A Linear Front End for Iterative Soft Interference Cancellation and ...

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005

507

A Linear Front End for Iterative Soft Interference Cancellation and Decoding in Coded CDMA Alberto Tarable, Member, IEEE, Guido Montorsi, Senior Member, IEEE, and Sergio Benedetto, Fellow, IEEE

Abstract—In this paper, after a description of the principles of iterative (turbo) multiuser detection for CDMA systems, a new structure, based on a linear user separator, is introduced and analyzed. The new user separator processes the matched filter outputs according to a minimum mean-square-error (MMSE) criterion for the first few iterations, while it bypasses the MMSE filter when the interferers’ bits are known with some reliability. This new receiver is tested by simulation and compared with other receivers previously introduced in literature. Also, a way of studying and designing iterative multiuser receivers is given by adapting the method of EXtrinsic Information Transfer (EXIT) charts. With the EXIT charts, the receiver performance (in an information-theoretic sense) can be evaluated as it evolves along the iterations. Thus, the receiver threshold can be reliably estimated. Index Terms—Coded CDMA, EXIT charts, iterative multiuser detection, turbo codes.

I

N THE last fifteen years, following the increasing success of CDMA systems for wireless applications, which culminated in the definition of the third generation CDMA-2000 standard for the USA, and of the UMTS standard for Europe and Japan [10], multiuser detection (MUD) has constantly been a powerful magnet for researchers. In fact, one of the most severe CDMA limitations is represented by MultiUser Interference (MUI), due to the fact that the active users transmit at the same time in the same bandwidth. Orthogonal spreading sequences cannot grant in general a perfect isolation of different users, neither in the uplink (users to base station), because users lack synchronization, nor in the downlink, because the CDMA channel shows the phenomenon of multipath propagation. As a consequence, the matched filter (MF) receiver suffers from the near–far effect [21], i.e., powerful users may overcome weaker users. Possible remedies are linear user separators (USs), like the decorrelator and the MMSE US [13], or nonlinear USs, essentially based on interference cancellation [8].1 A more recent field of research is constituted by iterative MUD, which combines MUD with the so-called turbo principle. Manuscript received December 21, 2002; revised July 14, 03 and January 18, 2004; accepted January 20, 2004. The editor coordinating the review of this paper and approving it for publication is X. Wang. This work was supported in part by Qualcomm, Inc., and in part by the Italian Ministry of Education and Research by CERCOM and FIRB funds. The authors are with the Center for Multimedia Radio Communications (CERCOM), Politecnico di Torino, I-10129 Torino, Italy (e-mail: [email protected]; [email protected]; sergio.benedetto@polito). Digital Object Identifier 10.1109/TWC.2004.843015 1The term “user separator” substitutes in this paper the more common “multiuser detector,” because “multiuser receiver” has been used for the whole receiver and using the term “multiuser detector” would have possibly been confusing.

Since its appearance, turbo decoding has proved to be a powerful decoding algorithm for parallel [6] and serial [5] concatenated codes, allowing performance close to the theoretical Shannon limit. On the basis of the analogy between coded CDMA systems and a serial concatenation of codes [11], an iterative multiuser receiver can be devised. In this kind of receivers, the US and the users’ channel decoders exchange information along a certain number of iterations. The exchanged information is exploited to improve the receiver performance at each iteration, provided that some necessary conditions are met. The goal of research is then to find a low-complexity US to be inserted in this iterative structure. To our knowledge, the first paper that deals with this subject dates back to 1997 [17]. The paper by Moher [15] introduces the concept of cross-entropy. The optimal US can be found, for example, in [22], while a suboptimal linear US, also suitable for application in multipath scenarios, is introduced in [23]. Another linear US inserted in a turbo receiver is presented in [1], while in [2] a suboptimal US is obtained by applying to the optimal, Viterbi-based US standard approximations of the Viterbi algorithm. A different design based on the MMSE criterion is proposed in [9]. In this paper, we introduce a new low-complexity US, suitable for an iterative multiuser receiver, based on linear filtering. Precisely, it works either as a fixed linear MMSE US or as an MF US, depending on the reliability of the a priori information that the channel decoders provide to the US. As simulations will show, the proposed receiver tends to single-user performance for sufficiently high signal-to-noise ratio (SNR). Also, it has a lower complexity than the receiver in [23], at the price of a higher convergence threshold. Another contribution of this paper is the application of the method of the EXtrinsic Information Transfer (EXIT) charts, invented by Ten Brink [20], to the analysis and design of iterative multiuser receivers. Although an approximate tool, EXIT charts are very useful to predict the behavior of an iterative structure like a turbo decoder or a turbo receiver and, consequently, can be used to obtain good design rules. The paper is organized in the following way. In Section I, the system is described. In Section II, the principles of turbo receivers are summarized in some detail. In Section III, the definition of linear SISO-US is introduced, and a new linear SISO-US is presented. In Section IV, simulations compare this new structure with others already in the literature. In Section V, the method of designing iterative multiuser receivers, based on the EXIT charts, is described. Finally, in Section VI, we draw some conclusions.

1536-1276/$20.00 © 2005 IEEE Authorized licensed use limited to: Politecnico di Torino. Downloaded on March 12,2010 at 12:54:55 EST from IEEE Xplore. Restrictions apply.

508

Fig. 1.


K users transmitting simultaneously on an AWGN channel.

In the following, the notation for all and . The notation for all and fixed .

means the collection of means the collection of

a sufficient statistics. The MF output in the th bit interval can be expressed in matrix form (3)

I. SYSTEM DESCRIPTION

where

Consider users transmitting simultaneously on an AWGN , the information bit channel. For the th user, stream , is input to a convolutional encoder, which outputs the corresponding coded bit stream . An interleaver reorders to obtain , where is the permutation that defines the th interleaver. BPSK is assumed throughout the paper as the modulation adopted by all users, and . After being with the correspondence modulated, each bit is multiplied by the th spreading sequence , where is the spreading factor, i.e., the ratio between bit and chip interval. Finally, the signal is shaped . For simplicity, will be assumed by the chip waveform to be a square impulse with duration , the chip interval. (See Fig. 1 for a picture of the transmitter side of the system). The signals from the users are added on the channel with Gaussian additive noise. The signal at the receiver can then be expressed in the following way:

(1) where is the th user’s received amplitude, is the user’s delay with respect to an arbitrary time origin, is the bit interval

(2) , and is Gaussian is the th signature, with support noise having power spectral density . In the following, we , i.e., a synchronous will suppose system. Although this is a rather unrealistic hypothesis for an uplink transmission, the extension to more general scenarios, like asynchrony or multipath, is not difficult. As it is well known [21], a bank of filters, matched to the users’ signatures, provides, for every symbol interval, a vector of observables, which we call the MF output, that constitute

correlation matrix with elements ;

vector of noise samples, with zero mean and covariance ; noise samples in different matrix bit intervals are independent. When considering a single bit interval, as in (3), the index will be often omitted unless needed. The objective of multiuser detection is to extract from all possible information to detect, say, user , with an affordable complexity. Optimum multiuser detection is practically unfeasible, since its complexity grows exponentially with . Linear and nonlinear suboptimal multiuser receivers have been designed to reach good performance with realistic computational loads. II. TURBO-RECEIVER It has been noticed [11] that the cascade (users’ encoders interleavers multiuser channel) can be viewed as a serial concatenation of codes, where the multiuser channel acts the role of the inner encoder. Extending this analogy to the receiver side, an iterative turbo-like receiver can be thought of, based on the same principles of turbo decoding. In this type of receiver, there are two soft-input soft-output (SISO) blocks. 1) The first, called in this paper SISO user separator (SISO-US), is basically a soft-output multiuser detector, which corresponds to the inner decoder in turbo decoding. 2) The second is actually a bank of SISO channel decoders, one for each user, working independently. It corresponds to the outer decoder. Let us consider in detail a given iteration, say the th, of the receiver. The SISO-US has two sets of inputs: the MF output , and what we call , which is used as a priori information, in the sense that the SISO-US assumes

Authorized licensed use limited to: Politecnico di Torino. Downloaded on March 12,2010 at 12:54:55 EST from IEEE Xplore. Restrictions apply.

(4)

TARABLE et al.: A LINEAR FRONT END FOR ITERATIVE SOFT INTERFERENCE CANCELLATION

Fig. 2. Iterative multiuser receiver. 5

509

= deinterleaver.

In the first iteration, , for every and , since no a priori information is actually available at the receiver on , the set of the transmitted bits. In the th iteration, is fed back from the channel decoders, which computed them th iteration. We refer to the set of as the in the current bit statistics. The optimal SISO-US outputs another set , where of variables, denoted by

Since the optimal SISO-US is exponentially complex in , it is important to find good approximations to (5) [and, consequently, to (6)] that are easy to compute. III. THE LINEAR SISO-US The SISO-US is said to be linear if in the th iteration, it performs a linear filtering on the MF output , obtaining the expressed by vector

(5) (8) subsets, the th of The set of outputs are divided into which will be . This th subset is then deinand sent as input to the terleaved according to permutation th SISO channel decoder, which performs the same algorithm, called forward–backward or BCJR, as in turbo decoding. Details on the algorithm can be found in [3]. The output of the th SISO channel decoder, after interleaving, will be the set of that will provide the current bit statistics for user in the th iteration. In the last iteration, each SISO decoder must also supply the final estimate of the corresponding information bit stream, which represents the output of the whole receiver. Actually, it is convenient to work in the logarithmical domain. So, instead of (5), the optimal SISO-US outputs (6) and the th SISO channel decoder, instead of (4), outputs (7) LLR stands for log-likelihood ratio. See Fig. 2 for a picture of the iterative receiver. It is worth noting that in the turbo code literature, what is exchanged between the SISO blocks is called Extrinsic Information (ExInf), which is equal to the output LLR from which the respective input LLR is subtracted. In this way, the output ExInf is made independent of the corresponding input. With our definitions, the LLRs output by the SISO-US do not depend on the respective input LLRs, so that no subtraction is necessary and (6) does coincide with the ExInf.

where is a matrix, and the LLRs are computed , meaning that (6) is approximated element-by-element from by (9) The above computation is made on the basis of the current bit statistics. By (3) and (8), the expression of has the following form: (10)

where and is a Gaussian noise sample with zero mean and variance . The exact calculation of (9) has a computational complexity growing exponentially with the number of users. To avoid this computational burden, we invoke the Gaussian approximation (GA), which consists in substituting the actual expression of the term with a Gaussian random variable with the same mean and variance. Clearly, the stronger the MUI relatively to the noise, the looser the GA. According to the GA, the LLRs output by the SISO-US are computed by making use of the following approximated expression: (11) where mean and variance of the Denoting for the th user and

, being

and , respectively term at the th iteration. , and supposing


510


Fig. 3. Linear S SISO-US. Labels on the switches denote the switching iteration.

that current bit statistics for different users are independent, a straightforward computation shows that (12) (13) Then, from (9) we obtain (14) Equation (14) reveals that the linear SISO-US performs a soft interferers’ cancellation. In fact, according to (12), is just the average value of the MUI, which is subtracted from in order to compute the corresponding LLR. The reliability of the LLRs in (14) is determined by , which decreases when the . It is worth noting that interferers’ bit estimates approach also a wrong statistics contributes to reduce the variance, though it really raises interference. This phenomenon is similar to the well-known error propagation that occurs in nonlinear multiuser detectors. Among the many linear SISO-USs that have been proposed, we outline in some detail the one introduced by Wang and Poor [23], which will be called hereafter the WP SISO-US. The WP defined by (8), solves SISO-US, in the th iteration, to find the following mean-square-error (MSE) problem:

(15) is the average value of the MUI in the th where iteration. , according to (14). The LLRs are then extracted from Since the MMSE filtering given by (15) maximizes the signal-to-interference-plus-noise ratio (SINR) in the filter output, it can be considered in this sense the best linear SISO-US. Unfortunately, implementing this receiver requires solving (15) for every user and every bit at each iteration, so operations per bit decoded per its complexity is at least iteration per user [23]. Starting from this consideration, we will introduce a new SISO-US, which is called hereafter switched

SISO-US (S SISO-US), which has a lower complexity, at a price of some loss in performance. In the new approach, only two cases are distinguished: when the interferers’ bit estimates are bad, including the limiting case of no interferers’ estimates available, as in the first iteration, and when the interferers’ estimates are good, including the asymptotic case of perfect knowledge of the interferers’ bits. Since our purpose is to have only one linear US for each of the two cases, it is natural to adopt for the first the standard linear MMSE filter , which maximizes the output [13], i.e., SINR when no a priori information can be used, and for the which is second case no linear filtering at all, i.e., optimal for a single-user AWGN channel. The latter case will be called MF SISO-US. It is worth noting that the WP SISO-US behaves like a standard MMSE SISO-US in the first iteration, while it ideally tends to the MF SISO-US as the iterations proceed. , denoted , Then, for the S SISO-US, the th row of will be (16) where is a vector with all elements equal to zero , the iteration at which but the th, set to one. The integer the SISO-US switches from MMSE to MF user detection, will be referred to as the switching iteration for user . Notice that, although the filter changes only once through the iterations, the LLRs given by (14) change at every iteration. In fact, and depend on the current bit statistics, which is updated by the channel SISO decoders at every iteration. iterations, the output LLR for the th user is In the first extracted from the MMSE filter output, according to (6). With this choice, the SINR after the filter is maximized at the first iteration. th iteration on, when the interferers’ estimates From the are supposedly good, the output LLR for the th user is extracted directly from the MF output. Notice the fundamental difference between the WP SISO-US and the S SISO-US. Although they are both based on the MMSE criterion, the former changes the filter matrix at each iteration, while the latter has a fixed filter matrix before switching. Thus, the linear filter is inserted in the iterative decoder loop for the WP SISO-US, while it is actually outside the loop for the S SISO-US, as shown in Fig. 3. Switching from the MMSE filter to the conventional one is necessary, because the MMSE SISO-US alone does not tend to



the single-user limit. In fact, to combat MUI, the MMSE filter lowers the user-of-interest SNR, so that, even if MUI is perfectly cancelled, single-user performance is not reached. After switching, the S SISO-US subtracts the estimated MUI directly from the MF output at no expense for the SNR on the channel. That grants single-user performance in the ideal case in which MUI is perfectly cancelled. The optimization of the parameter , the switching iteration for user , will be addressed later. It is easy to see that the proposed multiuser receiver is near–far resistant, thanks to the MMSE filter. This distinguishes the proposed SISO-US from that introduced in [1], consisting of a MF SISO-US from the first iteration, which is . That then a particular case of the S SISO-US, with receiver is not near–far resistant because, when the interferers’ amplitudes are particularly high, the LLRs output by the MF SISO-US are not good enough to start the process of improvement along the iterations. Let us finally consider the complexity of the S SISO-US. The S SISO-US performs the linear MMSE filtering, consisting in a matrix inversion, only at the beginning. This inversion operations by using the iterative can be performed with per method described in [23]. It gives then a complexity user. This can be made once and for all. In the subsequent iterations, it only subtracts the current MUI estimation. This task has a complexity growing linearly with the number of users; that is, operations per decoded bit per user per itit needs about eration. It is thus simpler than the WP SISO-US, which needs a different linear transformation for each user, for each bit, at each iteration. In Section IV, we will show that the lower complexity of the S SISO-US will have as a main consequence a higher convergence threshold with respect to the WP SISO-US. A. Switching Criterion For user , the S SISO-US performs iterations as an inMMSE SISO-US, before switching. The choice of fluences the performance of the receiver in terms of BER. However, choosing the best value of this parameter, i.e., the value that minimizes the system BER, is a challenging problem. , based on In this section, we propose a criterion to choose SINR maximization. Precisely, the S SISO-US switches when the output SINR for the MF SISO-US, with the current bit statistics, exceeds the output SINR for the MMSE SISO-US. Acis the minimum cording to this criterion, the best value of value of the iteration number that satisfies the inequality (17)

(17) . Since all paramewhere ters are supposed to be known, this relation can be checked on the fly by a processor, to find the best iteration for switching. Notice that, since , i.e., the switching iteration may be dependent on the considered bit interval. The problem of this optimization is the small additional amount of computation needed for it.

511

to a fixed value, As a simpler strategy, one can think to set independent of the bit interval, according to the th user’s power and to the amount of MUI. This strategy has generally a worse performance in terms of BER with respect to the SINR-maximizing approach (see Fig. 5). To find a possible fixed switching strategy, let us consider equal users, amplitudes a particular symmetric case, with , and constant correlation . Although this is not a realistic case, the example allows us to go on with computations and quantifies for a particular case the loss in terms of BER with respect to the SINR-maximizing switching strategy. By computing the expression of the quantities in (17) , and , we obtain the following relain terms of tion:

(18) Since the system is completely symmetric, let us suppose that and, as a consequence, (18) does not depend on the index . , we resort To estimate the behavior of , to simulation and find the loop gain with the MMSE SISO-US, making the simplifying hypothesis that this loop gain does not depend on . The simulation shows that a good approximation of the loop gain can be a first-order function like (19) with and suitable functions. Applying it, we obtain an eratively (19), with the starting point approximated expression for (20) This expression, substituted in (18), gives after a little bookand shown in (21) keeping the best value of , denoted

(21) denotes floor. A possible implementation of such a where strategy can be thought of by making use of a look-up table, which has entries for all possible combinations of the paramein (21). ters To find the solution in a particular scenario, we set and and, by simulation, we find and . is reproduced in Fig. 4. The figure also The value of when the SINR-maximizing shows the average value of strategy is chosen. This curve has been obtained by simulating the system. As it can be seen, the computed values are rather pessimistic with respect to the measured values. With this fixed switching strategy, more iterations with the MMSE SISO-US are performed than necessary.


512


Fig. 4. Fixed switching iteration, according to (21), in a four-user symmetric system with correlation 0.7.

K

Fig. 5. Results of simulation. = 4 synchronous users, all with the same power. Correlation between any couple of users is 0.7. Dotted lines refer to the receiver by Wang and Poor with the same system [23]. Dashed line refers to the receiver in [1].

IV. SIMULATIONS We exactly performed the same simulations as in [23], to permit direct comparison of the results. In the first simulation, we consider four users with the same SNR and with correlations . For all users, the convolutional code with channel code is the 16-state rategenerators 23, 35 (in octal form). Every user has a different random interleaver with length 256. The same set of interleavers

has been used for all simulations. The SISO channel decoders are based on the BCJR algorithm [3]. The switching iteration is chosen according to the SINR-maximizing switching strategy. See Fig. 5 for the results. It can be seen that the convergence threshold with the WP US is lower than that of the S SISO-US. . This is the The penalty is about 0.5 dB at BER of price to pay because of the reduced complexity. However, the curves show that the proposed receiver tends to the single-user as high as 5 dB. It is also represented in limit for



513

Fig. 6. Results of simulation. System is the same as Fig. 5. Solid and dashed lines refer to the receiver with the S SISO-US with the fixed and the SINR-maximizing switching strategy, respectively.

Fig. 7. Results of simulation. System is the same as Fig. 5, except that two users are 3 dB stronger than the other two. Performance of the weak users. SINR-maximizing switching strategy.

the figure the curve of the receiver in [1], which is clearly below threshold. For the same system, Fig. 6 shows the performance of the iterative receiver with the S SISO-US and the fixed switching strategy of (21), with respect to the performance of the same receiver with SINR-maximizing switching strategy. As it can . be seen, another half-dB penalty has to be paid at BER of dB. Single-user performance is reached for

Another simulation has been performed, again with four users, which considers two users 3 dB stronger than the other two. Otherwise, the system is identical to the previous one. Figs. 7–8 depict the performance for both the weak and the strong users. We observe that the receiver still performs well enough and converges to the single-user curve. Indeed, weak users take advantage of the power unbalance and their curves converge rapidly to single-user performance, while strong users


514

Fig. 8.


Results of simulation. System is the same as Fig. 7. Performance of the strong users. SINR-maximizing switching strategy.

Fig. 9. Results of simulation.

K = 8 equal users. System is the same as Fig. 7. SINR-maximizing switching strategy.

result to be anchored to the weak users’ . The same behavior can be observed for the WP US. users, with corA simulation has been performed for relation between each couple of users equal to 0.7. The other parameters are as in the previous simulations. Results for the

proposed and for the WP receiver, shown in Fig. 9, are compared. Again, the behavior is similar to the other cases already observed. Here the receiver with the S SISO-US converges when is above 6 dB and loses about 1.2 dB with respect to the receiver with the WP SISO-US.



515

Fig. 10. EXIT chart for the linear SISO-US receiver. System is the same as Fig. 7. E =N = 0 dB. Curves represent the loop gain of the receiver when different SISO USs are cascaded with the 16-state rate-1=2 convolutional code with generators 23, 35 (in octal form).

V. ITERATIVE MULTIUSER RECEIVERS AND EXIT CHARTS In this section, we analyze the proposed iterative receiver, as well as other receivers, with the method of the EXIT charts first introduced by Brink [20]. This method, although approximated, allows us to visualize the behavior of the receiver along the iterations and to compare different receivers. It also shows the con. In the following, we will vergence threshold in terms of suppose that the S SISO-US is used in conjunction with a fixed switching strategy. However, a generalization to the SINR-maximizing switching strategy is possible. Let us consider a symmetric channel with equal users. The receiver employs a S SISO-US with . Consider the LLR output in the th iteration by the SISO-US for the th user’s th bit, given by (14). We can define as the mutual information [7] between and the corresponding bit . By the definition of mutual information, quantifies in an information-theoretic sense the correlation between should converge to 1 as the itthe two quantities. Ideally, erations proceed, because that means perfect knowledge of the . Since all users are equal and all bits of the same user are bit treated in the same way, the average behavior of is not supposed to depend on the particular user and bit chosen. Define then , to take into account this invariance. Unlike the approach by Brink [20], which depicts input-output characteristics for each SISO block in the loop, we want to depict with the hypothesis that it does the loop gain not depend on the iteration number . Correlations among the LLRs along the iterations cause the loop gain being smaller and smaller as the iterations proceed. The analysis with the EXIT

charts, then, supposes perfect interleaving; that is, it neglects the phenomenon of saturation of the loop gain. The EXIT charts are obtained in the following way. The US output LLRs at the th iteration are introduced as input, in the following form: (22) where . The corresponding mutual information is computed by means of the following expression [20]:

(23) where is the considered frame length. The sum over all users and all bits is allowed because of the intrinsic symmetry of the depends on the free parameter . The system. The value of LLRs enter the SISO channel decoders, which output their new a priori LLRs. The new a priori LLRs enter the SISO-US th iteration, and the SISO-US at the beginning of the outputs , for every and . The mutual information is computed according to (23). The probabilities in (23) are computed based on the GA [see (11)]. We have plotted the loop gain instead of the input-output characteristics because the GA for the US output is tight when MMSE filtering is performed [16], so that using (22) instead of exact LLRs is a minor approximation, both with the MMSE SISO-US and the WP SISO-US. Figs. 10–12 show the EXIT charts for different values of the common user amplitude. In all charts, both axes represent the


516




input and output mutual information, denoted and in the figures, respectively, because the functions obtained by simulations are reproduced together with their own inverse. In this way, the evolution of the mutual information is followed by

tracing a zig-zag path, as shown in Fig. 11. This path must start . Since , the from the origin, because corresponds to the mutual information in the first value of iteration (first stroke of the dashed line, parallel to the axis).



Now becomes the abscissa and the value of must be read on the inverse loop gain (second stroke of the dashed line, parallel to the axis), and so on. After strokes of the zig-zag path, we can read . Figs. 10–12 show three pairs of curves, for the WP SISO-US, the MMSE SISO-US, and the MF SISO-US, denoted respec, and . For our receiver, strokes of the tively, path must touch the MMSE SISO-US curve, while, from the th stroke on, i.e., after switching, the reference curve is that for the MF SISO-US. To interpret the curves, notice that, considering the input mutual information on the axis and the output mutual information on the axis, where the curve lies over the bisector of the first quadrant, this corresponds to an improvement of the LLRs along the iterations. Where the curve lies under the bisector, instead, the LLRs worsen as the iterations proceed. Where the curve meets the bisector, the loop gain . has a fixed point, since dB, and it can Fig. 10 shows the EXIT chart for be observed that the receiver does not converge to single-user performance. In fact, the single-user value of the mutual in, because when the input mutual informaformation is tion is equal to 1, bits are perfectly known and their contribution can be subtracted from , perfectly cancelling MUI. In this case, the single-user limit is almost 0.5 bits, while both the MMSE and the WP curve converge to a value just over 0.3 bits, and switching to a MF SISO-US is actually harmful because it causes the mutual information to converge to a value between 0.2 and 0.3 bits. dB, and Fig. 11 shows the EXIT chart for both ours and the WP receiver converge to single-user limit, which is now about 0.9 bits. This chart shows clearly that, also with perfect interleaving, the MMSE SISO-US alone does not reach single-user performance, the fixed point of the loop gain being about 0.7 for this US. Also, this chart shows that the MF SISO-US, if used from the start, reaches a fixed point dB is between 0.3 and 0.4 bits. Its performance at dB. not substantially different from that at dB. As we Fig. 12 shows the EXIT chart for can see, this value can be considered as the threshold for convergence for our receiver. In fact, the MMSE curves converge to a point, with mutual information of about 0.6 bits, which is enough, when switching to the conventional US, to reach a value of about 0.8 bits, near to the single-user limit. The same value is reached by the WP receiver. Some observations can be added on the subject of EXIT charts. 1)

The EXIT charts show the exact loop gain only if the input LLRs are uncorrelated. In fact, the input LLRs are injected as given in (22). If, instead, the LLRs are correlated, the loop gain can be smaller. Intuitively, the higher the correlation, the smaller the loop gain. The correlation between the LLRs, in general, depends on the US type, on the value of users’ cross-correlations and amplitudes, on the interleavers. As we have seen by simulations, however, the real behavior for all three USs follows the ideal analysis based on EXIT charts.

517

EXIT charts give a suggestion on the best value of , which is in good agreement with the analysis in Section III-A. A final observation can be made: the EXIT charts for iterative multiuser receivers depend on users’ amplitudes and cross-correlations. If these parameters are somewhat controllable, then EXIT charts can be used to find the convergence threshold [20] and, consequently, the target received power. If, instead, due for example to asynchrony, cross-correlations are beyond control, the practical usefulness of EXIT charts becomes limited. 2)

VI. CONCLUSION In this paper, we have introduced a new linear SISO-US for iterative multiuser receivers, called S SISO-US, having a complexity per user per decoded bit per iteration linear with the number of users. We also have analyzed some issues that are related with this new receiver, such as the choice of the switching iteration. Simulations have shown that the receiver performance . Fiis reaching single-user limit for sufficiently high nally, a method of design for iterative multiuser receivers is given, by applying to them the method of EXIT charts, originally introduced for turbo codes. REFERENCES [1] P. D. Alexander, A. J. Grant, and M. C. Reed, “Iterative detection in code-division multiple-access with error-control coding,” Eur. Trans. Telecommun., vol. 9, pp. 419–425, Sep./Oct. 1998. [2] P. D. Alexander, M. C. Reed, J. A. Asenstorfer, and C. B. Schlegel, “Iterative multiuser interference reduction: Turbo CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1008–1014, Jul. 1999. [3] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” Inf. Theory, vol. IT-20, pp. 284–287, 1974. [4] S. Benedetto and G. Montorsi, “Iterative decoding of serially concatenated convolutional codes,” Electron. Lett., vol. 32, pp. 1186–1188, Jun. 1996. [5] , “Unveiling turbo codes: Some results on parallel concatenated coding schemes,” IEEE Trans. Inf. Theory, vol. 42, no. 3, pp. 409–428, Mar. 1996. [6] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correction coding and decoding: Turbo-codes,” in Proc. 1993 Int. Conf. Commun. (ICC’93), vol. 2, Geneva, Switzerland, May, 23–26 1993, pp. 1064–1070. [7] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [8] A. Duel-Hallen, “A family of multiuser decision-feedback detectors for asynchronous DS-CDMA systems,” IEEE Trans. Commun., vol. 43, no. 2, pp. 421–434, Feb. 1995. [9] H. El Gamal and E. Geraniotis, “Iterative multiuser detection for coded CDMA signals in AWGN and fading channels,” IEEE J. Sel. Areas Commun., vol. 18, no. 1, pp. 30–41, Jan. 2000. [10] European Telecommunications Standards Institute, TS 125 201 V3. 0.1: Universal Mobile Telecommunications System (UMTS); Physical Layer—General Description, 2000. [11] J. Hagenauer, “Forward error correcting for CDMA systems,” in Proc. IEEE Int. Symp. Spread Spectrum Techniques Appl., vol. 2, Mainz, Germany, Sep., 22–25 1996, pp. 566–569. [12] J.-M. Hsu and Ch.-L. Wang, “A low complexity iterative multiuser receiver for Turbo-coded DS-CDMA systems,” in Proc. ICC 2000, vol. 3, New Orleans, LA, Jun. 18–22, 2000. [13] R. Lupas and S. Verdú, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inf. Theory, vol. 35, no. 1, pp. 123–136, Jan. 1989. [14] R. J. McEliece, D. J. C. MacKay, and J.-F. Cheng, “Turbo decoding as an instance of Pearl’s “belief propagation” algorithm,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 140–152, Feb. 1998.


518


[15] M. Moher, “An iterative multiuser decoder for near-capacity communications,” IEEE Trans. Commun., vol. 46, no. 7, pp. 870–880, Jul. 1998. [16] H. V. Poor and S. Verdú, “Probability of error in MMSE multiuser detection,” IEEE Trans. Inf. Theory, vol. 43, no. 5, pp. 858–871, May 1997. [17] M. C. Reed, P. D. Alexander, J. A. Asenstorfer, and C. B. Schlegel, “Near single user performance using iterative multi-user detection for CDMA with turbo-code decoders,” in Proc. PIMRC 1997, Sep. 1997, pp. 740–744. [18] A. Tarable, G. Montorsi, and S. Benedetto, “A linear front end for iterative soft interference cancellation and decoding in coded CDMA,” in Proc. ICC, vol. 1, Helsinki, Finland, Jun. 2001, pp. 1–5. [19] A. Tarable, G. Montorsi, and S. Benedetto, “Analysis and design of interleavers for CDMA systems,” IEEE Commun. Lett., vol. 5, no. 10, pp. 420–422, Oct. 2001. [20] S. T. Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1727–1737, Oct. 2001. [21] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [22] M. C. Valenti and B. D. Woerner, “Iterative multiuser detection for convolutionally coded asynchronous DS-CDMA,” in Proc. PIMRC, vol. 1, Boston, MA, Sep., 8–11 1998, pp. 213–217. [23] X. Wang and H. V. Poor, “Iterative (Turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, Jul. 1999.

Alberto Tarable (S’01–M’01) received the Laurea degree (summa cum laude) and the Ph.D. degree in electronic engineering from Politecnico di Torino, Turin, Italy, in 1998 and 2002, respectively. From April 2001 to October 2001, he spent six months as a Visiting Scholar at the University of California, San Diego, working on the topic of space-time codes. Since March 2002, he has been working as a Researcher in the Dipartimento di Elettronica, Politecnico di Torino. His current interests include multiuser detection, CDMA systems, space-time coding, coding theory, and UWB systems.

Guido Montorsi (SM’03) was born in Turin, Italy, on January 1, 1965. He received the Laurea in ingegneria elettronica from Politecnico di Torino, Turin, Italy, in 1990 with a master thesis, concerning the study and design of coding schemes for HDTV, developed at the RAI Research Center, Turin. He received the Ph.D. degree in telecommunications from the Dipartimento di Elettronica, Politecnico di Torino, in 1994. In 1992, he spent the year as Visiting Scholar in the Department of Electrical Engineering, Rensselaer Polytechnic Institute, Troy, NY. In December 1997, he became an Assistant Professor at the Politecnico di Torino. From July 2001 to July 2002, he spent one year at Sequoia Communications, San Diego, CA, developing algorithm for 3G wireless receivers. In 2003, be became an Associate Professor at Politecnico di Torino. His interests are in the area of channel coding, particularly on the analysis and design of concatenated coding schemes and the study of iterative decoding strategies.

Sergio Benedetto (M’76–SM’90–F’97) received the “Laurea in Ingegneria Elettronica” (summa cum laude) degree from Politecnico di Torino, Turin, Italy, in 1969. Since 1981, he has been a Full Professor of digital communications at Politecnico di Torino. He has been a Visiting Professor at University of California, Los Angeles (UCLA), the University of Canterbury, New Zealand, and is an Adjoint Professor at Ecole Nationale Superieure de Telecommunications, Paris, France. He has coauthored books on probability and signal theory (in Italian) includingDigital Transmission Theory (Englewood Cliffs, NJ: Prentice-Hall, 1987), Optical Fiber Communications (Norwood, MA: Artech House, 1996), and Principles of Digital Communications with Wireless Applications (New York: Plenum-Kluwer, 1999), and over 250 papers in leading journals and conferences. He has taught several continuing education courses on the subject of channel coding for the UCLA Extension Program and for the CEI organization. Prof. Benedetto received the Italgas Prize for Scientific Research and Innovation in 1998. He has been Chairman of the Communications Theory Symposium of ICC 2001, and has organized numerous sessions in major conferences worldwide. He is the Area Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS for Modulation and Signal Design, a Distinguished Lecturer of the IEEE Communications Society, and is the Chairman of the IEEE Communication Theory Committee.
