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

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

Abstract—In this letter, we study differentially modulated,iteratively decoded CDMA. The iterative multiuser receiverproposed consists of an additional soft-input soft-output (SISO)differential decoder, when compared to turbo multiuser detec-tors for absolutely modulated systems. Algorithms for iterativedecoding with and without phase information at the receiver aredeveloped. The resulting turbo receivers with differential modu-lation outperform coherent receivers with absolute modulation atmoderate to high signal to noise ratios due to the interleaver gainassociated with recursive inner encoders in serially concatenatedencoding structures.

Index Terms—Code division multiaccess (CDMA), convolutionalcodes, differential modulation, multiuser detection, noncoherentdemodulation, turbo decoding.

I. INTRODUCTION

THE iterative or turbo decoding principle [1] has been ap-plied to multiuser detection ([2]–[5], just to name a few),

by viewing the code-division multiaccess (CDMA) channel asan inner encoder, and the forward error control (FEC) encodersat each transmitter as outer encoders of a serially concatenatedcoding scheme [6].

For iterative decoding of serially concatenated convolutionalcodes (SCCC), it is known that a recursive inner encoder re-sults in a so-called interleaver or turbo gain, which gives riseto steep drops in the bit error probability with every iteration inthe moderate to high signal-to-noise ratio (SNR) region. A re-cursive inner encoder is therefore preferable to a nonrecursiveone. This realization quickly led to the study of the performanceof iterative decoding with differential -ary phase shift keying(M-DPSK) as a rate-1 recursive inner encoder and a convolu-tional outer encoder [7]–[9]. These studies confirmed that codedDPSK, whether coherently or noncoherently demodulated usingan iterative decoder, performs better than coded coherent PSKat sufficiently high SNRs.

In this letter, we are interested in the iterative decoding ofa coded, DPSK-modulated, direct-sequence CDMA multiusersystem, because of the ease with which an absolutely mod-ulated PSK transmitter can be converted into a differentiallymodulated one. While the performance improvement obtainablewith DPSK over coherent PSK has previously been observedin [7], iterative multiuser detection with differential modula-tion has not been studied so far. Our main contribution is the

Manuscript received April 10, 2002; revised October 17, 2002 and December23, 2002; accepted January 31, 2003. The editor coordinating the review of thisletter and approving it for publication is X. Wang. This work was supported inpart by the Nortel Institute for Telecommunications and Bell Canada UniversityLaboratories.

Y. Wu is with the Centre for Wireless Communications, Oulu, Finland FIN-90014.

T. J. Lim is with the Department of Electrical and Computer Engineering,University of Toronto, Canada M5S 3G4 (e-mail: limtj@comm.utoronto.ca).

Digital Object Identifier 10.1109/TWC.2003.821218

derivation of a noncoherent CDMA soft-in soft-out (SISO) de-coder, and its integration into an iterative receiver with threeSISO component decoders: one each for the convolutional code,the CDMA channel, and the differential modulator. For compar-ison, well-known coherent CDMA SISO decoders are also dis-cussed briefly and simulated.

The rest of the letter is organized as follows. In Section II, weintroduce the convolutionally and differentially encoded CDMAmodel, including the transmitter and the channel. This is fol-lowed in Section III by a description of the iterative decodingprocess and details on the new receivers. The results of an in-vestigation into the performance of the systems based on com-puter simulation, their interpretation, and related discussion aregiven in Section IV. The results show that both receivers arecapable of very good performance, with only a slight nonco-herence penalty. Furthermore, the power of this class of systemis illustrated as they significantly outperform systems of abso-lutely encoded CDMA with coherent detection.

Finally, it should be noted that our approach can also be ap-plied to any interference-limited system, whether using spreadspectrum signalling or not.

II. SYSTEM MODEL

We consider a synchronous coded CDMA system of userswith binary DPSK modulation, signaling through an AWGNchannel. The block diagram of the transmitter of such a systemis shown in Fig. 1. The binary information data for user

, , are convolutionally encoded with code rate, and the code bits are block-interleaved.

The interleaved code bits of the th userare passed through a binary differential encoder to give1

(1)

Without loss of generality, the differential encoder is assumed tostart at the reference symbol . Each symbolis modulated by a spreading waveform , and transmittedthrough an AWGN channel. The received baseband equivalentcomplex signal can, therefore, be expressed as

(2)

where is the number of data bits per user per frame,is the symbol interval, and denote,respectively, the channel coefficients and normalized signalingwaveform of the th user, and is a zero-mean, circularlysymmetric, complex white Gaussian noise process with powerspectral density . It is assumed that is supported only on

1For simplicity, we consider binary modulation butM -ary modulation entailsonly slight and insignificant modifications to the proposed algorithms.

1536-1276/04$20.00 © 2004 IEEE

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 2, MARCH 2004 349

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

the interval and has unit energy. The phase error andamplitude are assumed to remain constant over one frame.

For the synchronous case, a sufficient statistic for demodu-lating the th code bits of the users is given by the -vector

whose th component is the output of a filter matched toin the th code bit interval, and

(3)

where denotes the normalized cross-correlation ma-trix of the signal set , i.e.,

, ,, and is a

white Gaussian noise vector sequence, independent of .

III. RECEIVER STRUCTURES AND ALGORITHMS

Turbo decoding for the system of Fig. 1. would split theoverall decoding task into three stages, with each stage de-coding one component code, respectively. The three componentdecoders have the same objectives regardless of whether thereceiver is coherent or noncoherent, and are easily understoodwith reference to Fig. 1. In each iteration

The CDMA Decoder provides inputs, with less multi-access interference (MAI) than the signals in the previousiteration, to a bank of DPSK decoders. Improved MAI re-moval is possible at each iteration because of judicious useof soft information from the previous iteration provided bythe DPSK decoder.The DPSK Decoder accepts inputs from the MAI decoderand the convolutional decoder in order to generate a betterestimate of the code symbols , .The Convolutional Decoder uses the log likelihood ratios(LLRs) of the code symbols, which come from the DPSKdecoder (after deinterleaving) in order to update LLR’s ofthe code symbols and information bits .

A. Noncoherent Receiver

1) CDMA Decoder: This receiver does not know the ma-trix and so its CDMA component decoder cannot performsymbol-by-symbol MAP decoding, nor can it use the subop-timal soft interference cancellation technique of [10]. Howeverwe can assume knowledge of the code correlation matrix andhence obtain the decorrelator output [11]

(4)

where is a Gaussian random vector with co-variance matrix .

While the th decorrelator output is MAI-free andtherefore a good candidate for the DPSK decoder input, thedecorrelator does not rely on soft information from any of theother component decoders and hence does not have the abilityto update its output with each iteration. This problem can besolved by noting that

(5)

where . The first equation fol-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)

in the th iteration, whereis the matrix of tentative decisions

of in the th iteration. These can be expressed interms of the LLR’s , obtained from the SISODPSK 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 thiteration outputs

(8)

which is then passed to the SISO DPSK component decoder.The SISO DPSK decoder requires the conditional variance

of , the th component of . Using (1), (3) and (4), the

covariance matrix of conditioned on , andcan be found to be

(9)

Furthermore, if , we have, where is a zero-mean, jointly Gaussian vector with

covariance matrix , and therefore

(10)

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

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

Equations (9) and (10) provide all the necessary information forthe SISO DPSK decoder.

2) SISO DPSK Decoder: The noncoherent SISO DPSK de-coder used in the proposed receiver is identical to the one in [7,Sec. II-B], so we will only provide a brief description here. Theinput to the DPSK decoder is , whose probability densityfunction (pdf) conditioned on user ’s channel and all codebits , , assuming complete MAI cancellation,is

(11)

where is the diagonal element of , and .The dependence on the unknown in (11) can be

removed by integrating over the pdf of to obtain. Letting denote the vector

, and be the vector ,we can find as a product ofterms, based on the independence of the additive Gaussiannoise in over time. Finally, by Bayes’ Rule we can obtainan expression for the conditional pdfnecessary in a MAP decoding algorithm (such as [12]).

However, computing requiresoperations, and it is desirable to reduce the complexity of thisoperation by making the approximation [7]

(12)

with

(13)

where is a quantity that does not depend onand is the modified Bessel function of order

zero. From (13) we know that there are different values ofwith respect to the set of all binary -tuples

. As such, we can construct a trellis struc-ture with the state at time as

, the state transition matrixand the “output symbol” generation matrix

. If we set the window size to besmall, then the difference in complexity between the exact ex-pression and its approximation is considerable.

With this trellis structure, branch metrics (12) and extrinsicinformation from the SISO convolutional decoder,we can use the standard APP algorithm described in [13] toimplement a noncoherent SISO DPSK decoder for each user.While the LLR of the data bits entering the differential encoder,i.e., , is fed back to the soft MAI canceller, the extrinsicinformation is fed forward to user s SISO convolu-tional decoder.

The block diagram of the noncoherent iterative multiuser re-ceiver is given in Fig. 2.

B. Coherent Receivers

Coherent receivers (which have perfect phase information onall users’ signals) can be easily derived from material in theliterature, and we will compare the performance of noncoherentdetectors with these. Briefly,

1) The CDMA Decoder can either be a full-complexity MAPdecoder (see [14, Sect. IV-A]) or a low-complexity inter-ference canceller [10];

2) The DPSK Decoder is described in [7, Sect. III-D] and isa straightforward application of symbol-by-symbol MAPdecoding to the DPSK trellis;

3) The Convolutional Decoder is a symbol-by-symbol MAPdecoder.

IV. SIMULATION RESULTS AND DISCUSSION

We simulated the performance of iterative receivers in ab-solutely encoded, coherently demodulated multiuser systems,

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 2, MARCH 2004 351

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510 -5

10-4

10 -3

10 -2

10 -1

100

Eb/No(dB)

Bit

Err

or R

ate

(BE

R)

iteration 1

2

3

4

5

Low-Complexity Rx'er, 6 iterationsAbsolutely Encoded, 5 iterationsFull-Complexity Rx'er, 1--5 iterations

Fig. 3. Performance of coherently demodulated iterative multiuser detectorwith � = 0:3, K = 4 and equal received signal power for all users.

and compared that with the proposed iterative receivers for dif-ferentially encoded ones. Unless otherwise indicated, the fol-lowing system parameters were used in the simulation. Thereare four users in the system, where all users have equal powerand employ the same rate-1/2 convolutional code with gener-ator . Each user uses its own random interleaver. Thesame set of interleavers are used for all simulations. The blocksize of the information bits for each user is . The non-coherent SISO DPSK decoder uses in the algorithm ofSection III-A-2. All simulations were performed on the -sym-metric channel [5] that is characterized by the spreading-codecorrelation matrix

...(14)

where is the cross-correlation parameter.The bit error rate (BER) curves obtained with coherent de-

modulation and DPSK are plotted in Fig. 3 for . Turbodetection of the absolutely encoded (BPSK) multiuser systemafter five iterations was simulated as a baseline for compar-ison.2 As can be seen, the performance of the full-complexitydetector significantly improves with the first few iterations, al-though gains appear to be marginal after four iterations. Moresignificantly, both low- and full-complexity detectors with dif-ferential encoding outperform the absolutely encoded system,by about 1.8 dB at a BER of .

These results, at first glance, may seem quite surprising be-cause differential encoding is typically associated with degra-dation in system performance. However, it is entirely consistentwith [7] and shows just how substantial interleaver gain can be.Comparing Fig. 3 to Fig. 3 in [7], we see that the proposed it-erative multiuser system has a BER after five iterations roughlyequivalent to the BER of a single-user iterative DPSK detectorafter three iterations. At a BER of , the proposed receiver

2This is essentially the same performance as the G(23; 35) convolutionalcode in a Gaussian channel.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510 -5

10 -4

10 -3

10 -2

10 -1

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iteration 1

2

3

4

5

coherent demodulation

Eb/No(dB)

Bit

Err

or R

ate

(BE

R)

Absolutely Encoded, 5 iterationsNon-Coherent Rx'er, Diff. Encoding

Fig. 4. Performance of noncoherently demodulated iterative multiuserdetector with � = 0:3, K = 4 and equal received signal power for all users.

0 1 2 3 4 5 6 7 810 -5

10 -4

10 -3

10 -2

10 -1

100

Non-coherent

Coherent

Eb/No(dB)

Bit

Err

or R

ate

(BE

R)

Low-Complexity Rx'er, 6 iterationsAbsolutely Encoded, 5 iterationsDifferential Encoding, 5 iterations

Fig. 5. Performance with � = 0:7, K = 4 and equal powers for all users.

after five iterations has an approximately 1 dB loss compared tothe single-user DPSK detector after 20 iterations.

The same relative performance found with coherent demod-ulation also extends to noncoherent demodulation. Fig. 4 showsthe performance of the proposed noncoherent receiver for thefirst five iterations with . Also shown are the BER curvesfor the absolutely encoded system, and the differentially en-coded, coherently demodulated scheme, all after five iterations.There is only a 0.8 dB noncoherence penalty after five iterationsat . However, the gain of the noncoherent receiverover absolute encoding is still 1.0 dB at .

When the cross-correlation increases to , results areshown in Fig. 5. In this example, results are less encouraging:The differentially encoded system with the full-complexity co-herent receiver has the same performance as the absolutely en-coded one at , after five iterations. The low-com-plexity coherent receiver with 6 iterations suffers a 2 dB losscompared with the full-complexity system at . Forboth the full- and the low-complexity receivers, performance isfar from the single-user DPSK case with coherent demodulation

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

shown in Fig. 2 of [7]. This is a manifestation of the well-knownphenomenon (see, e.g., [2]) that turbo multiuser detection doesnot always converge to single-user performance, especially athigh interference levels.

V. CONCLUSION

In this letter, we studied turbo detection for convolutionallyand differentially encoded CDMA systems, with either coherentor noncoherent demodulation. The proposed detectors are iter-ative decoders for a three-stage serially concatenated system ofconvolutional code, differential code and CDMA channel.

The noncoherent SISO multiuser detector and the low-complexity coherent SISO multiuser detector are based onMAI cancellation, while the full-complexity coherent SISOmultiuser detector uses a maximum a posteriori (MAP)algorithm. Simulation results show that coded, differentiallymodulated CDMA with interleaving can perform better thanits coherently modulated (or absolutely encoded) counterpartwhen full complexity coherent demodulation is used. Withnoncoherent demodulation, the performance gain is smaller,but the interleaver gain obtained from the recursive nature ofdifferential encoding is still evident.

Finally, we note that recently, Shi and Schlegel [2] also at-tempted to introduce a recursive code before the spreading op-eration in a CDMA channel, to obtain turbo gain. Our methodarose from the same idea, but is much simpler to implement. Itwould be interesting to compare the relative advantages and dis-advantages of these two techniques.

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