On The Performance of Iterative Noncoherent Detection ... - IEEE Xplore

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On the Performance of Iterative Noncoherent Detection of Coded -PSK Signals

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I. D. Marsland, Member, IEEE, and P. T. Mathiopoulos, Senior Member, IEEE

Abstract—Differential encoding is often used in conjunction with noncoherent demodulation to overcome carrier phase synchronization problems in communication systems employing -ary phase-shift keying ( -PSK). It is generally acknowledged that differential encoding leads to a degradation in performance -PSK systems with perfect carrier over absolutely encoded synchronization. In this paper, we show that when differential encoding is combined with convolutional encoding and interleaving, this degradation does not necessarily occur. We propose a novel noncoherent receiver for differentially encoded -PSK signals that is capable of significantly outperforming optimal coherent receivers for absolutely encoded -PSK using the same convolutional code. This receiver uses an iterative decoding technique and is based on a multiple differential detector structure to overcome the effect of the carrier phase error. In addition, to better illustrate the benefits of the powerful combination of convolutional encoding, interleaving, and differential encoding, we also present an iterative coherent receiver for differentially encoded -PSK. Index Terms—Differential phase-shift keying, iterative decoding, MAP estimation, multiple-symbol differential detection, noncoherent demodulation, noncoherent detection.

I. INTRODUCTION N COMMUNICATION systems employing -ary phase-shift keying ( -PSK)1 transmitted information possible phases of a carrier is represented by one of signal during a symbol interval. Coherent demodulation, which requires the maintenance of accurate carrier phase synchronization across the channel, is typically used with this technique. Alternatively, with differentially encoded -PSK, the information is represented by the change in carrier phase over two successive symbol intervals. Coherent demodulation can still be used, and an optimal receiver was recently proposed by Simon and Divsalar [1] that provides hard decisions of the symbols with maximum a posteriori probabilities (APP’s). This system yields a bit-error rate (BER) that is

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Paper approved by S. S. Pietrobon, the Editor for Coding Theory and Techniques of the IEEE Communications Society. Manuscript received February 26, 1998; revised December 11, 1998 and September 6, 1999. This work was supported in part by the Natural Science and Engineering Research Council (NSERC) of Canada under Grant OGP-443212, a BC Advanced Systems Institute (ASI) Fellowship, and a Killam Faculty Research Fellowship. The paper was presented in part at the International Conference on Telecommunications (ICT'98), Porto Carras, Greece, June 1998. I. D. Marsland is with the Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada (e-mail: [email protected]). P. T. Mathiopoulos is with the Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC V6T 1Z4, Canada. Publisher Item Identifier S 0090-6778(00)03634-5. 1Referred

to hereafter in this paper as absolutely encoded

M -PSK.

roughly twice that of absolutely encoded -PSK with Gray coding. Differential encoding is attractive because it facilitates noncoherent demodulation by removing the need for explicit phase synchronization, which can be difficult to maintain over time-variant channels. Traditionally, differential detection is used in conjunction with noncoherent demodulation, where the transmitted information is extracted by taking the phase difference between two successive samples of the matched filter output. As such, phase synchronization is maintained implicitly by using the previous sample to estimate the carrier phase error affecting the current sample. While its simplicity makes this approach attractive, its use leads to a substantial noncoherence penalty compared to coherently-demodulated differentially encoded -PSK. However, it has been shown that much of this degradation can be overcome by using a multiple differential detector (MDD) receiver structure [2]–[5].2 MDD receivers samples to estimate the phase implicitly use the previous error, thereby increasing the accuracy of the estimate. To further improve system performance, some recent research has focused on the class of error correcting codes generated by the serial concatenation of a convolutional encoder, an interleaver, and a differential encoder [6]–[10]. These codes are attractive because they combine the performance gains of error correction with the robustness of noncoherent demodulation. Although the noncoherent MDD receivers described above can be used with convolutionally encoded data, they are not appropriate for systems employing interleaving. However, by observing that differential encoding is a form of recursive convolutional encoding, it is apparent that these codes belong to a subset of the class of serially concatenated convolutional codes (SCCC's) [11], which are characterized by an inner and an outer convolutional code, separated by an interleaver. The iterative decoding structure presented in [12], which consists of a softoutput decoder for the inner code followed by a deinterleaver and a soft-output decoder for the outer code, provides an effective means for decoding SCCC's. Soft-output extrinsic information from the outer decoder is fed back to the inner decoder, and the entire decoding operation is repeated in an iterative fashion. When differential encoding is used as the inner code, a similar iterative structure can be used. In this case, the inner decoder must reverse the differential encoding instead of decoding a convolutional code. When used with noncoherent demodulation, the inner decoder must also be designed to provide channel 2The hardware structure of an MDD receiver consists of a combination of more that one distinct differential detector, with elements of time delay equal to progressively increasing multiples of the symbol duration . The number of distinct differential detectors will be denoted as . In some papers on the same subject, notably [4], this structure is referred to as multiple-symbol differential detection (MSDD).

0090–6778/00$10.00 © 2000 IEEE

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estimation. The use of iterative decoding for these concatenated codes with noncoherent demodulation has recently been proposed for the additive white Gaussian noise (AWGN) channel [6], [7], [13], [14], frequency-flat fading [8], [9], [15], and frequency-selective fading [10]. These receivers all use different inner decoders, depending on the channel under consideration. In this paper, we present two new inner decoders for the AWGN channel. One is for use with noncoherent demodulation and the other for coherent demodulation. The second is useful for providing comparisons between different systems as it provides a measure of the performance of the codes without the effect of the noncoherence penalty. The noncoherent receiver presented here differs from the one presented by Peleg and Shamai [6] and the one presented by Peleg et al. [7] in the algorithm used by the inner decoder, even though they all serve the same purpose. These differences, which are highlighted in this paper, are not by design but rather a result of the fact that the research was done independently. Additional research is required to compare and contrast the differences in terms of performance and complexity of the proposed decoders, but an adequate comparison is beyond the scope of this paper. It should also be noted that the coherent receiver presented here is also presented in [9], highlighting how active this research area is. The organization of this paper is as follows. The model of the investigated system, including the transmitter and channel, is presented in Section II. This is followed in Section III by a description of the iterative decoding process and details on the new inner decoders. 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 codes is illustrated as they significantly outperform systems using only an outer convolutional code with absolutely encoded, coherently demodulated -PSK.

II. TRANSMITTER AND CHANNEL MODEL DESCRIPTION A block diagram of the concatenated encoder used with the communication system presented in this paper is shown in Fig. 1. Digital data is convolutionally encoded, interleaved, and then differentially encoded prior to transmission over -PSK. A transmitted message an AWGN channel using symbols with each symbol conveying word consists of information bits. The message word is denoted by , where refers to the sequence . The the notation -state rate convomessage is first encoded with an is composed lutional code, and the resulting codeword symbols of bits, where is the number of of additional symbols required to drive the convolutional encoder (CE) to the zero state at the end of the word. The encoder is characterized by a code symbol generation matrix and a state transition matrix . If the encoder's state prior to encoding the th message symbol is , is and the encoder's response to input . the encoder advances to state

Fig. 1.

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Block diagram of the concatenated encoder.

The symbols of the codeword are rearranged using a nonuniform interleaver [16] that operates on a symbol-by-symbol basis, so each code symbol remains intact with only the symbol order changed. This differs from the more common bit-by-bit interleaving used by other researchers of this topic and leads to a very slight improvement in performance. The symbols of are referred to here as data symbols the interleaved word to distinguish them from the message and code symbols. The th data symbol is , while is the interleaver mapping. The data symbols are passed through a differential encoder -PSK modulation, can be implemented (DE), which, for adder and a delay element with digitally with a modulobits, where . In this paper, a depth of , we focus our investigation on the case where so the code symbols remain intact through interleaving and modulation. It is evident that differential encoding can be viewed as a form of recursive convolutional encoding, with a coding rate of unity and a memory length of one symbol. If the state of the differential encoder prior to encoding the th data symbol is , then the encoder is characterized by the code symbol generation and state transition matrices for all , where . The encoder is assumed to start in the zero state prior to encoding ), but is not terminated to the zero each data word (i.e., state at the end of the word. For transmission, the differentially encoded symbols are mapped to points in the -PSK signal constellation with natural mapping.3 The transmitted symbols for , where are is the transmitted energy per symbol. The transmitted energy per message bit is (1) The first transmitted symbol carries no data and is used as a reference symbol by the decoder. The combined effect of the transmitter’s pulse shaping filter and modulator, the AWGN channel, and the receiver’s demodulator, matched filter, and symbol-rate sampler can be modeled as a discrete-time complex low-pass equivalent channel. With nonis coherent demodulation, an unknown carrier phase error introduced so that the th received sample is (2) is an AWGN sample. The phase error is uniformly where and assumed to remain constant over the distributed over 3If the use of Gray mapping is desired, it should be implemented by directly modifying the data symbols prior to differential encoding.

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Fig. 2. Block diagram of the iterative decoder structure.

entire sequence of received samples.4 The noise samples have , zero mean and covariance denotes the expected where denotes complex conjugation, is the single-sided noise power spectral density, and value, is the Kronecker delta function. Using the received sample sequence , the decoder attempts to determine the transmitted . message word III. DECODER STRUCTURE AND ALGORITHMS To minimize the probability of a symbol error, ideally one would like the decoder output to be those symbols with the would be maximum APP's. That is, the th output symbol that symbol for which the conditional probability is largest. To compute the APP's, an optimal decoder must jointly take into consideration the convolutional encoding, the differential encoding, the phase mapping, and the channel. In practice, exact computation of the APP's is computationally prohibitive for interleaved systems. However, because interleaving reduces the statistical dependency between data symbols in the same vicinity, it is possible to split the overall decoding into two stages: an inner decoder, which reverses the differential encoding and compensates for the phase error, and an outer decoder, which decodes the convolutional code. Although splitting the decoder in this fashion is inherently suboptimal, much of the performance loss can be recovered by using iterative decoding. The proposed decoder, shown in Fig. 2, consists of the inner and outer decoders connected in an iterative decoding structure. Although similar structures have been suggested for other applications involving iterative decoding, particularly decoding of serially concatenated convolutional codes [12], the implementation of the inner decoder is quite different as it must perform a different task. Nonetheless, the overall iterative decoding process described here is very similar, if not identical, to most other iterative decoders, including [6], [7], and [9]. The operation of the iterative decoding process is presented in Section III-A. This is followed by descriptions of the inner decoder for use with noncoherent demodulation in Section III-B and coherent demodulation in Section III-C. A description of the outer decoder follows in Section III-D. A. Iterative Decoding Using the received samples, the inner decoder attempts to compute the a posteriori data symbol probabilities for all hypothetical and . These possible different APP's reflect the likelihood of each of the 4Although the decoder is designed with the constant phase assumption, it is fairly insensitive to modest phase variations, as illustrated in Section IV.

values for being transmitted, given that the samples are received. To calculate the APP's, either of the algorithms given in Sections III-B or III-C is used, depending on whether noncoherent or coherent demodulation is used. These algorithms and , intake into consideration the relationship between cluding the differential encoding, phase mapping, carrier phase error, and the AWGN. For simplicity, the algorithms all assume that the data symbols are statistically independent, although the are not necesa priori probability distributions sarily identical for each . Because the data symbols are actually the output of a convolutional encoder, this assumption is not valid. Although interleaving the output of the convolutional encoder does reduce the statistical dependence between data symbols in the same vicinity, the inner decoder is nonetheless capable only of producing estimates of the APP's, denoted by . For proper use by the outer decoder, the actual output of the inner decoder are these estimates with the a priori information removed by division. This so-called extrinsic information (3) is deinterleaved and passed to the outer decoder. Using the inverse of the mapping used by the interleaver in the encoder, the deinterleaver keeps the extrinsic information quantities associated with each data symbol together as a single unit, changing only the order of the units. The deinterleaved extrinsic inforand reflects the likelihood of the mation is different values for each code symbol to have been transmitted based on the received samples. It is information gained about solely from observation of the channel, as extracted by the introduced by the convoluinner decoder. The constraints on tional code, in the form of statistical dependency, have not been exploited in determining the extrinsic information. These constraints are taken into consideration by the outer decoder, which decodes the convolutional code, to produce estimates of the a posteriori message symbol probabilities . As noted, separating the decoding process into these two stages is suboptimal and leads to system performance that is quite poor. By assuming that the data symbols are independent in the inner decoder, considerable information is ignored. To compensate, the outer decoder also computes estimates of the a posteriori code symbol probabilities . After the extrinsic information from the inner decoder has been removed by division, the resulting quantities (4) are interleaved and passed back to the inner decoder. The inner decoder repeats its operation, using the same received , but using the interleaved quantities from the outer samples in place of the a priori probabilities decoder . Although the inner decoder is designed without knowledge of the constraints imposed by the convolutional code, using feedback in this fashion is an effective substitute.

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Using this additional information, the inner decoder recomputes the APP's, passing the new extrinsic information back to the outer decoder. This process is repeated several times in an iterative fashion, with the reliability of the APP's hopefully improving with each iteration, recovering some of the loss arising from the faulty independence assumption. After some fixed number of iterations, decoding terminates, and the outer produced on the final iteration to a decoder passes decision device where hard decisions are made.

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the argument to the Bessel function by yields

(8) For notational convenience, we define (9)

B. Inner Decoder (Noncoherent Demodulation) As mentioned, the purpose of the inner decoder is to calculate the APP’s of the data symbols entering the differential encoder and in doing so compensate for the carrier phase error. In this section, it is shown that these APP’s can be calculated using the standard APP algorithm developed by Bahl et al. [17]. Derivation of the inner decoder begins with finding an expression for the probability density function (pdf) , for received sample conditioned on the data symbols and the previously received samples. A reasonable approximation to this expression is then presented which provides computationally feasible branch metrics. The implementation of the APP algorithm using these metrics is then described. Since the noise is Gaussian and there is a one-to-one mapping and the transmitted symbols , it is between the data word conditioned on the evident from (2) that the pdf of sample data symbols and phase error is

and note that

(10) By using Bayes' rule with (8), it follows that

(5) (11)

and because the noise samples are independent

. Some insight into this pdf can be where is gained by observing that the conditional mean of (6) The dependence on

can easily be removed with [18]

(12)

(7)

where is a quantity that does not depend on , since and is the modified Bessel function of order zero. Multiplying

can be used as an unbiased estimator of the unTherefore, wanted phase affecting sample due to the differential encoder and the carrier phase error . This estimator is state , as well as based on all the previously received samples . the previous data symbols Although (11) provides an expression for the desired pdf, its usefulness is limited because different values of the estimate different possible realizations of arise for each of the . Since is not known at the receiver, optimal decoding requires that all the possibilities be explored. Clearly, this is not

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feasible when is large. To avoid this problem, in this paper we follow the approach taken in [3] and limit the number of previous samples used by the estimator to some small number with typically less than five. The new estimator depends and , and is given by only on

Using this trellis structure, with the branch metrics given above, it is straightforward to calculate the APP's. Note that

(16) (13) where the recursive definition of differential encoding has been used in the last line.5 The new estimator is also unbiased, but with a larger variance than if truncation was not performed. Although truncation does degrade system performance, by limiting the size of the observation window, it does allow the estimator to better track slow variations in . Note that in truncating the number of samples, it has implicitly been assumed , which is given that in place of . by (11) with By using truncation, different values of the estimate arise only possible values of . As such, trellisfrom the based decoding is suggested, and an obvious trellis structure is , readily constructed. Define the state at time as -tuples. so the state space is the set of all -ary states, there are branches correFrom each of the possible values of , for a total of sponding to the branches in a trellis section. The trellis structure is defined by and the the state transition matrix . “code symbol” generation matrix Associated with the branch corresponding to the specific se, quence of data symbols, , is the branch metric with

where, from left to right, the three multiplied terms correspond to the contribution to the APP from the past, present, and future received samples, respectively. By making use of the fact that differential encoding and the channel are causal, and that , the assumption that the data symbols are independent implies

(17) The assumption regarding the truncated estimators implies

(18) and

(19) By defining and (20) the expression for the APP is

(14) (21) with (15)

which is equivalent to the core expression in the standard APP algorithm [17]. By using the same assumptions used above, it is and can be computed recurshown in [19] that sively using the APP algorithm. That is

Note that this trellis structure is purely artificial; it is not to be confused with the trellis structure associated with the differential encoder.

(22) where

5The value of the k -indexed

summation is defined equal to zero when z = 1.

equals one if and zero otherwise. The recursion is initialized with

MARSLAND AND MATHIOPOULOS: ITERATIVE NONCOHERENT DETECTION OF CODED

for and otherwise, since the differis ential encoder is initialized to the zero state. Similarly, computed with the reverse recursion given by

(23) for all with the initial condition of , since the differential encoder is not terminated to the zero state. Therefore, the standard APP algorithm can be used to calculate the APP's if the data symbols are independent and the truncated estimators are used. The APP algorithm uses a simple states and the branch metrics given trellis structure with by (14). On the first iteration of the decoder, the data symbols are assumed to have equal a priori probabilities, so . On subsequent iterations, the interleaved values from the are used in place of . outer decoder The approach outlined above for the inner decoder differs from both [6] and [7]. In [6], the authors follow the MSDD approach taken in [4], and instead of truncating the estimators, samples into small subblocks and expartition the block of haustively search each subblock to estimate the APP using (7). The more interesting approach taken in [7] involves calculating (24) by approximating the integral with the composite trapezoidal points over the interval , where is rule using a system parameter. The resulting expression can then be manipulated into a form suitable for use with the APP algorithm states. When , this approach yields perwith formance virtually identical to the coherent receiver presented is not completely below. However, it is very unforgiving if constant over the entire block, whereas the approaches outlined here and in [6] are much more robust. To overcome this shortfall, the authors of [7] proposed using block partitioning, which leads to some performance degradation. They also proposed another modification which gives some leeway to variation in , but also increases the decoder complexity. Without additional research, more detailed discussion of the merits of the various proposals cannot be provided. C. Inner Decoder (Coherent Demodulation) When coherent demodulation is employed, there is no carrier phase error affecting the received samples, so the inner decoder need only take into consideration the differential encoding when computing the data symbol APP’s. Since differential encoding is the same as recursive convolutional encoding, the APP algorithm can be used directly. The algorithm is implemented with states corresponding to , using and as defined in Section II. The algorithm uses

(25)

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as the branch metrics. The forward recursion is initialized with for and otherwise, and the reverse since the differrecursion is initialized with ential encoder is not forced back to the zero state at the end of the word. The first received sample is not needed in this case as it carries no information. It is worth pointing out that this same decoder has a more practical application. If coherent demodulation is achieved -ary phase tracking loop to derive through the use of an the local carrier reference signal, a discrete phase error which will result. Denoting this is some integer multiple of for some , the received phase error by samples are (26) The discrete phase error is readily absorbed by the algorithm . merely by redefining the states as The only change required to the algorithm is that the forward for all recursion be initialized with to reflect the phase error and to begin decoding with . For block sizes of practical interest, there is negligible difference in performance vis-à-vis the case with perfect phase reference. D. Outer Decoder The outer decoder, which decodes the convolutional code, also makes use of the APP algorithm, implemented with states corresponding to the states of the convolutional encoder, and as defined in Secand using and tion II. The recursions are initialized with equal to one for and zero otherwise. In place of the branch metrics, the extrinsic information from the inner decoder is used. IV. SIMULATION RESULTS AND RELATED DISCUSSION The performance of the proposed coherent and noncoherent systems was investigated by means of computer simulation. Performance is compared with coherently demodulated, absolutely encoded -PSK and traditional differentially detected, -PSK. The effects of the memory differentially encoded length of the convolutional code and the interleaver block size were investigated. In addition, a slight carrier frequency offset was added and the performance of the noncoherent system was observed. Unless otherwise indicated, the following system parameters were used for the simulations. The , convolutional code system was tested with a 16-state, rate , so and . with generator symbols were produced by Codewords of length the encoder. A nonuniform symbol-by-symbol interleaver was 32 matrix used, where the symbols were stored in a 32 and read according to the technique described in [16], with . The interleaved symbols were differentially encoded and transmitted using quadrature . Gray mapping was used phase-shift keying (QPSK) and implemented prior to differential encoding. To illustrate the effectiveness of iterative decoding, the concatenated code was first tested with coherent demodulation. As can be seen by the results shown in Fig. 3, the results signifi-

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Fig. 3. Coherent demodulation: performance after the first 20 iterations for the concatenated code. Also shown is the performance of absolutely encoded, coherently demodulated QPSK, with 16-state and 256-state convolutional codes (CC’s).

Fig. 4. Noncoherent (NC) demodulation: performance after five iterations with Z = 2; 3; and 4 differential detectors. Also shown is the performance with coherent demodulation after five iterations and a 256-state convolutional code (CC) with a single differential detector followed by a Viterbi decoder (noniterative).

cantly improve with the first few iterations, although gains appear to be marginal after about six iterations. For comparison, also shown is the performance of a stand-alone convolutional code (CC) with absolutely encoded, coherently demodulated QPSK. Results for both the same 16-state code used with the concatenated scheme and a more powerful 256-state code with generator (561 753) are given. The concatenated code actually outperforms both stand-alone codes at a BER of 10 , by about 2.8 and 1.3 dB, respectively. At first glance, these results may seem quite surprising because differential encoding is typically associated with a degradation in system performance. However, it is actually a very positive reflection on the power of serial concatenation when this much gain can be realized even when one of the constituent codes (the differential encoding) has a coding rate of nearly one , i.e., it does not introduce any redundancy and yet significantly improves the BER performance. A precise theoretical explanation for this counter-intuitive phenomenon is difficult to provide for several reasons. Theoretical analysis is often

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 4, APRIL 2000

based on knowledge of the codeword weight distribution of the code, but finding the weight distribution for the concatenated code, based on any one specific interleaver, is computationally infeasible for large . By using the concept of the “uniform interleaver,” which considers the weight distribution of the codepossible permutations of the interwords averaged over all leaver, as is done in analysis of turbo codes and SCCC's [20], [21], it is possible to perform some limited theoretical analysis (see [22]). Another problem with theoretical analysis is that the concatenated code is not linear when differentially encoded QPSK modulation is used. As such, the uniform error property does not hold (i.e., the probability of error is a function of the transmitted message word). However, since the same surprising differences in performance occur when differentially encoded binary phase-shift keying (BPSK) is used,6 but the uniform error property does apply, it is useful to consider this case for theoretical purposes instead. Using the techniques given in [21]–[23], we computed the codeword weight distribution of the concatenated code for differentially encoded BPSK. Examination of the results showed that, on average, the fraction of codewords with any specific small weight is much less for the concatenated code than for the convolutional code by itself.7 That is, by using interleaving and differential encoding, the number of low-weight codewords is significantly decreased, thereby improving system performance. The same surprising performance found with coherent demodulation also extends to the noncoherent case when iterative decoding is used with the new inner decoder. Results shown in and . Increasing Fig. 4 are for five iterations with leads to performance gains as expected, and with , there is only a 0.5-dB noncoherence penalty compared with coherent demodulation after the same number of iterations. However, the benefits of using this system are obvious if the results are compared to the performance of a traditional differential detector , followed by a deinterleaver and a Viterbi decoder. When as in this case, the inner decoder has only a single state and therefore cannot exploit the information fed back from the outer decoder, preventing iterative decoding. So, to provide a more fair comparison in terms of decoder complexity, the memory length of the convolutional code has been increased. It is clear that traditional differential detection, even with a powerful 256-state convolutional code, is not at all competitive, with performance inferior by more than 2 dB at a BER of 10 . Another point to make about serial concatenation is in regard to the choice of the component convolutional code. While one may expect that by increasing the power of the component code, the power of concatenated code will also increase, this is not necessarily the case, as indicated by the results shown in Fig. 5. The performances of concatenated codes based on conand volutional codes with memory lengths of are shown after five iterations.8 For BER's above 10 , performance actually deteriorates with increasing memory length when coherent demodulation is used. Similar results hold with 6With BPSK, bit-by-bit interleaving must be used since m 6= n . This requires some minor modifications to the system described in this paper, but the overall decoding approach is similar (see [19]). 7Note that similar results are reported in [22]. 8These codes have been taken from [18, Table 8-2-1].

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Fig. 5. Memory length: performance with coherent demodulation after five iterations with concatenated codes based on convolutional codes with various differential detectors. memory lengths L and Z

( )

=3

noncoherent demodulation, although the decoder performs is too small (e.g., ). Once again, theopoorly if retical explanation of this surprising result is hampered by the nonbinary nature of the concatenated code. Furthermore, uniform-interleaver analysis is unable to provide an explanation for this behavior, even when differentially encoded BPSK is used. It seems likely that some artifact of the suboptimal iterative decoding strategy, combined with the codeword weight distribution, governs actual system performance. The nature of this artifact remains unknown and requires future research. As mentioned throughout this paper, it is expected that the noncoherent receiver should be fairly insensitive to modest variations in the carrier phase error. To test this hypothesis, we , making the readded a slight carrier frequency offset ceived samples [see (2)] (27) where is the symbol duration. The results presented in Fig. 6, decoder, show that for frequency offsets up to for the , there is very little performance degradation. The receiver is unable to compensate effectively for higher offsets, however, and the beginning of an error floor is visible in and . the curves for Although the discussion to this point has focused on QPSK , the same general results extend to higher order . Both coherent modulation schemes, such as 8-PSK and noncoherent demodulation were also tested with a 16-state, , trellis-coded modulation (TCM) code [24, Fig. 9], so rateand . Codewords of length symbols were used with the same interleaver described above, and natural mapping was used. As can be seen in Fig. 7, shares the same characteristics as the performance with system, although in a less pronounced manner. Iterative decoding still works well, but the advantages over regular TCM with absolutely encoded, coherently demodulated 8-PSK are not as large as with QPSK. There are two additional points worth mentioning regarding the performance of the proposed decoders. One is that, by far, the most important factor controlling performance is the size of the interleaver. Lengthening the interleaver size improves performance, while shortening it causes degradation, a finding that

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Fig. 6. Frequency offset: performance after five iterations of noncoherent demodulation Z with various frequency offsets f T .

( = 3)

(1

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Fig. 7. 8-PSK: performance after five iterations of coherent and noncoherent demodulation when 8-PSK is used with a rate = TCM code.

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is generally consistent with other interleaved systems using iterative decoding, such as turbo codes and serially concatenated convolutional codes. However, the noncoherence penalty also increases with increased interleaver size [19]. That is, not all of the gains realized with coherent demodulation when the interleaver size is increased are achieved with noncoherent demodulation. The second point worth noting is that there appears to be little difference between symbol-by-symbol interleaving and bit-by-bit interleaving. Most of the experiments presented in this paper were also done using bit-by-bit interleaving, and although symbol-by-symbol interleaving consistently performed better, the difference was typically about 0.1 dB in terms of signal-tonoise ratio. This minor difference must be weighed against the fact that bit-by-bit interleaving requires less decoder memory than symbol-by-symbol interleaving, but requires some additional processing in the inner and outer decoders. V. CONCLUSION We have introduced and evaluated two new iterative decoders for convolutionally and differentially encoded -PSK signals, one for use with coherent demodulation and the other for noncoherent demodulation using an MDD structure. It is shown that the concatenated code is substantially more powerful than the

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convolutional code by itself, and the advantages of this code apply not only to coherent demodulation, but to noncoherent demodulation as well. It was also shown that increasing the complexity of the convolutional code does not necessarily improve performance at low signal-to-noise ratios, with the more simple codes giving the best performance. In addition, the concatenated code was also extended to a more spectrally efficient TCM-type code and was shown to be effective. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers and the editor, Dr. S. S. Pietrobon, for their constructive suggestions. In addition, they would like to thank Prof. S. Shamai (Shitz) for his comments on this manuscript. The assistance provided by these individuals has greatly enhanced the quality of this paper. REFERENCES

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Ian D. Marsland (S’98–M’00) received the B.A.Sc. degree (with honors) in mathematics and engineering from Queen's University, Kingston, Canada, in 1987, and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of British Columbia, Vancouver, Canada, in 1994 and 1999, respectively. 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 1999, he has been an Assistant Professor with the Broadband Communications and Wireless Systems (BCWS) Centre in the Department of Systems and Computer Engineering at Carleton University, Ottawa, Canada. His research interests include error correction techniques for wireless communication systems.

P. Takis Mathiopoulos (S’80–M’89–SM’94) was born in Athens, Greece. He graduated from the German High School of Athens (Dorpfel Gymnasium) in 1974, and received the Diploma from the University of Patras, Patras, Greece, in 1979, the M.Eng. degree from Carleton University, Ottawa, ON, Canada, in 1989, all in electrical engineering. From 1981 to 1983, he was with Raytheon Canada Ltd., where he was involved with the analysis, design, implementation, and evaluation of a new generation of air-navigational equipment (DVOR and DME), and he was also responsible for preparing and teaching courses for the operation and maintenance of this equipment. 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, Ltd. 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. 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, as well as information technology and space science. 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. He has co-authored more than 25 IEEE journal papers and more than 50 papers published in various conference proceedings. Since 1993, he has been the Editor of Wireless Personal Communications for the IEEE TRANSACTIONS ON COMMUNICATIONS. He is serving also on the editorial boards of several other archival journals in the field of telecommunications. He has been a member of the Technical Program Committee for more than 40 international conferences. From 1992 to 1995, he was appointed 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.