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Performance of Trellis-Coded CPM with Iterative Demodulation and Decoding Krishna R. Narayanan and Gordon L. Stüber, Fellow, IEEE

Abstract—Interleaved trellis-coded systems with full response continuous-phase modulation (CPM) are considered. Upper bounds on the bit-error rate performance are derived for coherent detection on the additive white Gaussian noise and flat Rayleigh fading channels by considering the trellis code, interleaver, and CPM modulator as a serially concatenated convolutional code. A coherent receiver that performs iterative demodulation and decoding is shown to provide good bit error performance. Finally, a noncoherent iterative receiver is proposed and is shown to perform close to the coherent iterative receiver.

I. INTRODUCTION

M

ANY communication systems employ nonlinear power amplifiers and, hence, require a low peak-to-average power ratio for the modulated signal. Examples of such systems include satellite and mobile communications systems. Continuous-phase modulation (CPM) is a constant envelope modulation technique and, hence, a good choice for such systems. Trellis-coded modulation (TCM) is a technique that combines modulation and coding in order to achieve coding gains without sacrificing data rate or bandwidth. When TCM is combined with CPM, high power and bandwidth efficiency can be achieved. Convolutionally encoded CPM on additive white Gaussian noise (AWGN) channels has been considered by many researchers [1]–[3]. Upper bounds on the bit-error rate (BER) have been derived for such systems [1], [3]. A detailed analysis of the distance spectrum and BER has been considered in [4]. The design and analysis of trellis-coded CPM (TCCPM) schemes for AWGN channels has also been considered in [5]. However, none of the aforementioned work has considered the use of an interleaver between the trellis code and the CPM modulator for AWGN channels. Similarly, they have also ignored the effect of the recursive nature of CPM modulator in designing TCCPM schemes. The performance of TCCPM for fading channels has been evaluated in [6]–[9]. A fading channel typically exhibits correlation in time and, when the correlation is high, the channel Paper approved by A. Ahlen, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received June 29, 1999; revised August 4, 1999 and May 31, 2000. The work of G. L. Stüber work was supported in part by the National Science Foundation under Grant CCR-9903297. This paper was presented in part at the IEEE GLOBECOM Communication Theory Mini-Conference, Rio de Janeiro, Brazil, December 1999. K. R. Narayanan is with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843 USA (e-mail: [email protected]). G. L. Stüber is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA. Publisher Item Identifier S 0090-6778(01)03140-3.

exhibits prolonged deep fades. Therefore, some form of interleaving has to be used to combat the fading correlation. When TCM is used with phase-shift keying (PSK), pulse amplitude modulation, or quadrature amplitude modulation, the modulated symbols are interleaved using a block or convolutional interleaver [5]. Since interleaving attempts to destroy the correlation in the fading process, performance bounds can be derived by assuming that the channel is uncorrelated [5], [10]. However, when CPM is used, the modulated signal cannot be interleaved because interleaving destroys the continuous phase property of the modulated signal. Therefore, a system was proposed in [7] where an interleaver is used between the trellis code and the CPM modulator. At the receiver, a soft-output CPM demodulator generates metrics for the coded symbols. These metrics are then deinterleaved and used to derive the branch metrics for the Viterbi decoder that is used to decode the trellis code. This deinterleaving makes the channel “appear” uncorrelated to the trellis code. Performance bounds can then be developed by suitably modifying the transfer function of the trellis code and assuming that the channel is uncorrelated [8]. All of the aforementioned approaches completely ignore the contribution of the interleaver to the overall distance spectrum of the transmitted signals. From the recent advances in concatenated schemes, we know that the interleaver drastically influences the distance spectrum and, hence, will significantly change the system performance. It should be noted here that in contrast to conventional coded CPM schemes, the use of an interleaver introduces a delay. It is assumed throughout this paper that such a delay is tolerable. Our approach is to treat TCCPM as a serial concatenation of a trellis code and a CPM modulator, which represents a rate-1 recursive inner code. In contrast to the other papers, we explicitly consider the effect of the interleaver on the distance spectrum of the transmitted signals. We develop upper bounds on the BER performance under the assumption of a uniform interleaver and a truncated union bound for the exact interleaver used. The same approach was used in [11] to analyze the performance of interleaved convolutionally encoded systems with differential phase-shift keying (DPSK). In [11], performance bounds were derived for Rayleigh fading channels by assuming perfect interleaving of the modulator output. In this paper, we focus on TCCPM schemes for fading channels. As mentioned before, unlike the case of PSK, for CPM the modulated symbols cannot be interleaved prior to transmission. Consequently, we cannot assume an interleaved fading channel in the case of CPM signals. This makes a significant difference in the performance bounds developed in this paper in comparison to the ones in [11].

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NARAYANAN AND STÜBER: PERFORMANCE OF TCCPM WITH ITERATIVE DEMODULATION AND DECODING

Fig. 1.

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Block diagram of interleaved TCCPM system.

Finally, we consider iterative receivers for coherent and noncoherent reception of TCCPM. The coherent receiver is identical to the one in [12]. Noncoherent reception of uncoded CPM signals have been considered in [9] and [13]. A noncoherent receiver for TCCPM was proposed in [9] where a noncoherent front end is used to generate metrics for the coded bits and a Viterbi decoder is used for the trellis code. In this paper, we derive a noncoherent detector for iterative demodulating and decoding of TCCPM. Our receiver structure is similar to the one in [14] and based on the maximum-likelihood (ML) block detection principle in [15]. II. SYSTEM MODEL The TCCPM system considered in this paper is shown in Fig. 1. It is similar to the system in [7] and [8], but with a few differences. At each time instant, bits of the binary data seconvolutional code and biquence are encoded by a ratenary outputs of the convolutional encoder are multiplexed into the sequence . The binary sequence is then interleaved in to the sequence . Then, -tuples of the sequence are mapped on and the sequence to symbols is input to the CPM modulator. If , the symbol rate is the same as the uncoded rate of the -tuples taken from the se, the quence . We refer to such systems as TCCPM. If data rate is lower for the coded system than the uncoded system and, we refer to such systems as convolutionally encoded CPM. Since bit interleaving provides a higher diversity advantage than symbol interleaving [16], [17], we consider bit interleaving of the outer code words in contrast to the symbol interleaving use in [7] and [8]. The results in Section VIII will substantiate this claim. A. CPM Modulator The low-pass equivalent CPM signal is given by

prime. If , the CPM scheme is called full-response CPM, it is called partial-response CPM. In this paper, and if we will only consider full-response CPM. A full response CPM to emphasize signal can also be represented by the finite-state nature of CPM and, hence, suggest a trellis representation similar to that of convolutional codes. B. Channel Model When the modulated signal is transmitted over a frequencynonselective channel, the received signal can be expressed as where is the instantaneous channel remains constant over one modugain. We will assume that lated symbol interval and, consequently, the received signal over the th epoch can be written as (2) . For the AWGN channel, . where We also consider a land mobile radio channel characterized by frequency-nonselective (flat) Rayleigh fading and two-dimenis sional isotropic scattering. For this channel, and are zero-mean a complex random variable, where Gaussian random variables. The auto and cross-correlations of and are [18]

(3) is the zeroth-order Bessel function of the first kind where is the maximum Doppler shift. It should be noted that and since the complex channel gains are normalized to have unit energy, the average received signal energy is . III. SERIAL CONCATENATION OF CONVOLUTIONAL CODES

for

(1) is the energy per CPM symbol, which is related to the where according to . energy per information bit The information is contained in the phase , where is the phase response such that for , is the accumulated phase at the beginning is the modulation index. We will of the th epoch, and is rational and and are relatively assume that

In this section, we briefly review some results on serial concatenation of convolutional codes (SCCC). Consider the conwith rate , a pseudocatenation of an outer code with rate random interleaver, and an inner code ( and are relatively prime). Let us assume that each data bits in length (including appropriate tail bits) and, block is . For the outer and inner hence, the interleaver size is and denote the number of codes, respectively, let codewords of weight generated by input sequence of weight which is the result of concatenating error events such that . the code word diverges from the all zero state at time The distance spectrum of the overall code is extremely difficult to evaluate for a given interleaver and, hence, Benedetto et al.

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have suggested the uniform interleaver [19]. With the uniform interleaver, the number of codewords of the overall SCCC with can be apcode word weight and input weight , proximated by

(4) is the free distance of the outer code, and and where refer to the maximum number of error events possible for the outer and inner codes, respectively. It was shown in [19] that the exponent of in (4) is always negative when the inner code is recursive. Consequently, the number of codewords of the SCCC decreases exponentially with the block length and this was termed as interleaving gain. This phenomenon is precisely the reason for the excellent performance of the SCCC. Further, it can be seen that the interleaving gain is independent of the inner code parameters as long as the inner code is recursive. IV. CPM MODULATOR AS RECURSIVE INNER CODE Consider the low-pass equivalent representation of -ary CPM schemes in Section II. It was shown in [20] that an equivalent way to represent the CPM signal is in terms of the physical tilted phase, defined as . The CPM modulator can then be represented as a finite and state state machine with inputs as shown below

(5) , . It was shown in [20] that is time-invariant and can be synthesized as a differential encoder with arithmetic operations in the ring of integers modulo . where is given by That is, Note that since

Fig. 2.

Equivalent convolutional code for CPM.

consider the combination of the trellis code, interleaver, and the recursive CPE as an SCCC. We should be able to achieve a large interleaving gain similar to conventional SCCCs. In the rest of the paper, we follow this approach. V. PAIRWISE-ERROR PROBABILITY The combination of the convolutional code, interleaver and the modulator can be considered as an equivalent block code, whose code words are transmitted over the correlated fading channel. Therefore, we need to derive the pairwise-error probability for a correlated fading channel with CPM modulation. In this section, we derive this pairwise-error probability for a coherent receiver that performs ideal ML sequence detection. When the pairwise-error probability is combined with the union bound developed in Section VI, upper bounds on the BER of the overall concatenated code can be derived assuming ML detection. The pairwise-error probability was derived for binary modulation schemes in [23]–[25] and for TCM schemes in [26]. Here, we extend the analysis to CPM signals by noting that CPM can be considered as a form of coded modulation. Since we have considered the CPE as part of the SCCC, the CPM modulation is memoryless. This permits the use of the derivation in [26] for the case of TCCPM also. Here, we omit the detailed derivation and only show the changes that are required to use the results in [25] and [26]. and be the CPM signals correLet and the erroneously sponding to the transmitted code word decoded code word , respectively. Then

(6) where denotes modulo- addition. It was also shown in [20] , is a memoryless mapper that for on to any one of the possible CPM signals that maps from the CPM signal set . From (5) and (6), we can see that the CPM modulator is equivalent to a recursive rateconvolutional encoder followed by a memoryless mapper. The convolutional code, also called the continuousratephase encoder (CPE) is shown in Fig. 2. When designing TCCPM schemes, the inherent memory in the CPM scheme or, equivalently, in the CPE should be exploited. By carefully matching the trellis code to the CPE, the minimum Euclidean distance can be increased. Such approaches have been undertaken in [2], [5], [21], and [22]. However, none of these approaches have tried to exploit the recursive nature of the CPE. From Section III, we can see that if an interleaver is used between the trellis code and the CPM modulator, we can

(7) Let Then

and denote can be expressed as

and

, respectively.

(8) By using (7), the Euclidean distance between over one time interval

and is given by

(9)

NARAYANAN AND STÜBER: PERFORMANCE OF TCCPM WITH ITERATIVE DEMODULATION AND DECODING

Let be a vector of symbol positions at differs from (modulo ). Then, the time diwhich and is . Further, let versity between be the vector of channel gains corresponding and differ to the time instants for which and be a diagonal matrix with entries . In order to compute the pairwise-error probability between and , we first consider the received signals and corresponding to the transmitted signals in the absence of noise given by

The squared Euclidean distance between and is , where . For a coherent receiver with perfect knowledge of and the noise power spectral density , the pairwise-error probability is given by

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in [25] and [26]. This is due to the fact that the real and imaginary parts of the fading coefficients are normalized to have energy of 1/2 each and that entries of in this paper are 1/2 the value of those in [26]. In Section VI, we will use the exact pairwise-error probability to compute a lower bound on the BER and the Chernoff bound to compute an upper bound the BER. VI. PERFORMANCE ANALYSIS Consider the TCCPM system with bit interleaving shown in corresponding to Fig. 1. Let us assume that the CPM signal is transmitted. Let denote the th the input sequence in exactly epochs. If CPM signal which differs from corresponds to the input sequence , then the difference seis called the error sequence. Let quence denote the input weight of the error sequence. The union bound on the probability of bit error for correlated flat fading channels is (13)

(10) (14) where

The above integral can be evaluated by using the result of Pierce and Stein [25]. In order to evaluate (10), we first evaluate the , where poles and zeros of the function is the identity matrix and denotes the correla. If tion matrix with and denote the poles and residues of , then the pairwise-error probability is given by (the reader is referred to [25] for details of the derivation)

(11)

As can be seen from (11), in order to evaluate the exact pairhave to be wise-error probability, the poles and zeros of evaluated first. In general, for large , evaluating the poles and zeros is a numerically unstable problem. However, a Chernoff upper bound on the pairwise-error probability can be found as in [26]. For the Rayleigh fading channel, the Chernoff bound is given by

denotes the pairwise-error probability bewhere tween two sequences that differ in time instants, with and defined in Section V. It should be emphasized here that and depend on the transmitted code word and the reference . In general, for CPM schemes code word depends on and, hence, the transmitted code word cannot be assumed to be the all-zeros code word. Since depends on the particular and , to obtain the average performance we need to average over all possible and . This is computationally intensive and, hence, we introduce an approximation that results in a looser bound that can be more easily evaluated. A. Approximation for The components of the matrix are the squared Euclidean distances along each branch of an error event. In general for CPM schemes, for a given error sequence, the squared Euclidean distance depends on the exact information sequence transmitted and, hence, the union bound becomes difficult to with compute. A looser bound can be derived by replacing where the entries of are given by (15) which is fairly easy to evaluate.

(12)

The exact pairwise-error probability and the Chernoff bound differ slightly from their counterparts in [25] and [26] in that in the denominator instead of as (11) and (12) have a term

B. Approximation for An upper bound can be obtained on the pairwise-error prob. The worst case ability by assuming the worst case corresponds to the case when the correlaare the largest entries. tion corresponding to the entries of Since the autocorrelation of the channel

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,

is the lag corresponding to the th largest value . With these two approximations, we can express the upper as bound on

of

(16) is the pairwise-error probability where between two CPM signals with a time diversity of , and is corresponding to the worst case and , and the number of input error sequences with input weight , that results in a CPM signal having a time diversity of from the transmitted signal. This expression is still difficult to compute. However, we can reduce the complexity by noting that for the CPM schemes considered in this paper, the time diversity between two CPM signals is independent of the reference signal and depends only on the input error sequence. Therefore, we can assume that the reference signal corresponds to the all-zero input sequence and, hence, the upper bound can be written as (17) Similarly, if the number of error sequences with input weight and squared Euclidean distance is independent of the reference sequence, the union bound on bit-error probability for AWGN channels can be expressed as (18) where

can be evaluated from (4).

C. Uniform Interleaving Under the assumption of a uniform interleaver, we can obtain by using (4) as

of error sequences at a given Euclidean distance and time diversity from a reference path is independent of the reference path and, hence, no approximation or averaging is required. For quaternary CPFSK with , the time diversity is independent of the reference path, while the Euclidean distance is dependent on the reference path. Equation (17) is still difficult to evaluate for all possible . Consequently, we can derive an approximation, namely the truncated union bound by considering only terms corresponding to small values of . Such a bound will be reasonably tight for high . We now discuss the evaluation of the truncated values of union bound through the following two examples. Example 1: Consider a rate-1/2, two-state outer code with and an MSK modulator. generator polynomials The transfer function of the MSK modulator is given by (20) where the powers of refer to normalized squared Euclidean distances instead of Hamming distances. We can see that all error events are of input weight 2, corresponding to input error . The time diversity of the sequences of the form and the minimum time diversity is 2, correerror event is . sponding to The free distance of the outer code is 3. However, no input error sequence of weight 3 will produce a finite time-diversity error event for the CPM modulator. An input sequence produces a codeword . With a uniform interleaver, can be interleaved into a sequence of the , which produces two error form and . Clearly, the minimum events with time diversity . The corresponding is time diversity occurs when . If we let , we see given by for that the possible values of are , and the corresponding is 2 and for all . The worst case occurs when and is . Example 2: Consider a rate-1/2, two-state, outer code with and quaternary CPFSK modugenerator polynomials . Assume that the outputs of the outer encoder lator with are multiplexed and the multiplexed bit sequence is interleaved. The transfer function for the CPFSK modulator in terms of the input weight and error event length is given by

(19) is the number of error events for the CPM modwhere ulator with input error weight , and time-diversity that can be expressed as the concatenation of simple error events. To de, we first need to evaluate the error enumerating termine for the CPM modfunction ulator which enumerates all possible error events for the CPM modulator with input weight and time diversity . This can be done by using the state transition diagram and conventional transfer function bounds [27]. , where is the unit For CPFSK, ) step function. In this paper, we will consider MSK ( . It can be seen that and quaternary CPFSK with is independent of the reference sequence for both of the CPM schemes considered here. For MSK, the number

We have not shown the Euclidean distances as it complicates the expression due to the noninteger Euclidean distances, and, also since we can use the worst case distance in computing the upper bound. As for Example 1, we can compute the truncated union . bound considering all D. Block Interleaving In [7] and [9], a block interleaver was used between the TCM outer code and the CPM modulator to make the channel appear block interleaver, uncorrelated to the TCM code. With a the fades corresponding to two adjacent coded bits can be guarseconds and, hence, the block interanteed a separation of leaver decorrelates the channel. We now evaluate the BER upper

NARAYANAN AND STÜBER: PERFORMANCE OF TCCPM WITH ITERATIVE DEMODULATION AND DECODING

bound for a block interleaver by considering the TCCPM scheme as an equivalent block code. Consider the system in Example 1. An input sequence produces an outer codeword of the , which after multiplexing results in form or, equivalently, . This sequence . The output is block interleaved into of the MSK modulator corresponding to this sequence is of infinite weight. However, now consider a sequence of the form . This results in an outer code word of the form , which after interleaving produces . The sequences , , and independently produce error events each having time diversity 2. Notice that produce all input sequences of the form small time-diversity error events for all and . Consequently, the multiplicity of this error event is . This is precisely the problem with block interleaving. Although the minimum time diversity is high, the multiplicity is very high also. For this error . The correlation matrix event, can be constructed using and the pairwise-error probability evaluated from (12). The minimum time diversity occurs for , and equals 6 for this example. We can compute the upper bound by considering error events corresponding to or the truncated union bound by considering . To compute a lower bound, we consider only one term in the union bound. That is, we consider only one error event among all possible error events. Although we obtain a lower bound by considering any one error event, a tight bound is obtained by considering the error event that contributes most to the probability of error. The lower bound is then given by the exact pairwise-error probability for this error event. To compute a lower bound for AWGN channels, we consider only the error . That is, the error event corresponding to an event with . If the squared input error sequence of Euclidean distance of the error event corresponding to this error is , then, for AWGN channels event with

(21)

results in the The input error sequence error event with minimum time diversity also and, hence, the lower bound for Rayleigh fading channels can be found using (11) where and correspond to the input error sequence . We can extend the same argument to Example 2. For the CPFSK modulator, note that an input error sequence of the form or produces an error event of time diversity 2 for any reference sequence. Consider an input se. The outer encoder output corresponding quence to this sequence after multiplexing is , which after interleaving results in . This corresponds to three error events at the input to the CPM modulator and, hence, of the form results in a time diversity of 6. The same argument applies to which results in the input sequences of the form

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three error events of the form . In general, input sequences and of the form produce error events with time diversity for all and . The lower bound on is computed by cononly, that is the input error sidering one error event with and then using (21) and (11) for AWGN sequence channels and Rayleigh fading channels, respectively. For other classes of CPM signals, it may be difficult to determine the minimum diversity error event. However, for any full response CPFSK scheme, it can be seen from Fig. 2 and (6) that a weight-2 input sequence produces an error event of time diversity 2. If the free distance of the outer decoder is , then an input sequence which is the minimum free distance error event . of the outer code repeated twice results in time diversity of Therefore, the minimum time diversity for full-response CPFSK . can be upper bounded by E. Pseudorandom Interleavers The union bound derived by using the uniform interleaver is typically loose, and, much better interleavers can be designed. The pseudorandom interleaver obtained by generating random without replacement typically numbers between 0 and performs reasonably well. While for parallel concatenated systems, the use of such interleavers usually mimics the perfor, for serial conmance of the uniform interleaver at high catenated schemes, pseudorandom interleavers often perform significantly better than the bounds predicted by the uniform interleaver. The spread random interleaver, also known as the -random interleaver [28] is a pseudorandom interleaver that . To derive the union bound on the performs well at high BER of such systems, we need to evaluate the distance spectrum of the SCCC by considering the exact interleaver being used. It is almost impossible to compute the entire distance spectrum for reasonably large block sizes. We can compute a lower bound on the BER for AWGN channels by computing the minimum distance of the code. One cannot guarantee that computing the minimum distance can be accomplished easily, because we may have to search for an unacceptably long time to find the minimum distance codeword. To circumvent these problems, we have computed a truncated version of the distance spectrum by considering only those input sequences that are most likely to produce the error events that dominate the BER performance. For the code in Example 1, weight-1 input sequences produce a weight-3 outer code word and, since weight-3 input sequences to the inner encoder (CPM modulator) typically produce error events with very high time diversity. However, since the data block is terminated after input bits, sometimes such weight-3 input sequences are artificially terminated at the end of the block and, hence, produce error events with small time diversity. However, weight-2 input sequences produce even weight code words and, hence, after interleaving they can produce input sequences to the CPM modulator that produce error events with small time diversity. Therefore, we have considered all possible weight-2 input sequences and evaluated the resulting distance spectrum. We have and discarded error events with time diset a threshold . The choice of depends on the particular versity was set to system being considered. For Example 1 and 2,

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30. The truncated union bound on the BER can be computed by using (16) and considering only the above mentioned error events. For Example 2, we used the approximation in (15) to compute the upper bound. It should be noted that the upper bound computed this way is only approximate in that it is not always valid. The actual error event corresponding to minimum time diversity may correspond to an input weight larger than 2. However, the probability of such an event occurring is very small and, hence, the bounds developed above will be useful in most situations. We can compute a lower bound on the BER performance for AWGN channels by considering only the codeword with the least squared Euclidean distance corresponding to weight-2 input sequences. For the Rayleigh fading channel, the lower bound is computed by considering only the error event with lowest time diversity. VII. RECEIVER STRUCTURES A. Coherent Receiver The coherent receiver considered is identical to the one in [29]–[31]. It is assumed that perfect knowledge of the channel and the noise power spectral density is available gains at the receiver. The receiver then employs a soft-output demodulator and a soft-output decoder in an iterative fashion. During the th iteration, the demodulator produces log-likelihood ratios (LLRs) for each bit in the sequence , given by

based on the received signal and the a priori probabilities procan be written vided by the outer decoder. The LLRs as (22) is deinterleaved and used as The extrinsic term input to the outer decoder that produces a posteriori LLRs based on the code constraints of the outer code. The can be expressed as LLR

decoder can be thought of as applicable to the iterative receiver also at high . B. Non-Coherent Receiver From Sections VI and VIII we can see that a very significant interleaving gain is possible for the TCCPM scheme and, hence, . At such low TCCPM schemes can operate at very low , channel parameter estimation becomes very difficult. Non-coherent receivers do not require channel state informa. Itertion and, hence, are attractive for use at such low ative noncoherent detection is a rather new topic and there is very little work in this area. An iterative noncoherent receiver structure for iterative demodulation and decoding of DPSK signals in AWGN channels was proposed in [14]. The overall receiver structure is identical to the coherent receiver except that a noncoherent soft-output demodulator is used instead of the during iteration . The BCJR algorithm to generate resulting receiver structure is identical to the one in [14]. Unlike in the case of coherent reception, the BCJR algorithm cannot be used for optimum soft-output demodulation of the CPM signals. This is mainly due to the fact that the optimum metric for noncoherent demodulation is not additive and, hence, cannot be used in the BCJR algorithm. In general, noncoherent demodulation of CPM signals is accomplished either by using a suboptimum metric in conjunction with an optimum trellis-based algorithm or by using the optimum metric with a suboptimum trellis-based algorithm. One such approach is block detection, where a small window of data is used to generate soft-output instead of the entire data sequence [15]. A suboptimal noniterative noncoherent demodulator using this principle was used in [9] for Rayleigh fading channels. Here, we derive the optimum soft-output noncoherent demodulator that does not require channel state information (neither amplitude nor phase) for CPM signals for Rayleigh fading channels using block detection. Our approach and our derivation here is based on noncoherent block detection of CPM signals proposed by Simon and Divsalar in [15]. denote the transmitted CPM signal corresponding Let to the sequence . Then, the received signal is given by (24)

(23) remain constant over the interval . Further, let and denote the magnitude of the channel gain and phase of the channel gain, , the received signal over the interval respectively, and . , the noncoherent reTo generate the LLRs for ceiver observes the received signal over the window and evaluates the conditional for all possible data sequences probability of length , i.e., . -ary CPM there are possible In general, for values for vectors . For MSK, there are denoted by for We assume that

The extrinsic information can be expressed as and . For the coherent receiver, the functions and are implemented using the Bahl, Jelinek, Cocke and Raviv (BCJR) algorithm. At the end of each iteration, estimates are produced and the decisions are checked of the data bits for errors using a cyclic redundancy check (CRC). If the check is satisfied the iterations are stopped; otherwise the iterative iterations. Several studies have process continues up to shown that the performance of this iterative receiver is close to [19] and, hence, the that of the ML receiver for high performance bounds developed in Section VI based on the ML

and

NARAYANAN AND STÜBER: PERFORMANCE OF TCCPM WITH ITERATIVE DEMODULATION AND DECODING

. The LLR for bit

683

is then given

(28)

by where formly distributed over

. The phase and, hence [15]

is uni-

(25) (29) denotes the a priori probability for , which is where typically provided by the soft-output outer decoder. Due to the presence of the interleaver between the outer code and the CPM modulator, it is reasonable to assume that

is the zeroth-order modified Bessel function of the where and, hence first kind. For the AWGN channel,

(30) (26)

With iterative decoding, the soft-output demodulator should generate and pass only the extrinsic information, which . The exdoes not include the contribution of trinsic information can be obtained directly by using instead of in (25). , the conditional Using to denote the CPM signal can be evaluated as using probability

in [32] can be The piecewise-constant approximation for the phase used to implement this receiver. To evaluate th epoch, , of the CPM signal at the is indeshould be known. However, it can be seen that and, hence, can be set to zero. pendent of can be found in [15]. Techniques to recursively evaluate For the flat Rayleigh fading channel, (29) should be avergiven aged over the probability density function (pdf) of and, therefore by

(27)

(31)

The above integral can be rewritten as

(32) where

is a constant and

The term in the above equation depends only on the received signal and, hence, is a constant. Therefore The integrand in (32) is a Rician pdf that integrates to unity. Thereforea

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Fig. 3. BER performance of trellis-coded 4-ary CPFSK on an AWGN channel; h = 1=4; N = 512.

Fig. 4. Comparison of coherent and noncoherent receivers for MSK on AWGN channel; N = 1024.

(33)

cancels out in the evaluation of (25) and, The constant is evaluhence, need not be evaluated. Once ated, (25) can be used to evaluate the LLRs. An important feaneeds to be comture of this technique is that puted only during the first iteration. In the following stages, only needs to be evaluated and the LLRs updated accordat most involves only mulingly. Computing tiplications and, therefore, the complexity of this noncoherent receiver is rather low. VIII. RESULTS AND DISCUSSION A. AWGN Channel The performance of a TCCPM scheme with a rate-1/2 nonrecursive convolutional outer code with generator polynomials and quaternary CPFSK with is shown in Fig. 3. The truncated union bound evaluated by trunand 1024. cating the summation in (18) is plotted for For comparison, the performance of the optimum four-state convolutional code with CPFSK and without interleaving from [3] is also shown. Note that the BER upper bounds derived here are drastically different from those in [3]. It should be emphasized here that the plotted upper bound assumes a uniform interleaver. is The performance with an -random interleaver for also plotted in Fig. 3. The flattening of the BER curve is not very prominent for the -random interleaver, due to the increased free distance. The performance of the noncoherent receiver is shown in for a block length of . Fig. 4 for The CPM scheme is MSK and the generator polynomials of the . The performance of the coouter code are herent receiver for the same block length and interleaver is also shown for comparison. Observe that the noncoherent receiver performs to within 1 dB of the coherent with receiver. The maximum number of iterations was restricted to

Fig. 5. BER “upper” bound for trellis-coded MSK and 4-ary CPFSK (h = f T = 0:01.

1=4) on a Rayleigh fading channel with uniform interleaver;

8. The complexity of the noncoherent receiver may be slightly larger than that of the coherent receiver depending on the actual number of iterations used. B. Rayleigh Fading Channel The truncated BER union bound on the performance of the system in Example 1 is shown in Fig. 5 for a block interleaver and a uniform interleaver of different sizes. The truncated BER upper bound with the uniform interleaver is almost identical for . Notice that a huge improveMSK and CPFSK with ment is possible for the uniform interleaver by increasing . The performance of the system with block interleaving is independent of . These bounds suggest that uniform interleaving is significantly better than block interleaving. The performance of the system in Example 1 with is shown in Fig. 6 for a block interleaver and the pseudorandom interleaver. The truncated BER union bound, and the simulation results are also shown. Observe that the bounds are tight

NARAYANAN AND STÜBER: PERFORMANCE OF TCCPM WITH ITERATIVE DEMODULATION AND DECODING

Fig. 6. BER performance of trellis-coded MSK on a Rayleigh fading channel; f T = 0:01; N = 512.

Fig. 7. BER performance of trellis-coded MSK on a Rayleigh fading channel with an S -random interleaver; S = 15, N = 512.

at reasonably high . Similar results for an -random inis shown in Fig. 7. At low , the terleaver with bounds are loose. Since the BER upper bound is computed using a truncated union bound, the simulation results seem worse than . However, if we consider all the the upper bound at low error events, the union bound will be a true upper bound. Observe that the bounds computed by using the uniform interleaver are pessimistic. For MSK the approximations for and introduced in Sections VI-A and VI-B need not be used. This confirms that the looseness in the bounds with the uniform interleaver is mainly due to the assumption of a uniform interleaver. Random and pseudorandom interleavers perform significantly better than the uniform interleaver. The system in [7] and [8] uses symbol interleaving as opposed to bit interleaving. The performance of symbol and bit is shown in Fig. 8. interleaving for 4-ary CPFSK with It can be seen that bit interleaving significantly outperformed

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Fig. 8. Comparison of symbol and bit interleaving for 4-ary CPFSK on Rayleigh fading channel; h = 1=4, f T = 0:01, N = 1024.

Fig. 9. Comparison of iterative and noniterative receivers of equal complexity for MSK; Rayleigh fading; f T = 0:01, N = 1024.

symbol interleaving. This is due to the additional diversity advantage gained by interleaving coded bits rather than grouping bits in to symbols and then interleaving the symbols. The performance of the proposed iterative receiver and a noniterative receiver based on [7] and [8] is shown in Fig. 9. The noniterative receiver uses the BCJR algorithm for soft-output demodulation and a Viterbi decoder for the trellis code. Both the iterative and the noniterative receivers assume perfect channel state information. Fig. 9 compares the performance of the iterative and the noniterative receiver for the same latency and almost equal complexity. Specifically, a TCCPM scheme with a two-state outer with an iterative receiver and a TCCPM with a 64-state outer code and a noniterative receiver is compared for the same block length of 1024 bits. Maximum free distance convolutional codes are selected for the outer codes. Assuming that the complexity of the BCJR algorithm is approximately four times that of the Viterbi algorithm for the same number of states, and that four iterations are required for the iterative

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REFERENCES

Fig. 10. Comparison of coherent and noncoherent receivers for MSK; Rayleigh fading, f T = 0:01; N = 512.

receiver, the complexity of the noniterative receiver is slightly higher than that of the iterative receiver. Fig. 9 shows that iterative demodulation and decoding can outperform a noniterative , receiver of same complexity. For a block length of , the iterative receiver is better by about 2 dB. and The performance of the iterative receiver will be better if the block length is increased, while that of the noniterative receiver remains unchanged. The performance of the noncoherent receiver on a flat Rayleigh fading channel is shown in Fig. 10 for the system in Example 1. The performance of the coherent receiver is also shown for comparison. Observe that the performance of the noncoherent receiver is within a few decibels of the coherent receiver. The performance of the noncoherent receiver for the fading channel did not appear to improve after the first 2 or 3 iterations. This is mainly due to the fact that only a finite samples is used to generate the window of soft-output for each coded bit, as compared to the coherent receiver which uses the entire block to update the soft-output at each iteration. IX. CONCLUSIONS We have shown that iterative demodulation and decoding can significantly improve the performance of TCCPM on AWGN and flat fading channels. It is quite difficult to evaluate the performance exactly, because of the difficulty in computing the distance spectrum for a given interleaver. Approximate upper , bounds are developed in this paper that are tight at high and suggest that significantly better performance than conventional noniterative demodulation and decoding is possible. Simulation results have confirmed our claims. The optimum iterative noncoherent receiver based on the block detection principle was derived and shown to perform within 1 dB of the performance of the ideal coherent receiver on AWGN channels. The performance on flat Rayleigh fading channels was shown to be within a few decibels of that of the ideal coherent receiver.

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[27] A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding. New York: McGraw-Hill, 1979. [28] S. Dolinar and D. Divsalar, “Weight distributions for turbo codes using random and nonrandom permutations,” JPL, TDA Prog. Rep. 42-121, Aug. 1995. [29] M. Gertsman and J. Lodge, “Symbol-by-symbol map decoding of CPM and PSK signals on Rayleigh flat-fading channels,” IEEE Trans. Commun., vol. 47, pp. 788–799, 1997. [30] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Serial concatenation of interleaved codes: Design and performance analysis,” IEEE Trans. Inform. Theory, vol. 42, pp. 409–429, Apr. 1998. [31] K. R. Narayanan and G. L. Stüber, “A serial concatenation approach to iterative demodulation and decoding,” IEEE Trans. Commun., July 1999. [32] E. K. Hall and S. G. Wilson, “Turbo codes for noncoherent channels,” in Proc. Communications Theory Mini-Conf., GLOBECOM, Nov. 1997, pp. 66–70.

Krishna Narayanan was born in Madurai, India. He received the Ph.D. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in December 1998. He is currently an Assistant Professor in the Department of Electrical Engineering at Texas A&M University, College Station. His interests are broadly in communication theory with emphasis on coding and signal processing for wireless communications and digital magnetic recording. He is currently an Associate Editor for IEEE Communications Letters and the recipient of the CAREER Award from the National Science Foundation in 2001.

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Gordon L. Stüber (S’81–M’82–SM’96–F’99) received the B.A.Sc. and Ph.D. degrees in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 1982 and 1986, respectively. Since 1986, he has been with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, where he is currently a Professor. He spent the summers of 1993 and 1996 with Nortel Networks, Richardson, TX, and Ottawa, ON, Canada, respectively. He is also the Vice President for Research with WiLAN Wireless Data Communications (WWDC), Inc. He has over 140 refereed publications in books, journals, and conferences. He is author of the textbook, Principles of Mobile Communication (Norwell, MA: Kluwer Academic Publishers, 1996, 2nd edition 2000). His research interests include wireless communications and communication signal processing. Dr. Stüber served as Technical Program Chair for the 1996 IEEE Vehicular Technology Conference (VTC’96) and the 1998 IEEE International Conference on Communications (ICC’98). He is a past Editor for Spread Spectrum with the IEEE TRANSACTIONS ON COMMUNICATIONS (1993–1998). He is currently a member of the IEEE Communication Society Awards Committee, a member of the IEEE Vehicular Technology Society Board of Governors, Chair of the Fifth IEEE Workshop on Multimedia, Multiaccess and Teletraffic for Wireless Communications (MMT’00), and Chair of the Fifth YRP International Symposium Wireless Personal Multimedia Communications (WPMC’02). He was co-recipient of the Jack Neubauer Memorial Award for the Best Paper of the Year published by the IEEE Vehicular Technology Society on the subject of Vehicular Technology Systems.