Iterative Multi-User Detection for CDMA with FEC: Near ... - CiteSeerX

Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance Mark C. Reed, Paul Alexander, John Asenstorfer

Christian Schlegel

Mobile Communications Research Centre Department of Electrical Engineering Institute for Telecommunications Research 3262 Merrill Engineering Building University of South Australia University of Utah SPRI Building, Warrendi Road Salt Lake City UT 84112, U.S.A The Levels SA 5095, Australia FAX: +1 801 581 5281 E-Mail: [email protected] FAX: +61 8 8302 3873 Submitted to IEEE Transactions on Communications April 11, 1997

Abstract

This paper introduces an iterative multi-user receiver for direct sequence code division multiple access (DS-CDMA) with forward error control (FEC) coding. The receiver is derived from the maximum a-posteriori (MAP) criteria for the joint received signal, but uses only single user decoders. Iterations of the system are used to improve performance, with dramatic eects. Single user turbo code decoders are utilised as the FEC system and a complexity study is presented. Simulation results show that the performance approaches single user performance even for moderate signal to noise ratios.

Keywords: Code division multiaccess, Decoding, Random Codes, Multiuser Channels, Turbo Codes

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Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

1 Introduction

Page 2

With the standardisation of Direct Spread Code Division Multiple Access (DS-CDMA) for mobile communications [1], a number of vendors have introduced their products onto the world market. This has raised a lot of interest on the potential capabilities and capacity of this multiple access technology [2{10]. In this paper we study the uplink, or the base station (BS) receiver. In the design of these systems most are currently symbol synchronous or quasisymbol synchronous so that orthogonal codes can be utilised. When orthogonal codes are used the currently used BS linear lter receivers performs well in decoding the signal sent by taking advantage of this orthogonality. This orthogonality gives performance equal to single user performance. In a true mobile wireless system synchronisation is dicult to maintain and needs tight closed loop timing control between the base station (BS) and the mobile station (MS). If this timing control isn't maintained then the orthogonal properties are lost and performance degrades severely. Multipath eects, common in mobile radio channels, also destroy this orthogonal property. If the codes are randomly selected however, the performance of a synchronous system is on average the same as that of an asynchronous system. Work done by Grant et al. [11] shows that the capacity penalty vanishes, for a large number of users, using randomly selected spreading codes, as the ratio between number of users and spreading length becomes large. Jana et al. [12] have shown a slightly dierent result; they showed that the upper bound of the normalised minimum distance for a trellis coded multiuser CDMA system with nonorthogonal spreading is identical to that of the single-user case. This means that asymptotically, using non-orthogonal codes, or random codes, single user performance should be possible. With such a receiver the performance under asynchronous conditions will be the same as that under synchronous conditions. We are therefore motivated to look for new multi-user receiver structures that use random codes to achieve near single user performance. In [10] Giallorenzi et al. reviewed suboptimal multi-user receivers for convolutionally coded asynchronous DS-CDMA systems. They described three categories of receivers, the linear equaliser, the decision feedback equaliser, and trellis based systems. This paper describes a partitioned trellis based receiver with separate equalisation and decoding. We develop a multiuser receiver (or equalizer) from the maximum-a-posteriori (MAP) criteria. The MAP criteria maximises the probability of a correct decision and, hence, minimises the probability of error [13, p. 245]. Recently a new coding method, called Turbo Codes, was introduced [14]. This technique achieves reliable transmission while operating close to the Shannon limit. Turbo codes combine the concept of soft-in/soft-out decoding, iterative decoding, non-uniform random interleaving, and parallel concatenated convolutional codes (PCCC). Further to this paper, work has been done by Benedetto et al. [15{17] on serial concatenated convolutional codes (SCCC). Several authors have proposed using turbo codes for DS-CDMA systems [5, 6]. These papers discuss system implementations but show no performance results. In [7], turbo code decoders are used as single user decoders to a DS-CDMA multi-user system. In this paper we concatenate the MAP derived multi-user receiver and soft-in/soft-out single user trellis decoders to produce a type of serial concatenated convolutional code. The proposed structure diers from successive interference cancellation [8, 9], because no signal cancellation takes place. Our system can be viewed as a soft parallel interference canceller or more correctly as a SCCC. A paper by Douillart et al. [18] discusses the use of a SCCC to cancel inter-symbol interference. The technique described in [18] has similarities to this work but the end application is very dierent. The soft-in/soft-out single user decoders are implemented by using turbo code decoders

Page 3 which give very low error rates. A complexity analysis is presented and complexity reduction techniques are discussed as a way of minimising the computational load. The paper is organised as follows. In Section 2 we describe the symbol synchronous channel model. In Section 3 we introduce the iterative multi-user receiver and derive the metric for the single user decoders. We also describe the iterative process and assumptions made. In Section 4 we describe the modi cations to the single user turbo code decoder before detailing a complexity analysis in Section 5. Section 6 shows simulation results, and Section 7 contains a concluding discussion. Throughout this paper scalars are lower case, vectors are underlined lower case, and matrices are underlined upper case. The symbols (
) ; (
)? and j
j are the transposition, inversion and determinant operators respectively. Variables have subscripts that refer to the time increment and superscripts that refer to the user, except if stated otherwise. Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

1

>

2 System Model We model the uplink of a DS/CDMA communication system, as a coded, discrete-time system. The channel adds zero-mean white Gaussian noise with variance  = N =2, where N is the single sided noise power spectral density. The channel model is chip and symbol synchronous. k K users each transmit L coded symbols dt 2 f+1; ?1g, where k 2 f1;


; K g is the user number, and t 2 f0;


; L ? 1g identify the symbol interval. The spreading p code employed p N by k user k at symbol interval t consists of N chips and is denoted st 2 f?1= N; +1= N g . The matched lter output y t at time t can therefore be expressed as 2

0

0

( )

( )

y t = H t dt + nt

(1)

where dt is the data vector, H t is the cross correlation matrix of the spreading sequences, and nt is the noise vector. For a synchronous system

y = (y ;


; y L? ) 1

(2)

At = (st ;


; sKt )

(3)

>

>

>

0

and H t = At At 2 RK;K where >

1

is the bank of spreading-code matched lters. The noise samples nt are Gaussian distributed and have correlation Efnt ntg = H t . The coding method we use is limited to trellis codes and FEC is provided by convolutional codes. >

2

3 The Iterative Multi-user Receiver In this section we will state the problem faced in determining the iterative multi-user receiver. We then overview the system before deriving the receiver and the decoder in terms of the MAP criteria. A discussion of iterating the system concludes this section.

3.1 Deriving the Iterative Multi-User Receiver

The problem faced when designing a multi-user receiver with a turbo code structure is that of generating the correct probability information for the soft-in/soft-out decoders and of supplying

Page 4 appropriate a-priori information to the multi-user receiver on each iteration. Fortunately both these problems can be solved simultaneously. Figure 1 shows the iterative multi-user receiver system. The receiver takes the matched lter channel output as described in (1) and generates the conditional channel probabilities p(y tjdt), which are well known multivariate Gaussian conditional probabilities [13]. The metric generator then calculates the likelihood p(y tjdtk ) for each user k. This is the maximum-a-posteriori (MAP) result as discussed below. Single user soft in/soft out decoders generate the a-posteriori symbol probabilities Prfdtk = djy k g which are then used as a-priori information for the metric generator on the next iteration. When the required number of iterations have been completed a decision (^bj ) is output by the single user decoders. Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

( )

( )

Encoder (K users)

dt - Multiuser Channel

yt

Likelihood - Calculation

p(y tjdt) -

-

-

Metric Generator

p(y tjdtk )

Prfd(tk)j y (k) g ^b j

( )

( )

K Single  User



Decoders

Figure 1: Iterative Multi-User Model From the matched lter channel model (1), we know that the conditional probability of y t is [13, p.49]  
? 1 1 ? (4) p(y tjdt) = exp ? 2 y t H t y t ? 2y t dt + dt H t dt (2) K jH t j which is the multivariate Gaussian distribution. The MAP criteria [13] for the multiuser receiver can be stated as p(d ; y ) Prfdtjy tg = p(ty )t t = Cy p(d t; y t) = Cy p(y tjdt)Prfdtg (5) where Cy is a normalisation constant. This operation is symbol based and does not take into account the FEC code. (This lowers the number of states in the synchronous decoder from 2K (2K? + 2K ? 2), to 2K? + 2K ? 2 per coded symbol, where  is the memory length of the convolutional code.) The above MAP criteria (5) was also used in [19] to compute sub-optimal MAP metrics for single user decoders in multi-user CDMA. >

2

1

1

1 2 2

2

1

>

>

Page 5 In this work we constrain the system to use K single user decoders. The K single user decoders calculate the a-posteriori probabilities (APP) for the multi-user receiver according to the algorithm due to Bahl et al. [20]. Assuming a rate R = n convolutional code, there are n channel symbols dtk for every uncoded information bit bj , or encoder state transition. Denote these channel symbols by (dtk ;


; dt n? ) = d k . We then calculate X X Prfdtk = djy k g = PrfSj? = m0; d k jy k g (6) Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

1

( )

( )

( )

+

( )

1

( )

0

m

dk k (d t =d)

0

( )

1

( )

( ) 0

where Sj is the state at time j = bt=nc and bxc is the largest integer not greater than or equal to x. Vector d is the hypothesised channel symbol vector for a particular trellis transition, and t  t0  t + n ? 1. The APP (6) for the k MAP decoder output can be expressed [20] using three probability variables. They are the forward state probability, the reverse state probability, and the transition probability. The transition probability accepts the input information p(y tjdtk = d) and is equal to th

( )

j

(m0; m)

t+Y n?1 0 p(y t jd(tk) = d) PrfSj = mjSj?1 = m g t =t

=

0

0

(7)

0

when the transition exists. The state transition probability or a-prior probability equals PrfSj = mjSj? = m0g. We now address the task of calculating p(yt k ; dtk ) before calculating the conditional probability required. The metric for the single user decoders is generated by manipulating the conditional probability (4) to obtain the joint probability in terms of the dtk . Using X p(y tjdt)Prfdtg; (8) p(y t; dtk ) = 1

( )

( )

( )

( )

dt d(tk) =d)

(

the metric generator calculates

p(y tjdtk

p(y t; dtk ) ) = Prfdtk g ( )

( )

( )

=

X

p(y tjdt)

dt (k ) (dt =d) symbols d(ti) are

K Y

Prfdti g;

(9)

( )

i=1 (i= 6 k)

We have assumed that the coded independent among the dierent users (i.e. there is no transmission cooperation), allowing us to write Prfdt ;


(1)

; dtk?1); d(tk+1);


(

; dt K ) g (

=

K Y i i6 k

Prfdti g: ( )

(10)

=1 ( = )

We use the result of (9) as our conditional input information described in (7). ie. (11) p(yt k jdtk = d) = p(y tjdtk ) This statement is true for the multi-user case because the probability of dtk = d is dependent on the whole vector y t due to multiple access interference. ( )

( )

( )

( )

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Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

3.2 Iterating the Receiver

Multiuser systems describe receivers where users share information. If this is done correctly a joint detection process results with improved performance over systems without joint detection. In Turbo Code decoding [14], for example, the output probability of the rst MAP decoder, Prfbt = bjy g, is used as a-priori information for a second MAP decoder. In a similar fashion we assign the single-user MAP decoder output probabilities from iteration i to the a-priori input probabilities to the metric generator for iteration i + 1 in (9), i.e., we set Prfdtk = dg = Prfdtk = djy k g: (12) As in [14] this is justi ed due to the weak correlation between the single user convolutional codes and the spreading codes. On the rst iteration the multiuser receiver a-priori information is Prfdtk = dg = 1=2, i.e., all symbol sequences are assumed to be equi-probable. Due to the fact that the a-priori information cannot be factorised from the output of the metric generator th a-priori probability in uence cannot be removed as done in [14] where the extrinsic information is determined. The K single user decoders also do not cancel a-priori information as shown in [16]. If however systematic convolutional coding is used then the systematic information could be removed from the systematic bit probabilities to produce the extrinsic output. The iteration step discussed here highlights the fact that the receiver design is not a successive interference canceller, like for example [21], as no hard decisions are made. Instead soft APP's are passed between the soft-in/soft-out decoders and the metric generator. Interleaving between the multi-user receiver metric calculation and the single user decoders isn't required because the multi-user receiver is decoding across users and the single user decoders are decoding over time. ( )

( )

( )

( )

4 Single User Turbo Code Decoder In this section we describe the use of a turbo-code decoder [14] as the single user decoder in the structure shown in Fig. 1. We describe just the k decoder. The use of turbo codes for error correction gives us a very powerful decoding stage. As with single user turbo code results we expect and obtain better error performance than using a single MAP decoder as our single user decoder. The turbo code decoders need to be modi ed slightly to produce suitable output probabilities for the a-priori input to the multiuser metric generator. Single user encoders as shown in Fig. 1 are implemented in the same way as those used in [14] except that there are K of them, one for each user. Each user implements a turbo code decoder which receives likelihood values p(y tjdtk ). This data is used directly in the MAP decoder's branch metric calculation [20] which is modi ed from (7) to th

( )

j

(m0; m)

= PrfSj = mjSj?

1

= m0 g

t+Y n?1

p(y t jdtk ) 0

t =t 0

( ) 0

(13)

where the product is over all the dtk values that produce the transition of the MAP decoder from state m to state m0. Like the SCCC solution [16] the turbo code decoder takes no apriori information PrfSj = mjSj? = m0g = from the multi-user receiver, although the MAP decoders within the turbo code decoder pass a-priori information between each other. The iterative decoding step within the turbo code decoder is done slightly dierently to that of [22]. The factorisation of the log likelihood output of the rst MAP decoder for user k needs ( )

1

1 2

Page 7 to be modi ed as we are provided with a likelihood measure of p(y tjdtk ) not a bit b 2 f+1; ?1g with additive white Gaussian noise (AWGN). In a way similar to the factorisation of the output of the MAP decoder shown in [22, 23] we express the log likelihood ratio as  k (bj ) = La k;j + Lb k;j + Le k;j ; (14) where ( ) Prfbj = 1jy k g k  (bj ) = ln (15) Prfbj = 0jy k g Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

( )

( ) 1

( ) 2

( ) 1

( ) 1

( )

( ) 1

( )

is the log likelihood ratio output of the rst MAP decoder of user k at time j . La k;j is the a-priori log likelihood ratio from the second MAP decoder for user k at time j , and ( ) Prfbjk j^b = 1g Lb ;j = ln (16) Prfbjk j^b = 0g is the log likelihood ratio of the a-posteriori probabilities of the systematic (information) bits and Le k;j is called the extrinsic information, which after being computed from (14) and interleaved forms the a-priori information for the second MAP decoder for user k. In [14] after the desired number of iterations of the turbo code decoder were performed a hard decision on the information bits was output as the nal result. We however want to produce a soft a-posteriori probability output which can be used as a-priori information for the metric generator. The turbo code decoder is modi ed to produce systematic (uncoded) and redundant (coded) conditional bit probabilities as a-priori information to the metric calculation (Prfdtk jy k g). The output probabilities are calculated using (6). The coded bit probabilities are taken from the output of the rst and second MAP decoders because the two codes are punctured and because each turbo decoder uses a dierent coded sequence. The uncoded bit probabilities are taken from the output of the second kMAP decoder for user k. The turbo code decoder outputs a block containing L values of Prfdt jy k g in the same sequence as the input vector that it received from the metric generator on the previous iteration, p(y tjdtk ). After the nal multi-user system iteration a hard decision is output (^bj ) by the K decoders as in [14]. ( ) 2

( )

1

( )

( ) 1

( )

( )

( )

( )

( )

5 Complexity Analysis In this section we study the complexity of the system described. This is important so we can determine which parts of the system will consume most resources. Knowing this we can concentrate on reducing the complexity of these parts. We determine the number of oating point operations required per information symbol transmitted for the likelihood calculation, metric generator and turbo decoder. We assign the variable S to be the number of states in the decoder, the number of users to equal K , the number of paths out of each state to equal P , and the rate of the FEC code to equal R. Table 1 shows the complexity in terms of those variables. Fig. 2 shows the complexity of the three components of the system for a varying number of users. For more than six users (the case of interest for practical implementation) the 2K term in the likelihood generation and metric generation dominates complexity. Figure 2 shows the complexity of the likelihood function (4) as \like", the complexity of the metric generation (9) as \metric", and the complexity of the turbo code decoder as \tc", against users. The curve \total" is the sum of these three complexities. The number of turbo code iterations was set to four and the number of multiuser iterations was set to three, the same as used in the nal performance tests.

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Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

Procedure Likelihood Calc. Metric Gen.(1 Iter.) Turbo Decoder(1 Iter.)

Complexity(FLOP per Information Symbol) (2K (7K 3=6 + 4K 2 + 6K + 4))=(RK ) (2K (K ? 1))=(RK ) 2(PS (R?1 =2 + 10) ? 2S + 16)

Table 1: Complexity of Iterative Multiuser Receiver Components

Complexity (FLOP per Information Symbol)

1e+08 "like" "metric" "tc" "total" "malg_like" "malg_metric" "malg_total"

1e+07 1e+06 100000 10000 1000 100 10 1 0

2

4

6

8

10 12 Users (K)

14

16

18

20

Figure 2: Complexity in FLOP per symbol against number of users.

5.1 Complexity Reduction Techniques

From the complexity analysis it is clear that our solution is exponentially complex with the number of users, like the optimal decoder [24]. In this section we propose techniques to reduce this complexity. Equation 4 contains a number of complex linear algebraic tasks. Due to independence from the hypothesised vector d we can simplify this to   ?
1 p(y tjdt) = Ch exp ? 2 2y t dt + dt H t dt : (17) where Ch is a constant and does not have to be computed. This is a signi cant reduction as we have removed the need to compute the inverse of the cross correlation matrix (H t). Even with this scheme the dominant 2K term still exists in the likelihood calculation and the metric generation. Another technique, suggested independently by Hoeher [25] and Nasiri-Kenari [26], is to calculate likelihoods p(y tjdt) based on a one bit dierence from the previous likelihood. This >

2

>

Page 9 means that the calculation (4) only has to be done once. From there after a step-wise dierence calculation is required to determine all the possible likelihood values. The technique of [26] reduces the complexity but still requires the computation of 2K likelihoods, this method can however be used with a M algorithm requiring only linear complexity. Using the two reduction techniques the complexity of the likelihood calculator and the metric generator are now Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

Procedure Complexity(FLOP per Information Symbol) Likelihood Calc.(1 Iter.) (2K 2 + 3K + 5)=R Metric Gen.(1 Iter.) (K 2 ? K )=(RK )

Table 2: Reduced Complexity of Iterative Multiuser Receiver Components The complexity of the likelihood calculation (17) is shown in Fig. 2 as \malg like" and the complexity of the metric calculation is \malg metric", when M = K . The total sum complexity is now \malg total" which is linear with users, meeting the reduced complexity requirement. If the likelihood dierence calculation is used to calculate only a list of M outputs we no longer have all the possible likelihoods available. To try to maximise performance under these conditions a number of techniques are used. The likelihood calculation is run with a dierent starting vector on the second iteration generating a second likelihood list. This starting vector is selected based on making a hard decision on the likelihood output from the single user decoders. The likelihood lists from previous iterations are combined to increase the likelihood data available for metric generation.

6 Simulations The simulation result in Fig. 3 and 4 shows the average performance over users. For each simulation a turbo-code encoder of rate R = 1=2, consisting of two parallel recursive systematic encoders, G(37,21), memory  = 4, separated by random interleaving, was used as the error control code. The block size was set to RL = 200 as is typical in mobile applications. We assume the receiver and decoder know the noise variance ( ) exactly. Fig. 3 shows performance for 1,2, and 3 multi-user iterations, with the turbo decoder iterations xed to four. \tc single user" is single user performance. The simulation was done with (pseudo) random spreading codes and values N = 7, K = 5. All of the possible 2K likelihood values p(y tjdt) were calculated. We see that most of the improvement occurs in the second iteration. It is also apparent that after 2 or 3 iterations in this highly loaded random code case near single user performance is obtained. Fig. 4 shows the simulation results for an M algorithm reduced complexity solution as discussed in Sec. 5.1. Four turbo code iterations and three multi-user iterations were used to produce these results. The performance of a K = 5 user system, (pseudo) random spreading codes of length N = 7, with M = K is shown as \sub k5" and a larger system with N = 15, and K = 10 is shown as \sub k10". In this result we can see even with a large, highly loaded, sub-optimal, linear complex, system the performance in \sub k10" is only 1 to 2 dB away from single user performance. 2

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Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

"tc_single_user" "opt_tc_iter1" "opt_tc_iter2" "opt_tc_iter3"

0.1

Pe

0.01

0.001

0.0001 0

0.5

1

1.5

2 2.5 Eb/No (dB)

3

3.5

4

Figure 3: Iterative Multi-User receiver with Turbo Code Decoder performance.

7 Conclusions With the commercialisation of spread spectrum systems using DS-CDMA the question as to what performance is possible has been raised by a number of researchers, some answers have come, for the random spreading code case, from work done on bounds [12] and the capacity of such a system [11]. The main contribution of this paper is the derivation of the maximum a-posteriori criteria of a synchronous multi-user receiver which approaches single user performance even for large system loads i.e., spectrally ecient scenarios. We described the implementation of the iterative multi-user receiver and the modi cations necessary to apply single user turbo code decoders. Simulation results show the performance of the system, which indicate that the iterative multi-user receiver design combined with turbo code decoding approach turbo code single user performance. This is the case even with random spreading codes and a large number of users relative to the spreading factor. The complexity of the joint receiver is well known to be exponential with the number of users. We apply the M -algorithm to the likelihood calculation to obtain O(K ) complexity. The performance degradation was approximately 1dB to 2 dB away from single-user performance, for large systems. Because we are interested in mobile radio applications we used a small block size. Larger block sizes will however improve the performance of the decoders and therefore should improve the performance of the entire system.

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Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

"tc_single_user" "opt_tc_iter3" "sub_k5" "sub_k10"

0.1

Pe

0.01

0.001

0.0001 0

0.5

1

1.5

2 2.5 Eb/No (dB)

3

3.5

4

Figure 4: Reduced Complexity Iterative Multi-User receiver with Turbo Code Decoder performance.

8 Acknowledgements

The authors would like to thank system administrator Mr B. Cooper who has been invaluable in providing a reliable computer network to produce these results.

References [1] TIA/EIA IS-95: Mobile Station-Base Station Compatibility Standard for Dual-Mode Wideband Spread Spectrum Cellular System, July 1993. [2] K. S. Gilhousen, I. M. Jacobs, R. Padovani, A. J. Viterbi, L. A. Weaver, and C. E. Wheatley, \On the capacity of a cellular CDMA system," IEEE Trans. Veh. Technol., vol. 40, pp. 303{312, May 1991. [3] A. J. Viterbi, \Very low rate convolutional codes for maximum theoretical performance of spreadspectrum multiple-access channels," IEEE Trans. Veh. Technol., vol. 8, pp. 641{649, May 1990. [4] T. R. Giallorenzi and S. G. Wilson, \Multiuser ML sequence estimator for convolutionally coded asynchronous DS-CDMA systems," IEEE Trans. Commun., vol. 44, pp. 997{1008, Aug. 1996. [5] J. Hagenauer, \Forward error correcting for cdma systems," in IEEE Int. Symp. on Spread Spectrum Techniques and Applications, (Mainz, Germany), pp. 566{569, Sept. 1996. [6] J. Li and H. Imai, \Turbo coding for spread spectrum," in IEEE Int. Symp. on Inf. Theory Applications, (Victoria, B.C., Canada), pp. 665{668, Sept. 1996.

Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

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[7] P. Jung, M. Nahan, and J. Blanz, \Application of turbo{codes to a CDMA mobile radio system using joint detection and antenna diversity," in IEEE 44th Vehicular Technology Conference, (Stockholm, Sweden), pp. 770{774, June 1994. [8] A. Saifuddin, R. Kohno, and H. Imai, \Cascaded combination of cancelling co{channel interference and decoding of error{correcting codes for CDMA," in IEEE Int. Symp. on Spread Spectrum Techniques and Applications, (Oulu, Finland), pp. 171{175, 1994. [9] J. M. Holtzman, \DS/CDMA successive interference cancellation," in IEEE Int. Symp. on Spread Spectrum Techniques and Applications, (Oulu, Finland), pp. 69{78, July 1994. [10] T. R. Giallorenzi and S. G. Wilson, \Suboptimum multiuser receivers for convolutionally coded asynchronous DS-CDMA systems," IEEE Trans. Commun., vol. 44, pp. 1183{1196, Aug. 1996. [11] A. J. Grant and P. D. Alexander, \Randomly selected spreading sequences for coded CDMA," in IEEE Int. Symp. on Spread Spectrum Techniques and Applications, vol. 1, (Mainz, Germany), pp. 54{57, Sept. 1996. [12] R. Jana and L. Wei, \Bounds for minimum euclidean distance for coded multiuser CDMA systems," in IEEE Int. Symp. on Spread Spectrum Techniques and Applications, (Mainz, Germany), Sept. 1996. [13] J. G. Proakis, Digital Communications. McGraw{Hill, 3 ed., 1995. [14] C. Berrou, A. Glavieux, and P. Thitmajshima, \Near Shannon limit error-correcting coding and decoding:turbo-codes," in IEEE Int. Conf. on Communications, (Geneva, Switzerland), pp. 1064{ 1070, May 1993. [15] S. Benedetto, G. Montorsi, D. Divsalar, and F. Pollara, \Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding," JPL TDA Progress Report, vol. 42-126, Aug. 1996. [16] S. Benedetto and G. Montorsi, \Iterative decoding of serially concatenated convolutional codes," IEE Electron. Lett., vol. 32, pp. 1186{1188, June 1996. [17] S. Benedetto and G. Montorsi, \Serial concatenation of block and convolutional codes," IEE Electron. Lett., vol. 32, no. 10, 1996. [18] C. Douillard, C. B. Michel Jezequel, A. Picart, P. Didier, and A. Glavieux, \Iterative correction of intersymbol interference: Turbo-equalisation," European Transactions on Telecommunications, vol. 6, pp. 507{511, Sept. 1995. [19] P. D. Alexander and L. K. Rasmussen, \Sub{optimal MAP metrics for single user decoding in multiuser CDMA," in Int. Symp. Info. Theory and Apps., vol. 2, (Victoria, Canada), pp. 657{660, Sept. 1996. [20] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, \Optimal decoding of linear codes for minimising symbol error rate," IEEE Trans. Inform. Theory, vol. 20, pp. 284{287, 1974. [21] A. Hafeez and W. E. Stark, \Combined decision{feedback multiuser detection/soft{decision decoding for CDMA channels," in IEEE 46th Vehicular Technology Conference, (Atlanta, USA), pp. 382{386, 1996. [22] P. Robertson, \Improving decoder and code structure of parallel concatenated recursive systematic (turbo) codes," in Proc. IEEE Int. Conf. on Universal Personal Communications, (San Diego, California), pp. 183{187, Sept. 1994.

Iterative Multi-User Detection for CDMA with FEC: Near Single User Performance

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[23] C. B. Schlegel, Trellis Coding. IEEE Press, 1997. [24] S. Verdu, \Minimum probability of error for asynchronous Gaussian multiple{access channels," IEEE Trans. Inform. Theory, vol. 32, pp. 85{96, Jan. 1986. [25] P. Hoeher, \On channel coding and multiuser detection for DS-CDMA," in Proc. IEEE Int. Conf. on Universal Personal Communications, (Ottawa, Canada), pp. 641{646, Oct. 1993. [26] M. Nasiri-Kernari and C. K. Rushforth, \An ecient soft{decision algorithm for synchronous CDMA communications with error{control coding," in Proc. IEEE Int. Symp. on Information Theory, (Trondheim, Norway), p. 227, June 1994.