symbol asynchronism will occure in the uplink transmission. The classical methods

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1 Gallager Codes for Asynchronous Multiple Access. A. de Baynast y and D. Declercq Laboratoire ETIS - ENSEA 6, avenue du Ponceau F95014 Cergy-Pontoise Cedex France debaynas, declercq@ensea.fr Abstract. In the next wireless communication systems (UMTS or IMT-2000), the symbol asynchronism will occure in the uplink transmission. The classical methods (matched lter or RAKE receiver) are unsuitable in that case, and a joint decoding is necessary. In this paper, we propose a robust method to the symbol asynchronism for a multiple access system based on the Gallager codes. As in the synchronous case (de Baynast and Declercq, 2002), we show that we can bypass the spreading sequences in an asynchronous multiple access system. The "algebraic" orthogonality is replaced by a "statistical" orthogonality which is provided by the randomness of the Gallager codes. The performance of our system are close to the optimum achievable bound (2dB or less at BER= 1e 3 for a block length N = 2000) for many congurations. Keywords: Joint multiple access detection, symbol asynchronism, Gallager codes, capacity for an asynchronous multiple access system. Introduction In the scope of multiple access techniques, spread spectrum has been proposed for code division multiple access (DS-CDMA) and is successfully used in IS-95 system. Spread spectrum multiple access employs a set of spreading sequences which provide an "algebraic" orthogonality between all users in a cell. This orthogonality helps to mitigate the multiple access interference (MAI) at the receiver. For third generation systems, IMT-2000 or UMTS, the increase in rates implies a smaller spreading factor: 1 S F 16 instead of 1024 in IS-95. In the uplink transmission, asynchronism chip rate or symbol rate between the users dramatically destroys the orthogonality of the spreading sequences; interference penalizing too much the performance of the RAKE receiver, e.g. (Johansson and Svensson, 1995). For the asynchronous DS-CDMA system, (Scaglione et al., 2000) allow each user to activate at each symbol interval just one of N f frequencies (N f > N u, the number of users). This family of codes y A. de Baynast is supported by a DGA (French Defense) PhD grant. c 2002 Kluwer Academic Publishers. Printed in the Netherlands.

2 2 A. de Baynast and D. Declercq for CDMA systems eliminate MAI completely. However, it requires the introduction of trailing zeros also called guard chips at the end of the spreading sequences. The number of these guard chips is at least equal to the maximum oset between the users in terms of chips. Because this number should not be too large, the users have to be quasi-synchronous. This method is more adequate to IS-95 than to IMT More generally, none set of spreading sequences can achieve the Welsh's bound in a completely asynchronous transmission even with strong assumptions like quasi-synchronous transmission, Gaussian interference, etc. Since the asynchronism penalizes too much the performance of a DS- CDMA system, it seems to be judicious to consider an asynchronous multiple access scheme which favors coding at the expense of spreading: the "algebraic" orthogonality between the spreading sequences is then replaced by a kind of orthogonality in the code structures which we will denote "statistical" orthogonality. Such a method has been successfully proposed in the synchronous case (de Baynast and Declercq, 2002) and would intuitively not be aected by the asynchronism (symbol or frame asynchronism). That will be conrmed by simulations means (see section 4). We describe now the proposed approach: we assign at each user a Gallager code also described in the literature as low density parity check code (LDPC), (Gallager, 1962; MacKay, 1999) and at the receiver we adopt a joint decoding algorithm. This random coding scheme with long enough block can reach the capacity of the AWGN synchronous MAC, (Cover and Thomas, 1991) and (Chung et al., 2001). Whereas this result has not been extended to the case of the AWGN asynchronous MAC, we will show with simulations that the performance achieved by such a system are very close to the capacity. We suppose in the sequel that the osets between all users are known from the receiver. Since the type of synchronism that is dicult to achieve in many practical situations is symbol synchronism, we limit our study on the symbol asynchronous multiple access system. The paper will proceed as follows: after describing the model in the second part, we present the theoretical performance bounds of the AWGN asynchronous multiple access channel from information theory. In the fourth part, we describe the proposed joint multiple user decoding algorithm. Finally, we present some simulation results.

3 Gallager Codes for Asynchronous Multiple Access 3 1. Channel model We supposed without loss of generality that we are frame-synchronous and symbol asynchronous. The simultaneous N u users are sorted in ascending order of their respective osets 1 2 : : : Nu with 1 = 0. Furthermore, we supposed that the sampling clock at the receiver is able to synchronize at any oset n and this for any value of n;n+1 = n+1 n. This synchronization can be realized by the use of scrambling sequences and the use of a powerful digital downconverter. Since the receiver knows the assigned rectangular pulse waveforms as well the symbol period T and the relative osets n;n+1, it can compute y[l] = y (1) [l]; y (2) [l]; : : : ; y (Nu) [l] by sampling the continuous n o N 1 l=0 observation data at each instant lt + n (see gure 1). Because of the s 1 [l 1] s 2 [l 1] symbol period s 1 [l] s 2 [l] s 1 [l + 1] s Nu [l 1]. s Nu [l] : : : y (Nu) [l+1] y (1) [l] : : : y (Nu) [l] y (1) [l+1] : : : 1;2 : : : 1 Nu Figure 1. Asynchronous model channel factorization theorem (Lehman, 1959), y (1) [l]; y (2) [l]; : : : ; y (Nu) [l] l=0 are sucient statistics to estimate the transmitted messages. This implies that the channel output fy(t)g enters in the computation of the posterior probability of each message only through y (n) [l]. See (Verdu, 1989a) for more details. The basic discrete time AWGN asynchronous multiple access channel with total input power constraint, rectangular pulse and noise variance N 0 =2 can also be modeled by: y = XN u n=1 a n M ;n s n + n = h XN u n=1 n o N 1 M ;n x n + n = M x + n = z + n (1) T where y = y T [0]y T [1] : : :y T [N 1]i are the observed data with h i y[l] = y (1) [l] y (2) [l] : : :y (Nu) [l], a n the real fading coecients, M ;n

4 4 A. de Baynast and D. Declercq the matrix which contains the asynchronism coecients n;n+1, s n = fs n [l]g l=0;:::;n 1 the transmitted codeword of length N and power IE[s 2 n[l]] = 1 from the n th user and n = fn[l]g l=0;:::;n 1 the additive white noise following the normal distribution. Let C n be a binary linear (N; K) code, i.e., a code of block length N, dimension K. Let M = N K (the code rate R is equal to R = K=N); then C n is dened by a M N parity-check matrix H n and every codeword s n 2 C n satises the parity-check equation H n s n = 0. Together with the parity-check matrix H n, we associate a generator matrix G n (of full column rank) of size N K which satises: G n b n = s n (2) where b n is a vector of K information bits. We suppose in the sequel that the frame length N, the code rate R are equal for all users, we are frame synchronous and one trailing zero is added at the beginning of each frame to ensure the causality of the transmission. 2. Performance limits of a asynchronous multiple user system The aim of this section is to recall the performance limits of the AWGN asynchronous MAC for BPSK sources (2 f 1; 1g). This analysis will be useful to evaluate the performance of the proposed method. Under the assumptions of the previous section, we adopt the following denition according to (Gilhousen et al., 1991; Verdu and Shamai, 1999): DEFINITION 1. The spectral eciency D [bits.s 1.Hz 1 ] of the global system (sometimes referred also as the system load) is equal to the sum of achievable rates: D = N u R (3) 2.1. Capacity of the asynchronous AWGN-MAC with Gaussian sources Before we derive the capacity, i.e. the maximal spectral eciency, of the asynchronous AWGN-MAC with BPSK sources, we recall the formula of the capacity of the asynchronous AWGN-MAC with Gaussian sources. Under the global power constraint, i.e. the average power per symbol

5 E x = 1 dened as E x = 1 N u (Verdu, 1989b): Gallager Codes for Asynchronous Multiple Access 5 C = lim N!+1 P Nu n=1 IE x 2 n[l], the capacity is given by " 1 2N log 2 I + 2C E b M M T N 0 where I denotes the identity matrix, j:j the determinant operator and N the code block length. With two users, it has been shown in (Verdu, 1989a) p.737 that the worst case oset between the signals is zero, i.e. in which case the channel is symbol synchronous. The most favorable case occurs when the symbol oset is equal to half the symbol period. We observe the same results with two BPSK sources Capacity of the asynchronous AWGN-MAC with binary sources Under the following assumptions, A1) global power constraint: the average power per symbol Ex = 1 dened as Ex P = 1 Nu N u n=1 IE s 2 n[l], A2) all users are equal power: IE s 2 n[l] = 1; 8n = 1; : : : ; N u A3) distinct osets: n;n+1 6= n 0 ;n 0 +1; 8n 0 6= n, the capacity of the system is given by (5). # (4) C = lim N!+1 X log 2 exp s 0 1 N n X N u s Z 1 2 NuN 1 exp (2 2 n )(NNu)=2 [2y T M (s 0 s) (s 0 s) T M T M (s 0 s)] with 2 n, the noise variance dened as: 2 n = 1 2CE b =N 0. jjyjj n PROPOSITION 1. Under the conditions A1 and A2, the capacity for BPSK sources is: maximized if all osets are distributed uniformly over a period symbol ( n;n+1 = 1 N u ), dy!! : (5)

6 6 A. de Baynast and D. Declercq minimized if the users are synchronous ( n = 0; 8n = 1; : : : ; N u ). As shown on gure 2, lim Eb =N 0!1 C = 1:5 for 2 synchronous users (o dashdotted line) and lim Eb =N 0!1 C = 2 when the oset 6= 0 ( solid line). Indeed, in the noiseless case, if the users are synchronous and have equal power, an ambiguity occurs when 0 is observed: it may come from f 1; +1g or f+1; 1g and a coding step is necessary even in the noiseless case. The asynchronism suppresses this ambiguity by taking account of the values of the neighbouring samples BPSK Nu=2 τ=0t (sync.) BPSK Nu=2 τ=0.25t BPSK Nu=2 τ=0.50t Async. Gaussian E b /N 0 [db] Capacity: R mean number of users Nu = 2 Figure 2. Eb=N 0 vs. capacity: Gaussian sources, BPSK 2 users with equal power (asynchronous case: = 0:25T and 0:50T, synchronous case: = 0T ) Achievable BER for Gaussian asynchronous MAC with binary sources In practice, the system works at a certain non-vanishing BER depending of the required Quality of Service (QoS). The aim of this section is to derive the minimum E b =N 0 required with respect to the expected BER. The rate R 0 achieving a certain non-zero BER will obviously be greater that the rate R achieving a transmission without error. In a multiple user context (N u BPSK modulations), in order to derive the new rate R 0, we assume that each user has the same BER for a given E b =N 0. This assumption is veried if all users have the same coding rate

7 Gallager Codes for Asynchronous Multiple Access 7 R and the same power. In that case, the new code rate R 0 (R) is given by (6). R 0 = R " N u + XN u n=1 n N u BER Nu n (1 BER) n : log 2 BER Nu n (1 BER) n # (6) Figure 3 shows the achievable BER as a function of E b =N 0 for rates R = 1 2 and 1 4, 2 users with dierent oset (asynchronous case: = 0:25T and 0:50T, synchronous case: = 0T ) Minimum BER R=1/4 τ=0.5t R=1/4 τ=0.25t R=1/2 τ=0.5t R=1/4 τ=0t R=1/2 τ=0.25t R=1/2 τ=0t E b /N 0 [db] Figure 3. BER vs E b=n 0 for dierent rates (R = 1 ; 1 ) - Equal power BPSK async user ( = 0T; 0:25T; 0:50T ). These abacuses will be useful to compare the performance of our simulation results with the theoretical bounds with respect to the parameters of the system. After having given the performance bounds for AWGN MAC with respect to the signal to noise ratio Eb =N 0, the rate R and the BER, we describe the proposed algorithm based on Gallager random-like codes.

8 8 A. de Baynast and D. Declercq 3. Joint multiple user detection based on Gallager codes Since the introduction of turbo-codes in (Berrou et al., 1993), many new coding and decoding techniques have been proposed (MacKay, 1999). It turns out that all good codes are random-like codes and that they share a common decoding algorithm: the belief propagation on graphical representations (Kschischang et al., 2001). In this paper, we use the factor graphs that are powerful tools to develop decoding algorithms. Factor graphs have been proposed by (Wiberg, 1996) as a generalization of Tanner graphs in coding theory (Tanner, 1981). They are bipartite representations of systems composed of data nodes and functional nodes. The data nodes represent observations and input symbols while the function nodes describe how their adjacent data nodes interact. The branches of the graph carry probability weights that comes in and out the data nodes. Belief propagation in a graph depicts how the weights are updated until a xed point has been reached (Kschischang et al., 2001). It can be shown that exact a posteriori weights can be computed if the factor graph is indeed a tree, that is there is no cycles in the graph. Besides, if the cycles in the graph are "suciently" long, iterative decoding with probability propagation yields excellent (though approximate) results, close to optimum performance. First, we briey describe the decoding problem for Gallager codes in the single user case and in the second paragraph, we derive the joint asynchronous multiple user decoding problem for Gallager codes Gallager Codes in the single user case These block codes have been proposed by Gallager in 1963, together with a stochastic decoding algorithm which is very close to belief propagation. Mc Kay & al. have rediscovered and extended LDPC Gallager codes recently (MacKay, 1999) and have shown that Gallager codes can be easily decoded with iterations of belief propagation on their factor graph (cf. gure 4). First, we describe the functional nodes (black square). Since it is the single user case, we have: x[l] = s[l]. Each channel node calculates the conditional probability densities: p(x[l]jy[l]) / 1 p 2 2 exp (y[l] x[l]) 2 =2 2 (7)

9 Gallager Codes for Asynchronous Multiple Access 9 Each parity-check node indicates that the set Q k of the codeword bits fs[l]g 2 Q k to which the parity-check is connected have even parity: X Q k s[l] = 0 mod 2 (8) When a channel output vector y is observed, the iterative probability propagation decoder begins by sending messages P (x[l]jy[l]) according to (7) from y to x. Then messages Q (1) (x[l]) are sent from the codeword x k to the parity-check constraints and messages R (1) (x[l]) from the paritycheck constraints back to the codeword x according to (8). Each time an k iteration i is completed, new estimates of APP(x[l]jy) for l = 1; : : : ; N are obtained. After a prespecied stopping rule such as the maximum number of iteration or no change in the estimated codeword has been reached, the iterative decoding stops. For more details on the Gallager codes decoding, refer to (MacKay, 1999; Kschischang et al., 2001). channel output y[1] y[2] : : : y[n] P (sjy) channel node codeword s[1] s[2] : : : s[n] Q R parity-check node Figure 4. A factor Graph for a Gallager code C(N; M) 3.2. Gallager codes in the asynchronous multiple user case In the multiple user case, we rewrite the model described by (1): 8 >< >: y (1) [l] = [0 1;2 1;2 0 1;2 0] x[l] + n (1) [l] y (2) [l] = [0 1;2 0 1;2 1;2 0] x[l] + n (1) [l]. y (Nu) [l] = [0 1;2 0 1;2 0 1;2 ] x[l] + n (1) [l]

10 10 A. de Baynast and D. Declercq or equivalently in a vectorial form with y[l] = M 0 x[l] + n[l] = z[l] + n[l] (9) x[l] = [x 1 [l 1] x 1 [l] x 2 [l 1] x 2 [l] x Nu [l 1] x Nu [l]] (10) Then, each codeword x n [l] is connected to both variables z[l 1] and z[l]. Figure 5 shows the factor graph for a joint asynchronous multiple user system using Gallager codes. The fading coecients a n are supposed to be perfectly known at the receiver. As in the single user case, each channel node calculates the conditional probability densities for the channel: p(z[l]jy[l]) / exp jjy[l] z[l]jj 2 2=2 2 (11) In the multiple user case, z[l] is described by (10). The variable z[l] has 2N u components and then 2 2Nu possible states. We dene, as the "spine-check" node, the functional node to which z[l], z[l 1] and s n [l]; 81 n N u are connected. Using (10), this functional node is described by: XN u n=1 a n M ;n s n [l] = z[l] (12) with s n [l] = [s n [l 1]s n [l]] T Such as in the single user case, once a channel output vector y is observed, the iterative probability propagation decoder begins by sending messages P (z[l]jy[l]) from y[l] to z. Messages O 00 [l 1] are sent from z[l 1] to the spinal-check and T 0 [l] are sent from the spinal-check to z[l]: the forward step. Messages O 0 [l] are sent from z[l] to the spinal-check and T 00 [l 1] are sent from the spinal-check to z[l 1]: the backward step. Then messages P (1) l (s 1 [l]) (resp. S (1) l (s 1 [l])) are sent from the spinecheck node to s 1 [l] (resp. from s 1 [l] to the spinal-check node). The user index is arbitrary in the case of all users with equal power. As in the single user case, messages Q (1) (s k 1[l]) are sent from the codeword s 1 to the parity-check constraints and messages R (1) (s k 1[l]) from the parity-check constraints back to the codeword s 1 according to (8). For the others users (2 n N u ), the procedure is exactly the same. Each time an

11 Gallager Codes for Asynchronous Multiple Access 11 Gallager code for user 2 " y (1) [1] y (2) [1] # " y (1) [2] y (2) [2] # " y (1) [N] y (2) [N] # P (zjy) O 00 T 0 z[1] z[2] z[n] T 00 O 00 S P s1[1] s1[2] s1[n] Q R Gallager code for user 1 Figure 5. A factor Graph for a joint asynchronous multiple user decoding algorithm using Gallager codes C(N; M) iteration i for all users is completed, new estimates of APP(m n [l]jy) for l = 1; : : : ; N are obtained Encoding and decoding computational complexity Fast-encoding Gallager codes One of the drawbacks of Gallager codes is that their encoding time generally scales as O(N 2 ) because the generator matrix G is not generally sparse. In fact, some methods exist to ensure the generator matrix is sparse (see for instance (MacKay et al., 1998)). In our case, we compute the Gaussian elimination to calculate the generator matrix G n from the parity-check matrix H n using the Markowitz criterion, (Du et al., 1986). It ensures a good sparsity for G n, roughly say O(3N), despite it is a local criterion Decoding The major limitation of such joint multiple user detection algorithm is its exponential complexity in the number of users. Fortunately, there exist various means in order to reduce this complexity. A well-known result in the graph theory, see for instance (Frey, 2000) is that the exponential complexity at the spinal node can be reduced to a polynomial complexity o(n 3 u ) with no loss in performance. In most of cases, this remains too complex. Several suboptimal strategies are possible. For example,

12 12 A. de Baynast and D. Declercq a hybrid structure "joint decoding/successive interference cancellation (SIC)" can be applied. The users are gathered in subsets (4 to 8 users in each subset). A joint decoding algorithm for each subset is used while the SIC procedure (Patel and Holtzman, 1994) is used to go of a subset to another. In a future work, we plan to quantify the loss induced by this suboptimal structure. 4. Simulation results In this set of simulations, we present the performance of the proposed asynchronous multiple user joint decoding algorithm for several code rates R and several system loads over the AWGN multiple access channel. Each user has the same code rate R. The system load is dened as the ratio between the number of user N u and the processing gain R (see section 2 for more details). We run the decoder on 10 3 frames for a block length N = Although the Gallager codes are much better for a larger block length ( 20000), we limit our simulations to this length for realistic implementation issues. To limit the computation duration, we set the maximum number of iteration to 100 whereas the algorithm did completely not nish converging. To be in agreement with a UMTS system, the system should work at an high spectral eciency: we set the system load to values greater than 0:5 in all the cases. The transmission is BPSK modulated. Perfect knowledge of the channel fading coecients (equal to one in these simulations) and of the osets is assumed. We compare in presence of asynchronism our algorithm with a DS-CDMA system. To simulate the DS-CDMA system, the spreading sequences are OVSF sequences of length 2 ([1 1] and [1 1]). At the receiver, we use a matched lter followed by a single user Gallager decoding described in section 3.1. The simulations are reported in gure 6 and 7. Since the performance of all users are very close (less than 0:1dB), the average BER between all users is pictured. In the rst simulation, we set the system load to 1 with R = 1=2 and N u = 2 for several asynchronism osets: = 0T (sync.), 0:25T and 0:5T. The number of ones per row t r in the parity-check matrices H n is set to 5. Note that in this paper, we do not optimize the parity-check matrices H n. This optimization will be reported in future work. Additionally, we have also drawn the achievable BER(E b =N 0 ) given in section 2 for R = 1 2 and = 0T; 0:25T and 0:5T. These curves correspond to the lower achievable bounds of the transmission. Note that we do not compare the obtained performance to the single user performance since

13 Gallager Codes for Asynchronous Multiple Access 13 in a multiple access system the single user performance is not generally the optimum achievable bound (even it is true at high signal to noise ratio and unequal power or 6= 0). In the second simulation, the system load is equal to 0:5 i) with a rate R = 1=4, no spreading and N u = 2 (MAC based on Gallager codes, o solid line) and ii) R = 1=2, S F = 2 which gives a "total" redundancy factor equal to 4 and N u = 2 (DS-CDMA, dashed line). The number of ones per row t r in the parity-check matrices H n is set to 3. The results are pictured on gure 7. Additionally, we have drawn the performance of the synchronous multiple user joint decoding algorithm for a rate R = 1 4 (+ dashdotted line) τ = 0T (Sync.) τ = 0.25T τ = 0.5T BER E /N [db] b 0 Figure 6. BER vs E b=n 0 - R = Equal power BPSK async. 2 user ( = 0T; 0:25T; 0:50T ). We can notice that our asynchronous multiple user joint decoding algorithm provides good performance since the iterated process is close to the optimum curve (2dB or less at BER= 1e 3) for the rates R = 1=2; 1=4 (for R = 1=4, the theorical bounds are pictured on gure 3). These results are in agreement with the results on the coding theory of section 2 and conrm both following facts: i) a multiple access system do not necessarily need an "algebraic" orthogonality between the users to be powerful, ii) the proposed joint algorithm is robust to the asynchronous (moreover, the best performances are obtained for an asynchronism oset equal to an half symbol period whereas the worst performances are obtained in the synchronous case).

14 14 A. de Baynast and D. Declercq 10 0 τ=0.5t (DS CDMA) τ=0t (sync.) τ=0.5t (Gallager MAC) BER E b /N 0 [db] Figure 7. BER vs E b=n 0 - BPSK 2 async. user ( = 0:5T ) - : DS-CDMA (R = 1 2, S F = 2), o: the proposed method based on Gallager codes (R = 1 4 ) As expected, the performance of the DS-CDMA system using a matched lter and a Gallager single user decoder is badly aected by the asynchronism. Note that the oset = 0:5T is is not the worst. Indeed the average multiple access interference is equal to 1 2 IE[s2 ] in this case and for an oset = 1T (i.e. a chip period), the average MAI is equal to IE[s 2 ]. However, the complexity of the DS-CDMA system is linear with the number of users whereas the complexity of the proposed algorithm is exponential in the number of users. Conclusion In a symbol asynchronous DS-CDMA system at high spectral eciency (uplink transmission in IMT-2000 with a small spreading factor), the performance of the decoder are too much penalized by the MAI after the despreading step. In this paper, we have also proposed a joint asynchronous multiple user decoding algorithm using Gallager codes. We have shown that i) the asynchronism can improve the performance if a joint decoding algorithm is used, ii) we can bypass the spreading sequences in an asynchronous multiple access system as in a synchronous system, see (de Baynast and Declercq, 2002). To make a comparison with the theorical AWGN MAC capacity, we have derived the performance

15 Gallager Codes for Asynchronous Multiple Access 15 bounds with respect to the code rate R, the number of users N u, the signal to noise ratio E b =N 0 and the BER. Note that these bounds generally are not equal to the single user bounds. The performance of our system are close to the optimum achievable bound (2dB or less at BER= 1e 3 for a block length N = 2000) for a system load RN u equal to 1 and 0:5 (R = 1=2 and 1=4). The major problem is still the complexity of the algorithm exponential in the number of users. As described in section 3.3.2, a suboptimal method combining multistage and iterative detection is possible. References Berrou, C., A. Glavieux, and P. Thitimajshima: 1993, `Near Shannon Limit Error- Correcting Coding and Decoding: Turbo Codes'. In: ICC. pp. 1064{1070. Chung, S.-Y., T. Richardson, and R. Urbanke: 2001, `Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation'. IEEE Trans. Inform. Theory 47, 657{670. Cover, T. and J. Thomas: 1991, Elements of information theory. New york : Wiley edition. de Baynast, A. and D. Declercq: 2002, `Gallager codes for multiple access'. accepted to IEEE Symposium on Information Theory. Du, I. S., A. M. Erisman, and J. K. Reid: 1986, Direct methods for sparse matrices. Clarendon Press ; Oxford University Press, oxford : New-York edition. Frey, B.: 2000, Graphical Models for Machine Learning and Digital Communication. Cambridge, Massachussets: The MIT Press. Gallager, R.: 1962, `Low-Density Parity-Check codes'. IRE Transactions on Information Theory. Gilhousen, K. S., I. Jacobs, R. Padovani, A. Viterbi, L. Weaver, and C. Wheatley: 1991, `On the Capacity of a Cellular CDMA System'. IEEE Trans. on Vehicular Technology 40(2). Johansson, A. and A. Svensson: 1995, `Multi-stage interference cancellation in multirate DS/CDMA systems'. In: PIMRC`95. Toronto, Canada. Kschischang, F., B. Frey, and L. H.-A.: 2001, `Factor graphs and the sum-product algorithm'. IEEE Trans. Inform. Theory 47(2), 498{519. Lehman, E.: 1959, Testing Statistical Hypotheses. New york: Wiley edition. MacKay, D.: 1999, `Good Error-Correcting Codes Based on Very Sparse Matrices'. IEEE Transactions on Information Theory 45. MacKay, D., S. Wilson, and M. Davey: 1998, `Comparison of constructions of irregular Gallager codes'. Patel, P. and J. Holtzman: 1994, `Analysis of a simple successive interference cancellation scheme in a DS/CDMA system'. IEEE Journal on Selected Areas in Communications 12(5). Scaglione, A., G. Giannakis, and S. Barbarossa: 2000, `Lagrange/Vandermonde MUI Eliminating User Codes for Quasi-Synchronous CDMA in Unknown Multipath'. IEEE Transactions on Signal Processing 48(7), 2057{2073.

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