Parallel Concatenated Turbo Codes for Continuous Phase Modulation

Transcription

1 Parallel Concatenated Turbo Codes for Continuous Phase Modulation Mark R. Shane The Aerospace Corporation El Segundo, CA Richard D. Wesel Electrical Engineering Department University of California, Los Angeles Abstract There are several ways to use iterative decoding techniques with coded continuous phase modulation (CPM). This paper presents a method using parallel concatenation of convolutionally-coded CPM. All CPM schemes can be decomposed into a ring convolutional code and a memoryless signal mapper. This allows a properly designed convolutional code trellis to be combined with the CPM trellis, producing the joint trellis of a constituent code. Parallel concatenation of such constituent codes combined with iterative decoding yields very good performance. We present the results of code searches for two binary CPM schemes, each using bit interleaving and symbol interleaving. Simulation results using various length interleavers are also provided. Low complexity encoders are shown to perform better using bit interleaving, whereas encoders with higher complexity perform better with symbol interleaving. I. INTRODUCTION Continuous phase modulation (CPM) is widely used on radio channels because of its good spectral properties and ability to allow nonlinear amplifiers to be operated in saturation. CPM is by itself a form of coded modulation, due to the memory created by the continuous phase of the signal and possibly by additional memory introduced by partial response signaling. In [1] it was shown that CPM can be decomposed into a continuous phase encoder (CPE) and a memoryless modulator (MM). The CPE is a ring convolutional encoder (not necessarily binary) which produces codeword sequences that are mapped onto waveforms by the MM, creating a continuous phase signal. Once the memory of CPM was made explicit, it became possible to design trellis and convolutionally coded CPM systems ([2], [3], [4]) which allowed the trellis code and the CPE to be combined into a single joint convolutional code. Systems designed in this manner typically have larger Euclidean distances, and thus better performance, than systems designed using the traditional approach, for a given number of trellis states. To mitigate the effect of the correlation caused by fading channels, an interleaver can be placed between the trellis code and CPE [5], [6]. (An interleaver cannot be placed after the modulator since reordering the channel symbols destroys the phase continuity of the signal.) It was then recognized ([7], [8], [9]) that this serial concatenation of the trellis code with the CPE could be decoded using iterative decoding algorithms, due to the recursive nature of the CPE which serves as the inner code. In this paper we look at using convolutionally coded CPM in a parallel concatenated scheme. As with trellis coded CPM, This work was supported under U.S. Air Force Contract No. F C-0094 and in part by NSF CAREER Award CCR and the Xetron Corporation the convolutional encoder and the CPE can be combined into a single joint code. Two joint codes are used as the constituent encoders of a parallel concatenated code. This allows one to take advantage of simplifications which can result from considering a convolutional code and CPM jointly, together with the iterative decoding advantages of turbo codes. This paper is organized as follows. Section II briefly describes continuous phase modulation, and how it can be separated into a finite-state machine and a memoryless signal mapper. We also discuss the method of representing CPM waveforms in signal space through the use of an equivalent constellation. Section III describes in greater detail the system model and encoder structure. We illustrate our methods using minimum shift keying (MSK) and partial response Gaussian MSK (GMSK) to produce rate 1/2 encoders. Both symbol interleaving and bit interleaving are considered. Section IV presents the results of code searches for these modulation schemes. Good constituent codes with up to 5 memory elements are presented. Section V gives simulation results for several of these codes using information block lengths of 512, 1024, and bits and compares them with some previous results. We conclude with a few remarks in Section VI. A. Description II. CONTINUOUS PHASE MODULATION In [1] Rimoldi derived the tilted-phase representation of CPM, with the information bearing phase given by ψ(t, U) = 4πh U i q(t it ). (1) i=0 The modulation index h is equal to K/P, where K and P are relatively prime integers. U is an input sequence of (possibly coded) M-ary symbols, U i {0, 1, 2,..., M 1}. T is the channel symbol period. The phase response function q(t) is a continuous and monotonically increasing function subject to the constraints { 0, t 0 q(t) = 1, (2) 2, t LT where L is an integer. The phase response is usually defined in terms of the integral of a frequency pulse g(t) of duration LT, i.e., q(t) = t g(τ)dτ. For full response signaling L = 1, while for partial response systems L > 1.

2 Finally, the transmitted signal s(t) is given by s(t, U) = 2Es T cos(2πf 1t + ψ(t, U) + ψ 0 ). (3) The asymmetric carrier frequency f 1 = f c h(m 1)/2T, where f c is the true carrier frequency. E s is the energy per channel symbol and ψ 0 is the initial carrier phase. We assume that f 1 T is an integer; this condition leads to a simplification when using the equivalent constellation representation of the CPM waveform. B. Decomposition As shown in [1], any CPM scheme can be divided into a time-invariant finite state machine and a time-invariant memoryless signal mapper. Fig. 1 shows the CPEs for CPM schemes with L = 1 or 3 and M = P. Arithmetic is no longer done modulo-2, as for binary codes, but modulo-p, where P is the denominator of the modulation index h. Thus the CPE is a convolutional code operating on the ring of integers Z P. In order to join a convolutional code with the CPE into a single encoder, the convolutional code and the CPE must be algebraically compatible. For this reason, h and M are chosen such that M = P κ for some integer κ, often taken to be 1. This is the case considered here and thus M = P. As a result the CPEs for the CPM schemes of interest are rate 1/(L + 1). It is important to note that the decomposition is not unique. It is possible, for example, to find a feedback-free CPE which produces the same waveform. These systems are equivalent in the sense of [10]; i.e. the set of output waveforms is identical. However, since the mappings between inputs and outputs are different, they are not strictly equivalent, and the systems will have different probabilities of error. C. Equivalent Constellation For simulation and code search purposes we use the signal space representation of CPM. Complex baseband representations of the waveforms are used to convert the waveform channel into a discrete memoryless channel (DMC) [11]. Every T seconds the modulator outputs a waveform segment based on the L + 1 M-ary inputs it receives from the CPE. By assuming that f 1 T is an integer, the set of waveforms output by the modulator is the same for every symbol period. Thus there are M L+1 possible waveforms, which are in general correlated. The memoryless modulator, AWGN channel, and demodulator (see Fig. 3) can be lumped into a DMC, yielding an equivalent constellation. The waveform segments can be represented in signal space as points on a multi-dimensional sphere, and described with respect to an orthonormal basis which can be found using the Gram-Schmidt algorithm as described in [11]. Each signal is then represented by a d- dimensional vector, where d is the dimensionality of the signal space. In general a bank of 2M L correlators or filters are required by the optimal Viterbi CPM receiver [12], which gives an upper bound on d. (a) (b) Fig. 1. CPE with M = P for (a) L = 1 and (b) L = Fig. 2. MSK equivalent constellation If f 1 T was not an integer, but a rational number, the equivalent constellation would periodically cycle through a set of rotations. Fig. 2 gives a simple example of an equivalent CPM constellation for MSK. The constellation labels refer to the outputs of the CPE given in Fig. 1(a). III. SYSTEM MODEL For the remainder of this paper we consider rate 1/2 turbo codes using MSK-type modulation, i.e. binary CPM schemes with h = 1/2. Therefore M = P = 2, and the CPEs are regular binary convolutional codes. The dimensionality of the equivalent constellation is d = 2 L. Fig. 3 shows the parallel concatenation of convolutionally coded CPM. We call the cascade of the convolutional encoder (CE) and the CPE the constituent encoder, or joint code. The rate 1/(L + 1) CPE as derived in [1] has only a single input. However, in order for the CE and CPE to be combined into a single trellis, the number of CPE inputs must match the number of outputs of the CE. In order to accommodate multiple inputs, an equivalent CPE must be used, which has been called an N-consecutive CPE, or NCPE for short [13]. Thus for rate k/n CE s, an equivalent CPE of rate n/n(l + 1) is needed. Fig. 4 shows the 2CPEs which are used to construct the rate 1/2 turbo codes considered in this paper. Every nt seconds, k input bits are sent to the CE, which has m CE memory elements. The n output bits of the CE are input to the CPE, which has L memory elements. The total number of memory elements in the joint code is m = m CE + L. The n(l + 1) CPE output bits are then mapped to n constellation

3 joint code DMC CE CPE MM Demod IL n(t) n'(t) Iterative Decoder interleaver CE CPE MM Demod CPM Fig. 3. System model CE CPE Fig. 5. Rate 1/2 encoder for GMSK, L = 3 (a) of k information bits are interleaved as a single unit using a length /k symbol interleaver. With bit interleaving, the k information bits are interleaved independently of one another using k different bit interleavers of length /k. In both cases the overall interleaver length in bits is given by N = ; the difference between the two interleaver types lies in their structural constraints. Distinguishing between N and is useful in order to make comparisons with serially concatenated systems, for which N. (b) Fig. 4. 2CPE for (a) L = 1 and (b) L = 3 points, represented by n vectors of length d. White Gaussian noise with power spectral density N 0 /2 is added to each vector component. The noisy vectors are then input to the iterative decoder, which outputs estimates of the information bits. The encoder is similar in structure to the one described in [14], [15]. Fig. 5 shows an example for a CPM scheme with L = 3. Referring to the figure, the rate 2/2 CE has 4 states, bringing the total number of states for the constituent encoder to 32. The overall encoder is systematic; however, the constituent encoders themselves are not. (The constituent encoders can be thought of as systematic but with half of the systematic bits punctured.) Both the upper and lower constituent encoders require sending systematic bits for decoding purposes, but the bits should not be duplicated. The most significant bit (MSB) of the input to the upper encoder corresponds to the least significant bit (LSB) of the input to the lower encoder, and vice-versa. The example in Fig. 5 shows the state of the CPeing fed back to the CE. In an actual implementation, the state of the CPM modulator is not directly available. However, one can implement a finite state machine to shadow the content of the CPE, and use its state values for feedback into the CE. We consider two types of interleaving. Let N represent the interleaver length (in bits) and represent the number of information bits per block. With symbol interleaving, the group IV. CODE SEARCH The approach we use to find good codes for a given CPM scheme is to consider a fixed CPE and then search over all CEs of a given number of states. An alternative approach that was not considered would be to design a constituent encoder from scratch, subject to the constraint that it produce valid continuous phase output sequences when matched to an appropriate memoryless modulator. For each CPM scheme we perform two searches: one to find good codes for use with bit-interleaving, and one to find good codes for symbol-interleaving. Let d i (de si ) represent the normalized squared Euclidean distance when the bit-wise (symbol-wise) input Hamming weight is i. The distances are normalized by 2, where is the energy per information bit. We optimize the distance profile as done in [16], [14]. We successively maximize d 2 (or de s2), and minimize the number of nearest neighbors, then maximize d 3 (or de s3), and minimize its number of nearest neighbors, and so on, up to d 6. The two CPM schemes we consider are MSK (L = 1, with a rectangular frequency pulse) and GMSK (L = 3, with a truncated Gaussian frequency pulse) with BT = 1/3. In order to simplify the code search as in [4], the search is based on modified incremental normalized squared Euclidean distances (INSED). When the signals are represented as points in an equivalent constellation, the modified INSEDs are given by the minimum distances between constellation points for a given error vector. This simplifies the code search by allowing distances to be computed based on the all-zero trellis path. For MSK, the modified INSED is equal to the true INSED, but this is not the case for GMSK. Therefore the GMSK codes are not

4 necessarily optimal, and better codes may exist. Tables I-IV give the results for constituent encoders with k = 2 that produce a concatenated code of rate 1/2. Tables I-II give the results for MSK, and Tables III-IV give the results for GMSK. Up to m = 5 memory elements are considered for each modulation. When more than one code is found with the same distance profile, only one is reported. The code results are given in two parts: the first describes the codes and the second gives the distance profile for the codes. In the first part the codes are identified by the rows of their state-space matrices, using octal notation. A convolutional encoder can be described by the state- space equations s j+1 = s j A + u j B x j = s j C + u j D where s j is the 1 m state vector, u j is the 1 k input vector, and x j is the 1 n output vector, with j serving as the time index. The elements of these vectors and the matrices A, B, C, andd are binary, and addition is performed modulo- 2. The second part of each table gives the distance profile for each code, with the number of nearest neighbors for the distances in parentheses. As an example, the code given in Table IV for L = 3 GMSK with m CE = 2 has the state-space matrices A = , [ ] B =, C = , [ ] D = This encoder is shown in Fig. 5. V. SIMULATION RESULTS As stated in Section III, our simulations assume coherent demodulation. The decoder is composed of two APP modules which implement the forward-backward algorithm as described in [17]. A maximum of 10 iterations was allowed. Using the equivalent constellation allows the channel metrics to be computed in the usual way. Both upper and lower encoders are terminated to the zero state by the use of individual tail sequences. Extended spread random interleavers [15], characterized by parameters S, T, and X, are used and are described in Table V. The simulations use the codes given in Section IV. TABLE I FOR SYMBOL-WISE DISTANCE WITH MSK (L=1) 1 {3,1} {1,2} {2,5} {11,0} 2 {1,6,1} {1,2} {2,0,7} {11,0} 3 {4,2,11,5} {3,4} {0,0,2,5} {11,0} 4 {10,4,3,30,21} {23,4} {0,0,2,0,5} {11,0} DISTANCE PROFILE FOR CODES OPTIMIZED FOR SYMBOL-WISE DISTANCE WITH MSK (L=1) m CE d E s2 d E s3 d E s4 d E s5 d E s6 1 2(1) 2(1) 4(2) 4(1) 6(3) 2 6(2) 2(2) 4(2) 4(5) 6(10) 3 10(2) 2(1) 2(1) 4(4) 4(3) 4 22(1) 2(1) 2(1) 4(9) 4(2) TABLE II FOR BIT-WISE DISTANCE WITH MSK (L=1) 1 {3,1} {1,3} {2,5} {11,2} 2 {7,5,6} {7,5} {2,2,7} {11,2} 3 {5,7,6,17} {5,6} {2,2,0,5} {11,0} 4 {11,4,2,30,31} {13,30} {2,0,0,0,5} {11,0} DISTANCE PROFILE FOR CODES OPTIMIZED FOR BIT-WISE DISTANCE WITH MSK (L=1) m CE d 2 d 3 d 4 d 5 d 6 1 4(2) 2(2) 4(3) 6(6) 8(9) 2 8(2) 2(2) 4(7) 6(24) 4(2) 3 18(2) 4(2) 2(1) 2(1) 4(3) 4 34(2) 4(1) 2(1) 2(1) 4(4) The figures below compare our results with those of serially concatenated systems which use rate 1/2 outer codes and interleavers of length N = 1024 bits. For these systems = 512. On the other hand, for parallel concatenated systems the interleaver length and information block length are the same. Therefore, for constant, the serially concatenated systems inherently have twice the interleaver depth of our parallel concatenated system. The results in Figs. 6-9 show the performance of our system with = 512 (same as the serial method), = 1024 (same N as the serial method), and = Fig. 6 shows the performance of parallel concatenated convolutionally coded MSK with 4 state constituent encoders. Also shown is a result from [7] which uses serial concatenation. The system in [7] uses the 2 state MSK CPE as the inner code with a 4 state convolutional outer code, separated by a length N = 1024 bit interleaver. Fig. 6 shows that the parallel concatenated system with (approximately) the same number of states does not perform well with respect to the serial system. For the parallel concatenated system, bit interleaving has

5 TABLE III FOR SYMBOL-WISE DISTANCE WITH GMSK (L=3) 0 {5,1,1} {2,4} {112,41,21} {204,10} 1 {14,11,1,5} {16,14} {10,102,41,31} {214,10} 2 {4,20,15,1,5} {2,4} {10,0,112,41,31} {204,10} TABLE V PARAMETERS FOR EXTENDED SPREAD INTERLEAVERS /2 S T X DISTANCE PROFILE FOR CODES OPTIMIZED FOR SYMBOL-WISE DISTANCE WITH GMSK (L=3) m CE d E s2 d E s3 d E s4 d E s5 d E s (1) 3.48(1) 3.66(1) 5.31(1) 5.49(1) (1) 1.83(1) 3.66(1) 3.48(1) 5.13(1) (1) 1.83(1) 1.83(1) 3.48(1) 3.67(2) TABLE IV FOR BIT-WISE DISTANCE WITH GMSK (L=3) 0 {5,1,1} {2,4} {112,41,21} {204,10} 1 {14,5,5,1} {2,10} {10,112,51,21} {204,0} 2 {4,24,1,11,15} {12,4} {10,10,102,41,31} {204,10} DISTANCE PROFILE FOR CODES OPTIMIZED FOR BIT-WISE DISTANCE WITH GMSK (L=3) m CE d 2 d 3 d 4 d 5 d (2) 1.83(1) 3.48(1) 5.13(2) 3.66(1) (1) 1.83(1) 3.48(1) 5.13(1) 3.48(1) (1) 3.66(1) 1.83(1) 1.83(1) 3.48(1) an edge over symbol interleaving that increases with. For = 512, bit interleaving has a 0.1 db advantage, but for = 16384, bit interleaving is approximately 0.45 db better at a of. Fig. 7 shows the performance of parallel concatenated MSK with 8 state constituent encoders. For equal, the higher complexity encoders do not quite attain the performance of the serially concatenated system. However, for equal N, the parallel concatenated system is able to match the slope of the serial system, and converges at lower SNR. Thus to achieve the same performance as serial concatenation with MSK, the parallel concatenation method requires more than twice the number of encoder states and twice the, at least when using short interleavers together with small numbers of states. With the 8 state constituent encoders, symbol interleaving beats bit interleaving by about 0.25 db, and this gain is held constant for all at all SNR. Using the longest symbol interleaver allows the system to achieve a of at an just over 1 db. Figs. 8 and 9 plot simulation results for L = 3 GMSK, using 8 and 16 state constituent encoders, respectively. For the 8 state encoder, no additional memory is added to the CPE, although its distance profile is improved by allowing its state to =512, serial (from [7]) E /N b 0 Fig. 6. Simulated for rate 1/2 MSK, L=1, m CE = 1 be fed back. Both figures show for comparison a result from [8], which uses serial concatenation. The system in [8] uses a 16 state outer convolutional code with the 8 state inner CPE code, separated by a length 1024 bit interleaver. Although [8] uses a raised cosine frequency pulse instead of a Gaussian pulse, it is still a good comparison, as the two CPM schemes have very similar uncoded distances. Fig. 8 shows that for the low complexity (8 state) constituent encoders, symbol interleaving performs significantly worse than bit interleaving for all interleaver lengths. Due to the error floor of the bit interleaved parallel scheme, for s less than 5 the system from [8] achieves better performance. However, as shown in Fig. 9, when the number of states in the constituent encoders is increased from 8 to 16, the parallel concatenation method with symbol interleaving outperforms the serial method by about 0.5 db for equal, and 0.85 db for equal N. Using bit interleaving also yields better performance than the serial system, but with smaller margins. To be fair it must be noted that the complexity of the parallel system is slightly greater for one of the constituent codes. With the length = interleaver, a of is attained with an of 1.2 db. VI. CONCLUSIONS Iterative decoding of parallel concatenation of convolutionally coded CPM performs very well and is a viable alternative to serial concatenation on the AWGN channel. Based on the

6 =512, serial (from [7]) =512, serial (from [8]) Fig. 7. Simulated for rate 1/2 MSK, L=1, m CE = 2 Fig. 9. Simulated for rate 1/2 GMSK, L=3, m CE = 1 =512, serial (from [8]) Fig. 8. Simulated for rate 1/2 GMSK, L=3, m CE = 0 examples given in this paper, it is not clear if one of the methods (serial vs. parallel) will give better performance in general for a given complexity. Perhaps, as suggested by the examples, serial concatenation performs better in a lower complexity region, while parallel concatenation is superior when constituent encoders with a greater number of states must be used, e.g. when employing partial response CPM. It also appears that for parallel concatenation of convolutionally coded CPM, using bit interleaving gives better results for low complexity encoders when compared with symbol interleaving. For encoders with larger numbers of states, symbol interleaving performs best, except for cases which manifest an unacceptably high error floor. Although we use binary MSK-type modulation for illustrative purposes, the design methodology described in this paper is applicable to any CPM scheme. REFERENCES [1] B.E. Rimoldi, A decomposition approach to CPM, IEEE Trans. Inform. Theory, vol. 34, pp , Mar [2] B.E. Rimoldi, Design of coded CPFSK modulation systems for bandwidth and energy efficiency, IEEE Trans. Commun., vol. 37, pp , Sept [3] R.H.-H. Yang and D.P. Taylor, Trellis-coded continuous-phase frequency-shift keying with ring convolutional codes, IEEE Trans. Inform. Theory, vol. 40, pp , July [4] B.E. Rimoldi and Q. Li, Coded continuous phase modulation using ring convolutional codes, IEEE Trans. Commun., vol. 43, pp , Nov [5] F. Abrishamkar and E. Biglieri, Suboptimum detection of trelliscoded CPM for transmission on bandwidth- and power-limited channels, IEEE Trans. Commun., vol. 39, pp , July [6] R.W. Kerr and P.J McLane, Coherent detection of interleaved trellis encoded CPFSK on shadowed mobile satellite channels, IEEE Trans. Veh. Technol., vol. 41, pp , May [7] K.R. Narayanan, Iterative demodulation and decoding of trellis coded CPM, in Proc. of the 1999 IEEE Military Communications Conf., Oct.- Nov. 1999, pp [8] C. Brutel and J. Boutros, Serial concatenation of interleaved convolutional codes and M-ary continuous phase modulations, Annales des Télécommunications, vol. 54, pp , Mar.-Apr [9] V.F. Szeto and S. Pasupathy, Iterative decoding of serially concatenated convolutional codes and MSK, IEEE Commun. Lett., vol. 3, pp , Sept [10] G.D. Forney, Convolutional codes I: Algebraic structure, IEEE Trans. Inform. Theory, vol. 16, pp , Nov [11] J.M Wozencraft and I.M. Jacobs, Principles of Communication Engineering, Wiley, New York, [12] J.B. Anderson, T. Aulin, and C.E. Sundberg, Digital Phase Modulation, Plenum, New York, [13] H.-K. Lee, D. Divsalar, and C. Weber, Multiple symbol trellis coding of CPFSK, IEEE Trans. Commun., vol. 44, pp , May [14] C. Fragouli and R.D. Wesel, Symbol interleaved parallel concatenated trellis coded modulation, in Proc. IEEE ICC Commun. Theory Mini- Conference, June 1999, pp [15] C. Fragouli and R.D. Wesel, Turbo encoder design for symbol interleaved parallel concatenated trellis coded modulation, Accepted for publication subject to revisions, IEEE Trans. Commun. [16] S. Benedetto, R. Garello, and G. Montorsi, A search for good convolutional codes to be used in the construction of turbo codes, IEEE Trans. Commun., vol. 9, pp , Sept [17] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, A soft-input soft-output APP module for iterative decoding of concatenated codes, IEEE Commun. Lett., vol. 1, pp , Jan

7 VII. ADDENDUM Section V contains an error which was not caught in time to be corrected for the Proceedings. The variable n [7] was misinterpreted as the interleaver length, not the input block length. As a result the curve from [7] in Figs. 6 and 7 is mislabeled. It should indicate that the input block length is 1024, not 512. Corrected plots are given below. have better asymptotic (i.e. high SNR) performance. However, increasing the constraint length of the outer code, which is necessary to increase the free distance, also increases the SNR required for the iterative decoder to converge [18]. Thus for values which can be simulated, more complex outer codes with larger free distances actually perform worse. Therefore, based on the examples considered in this paper, it appears that serial concatenation is unable to match the performance achievable using parallel concatenation, in this low SNR regime. REFERENCES [18] K.R. Narayanan, Low complexity turbo equalization with binary precoding, in Proc. of ICC 2000, Jun. 2000, pp.1 5. =1024, serial (from [7]) E /N b 0 Fig. 10. Corrected version of Fig. 6. Simulated for rate 1/2 MSK, L=1, m CE = 1 =1024, serial (from [7]) Fig. 11. Corrected version of Fig. 7. Simulated for rate 1/2 MSK, L=1, m CE = 2 A few comments are in order in light of this. From Fig. 11 it is clear that the parallel concatenated system using symbol interleaving outperforms the serial system from [7] by about 0.35 db. The trade-off is that a larger number of states are required by the constituent encoders (8 and 8 vs. 4 and 2). It might then be suggested that using an outer code with a larger free distance in the serial concatenated code could be used to make up this difference. Indeed, these codes will