give the single user performance, which is the Turbo code performance for the fading channel under consideration. The performance of the space-time block code using G2 without the Turbo coding is also shown. For = 0:1, single user performance is nearly achieved after just the rst stage decoding. The second stage decoding curve is indistinguishable from that of the single user performance. For = 0:3, the performance improvement obtained by employing the second stage of decoding is obvious from Fig. 3b. After the second stage decoding, single user performance is approached. By combining a Turbo code with a space-time block code, a performance gain
of about 2:5dB is achieved at BER=10?4 compared to using a space-time block code only. An iterative receiver structure can be easily constructed by feeding back the posterior information obtained after the second stage decoding to the soft estimators. We have carried out simulations using this iterative structure, but results show that the improvement over the second stage of decoding is marginal. In Fig. 3b, we plot the BER performance for the second iteration of the \iterative receiver" (denoted by \Ite 2"), which is almost indistinguishable from the second stage decoding curve. Thus the extra computations incurred by the iterative structure are not justi ed. Next we study the performance of our receiver in a near-far situation where two users are 20dB stronger than the other two users, all other parameters remain the same as in Fig 3. The BER performance for the strong user and weak user are given in Fig. 4a and 4b respectively. The performance, for both the weak and strong users, approaches single user performance after the second stage decoding. Finally, we point out that the received signal model in (2) is also valid for a CDMA system with space-time convolutional coding [1] replacing the combination of space-time block codes and Turbo codes. An iterative receiver can be constructed using the parallel interference cancellation scheme [10]. Fig. 1 gives the frame error rate performance for the rst two iterations of the iterative receiver for a CDMA system with space-time convolutional coding. It is seen that with 2 iterations, single user performance is achieved. Another observation is that the performance improvement obtained by employing the iterative structure is marginal. This is consistent with our previous observations for the spacetime block coded system.
[2] S. M. Alamouti, \A simple transmitter diversity scheme for wireless communications," IEEE JSAC, vol. 16, No. 8, pp. 1451-1458, Oct. 1998. [3] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, "Space-time block coding for wireless communications: performance results," IEEE JSAC, vol. 17, No. 3, pp. 451-460, March 1999. [4] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, "Space-time block codes from orthogonal designs," IEEE Trans. Info. Theo., vol. 45, No. 5, pp. 14561467, July 1999. [5] C. Berrou and A. Glavieux, \Near optimum error correcting coding and decoding: Turbo-Codes," IEEE Trans. Comm., vol. 44, No. 10, pp. 12611271, Oct. 1996. [6] S. Dennett, \The CDMA2000 ITU-R RTT candidate submission," V. 0.17, TIA, July 28, 1998. [7] J. G. Proakis, Digital Communications, 3rd Edition, McGraw-Hill, 1995. [8] S. Verdu, Multiuser Detection, UK: Cambridge University Press, 1998. [9] A. Papoulis, Probability, Random Variables and Stochastic Processes, New York: McGraw-Hill, 1984. [10] Yumin Zhang, Iterative and Adaptive Receivers For Wireless Communication and Radar Systems, Ph.D. Dissertation, Lehigh University, May 2000. 0
10
In this paper, we studied the application of Turbo codes and space-time block codes in CDMA systems. A multistage receiver is proposed using parallel interference cancellation schemes. Simulation results show that with reasonable levels of MAI ( 0:3), near single user performance can be achieved. The receiver developed in this paper was generalized to decode CDMA signals with space-time convolutional coding and similar performance was observed.
7. REFERENCES
[1] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time codes for high data rate wireless communication: Performance criteria and code construction," IEEE Trans. Info. Theo., vol. 44, No. 2, pp. 744-765, Mar. 1998.
Frame Error Rate
6. CONCLUSIONS
MMSE −1
1
10
2
single user
−2
10
5
5.5
6
6.5
7
7.5 SNR(dB)
8
8.5
9
9.5
10
Figure 1: Performance of the iterative multiuser receiver for CDMA with space-time convolutional coding [10] with K = 4, = 0:3, 4-PSK S-T code with rate 2/b/s/Hz, 130 symbols per frame, 2 transmit and 2 receive antennas where MMSE is used in the rst stage decoding.
b 11 d1
S-T
Turbo Encoder
α11
y 11
α1M
S1(t)
Encoder
MF1
Posterior Probability -1
Tx1
y
Rx1
b 1K
MFK
R
^ Turbo Decoder
Diversity Combining
1K
Soft MMSE Esitmator
S-T Encoder
Soft MMSE Esitmator
S-T Encoder
Turbo Decoder
Soft MAI cancellation
d1
& SK(t) b N1
Diversity Combining
y MF1
Turbo Decoder
M1 -1
S1(t)
dK
Turbo Encoder
RxM
S-T
y MFK
TxN
Encoder
^
Diversity Combining
First stage decoding
R
Turbo Decoder
dK
Second stage decoding
MK
b NK SK(t)
Figure 2: Structure of our K user CDMA system (including our multistage receiver) with combined Turbo coding and space-time block coding, N transmit antennas and M receive antennas. −1
−1
10
10
−2
−2
Bit Error Rate
10
Bit Error Rate
10
S−T block code
−3
10
Single user
−3
S−T block code
10
S1 S2
S1
single user
S2
−4
10
Ite 2
−4
0
1
2
3
4
5 SNR(dB)
6
7
8
9
10
10
0
1
2
3
4
5 SNR(dB)
6
7
8
9
10
(a) = 0:1 (b) = 0:3 Figure 3: Performance of the multistage receiver for CDMA with Turbo coding and space-time block coding with K=4 users, 2 transmit and 2 receive antennas. Strong user
−1
Weak user
−1
10
10
−2
−2
Bit Error Rate
10
Bit Error Rate
10
S−T block code
−3
S−T block code
−3
10
10
S1 Single user
S1
S2
Single user S2
−4
10
−4
0
1
2
3
4
5 SNR(dB)
6
7
8
9
10
10
0
1
2
3
4
5 SNR(dB)
6
7
8
9
10
(a) Strong user (b) Weak user Figure 4: Performance of the multistage receiver for CDMA with Turbo coding and space-time block coding under a near-far situation with K=4, = 0:3, 2 transmit and 2 receive antennas. Two users are 20dB stronger than the other two users.
