Multiple Differential Detection Of Parallel ... - Semantic Scholar

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 1998

265

Multiple Differential Detection of Parallel Concatenated Convolutional (Turbo) Codes in Correlated Fast Rayleigh Fading Ian D. Marsland, Student Member, IEEE, and P. Takis Mathiopoulos, Senior Member, IEEE

Abstract— A new technique for iterative decoding of parallel concatenated convolutional (turbo) codes (PCCC’s) for the correlated fast Rayleigh fading channel is proposed and evaluated. This technique is based upon the use of a multiple differential detector (MDD) receiver structure which exploits the statistical characteristics of the fading process to overcome the effects of the rapid phase and amplitude variations. Since traditional MDD receivers cannot be used with PCCC’s because they do not produce soft output and are not compatible with channel interleaving, a novel MDD receiver structure is derived which overcomes these shortfalls. In addition, with careful use of extrinsic information related to the a posteriori probability distribution function of the transmitted symbols, the receiver is designed in such a fashion as to allow channel estimation to improve with each iteration. Evaluation of the proposed receiver by means of computer simulation has shown dramatic performance improvements in fast Rayleigh fading channels as compared to long constraint-length conventional convolutional codes using both single and traditional MDD receiver structures. Index Terms—Differential phase shift keying, fading channels, MAP estimation, multiple-symbol differential detection, timevarying channels, turbo codes.

I. INTRODUCTION

B

ECAUSE of the impressive near-Shannon-limit errorcorrection capability achieved by iterative decoding of parallel concatenated convolutional (turbo) codes (PCCC’s) in an additive white Gaussian noise (AWGN) channel [1], [2], researchers have begun to investigate their application for digital communication over fading channels. Early research on this topic is based upon simplifying assumptions for the channel model, including perfect knowledge of the phase and amplitude of the fading process, and uncorrelated fading (see, for example, [3]–[5]). More recently, in [6], the performance of PCCC’s in correlated Rayleigh fading channels was analyzed. In this work, cases where the amplitude of the fading process is both known and unknown were investigated. However, it was still assumed that its phase is perfectly known and constant over several symbol intervals. Typically, this assumption is

valid only for very slow fading. In fast fading, however, the phase of the fading process is changing rapidly, and therefore very difficult to track [7]. As there is no universally accepted definition of fast fading, for the purpose of this paper, 1 we consider fast fading to occur when the product is greater than about 0.1. Although for most existing mobile and/or cellular telecommunication systems (e.g., GSM [8]) fast fading does not occur, such products might arise in other mobile applications, such as those using low bit rate signals and/or higher carrier frequency bands (e.g., 20–30 GHz). More importantly, from a theoretical point of view, reliable communication over fast fading channels presents a challenging and interesting research problem. In [9] and [10], we considered the performance of PCCC’s in correlated fast Rayleigh fading channels, where it was shown that, for a product of up to 0.1, excellent biterror rates are possible with the use of a single differential detector and a channel interleaver. However, it was also shown that if the fading rate is even faster (i.e., ), reliable communication is not possible because of the single differential detector’s inability to track the very rapid phase variations. It should be noted that the performance could not be improved by merely increasing the number of iterations performed by the turbo decoder. Performance of differentially detected systems can be improved, however, by using a multiple differential detector (MDD)2 receiver structure (see, for example, [11], [13], [14], [16] and [17] for the AWGN channel, and [12] and [15]3 for the fading channel). The resulting decoders take into full consideration the correlation of the fading process. Two benefits are realized through the use of more than one differential detector. The first is that substantially lower error floors can be achieved because of the improved tracking of the phase variations. The second is a reduction in the noncoherence penalty, i.e., the increase in the signal-to-noise ratio (SNR) required for a noncoherent receiver to match the performance of an ideal coherent receiver which uses 1

Manuscript received September 11, 1996; revised April 24, 1997 and August 18, 1997. This work was supported in part by the Natural Science and Engineering Research Council (NSERC) of Canada, through a postgraduate scholarship and under Grant OGP-443212, a BC Advanced Systems Institute (ASI) Fellowship, and a Killam Faculty Research Fellowship. The authors are with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, B.C., V6T 1Z4 Canada. Publisher Item Identifier S 0733-8716(98)00166-8.

B is the Doppler spread of the channel and T is the symbol duration.

2 The

hardware structure of MDD receivers consists of a combination of more than one distinct differential detector with elements of time delay T equal to progressively increasing multiples of the symbol duration [11]–[15]. The number of distinct differential detectors will be denoted as Z . In other papers on the same subject which have appeared after the publication of [11], [12], the same structure is denoted as multiple-symbol differential detection receivers (see, for example, [14]). 3 A more comprehensive list of related references can be found in [15].

0733–8716/98$10.00  1998 IEEE

266


Fig. 1. Block diagram of the encoder.

perfect knowledge of the fading process. However, existing MDD receiver structures, which are designed for decoding conventional convolutional codes, are not appropriate for iterative decoding since they do not produce the necessary soft-output extrinsic information. Furthermore, they are incompatible with interleaving, an important component for achieving good performance in fading4 channels. To combine the advantages of multiple differential detection with the errorcorrection capabilities of PCCC’s, a novel decoder structure is derived and presented in this paper. This receiver yields dramatic performance improvements over single differential detection of PCCC’s and multiple differential detection of convolutional codes. In Section II of this paper, the components of the decoder are described, including the parallel concatenated convolutional encoder and the channel interleaver. The correlated Rayleigh fast fading channel model with which the performance of the system is investigated is also presented. Section III contains a description of the new decoder, along with a theoretical derivation. Performance evaluation results and related discussion can be found in Section IV, where comparisons with conventional convolutional codes are made in terms of both the bit- and block-error rates.

Encoding and transmission of data is carried out on blocks . The bits of binary information digits, each of length are assumed to be statistically independent, and each block shall be denoted by , where the notation refers to the sequence . The encoder, as shown in Fig. 1, is a slightly modified version of the one originally proposed by Berrou et al. [1]. Two identical recursive systematic convolutional (RSC) encoders are connected in parallel, separated by a parallel concatenation interleaver (PCI). A “simile odd–even” helical interleaver is used, which ensures that both RSC encoders can be simultaneously driven to the zero state with a common tail at the end of each block [18]. The length of the tail is equal to the constraint length of the RSC codes. To achieve an overall coding rate of 1/2, puncturing is used in such a fashion that only every second parity bit produced by each constituent encoder is transmitted. The resulting concatenated code can be considered to be composed of two constituent codes, one generated from

the noninterleaved information sequence (the noninterleaved code), and one from the interleaved sequence (the interleaved code). By modifying the approach suggested in [19], a parallel concatenation deinterleaver (PCI ) is connected to the output of the second RSC encoder, reversing the mapping of the helical interleaver. This ensures that the parity bit transmitted with each systematic bit is the same one produced by one of the RSC encoders when was encoded. This modification has the advantages that only one signal mapper is needed, and the deinterleaver operates on bits instead of symbols. For transmission, Gray mapping is used to convert each bit pair to a point in the quadrature phase-shift keying (QPSK) signal constellation, so one information bit is transmitted with each channel use. To provide additional protection against fading, a channel interleaver (CI) is used to reorder the transmission of the symbols, as suggested in [6]. As is well known, block interleavers provide a simple and effective means for breaking the correlation encountered in fading channels, and can be used for the CI since they use a different mapping from the helical interleaver used for the PCI. Throughout this paper, the interleaved symbols will be denoted by , where , with the number of information bits and the number of tail bits.5 The two parameters of a block interleaver are its depth and width. The depth determines the number of transmitted symbols between consecutively encoded bits, while the width determines the number of bits encoded between two consecutive transmitted symbols. A large depth is desirable to break up long fades, but the width must be kept sufficiently large, at least as long as the constraint length of the RSC codes. Performance of the decoder is improved if the parity bits transmitted in two successive symbol intervals are produced by different RSC encoders. To fulfill this objective, it is necessary that the channel interleaver map symbols from even-numbered positions in the uninterleaved sequence to even-numbered positions in the interleaved sequence, and symbols from odd positions to odd positions. This is achieved by choosing only odd values for the depth and width of the block [20]. To accommodate messages of arbitrary block length, it may be necessary to leave part of the interleaver block unused. A subblock which is two columns wide and an even number of rows deep must be discarded to maintain the above described odd–even mapping. As an illustrative example, consider the block interleaver shown in

4 Here, and for the remainder of this paper, unless otherwise indicated, the term “fading” refers to correlated fast Rayleigh fading.

5 Since a QPSK type signal and a rate-1/2 code are being used, each information or tail bit corresponds to one symbol.

II. ENCODER AND CHANNEL MODEL DESCRIPTION

MARSLAND AND MATHIOPOULOS: MULTIPLE DIFFERENTIAL DETECTION OF PCCC’S

267

[22], [23], in which , where is the zero-order Bessel function of the first kind, is the Doppler spread, and is the symbol duration. With this channel model, the fading process is correlated in time, with the amplitude having a Rayleigh distribution and the phase uniformly distributed over . III. DERIVATION AND DESCRIPTION OF THE NEW DECODER

Fig. 2. Example of a 41

2 23 odd–even block channel interleaver.

Fig. 2. This interleaver is designed to work on blocks of 934 symbols, which corresponds to 930 information bits6 and 4 tail bits, a size used for many of the simulations presented in Section IV. It has a nominal depth of 41 and width of 23, but since 41 23 943 is slightly larger than the required size of 934, the interleaver is reduced in size by leaving 12 symbol positions unused. Thus, the effective interleaver size is 931 symbols. Since this effective size is less than the desired size of 934, the remaining three symbols are left uninterleaved and appended to the end of the interleaved sequence. In general, for other odd–even block interleavers of different sizes, up to three symbols will not fit, and must be left uninterleaved. Finally, to assist the noncoherent detection used by the receiver, the interleaved symbols are differentially encoded prior to transmission over the channel. The differential QPSK (DQPSK) symbols are given by [21] (1) set equal to one. with the reference symbol The channel is represented by a discrete-time low-passequivalent model with correlated frequency-nonselective Rayleigh fading and AWGN. Samples of the received signal are given by

In general, decoding of PCCC’s is performed in an iterative fashion using two decoding units [1]. The noninterleaved code is decoded first, then information produced by this decoding is passed to the second decoding unit where decoding of the interleaved code is performed. Information from the second decoding is fed back to the first decoding unit, and the decoding process is repeated. It is expected that the information passed between the decoding units will improve the reliability of each decoding operation. After some fixed number of iterations, decoding terminates, and estimates of the a posteriori probabilities (APP), , are passed to a decision device which selects the value of each information bit that was most likely to have been transmitted given the received samples . This general approach is implemented in our new decoder, which is illustrated by the block diagram of Fig. 3. The received samples are used by the primary metric calculation unit (MCU) to generate a set of MDD metrics which reflect the likelihood of each sample being received given some set of transmitted symbols, as will be described in Section IIIA. Using a novel algorithm described in Section III-B, the secondary MCU’s transform these metrics into a form appropriate for use by the constituent decoders (CD’s), which in turn calculate the a posteriori probabilities using an algorithm developed by Bahl, Cocke, Jelinek, and Raviv [24] (referred to as the BCJR or MAP algorithm) that has been modified for use with PCCC’s [1], [19]. In Section III-C, it is shown that the modified BCJR algorithm is appropriate for use with noncoherently detected signals transmitted over fading channels. In Section III-D, the information produced by the CD’s that is passed to the next decoding unit is described. It is important to note that by including the secondary MCU’s within the iterative loop and feeding them with information produced by the CD’s, channel estimation, as well as the bit-error rate, improves with each iteration.

(2) A. Review of Metrics represents the AWGN, which is modeled by a where zero-mean complex Gaussian discrete random process with autocorrelation for all , where is the expected value of , denotes complex conis the single-sided noise power spectral density, jugation, and is the Kronecker delta function. The fading is also modeled by a zero-mean complex Gaussian discrete random process with the land-mobile correlation fading model 6 Since there are restrictions on the dimensions of “simile odd–even” helical interleavers, only certain block sizes can be considered. A block size of 930 30 information bits was selected since it corresponds to the size of a 31 helical interleaver.

2

When using coherent detection with perfect knowledge of the fading process at the receiver, there is no need to differentially encode the signal prior to transmission, and decisions are traditionally based on the metrics

(3) where is the conditional probability density function (pdf) of the th received sample given that the th transmitted symbol is .

268


Fig. 3. Block diagram of decoder structure.

On the other hand, however, for noncoherent systems, differentially encoded signals are often used. For differentially detecting convolutionally encoded signals with the Viterbi algorithm, it was shown in [12] and [15] that the optimal maximum likelihood sequence estimation (MLSE) metrics for correlated Rayleigh fading are

To reduce the complexity of the MLSE metrics, it is convenient to consider the approximation that the fading is correlated over only consecutive symbols (i.e., assume for all for some small value of (e.g., ). Under this approximation, it has been shown in [15] that

(4) denotes the magnitude of , are the th-order where , and is the correlinear prediction coefficients for sponding minimum mean-squared prediction error (MMSPE). is used as an estimate of In (4), , and depends not only on the prior received samples , but also on the transmitted symbols . Although these metrics provide the most accurate estimate of the fading in terms of MMSPE, they require an inordinate amount of work to compute for large (e.g., ) since they must be computed for all possible realizations of .

(5) where the linear prediction coefficients solving

.. .

.. .

..

.

are given by

.. .


.. . and the MMSPE,

(6)

.. .

269

and is computed in the previous step of the recursion, and , the a priori probability of symbol being transmitted. The independence of and is again a product of channel interleaving. Since the quantities

, is given by of

are normalized, avoiding direct calculation , so

(7) As previously mentioned, receivers employing these metrics are referred to as multiple differential detectors (MDD) [12], [15] since they can be implemented by employing a combination of distinct differential detectors, each with a time delay of progressively increasing multiples of the symbol duration . It is the purpose of the primary MCU in Fig. 3 to compute these metrics for all possible realizations of , for each . B. New MDD Metrics for PCCC’s in Fading Channels An efficient algorithm exploiting the metrics of (5) has been developed for decoding convolutional codes by means of the Viterbi algorithm [15], [25]. However, that algorithm is not appropriate for decoding PCCC’s since it is not compatible for use with interleaving. To use the MDD metrics given by (5) to find the a posteriori probabilities with the BCJR algorithm, it is first necessary to remove their dependence on . This is readily accomplished by using

(8) If the width of the channel interleaver is larger than the is essentially constraint length of the RSC codes, independent of , so

(11)

and the new metrics are (12) These new metrics are computed by the secondary MCU’s of the receiver shown in Fig. 3. Since these metrics depend on only one transmitted symbol, they can be used in systems employing interleaving. They are also appropriate for use with the modified BCJR algorithm, as the following discussion shows. C. The Modified BCJR for Fading Channels The constituent decoders of the receiver shown in Fig. 3 generate the APP of each information bit, based on either the interleaved or noninterleaved RSC code, depending on which decoding unit the CD is a part of. A modified version of the BCJR algorithm is used for this purpose in traditional PCCC decoders for the AWGN channel [1], [26], [27]. As is shown in this section, the same algorithm, but with the metrics described above, can be used for differential detection in fading channels. For either constituent code, let be the information bits in the order in which they were encoded, and let be the received samples in the same order, so that bit is received . The a posteriori probabilities of bit can be in sample expressed as (13)

(9) The calculation of recursive fashion with

can be performed in a

is the state of the constituent RSC encoder immediwhere ately before is encoded. In [24], it is shown that by applying Baye’s rule, (13) can be calculated with

(14)

(10) However, is given by (5),

and

Note that

cannot be simplified to as for AWGN channels because some

270


information regarding the fading affecting is carried in . However, since no information regarding the value of is stored in , , the a priori probability of . Also

where is the parity bit transmitted along with . If this parity bit was not punctured, i.e., it was produced by the RSC encoder corresponding to the constituent code being decoded, then

if PB otherwise

(15) note that To simplify (15), for depend although the information bits transmitted in and not or , the fading in only on state depends on . Prediction of the fading is more reliable if and are used, but the benefits of this information are quite insignificant since channel interleaving is used, and can therefore be ignored, simplifying the decoder structure. In addition, since state transitions depend only on the code under consideration

if NS otherwise is the state reached from state where NS input . Therefore

NS

(16) with

(17)

By defining (18) and

(19)

(22)

is the parity bit generated when encoding where PB from state . With traditional PCCC’s for the AWGN channel using coherently detected QPSK signals, the systematic bits are transmitted on the in-phase channel ( channel) and the parity bits are transmitted on the quadrature-phase channel ( channel) [1]. Traditional PCCC decoders can therefore deal with the effects of punctured parity bits by supplying values of zero in place of the channel output wherever parity bits have been punctured. However, this approach fails in noncoherently detected systems because the phase uncertainty implies that the parity bit is not isolated to only the channel. It is straightforward to show that an equivalent alternative is to average the metrics over the unknown bit by using in (21) when parity bit has been punctured. In place of in (21), it is convenient to use the new metrics described in Section III-B, namely, as given by (12), where SM is the QPSK symbol corresponding to the bit pair . This substitution is not entirely valid since depends on the received samples corresponding to the previously encoded bits, whereas depends on the received samples corresponding to the previously transmitted symbols. This substitution is nonetheless physically realizable by requiring that all of the samples for a given block be received before decoding commences. Of course, due to this substitution, the resulting decoder will be suboptimal. However, as shown by the simulation results presented in Section IV, this suboptimality should not be a serious concern since this receiver is capable of providing very significant performance improvements over other MDD receivers designed for fading channels. Therefore, we shall assume

(14) can be expressed as SM

NS

(20)

To evaluate (20), note that

(21)

(23)

used in (20) can be computed in a Values for recursive fashion. Starting with and , for is generated from by


(24) Furthermore, since

(25) Similarly, Starting with , for by

is computed recursively, but in reverse. and is computed from

271

a fashion that the reliability of the a posteriori probabilities improves with each iteration [1]. Information regarding the value of information bit can be considered to come from three sources: 1) the a priori pdf of , 2) the systematic information based directly on the received sample containing , and 3) from information about as reflected by the coding process on the other received samples. This third component, the “extrinsic” information, is what is passed from one decoding unit to the next, where it is used in lieu of the a priori probabilities , as was suggested by Robertson in [26]. To compute the extrinsic information from , it is necessary to remove, by division, the a priori and systematic information. Since the systematic information is embedded in the received metric SM , it can be extracted as SM . The extrinsic information is, therefore (27) SM

NS

(26)

To use in place of in the next decoding unit, it is necessary to have , so instead we normalize and use what is shown in (28), found at the bottom of the page, as the extrinsic information. Just as the extrinsic information contained in can be used in place of in the next decoding unit to improve system reliability, it is desirable to use similar information found in in place of the a priori symbol probabilities used in (11) to compute the new metrics. Observe what is shown in (29) at the bottom of the next page. As was done for calculating , the extrinsic information is found by dividing by the a priori and systematic information, namely, and SM , respectively. Normalizing with respect to and leaves

Equation (20), in conjunction with (23)–(26), defines the modified BCJR algorithm used by the constituent decoders to find the a posteriori probabilities for fading channels. The CD’s are also used to generate the information passed between decoding units, the nature of which is described in the following.

NS NS (30)

D. The Extrinsic Information The strength of turbo decoding lies in its ability to pass information from one decoding unit on to the next in such

which is used by the next decoding stage in place of SM .

SM NS SM (28) SM NS SM

272


As a result of puncturing, cannot be determined for symbols with punctured parity bits. For these symbols, the values of produced in the previous decoding unit are passed directly on to the next unit. Although these values are less reliable than the ones produced by the current unit (and are both produced and used by decoding units operating on the same constituent code), the odd–even nature of the interleavers ensures that these values are never used in place of when computing the metrics for some . This is important because the fading affecting received sample is more closely correlated with the fading in than with the other samples. IV. PERFORMANCE EVALUATION RESULTS AND DISCUSSION To evaluate the performance of the proposed coding scheme and decoder structure, we ran Monte Carlo simulations. Simulations with varying numbers of iterations ( ) were carried out over a range of signal-to-noise ratios for various fading rates ( products) and block sizes ( ). The system was implemented using a parallel concatenated convolutional encoder with generator (37, 21) for both constituent RSC encoders. An odd–even block channel interleaver was employed, and land–mobile fading was used to simulate the mobile channel. The decoders were implemented as described in Section III and their performance is compared with: 1) a conventional 256-state convolutional code7 (CC) with single differential detection and channel interleaving, 2) a conventional 64-state CC with MDD but without channel interleaving, and 3) an ideal coherent receiver using the same PCCC and channel interleaver. Although the third comparison is unrealistic since the fading process, especially for high products, is never perfectly known at the receiver, the performance of this receiver might be viewed as the best possible for the coding scheme. For , using finite length blocks of information bits and a channel interleaver with a nominal size of 41 23, fairly good performance, as illustrated in Fig. 4, is possible with the use of PCCC’s and differential detectors. The results improve with each iteration, but gains ). In fact, are negligible after about five iterations ( with all of the simulations carried out in the preparation of this paper, it was evident that exceeding five iterations was of little benefit, regardless of the SNR, the product, or the 7 The number of states of the conventional convolutional codes was chosen to be large so that the complexities of the resulting decoder structures are roughly comparable.

SM

=

Fig. 4. Bit-error rate performance for Rayleigh fading with Z 2 differential detection, with BT = 0:01; Nb = 930 bits, and 41 23 channel interleaving. For comparison, the performance of a 256-state convolutional code (CC) with the same channel interleaver and single differential detection is also shown.

2

number of differential detectors employed. For comparison, the performance of a conventional 256-state CC, with generator (561, 753) and the same channel interleaver, using single differential detection and a Viterbi decoder, is also shown in the same figure. Only two iterations are needed for the PCCC to perform better than the CC, and after five iterations, the PCCC is about 1.3 dB better at a bit-error rate (BER) of . This suggests that PCCC’s do not provide a significant advantage over classical CC’s in noncoherent Rayleigh fading with 0.01. It appears that the same observation is valid for . It is also interesting to compare the performance of the noncoherent receiver with the performance of the ideal coherent receiver. As can be seen in Fig. 5, at and for a BER of , after iterations the noncoherent penalty is 3.7 dB for a single differential detector8 (i.e., ). The use of a second differential detector ( ) reduces the noncoherence ) penalty to 3.2 dB, while three differential detectors ( have a noncoherence penalty of 2.9 dB. The advantages of using more than one differential detector in conjunction with PCCC’s and the iterative decoder are clear. As illustrated in Fig. 6, the benefits of multiple differential detection are even more pronounced at higher products. For example, at , with , performance is limited by an error floor at a BER of about , which 8 Note

that for

Z

= 1, the receiver investigated in [10] results.

NS (29) SM

NS


Fig. 5. Comparison of the performance with different number (Z ) of differential detectors, with BT 0:01, Nb = 930 bits, and 41 23 channel interleaving, after five iterations. Also shown is the performance of an ideal coherent detector, with the same PCCC and interleavers, after five iterations.

=

2

Fig. 6. Comparison of the performance with different number (Z ) of differential detectors, with BT = 0:125, Nb = 930 bits, and 41 23 channel interleaving, after five iterations. Also shown is the performance of a 256-state convolutional code (cc) with the same channel interleaver and single differential detection, and the performance of a 64-state CC with a traditional Z = 3 MDD receiver [15].

2

arises from the rapid phase variation of the fading process. By using multiple differential detectors, this error floor can be . At a BER of , performance reduced to well below instead gains of about 3.5 dB are obtained by using a decoder. At these higher products, the of a proposed decoders significantly out perform the 256-state CC with a single differential detector, which also suffers greatly from these phase variations. The proposed decoders also significantly exceed the already good performance of other sophisticated receiver structures, such as the MLSE decoders derived for fading channels [15], which support multiple differential detectors, but are not compatible with channel interleaving. For example, versus the decoder presented in [15] and a 64-state CC, the proposed decoder improves with . the performance by about 8 dB at a BER of product is increased to 0.2, a single differential If the ) is unable to make any reliable decisions detector (

273

Fig. 7. Comparison of the performance with different number of differential detectors, with BT = 0:2, Nb = 930 bits, and 41 23 channel interleaving, after five iterations.

2

Fig. 8. Comparison of the bit-error rate performance with different block sizes with BT = 0:01 and Z = 2, after five iterations.

about the transmitted bits, as illustrated in Fig. 7. Reliable communication is possible with , but a significant ) can savings in the SNR (of more than 3 dB at a BER of instead. Using more than three be realized by using ) provides additional, albeit differential detectors (i.e., minor, savings. As in the case of the AWGN channel, it has been found that the block size used by the encoder has a significant impact on performance of the proposed system. As an illustrative product of 0.01 that example, it is shown in Fig. 8 for a increasing the block size leads to substantial improvements. products. This trend This trend also holds for higher occurs, in part, because of the increased size of the CI which is better at breaking up long fades, and in part because of the fact that the PCCC becomes more powerful as a result of the increased size of the PCI. Although the performance of iterative decoding of PCCC’s is neither substantially better than for CC’s nor at particularly close to ideal coherent detection in terms of the BER, PCCC’s are more outstanding in terms of the block error , the rate (BKER), as shown in Fig. 9. For a BKER of

274


Fig. 9. Comparison of the block-error rate performance with different num0:01, Nb = 930 bits, and ber (Z ) of differential detectors, with BT 41 23 channel interleaving, after five iterations. Also shown in performance of an ideal coherent detector with the same PCCC and interleavers, again after five iterations.

2

=

PCCC receiver with and is 2.2 dB better than the 256-state CC, and the noncoherence penalty is only 2.3 dB. The low block-error rates for iterative decoding of PCCC’s with the MDD receiver suggests that they are an attractive candidate for use in conjunction with a communication system employing automatic repeat request (ARQ) techniques [28]. V. CONCLUSIONS We have introduced and evaluated, with the aid of computer simulation, a new and promising technique for decoding parallel concatenated convolutional (turbo) codes. The novelty of this technique is based upon the adaptation of a multiple differential detector receiver structure to an iterative decoding environment, providing soft-output decisions and supporting channel interleaving. With this novel decoder, channel estimation improves with each iteration, resulting in increased reliability of the received data. The proposed technique dramatically improves the performance of digital communication systems operating in correlated fast Rayleigh fading channels, as compared to the already good performance of traditional single and multiple differential detection of conventional convolutional codes. ACKNOWLEDGMENT The authors are very appreciative of discussions held with Prof. S. Kallel, University of British Columbia, and Prof. D. Makrakis, University of Western Ontario. Dr. Kallel was able to provide assistance with techniques for dealing with puncturing and extracting the systematic information, while Dr. Makrakis provided assistance with the multiple differential detector receiver structures. REFERENCES [1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo codes,” in Proc. IEEE Int. Conf. Commun. (ICC’93), Geneva, Switzerland, May 1993, pp. 1064–1070.

[2] C. Berrou and A. Glavieux, “New optimal error correcting coding and decoding: Turbo codes,” IEEE Trans. Commun., vol. 44, pp. 1261–1271, Oct. 1996. [3] S. Le Goff, A. Glavieux, and C. Berrou, “Turbo-codes and high spectral efficiency modulation,” in Proc. IEEE Int. Conf. Commun. (ICC’94), New Orleans, LA, May 1994, pp. 645–649. [4] J. Hagenauer, P. Robertson, and L. Papke, “Iterative (‘TURBO’) decoding of systematic convolutional codes with the MAP and SOVA algorithms,” in Proc. ITG Conf. “Source and Channel Coding,” Munich, Germany, Oct. 1994, pp. 21–29. [5] P. Jung, M. Nasshan, and J. Blanz, “Application of turbo-codes to a CDMA mobile radio system using joint detection and antenna diversity,” in Proc. IEEE 44th Veh. Technol. Conf. (VTC’94), Stockholm, Sweden, June 1994, pp. 770–774. [6] E. K. Hall and S. G. Wilson, “Design and performance analysis of turbo codes on Rayleigh fading channels,” in Proc. Conf. Inform. Sci. Syst. (CISS’96), Princeton, NJ, Mar. 1996, pp. 43–48. [7] W. Y.-C. Lee, Mobile Communications Engineering. New York: McGraw-Hill, 1982. [8] M. Mouly and M.-B. Pautel, The GSM System for Mobile Communications, 1995. [9] I. D. Marsland and P. T. Mathiopoulos, “Differential detection of turbo codes for Rayleigh fast fading channels,” in Proc. Workshop Multimedia Wireless Commun. Computing (WMWCC’96), Victoria, B.C., Canada, Sept. 1996, pp. 11–12. , “Differential detection of turbo codes for Rayleigh fast fading [10] channels,” IEEE Commun. Lett., 1997, to be published. [11] D. Makrakis and P. T. Mathiopoulos, “Trellis coded noncoherent QAM: A new bandwidth and power efficient scheme,” in Proc. IEEE 39th Veh. Technol. Conf. (VTC’89), San Francisco, CA, May 1989, pp. 95–100. , “Optimal decoding in fading channels: A combined envelope, [12] multiple differential and coherent approach,” in Proc. IEEE Global Conf. Commun. (GLOBECOM’89), Dallas, TX, Nov. 1989, pp. 1551–1557. [13] D. Makrakis and K. Feher, “Optimal noncoherent detection of PSK signals,” IEE Electron. Lett., vol. 26, pp. 146–155, Mar. 1990. [14] D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun., vol. 38, pp. 300–308, Mar. 1990. [15] D. Makrakis, P. T. Mathiopoulos, and D. P. Bouras, “Optimal decoding of coded PSK and QAM signals in correlated fast fading channels and AWGN: A combined envelope, multiple differential and coherent approach,” IEEE Trans. Commun., vol. 42, pp. 63–75, Jan. 1994. [16] S. Samejima, K. Enomoto, and Y. Watabe, “Differential PSK system nonredundant error correction,” IEEE J. Select. Areas Commun., vol. SAC-1, pp. 78–81, Jan. 1983. [17] F. Edbauer, “Bit error rate of binary and quaternary DPSK signals with multiple differential feedback decision,” IEEE Trans. Commun., vol. 40, pp. 457–460, Mar. 1992. [18] A. S. Barbulescu and S. S. Pietrobon, “Terminating the trellis of turbocodes in the same state,” IEE Electron. Lett., vol. 31, pp. 22–23, Jan. 1995. [19] P. Robertson and T. Wörz, “A novel coded modulation scheme employing turbo codes,” in Proc. URSI & ITG Conf. “Kleinheubacher Tagung,” Kleinheubach, Germany, Oct. 1995. [20] A. S. Barbulescu and S. S. Pietrobon, “Interleaver design for turbo codes,” IEE Electron. Lett., vol. 30, pp. 2107–2108, Dec. 1994. [21] J. G. Proakis, Digital Communications, 2nd ed. New York: McGrawHill, 1989. [22] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [23] L. J. Mason, “Error probability evaluation for systems employing differential detection in a Rician fast fading environment and Gaussian noise,” IEEE Trans. Commun., vol. COM-35, pp. 39–46, Jan. 1987. [24] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, Mar. 1974. [25] D. P. Bouras, “Advanced noncoherent receivers for mobile fading channels,” Ph.D. dissertation, Univ. British Columbia, Vancouver, B.C., Canada, Apr. 1995. [26] P. Robertson, “Improving decoder and code structure of parallel concatenated recursive systematic (turbo) codes,” in Proc. IEEE Int. Conf. Universal Personal Commun. (ICUPC’94), San Diego, CA, Sept. 1994, pp. 1298–1303. [27] D. Divsalar and F. Pollara, “Turbo codes for PCS applications,” in Proc. IEEE Int. Conf. Commun. (ICC’95), Seattle, WA, June 1995, pp. 54–59. [28] K. R. Narayanan and G. L. Stüber, “A novel ARQ technique using the turbo coding principle,” IEEE Commun. Lett., vol. 1, pp. 49–51, Mar. 1997.


Ian D. Marsland (S’96) received the B.A.Sc. (Honors) degree in mathematics and engineering from Queen’s University, Kingston, Ont., Canada, in 1987 and the M.A.Sc. degree in electrical engineering from the University of British Columbia, Vancouver, Canada, in 1994. From 1987 to 1990, he was with Myrias Research Corporation, Edmonton, Canada, and CDP Communications, Inc., Toronto, Canada, where he worked as a Software Engineer. Since 1991, he has been a graduate student at the University of British Columbia, where he is currently studying toward the Ph.D. degree in electrical engineering and working as a Sessional Lecturer. Mr. Marsland is the recipient of an NSERC postgraduate scholarship and a BC Science Council GREAT award.

275

P. Takis Mathiopoulos (S’80–M’81–SM’94) graduated from the German High School of Athens (Deutsche Schule Athen—Dörpfeld Gymnasium) in 1974. He received the Diploma from the University of Patras, Patras, Greece, in 1979, the M.Eng. degree from Carleton University, Ottawa, Canada, in 1982, and the Ph.D. degree from the University of Ottawa, Ottawa, Canada, in 1989, all in electrical engineering. From 1981 to 1983, he was with Raytheon Canada Limited, where he was involved in the analysis, design, implementation, and evaluation of a new generation of air-navigational equipment (DVOR and DME). He was also responsible for preparing and teaching courses for the operation and maintenance of both of these equipments. From 1983 to 1988, he was with the Department of Electrical Engineering, University of Ottawa, where, since 1985, he has served as a Research Engineer and twice as a Sessional Lecturer. During this time, he was also a Consultant to Raytheon Canada Limited and other companies. In 1989, he joined the Department of Electrical Engineering, University of British Columbia, where, since 1994, he has held the position of Associate Professor, and currently is on sabbatical. He has acted as a consultant for various industrial and governmental organizations, has taught numerous short courses, and has delivered several invited talks, including a plenary lecture. He has served as a member of various scientific and advisory panels of the European Commission in the technical fields of mobile, personal, and multimedia telecommunications. His research activities have been in the general area of RF and microwave digital telecommunications, with recent emphasis on terrestrial and satellite-based wireless personal communication systems and networks. Since 1993, Dr. Mathiopoulos has been the Editor of Wireless Personal Communications for the IEEE TRANSACTIONS ON COMMUNICATIONS. He has organized numerous technical sessions for several ICC and GLOBECOM conferences and has served as a member of the Technical Program Committees for more than 20 IEEE conferences. He has served on the Editorial Board of the IEEE PERSONAL COMMUNICATIONS MAGAZINE (1994–1996) and other technical journals. From 1992 to 1995, he was appointed as a Fellow of the Advanced Systems Institute, and from 1996 to 1997, he was a Killam Research Fellow. He is a Registered Professional Engineer in Greece and the Province of British Columbia, Canada.