Turbo Multiuser Detection for Differentially Modulated ... - IEEE Xplore

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 2, MARCH 2004

Turbo Multiuser Detection for Differentially Modulated CDMA Yi Wu and Teng Joon Lim, Senior Member, IEEE

Abstract—In this letter, we study differentially modulated, iteratively decoded CDMA. The iterative multiuser receiver proposed consists of an additional soft-input soft-output (SISO) differential decoder, when compared to turbo multiuser detectors for absolutely modulated systems. Algorithms for iterative decoding with and without phase information at the receiver are developed. The resulting turbo receivers with differential modulation outperform coherent receivers with absolute modulation at moderate to high signal to noise ratios due to the interleaver gain associated with recursive inner encoders in serially concatenated encoding structures. Index Terms—Code division multiaccess (CDMA), convolutional codes, differential modulation, multiuser detection, noncoherent demodulation, turbo decoding.

I. INTRODUCTION

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HE iterative or turbo decoding principle [1] has been applied to multiuser detection ([2]–[5], just to name a few), by viewing the code-division multiaccess (CDMA) channel as an inner encoder, and the forward error control (FEC) encoders at each transmitter as outer encoders of a serially concatenated coding scheme [6]. For iterative decoding of serially concatenated convolutional codes (SCCC), it is known that a recursive inner encoder results in a so-called interleaver or turbo gain, which gives rise to steep drops in the bit error probability with every iteration in the moderate to high signal-to-noise ratio (SNR) region. A recursive inner encoder is therefore preferable to a nonrecursive one. This realization quickly led to the study of the performance of iterative decoding with differential -ary phase shift keying (M-DPSK) as a rate-1 recursive inner encoder and a convolutional outer encoder [7]–[9]. These studies confirmed that coded DPSK, whether coherently or noncoherently demodulated using an iterative decoder, performs better than coded coherent PSK at sufficiently high SNRs. In this letter, we are interested in the iterative decoding of a coded, DPSK-modulated, direct-sequence CDMA multiuser system, because of the ease with which an absolutely modulated PSK transmitter can be converted into a differentially modulated one. While the performance improvement obtainable with DPSK over coherent PSK has previously been observed in [7], iterative multiuser detection with differential modulation has not been studied so far. Our main contribution is the Manuscript received April 10, 2002; revised October 17, 2002 and December 23, 2002; accepted January 31, 2003. The editor coordinating the review of this letter and approving it for publication is X. Wang. This work was supported in part by the Nortel Institute for Telecommunications and Bell Canada University Laboratories. Y. Wu is with the Centre for Wireless Communications, Oulu, Finland FIN90014. T. J. Lim is with the Department of Electrical and Computer Engineering, University of Toronto, Canada M5S 3G4 (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2003.821218

derivation of a noncoherent CDMA soft-in soft-out (SISO) decoder, and its integration into an iterative receiver with three SISO component decoders: one each for the convolutional code, the CDMA channel, and the differential modulator. For comparison, well-known coherent CDMA SISO decoders are also discussed briefly and simulated. The rest of the letter is organized as follows. In Section II, we introduce the convolutionally and differentially encoded CDMA model, including the transmitter and the channel. This is followed in Section III by a description of the iterative decoding process and details on the new receivers. The results of an investigation into the performance of the systems based on computer simulation, their interpretation, and related discussion are given in Section IV. The results show that both receivers are capable of very good performance, with only a slight noncoherence penalty. Furthermore, the power of this class of system is illustrated as they significantly outperform systems of absolutely encoded CDMA with coherent detection. Finally, it should be noted that our approach can also be applied to any interference-limited system, whether using spread spectrum signalling or not. II. SYSTEM MODEL We consider a synchronous coded CDMA system of users with binary DPSK modulation, signaling through an AWGN channel. The block diagram of the transmitter of such a system for user is shown in Fig. 1. The binary information data , , are convolutionally encoded with code rate , and the code bits are block-interleaved. of the th user The interleaved code bits are passed through a binary differential encoder to give1 (1) Without loss of generality, the differential encoder is assumed to . Each symbol start at the reference symbol is modulated by a spreading waveform , and transmitted through an AWGN channel. The received baseband equivalent can, therefore, be expressed as complex signal (2) is the number of data bits per user per frame, where is the symbol interval, and denote, respectively, the channel coefficients and normalized signaling is a zero-mean, circularly waveform of the th user, and symmetric, complex white Gaussian noise process with power is supported only on spectral density . It is assumed that

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1For simplicity, we consider binary modulation but -ary modulation entails only slight and insignificant modifications to the proposed algorithms.

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Fig. 1. Block diagram of the coded CDMA transmission system.

the interval and has unit energy. The phase error and are assumed to remain constant over one frame. amplitude For the synchronous case, a sufficient statistic for demodulating the th code bits of the users is given by the -vector whose th component is the output of a filter matched to in the th code bit interval, and

is MAI-free and While the th decorrelator output therefore a good candidate for the DPSK decoder input, the decorrelator does not rely on soft information from any of the other component decoders and hence does not have the ability to update its output with each iteration. This problem can be solved by noting that

(3)

(5)

denotes the normalized cross-correlation mawhere , i.e., trix of the signal set , , , and is a white Gaussian noise vector sequence, independent of .

. The first equation folwhere lows from the differential encoding (1). If we neglect the noise term in (4), we have . By averaging over two symbol intervals, the MAI component within can be estimated as (6)

III. RECEIVER STRUCTURES AND ALGORITHMS Turbo decoding for the system of Fig. 1. would split the overall decoding task into three stages, with each stage decoding one component code, respectively. The three component decoders have the same objectives regardless of whether the receiver is coherent or noncoherent, and are easily understood with reference to Fig. 1. In each iteration The CDMA Decoder provides inputs, with less multiaccess interference (MAI) than the signals in the previous iteration, to a bank of DPSK decoders. Improved MAI removal is possible at each iteration because of judicious use of soft information from the previous iteration provided by the DPSK decoder. The DPSK Decoder accepts inputs from the MAI decoder and the convolutional decoder in order to generate a better , . estimate of the code symbols The Convolutional Decoder uses the log likelihood ratios (LLRs) of the code symbols, which come from the DPSK decoder (after deinterleaving) in order to update LLR’s of and information bits . the code symbols A. Noncoherent Receiver 1) CDMA Decoder: This receiver does not know the matrix and so its CDMA component decoder cannot perform symbol-by-symbol MAP decoding, nor can it use the suboptimal soft interference cancellation technique of [10]. However we can assume knowledge of the code correlation matrix and hence obtain the decorrelator output [11] (4) where variance matrix

is a Gaussian random vector with co.

in

the

th

iteration, where is the matrix of tentative decisions in the th iteration. These can be expressed in of , obtained from the SISO terms of the LLR’s DPSK decoder at the th iteration, as

(7) Note that at every iteration is the decorrelator output , which does not require phase information. With the estimated MAI term in (6), the CDMA decoder at the th iteration outputs (8) which is then passed to the SISO DPSK component decoder. The SISO DPSK decoder requires the conditional variance , the th component of . Using (1), (3) and (4), the of conditioned on covariance matrix of can be found to be

,

and

(9) Furthermore, if , we have , where is a zero-mean, jointly Gaussian vector with , and therefore covariance matrix (10)

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Fig. 2. Block diagram of the noncoherent iterative receiver structure.

Equations (9) and (10) provide all the necessary information for the SISO DPSK decoder. 2) SISO DPSK Decoder: The noncoherent SISO DPSK decoder used in the proposed receiver is identical to the one in [7, Sec. II-B], so we will only provide a brief description here. The input to the DPSK decoder is , whose probability density and all code function (pdf) conditioned on user ’s channel , , assuming complete MAI cancellation, bits is

(11) is the diagonal element of , and . where in (11) can be The dependence on the unknown to obtain removed by integrating over the pdf of . Letting denote the vector , and be the vector , as a product of we can find terms, based on the independence of the additive Gaussian noise in over time. Finally, by Bayes’ Rule we can obtain an expression for the conditional pdf necessary in a MAP decoding algorithm (such as [12]). requires However, computing operations, and it is desirable to reduce the complexity of this operation by making the approximation [7]

(12)

with

is a quantity that does not depend on and is the modified Bessel function of order different values of zero. From (13) we know that there are with respect to the set of all binary -tuples . As such, we can construct a trellis structure with the state at time as , the state transition matrix and the “output symbol” generation matrix . If we set the window size to be small, then the difference in complexity between the exact expression and its approximation is considerable. With this trellis structure, branch metrics (12) and extrinsic from the SISO convolutional decoder, information we can use the standard APP algorithm described in [13] to implement a noncoherent SISO DPSK decoder for each user. While the LLR of the data bits entering the differential encoder, , is fed back to the soft MAI canceller, the extrinsic i.e., is fed forward to user s SISO convoluinformation tional decoder. The block diagram of the noncoherent iterative multiuser receiver is given in Fig. 2.

where

B. Coherent Receivers Coherent receivers (which have perfect phase information on all users’ signals) can be easily derived from material in the literature, and we will compare the performance of noncoherent detectors with these. Briefly, 1) The CDMA Decoder can either be a full-complexity MAP decoder (see [14, Sect. IV-A]) or a low-complexity interference canceller [10]; 2) The DPSK Decoder is described in [7, Sect. III-D] and is a straightforward application of symbol-by-symbol MAP decoding to the DPSK trellis; 3) The Convolutional Decoder is a symbol-by-symbol MAP decoder. IV. SIMULATION RESULTS AND DISCUSSION

(13)

We simulated the performance of iterative receivers in absolutely encoded, coherently demodulated multiuser systems,


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Fig. 4. Performance of noncoherently demodulated iterative multiuser detector with = 0:3, K = 4 and equal received signal power for all users. 100 Low-Complexity Rx'er, 6 iterations Absolutely Encoded, 5 iterations Differential Encoding, 5 iterations

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and compared that with the proposed iterative receivers for differentially encoded ones. Unless otherwise indicated, the following system parameters were used in the simulation. There are four users in the system, where all users have equal power and employ the same rate-1/2 convolutional code with gener. Each user uses its own random interleaver. The ator same set of interleavers are used for all simulations. The block . The nonsize of the information bits for each user is in the algorithm of coherent SISO DPSK decoder uses Section III-A-2. All simulations were performed on the -symmetric channel [5] that is characterized by the spreading-code correlation matrix

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where is the cross-correlation parameter. The bit error rate (BER) curves obtained with coherent de. Turbo modulation and DPSK are plotted in Fig. 3 for detection of the absolutely encoded (BPSK) multiuser system after five iterations was simulated as a baseline for comparison.2 As can be seen, the performance of the full-complexity detector significantly improves with the first few iterations, although gains appear to be marginal after four iterations. More significantly, both low- and full-complexity detectors with differential encoding outperform the absolutely encoded system, . by about 1.8 dB at a BER of These results, at first glance, may seem quite surprising because differential encoding is typically associated with degradation in system performance. However, it is entirely consistent with [7] and shows just how substantial interleaver gain can be. Comparing Fig. 3 to Fig. 3 in [7], we see that the proposed iterative multiuser system has a BER after five iterations roughly equivalent to the BER of a single-user iterative DPSK detector , the proposed receiver after three iterations. At a BER of 2This is essentially the same performance as the G(23; 35) convolutional code in a Gaussian channel.

after five iterations has an approximately 1 dB loss compared to the single-user DPSK detector after 20 iterations. The same relative performance found with coherent demodulation also extends to noncoherent demodulation. Fig. 4 shows the performance of the proposed noncoherent receiver for the . Also shown are the BER curves first five iterations with for the absolutely encoded system, and the differentially encoded, coherently demodulated scheme, all after five iterations. There is only a 0.8 dB noncoherence penalty after five iterations . However, the gain of the noncoherent receiver at . over absolute encoding is still 1.0 dB at , results are When the cross-correlation increases to shown in Fig. 5. In this example, results are less encouraging: The differentially encoded system with the full-complexity coherent receiver has the same performance as the absolutely en, after five iterations. The low-comcoded one at plexity coherent receiver with 6 iterations suffers a 2 dB loss . For compared with the full-complexity system at both the full- and the low-complexity receivers, performance is far from the single-user DPSK case with coherent demodulation

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shown in Fig. 2 of [7]. This is a manifestation of the well-known phenomenon (see, e.g., [2]) that turbo multiuser detection does not always converge to single-user performance, especially at high interference levels.

V. CONCLUSION In this letter, we studied turbo detection for convolutionally and differentially encoded CDMA systems, with either coherent or noncoherent demodulation. The proposed detectors are iterative decoders for a three-stage serially concatenated system of convolutional code, differential code and CDMA channel. The noncoherent SISO multiuser detector and the lowcomplexity coherent SISO multiuser detector are based on MAI cancellation, while the full-complexity coherent SISO multiuser detector uses a maximum a posteriori (MAP) algorithm. Simulation results show that coded, differentially modulated CDMA with interleaving can perform better than its coherently modulated (or absolutely encoded) counterpart when full complexity coherent demodulation is used. With noncoherent demodulation, the performance gain is smaller, but the interleaver gain obtained from the recursive nature of differential encoding is still evident. Finally, we note that recently, Shi and Schlegel [2] also attempted to introduce a recursive code before the spreading operation in a CDMA channel, to obtain turbo gain. Our method arose from the same idea, but is much simpler to implement. It would be interesting to compare the relative advantages and disadvantages of these two techniques.

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