Turbo-coded Multi-alphabet Binary CPM for Concatenated Continuous Phase Modulation

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1 no symbol mapping is required, and also the inner ode an be ombined with the CPE in the trellis oded modulation sense [4]. Simulation shows that the use of non-binary outer enoder an give typially.3db improvement for 4CPFSK, h =/4. A suboptimal design tehnique would be to maximize the minimum Eulidean distane of the ombined inner ode. Disadvantages of this sheme are ) optimization of the outer ode is diffiult, and 2) non-binary odes are more diffiult to handle. In this paper, we adopt the basi serial onfiguration in Fig. but with a turbo ode, i.e., parallel onatenated onvolutional ode (PCCC), as the outer ode. Also, a modified binary CPM enoder is used as the inner ode, illustrated in Fig. 2. This has the advantage of binary oding throughout, together with inreased power in the outer ode. Turbo oded CPM using soft iterative deoding has been desribed in [], [2]. However, multiple CPM hannels are required in [] to implement the turbo oded CPM. More speifially, [] onatenates two CPM hannels with a turbo ode and shows that some small improvement an be ahieved. The use of multiple CPM hannels makes it easier to onsider turbo odes. However, this approah has two drawbaks. Firstly, it ompliates the modulation sheme. Seondly, implementing multiple CPM hannels on a given frequeny band may not be realisti. The sheme in [2] is losely related to our sheme. However, we fous on binary CPM with full response whereas [2] is onerned with 8-ary CPM system with partial response. It turns out that the outer ode design depends signifiantly on different CPM systems. When employing a PCCC as an outer ode, a naive approah would be to modify a standard PCCC sheme by replaing the binary phase shift keying (BPSK) with a CPM. However, it is shown by simulation that suh an approah would not perform well. Even when an additional interleaver (alled a onatenation interleaver) is inserted before the CPM, the improvement in the performane is still limited. Motivated by this fat, the key question we address in this paper is how to make a PCCC based onatenated CPM sheme work. It turns out that this involves areful design of all the individual omponents of the system, inluding the outer ode, the inner ode and the onatenation interleaver, as desribed below. Firstly, due to the reursive property of CPM, the PCCC we use differs from a standard PCCC with BPSK modulation in that our onstituent enoders for the PCCC are nonreursive. It turns out that for serially onatenated CPM, non- This full text paper was peer reviewed at the diretion of IEEE Communiations Soiety subjet matter experts for publiation in the IEEE GLOBECOM 25 proeedings. Turbo-oded Multi-alphabet Binary CPM for Conatenated Continuous Phase Modulation Jun Ning, Minyue Fu and Graham Wade Shool of EECS, University of Newastle, NSW 238 Australia s: jun.ning@studentmail.newastle.edu.au; minyue.fu@newastle.edu.au Abstrat Constant envelope and high spetral effiieny are among the key properties whih make ontinuous phase modulation (CPM) a preferred hoie for many appliations. Numerous methods are available for designing onatenated CPM systems to ahieve additional oding gains. In this paper, we propose a new serial onatenated CPM sheme. Our design uses a multialphabet binary CPM as an inner ode, a modified turbo ode as an outer ode, and speially designed interleavers. The extrinsi information transfer (EXIT) hart tehnique has been employed to assist the outer ode design. Using soft deision deoding, we show that our design gives notieable improvements ompared with previous onatenated CPM shemes over additive white Gaussian noise (AWGN) hannel. Keywords: Continuous phase modulation; turbo ode; onatenated ontinuous phase modulation; serial onatenated ode. I. INTRODUCTION Continuous phase modulation (CPM) has several good properties whih make it a preferred modulation sheme for radio and satellite ommuniations. Among these properties are onstant envelope, spetral effiieny and inherent oding; see, e.g., [], [2]. CPM is a oded modulation sheme that an be deomposed into a reursive ontinuous phase enoder (CPE) and a memoryless mapper (MM) [3]. The CPE provides the inherent ode trellis, whih in turn depends upon the CPM design parameters. These inlude the modulation index, h =2k/p, where k and p are relatively prime positive integers. Other parameters are the frequeny pulse g(t), the duration, L, of the frequeny pulse, the size of the data alphabet, M, and the symbol period, T. Usually, M = 2, 4 or 8, and L is between and 3 symbol periods. These parameters are usually hosen to satisfy a given bandwidth speifiation, whih in turn limits the bit error rate (BER) performane of the inherent ode. A ommon solution is to improve the power effiieny by using the serial onatenated sheme in Fig. ; see, e.g., [5], [8]. If R o is the rate of the outer ode, the rate of this sheme is R = R o log 2 M bits/symbol. When the outer ode is a simple onvolutional ode, the sheme beomes the serial onatenated onvolutional oded (SCCC) system desribed in [6] and advantages an be taken from the reursive nature of the CPE. Iterative deoding an give large oding gains at the expense of the interleaver delay. Another approah is to enhane the distane spetrum of the CPM signal by ombining the CPE with an inner ode of rate R i,givingan overall rate R = R o R i log 2 M bits/symbol. Ideally, both inner and outer odes operate over the same algebrai field so that IEEE Globeom /5/$2. 25 Crown Copyright

2 This full text paper was peer reviewed at the diretion of IEEE Communiations Soiety subjet matter experts for publiation in the IEEE GLOBECOM 25 proeedings. U o Soure Outer Code Co π U I CPM CI Soure Outer DEC Fig.. π o G(D) π Iterations π CPM DEC Serial onatenated CPM sheme C o s π G(D) Pun. C o p π2 Fig. 2. Pun. C o p2 π3 π M U X Proposed onatenated CPM sheme Channel Multi- Alpha. CPM reursive onstituent enoders give a signifiant improvement ompared with reursive ones at low signal-to-noise ratio (SNR). Seondly, in order to math the binary outer ode, the inner ode (CPM) is hosen to be a binary CPM. However, to strengthen the Eulidean distane of the inner enoder, we deploy a multi-alphabet binary CPM. It is known [], [2] that multi-h CPM has better distane and memory properties than a single-h CPM. A multi-alphabet binary CPM is similar to, but somewhat more general than, a multi-h binary CPM. It offers more flexibility and gives similar improvements in these properties with same or lower omplexity. Finally, the onatenation interleaver we use is hosen to both deorrelate the inner and outer ode sequenes and maintaining the independene of the two onstituent ode sequenes in the outer ode. This is an important property for the suessful deoding of the PCCC outer ode. Indeed, although a generi interleaver may de-orrelate the inner and outer odes well, it would seriously orrelate the onstituent ode sequenes, weakening the purpose of using a PCCC. We have ompared our designs with previously available results on onatenated CPM. Simulations are onduted for AWGN hannels. Our sheme ahieves about.5 db improvement at bit error rate (BER) of 5 when overall oding rate R =/2. The rest of this paper is organized as follows: Setion II proposes the new onatenated CPM sheme. Setion III disusses the soft deision deoding algorithm used in our study. Setion IV shows some simulations and omparisons and Setion V onludes the paper. II. A NEW CONCATENATED CPM SCHEME The proposed sheme is depited in Fig. 2, where only the enoding part is shown. The outer ode is a binary PCCC. The systemati bit sequene and parity bit sequenes (possibly puntured) are sent to a onatenation interleaver π to produe bit sequene for the inner ode whih is a multialphabet binary CPM. In the following, we detail the design of eah omponent in the system. A. Inner Code Binary CPM is by far the simplest CPM sheme. It is easier to understand and simpler to implement than other CPM shemes with a higher M value. However, CPM shemes with a higher M value have the advantage of allowing the outer ode to have a lower oding rate, thus having the potential to offer a better overall oding gain. Indeed, sine the overall oding rate is given by R = R o log 2 M, a higher M permits a lower R o for the same R. The tradeoff is the design omplexity and deoding omplexity, i.e., a system with a higher M and lower R o is more omplex to design and deode. Reall that a baseband CPM signal an be expressed as where φ(t, I) =2π n k= s(t) = exp(jφ(t, I)) I k h k q(t kt), nt t<(n +)T T is the symbol period, {I k } is the sequene of M-ary information symbols taken from the alphabet set Λ = {±, ±3,...,±(M )}, M 2 is an even number, {h k } is a sequene of modulation indies and q(t) the phase pulse whih is the integrate of the frequeny pulse g(t), with q(t )=/2. A single-h CPM has a onstant h k (or h) whereas a multi-h CPM uses a periodi h k. Note that the average phase hange is given by Mh k π/2. A multi-alphabet CPM is a simple generalization of the multi-h sheme. Instead of varying h, we may vary the alphabet set, i.e., we use a periodi Λ k instead of a onstant Λ. As in the multi-h sheme, the purpose of this CPM is to inrease the Eulidean distane. But this is ahieved by varying the possible phase inrements in the alphabets rather than the modulation index. It is obvious that any multi-h CPM an be regarded as a multi-alphabet CPM. The onverse is not neessarily true. This is due to the fat that, for the multi-alphabet CPM, the differene between any two different symbols is not neessary to be an even number. Thus, multialphabet CPM shemes are more flexible than multi-h CPM shemes. We note that the multi-alphabet CPM sheme is somewhat similar to the so-alled generalized asymmetri multi-h CPM sheme in [3] where the h value depends on both the time and the value of the input symbol. The multi-alphabet binary CPM we propose to use in this paper has the following parameters: retangular frequeny pulse, h = 2k/p = /4, the alphabets alternate between Λ = {2, 2} and Λ 2 = {, 2}. It is obvious that this CPM has a periodi trellis and the ardinality of the multialphabet binary CPM (or the effetive value of M) is still equal to 2. As the value of a symbol may equal 2 or 2, the h is hosen to be /4 in order to remain the CPM bandwidth same or almost same with that of Minimum-Shift Keying () whih has a symbol alphabet Λ={, }. IEEE Globeom /5/$2. 25 Crown Copyright

3 This full text paper was peer reviewed at the diretion of IEEE Communiations Soiety subjet matter experts for publiation in the IEEE GLOBECOM 25 proeedings. Given Λ = {2, 2}, theλ 2 = {, 2} alphabet is used in preferene to the Λ = {, } in order to maximize the minimum Eulidean distane. For a full response CPM, the number of possible phase states in the time-varying phase state trellis equals p, over two adjaent symbol periods. A higher number of the states means a higher system omplexity. has 4 possible states in the trellis and its minimum Eulidean distane is equal to 2. (see [], [2]). It is verified that the multi-alphabet binary CPM above has 8 possible states and its minimum Eulidean distane roughly equals 3.. Obviously, the multi-alphabet binary CPM ahieves the higher minimum Eulidean distane at expense of the omplexity. The minimum Eulidean distane, 3., an also be ahieved by a multi-h binary CPM with same bandwidth. But the number of states required is doubled, i.e., p = 6, ompared with that of the multi-alphabet binary CPM. This demonstrates the advantage of using the multi-alphabet binary CPM when trying to inrease the minimum Eulidean distane without bandwidth expansion. It is known that the power spetrum of CPM is largely determined by the average phase hange per symbol period T, provided that L and the shape funtion are fixed [2], []. This is espeially true when the maximum phase hange is fixed. For a single-h CPM, the average phase hange is given by Mhπ/2. Sine has M = 2 and h = /2, the average phase hange is π/2 whih equals the maximum phase hange. For a multi-h binary CPM, the average phase is hπ, where h is the average value of h. For our multi-alphabet binary CPM, the average phase hange is easily verified to be 7π/6 whih is slightly smaller than the maximum phase hange, π/2. Thus, it is expeted that the generalized multih binary CPM possesses a slightly narrower bandwidth than. Indeed, simulations show that the former two shemes have very omparable power spetra (Details are not shown in this paper). B. Outer Code Design and EXIT Charts As mentioned above, the outer ode is a binary PCCC. However, there is a main differene in the onstituent enoder G(D). A normal PCCC uses a reursive G(D) to ensure that if one parity bit sequene exhibits a low weight, the other is likely to have a higher weight. Here, we find that at relatively high BER, a non-reursive ode G(D) an perform better. This may be due to the fat that for a normal PCCC the assoiated modulation is a memoryless BPSK, whereas in our ase the CPM is already a reursive enoder. In fat, the onatenation of a PCCC and CPM an be regarded as two serially onatenated odes onneted in parallel, and then merged into the inner CPM ode. Thus, the ombination of eah onstituent enoder G(D) and the CPE an be viewed as a loal serial onatenated ode and the CPE has the reursion property. To assist the design of the outer ode, we may evaluate the deoding performane of the onatenated sheme at low signal-to-noise ratios using the so-alled EXIT harts [5], [6]. With multi-alphabet binary CPM ating as the inner Sheme A :.8 multi alphabet CPM single nonreursive (7,5) Eb/No =. db.8.8 Sheme C ( proposed ) multi alphabet CPM PCCC, nonreursive (2, 3) Eb/No = 5 db.8.8 Sheme E PCCC, reursive (3, 2) Eb/No =.3 db.8 Fig Sheme B multi alphabet CPM PCCC, reursive (3, 2) Eb/No =.5 db.8.8 Sheme D single nonreursive (7, 5) Eb/No =. db.8.8 Sheme F PCCC, nonreursive (2, 3) Eb/No =. db.8 EXIT harts of onatenated CPM systems ode, we onsider three strutures for rate-/2 outer odes: a single onvolutional ode, a PCCC with a reursive onstituent ode and a PCCC with a nonreursive onstituent ode. These shemes are referred to as shemes A, B and C, respetively. For PCCC, the first onstitute ode is terminated whereas the seond one is not, and the parity bits are puntured to give an overall ode rate of /2. The interleaver in Sheme A and the interleavers π o in Shemes B and C are S-random. The onatenation interleavers for Shemes B and C will be disussed shortly. The input information blok has 2, bits. The EXIT harts shown in Fig. 3 are omputed using the approah in [4] whih is based on [5], [6], [7]. For eah sheme, the outer ode is optimized by varying its generator polynomial and memory size so that the E b /N threshold for deoding onvergene is minimized. For Sheme A, the best outer ode is a 4-state nonreursive onvolutional ode with generator polynomial (7, 5) (otal). For Sheme B, the optimal outer ode is a 2-state reursive onvolutional ode with generator polynomial (3, 2). These two shemes have onvergene thresholds around. db. For Sheme C, the threshold an be minimized to about 5 db when the outer ode is a PCCC with onstituent ode being the nonreursive onvolutional ode (2, 3). For omparison purposes, we also plot the EXIT harts in Fig. 3 for the shemes with as the inner ode. Replaing the multi-alphabet CPM by in the shemes A, B and C, we obtain the shemes D, E and F. We an see learly from IEEE Globeom /5/$2. 25 Crown Copyright

4 This full text paper was peer reviewed at the diretion of IEEE Communiations Soiety subjet matter experts for publiation in the IEEE GLOBECOM 25 proeedings. Fig. 3 that the proposed sheme (sheme C) offers the lowest (best) onvergene threshold. C. Conatenation Interleaver The primary role of the onatenation interleaver π is to de-orrelate the inner ode from the outer ode, thus giving some interleaving gain. However, we have observed that a generi interleaver does not work effetively. Indeed, if the two onstituent odeword sequenes of the outer ode are interleaved with eah other, they will be strongly orrelated after the inner oding. This would ause a serious problem for the outer deoding beause a PCCC heavily relies on the independene of the onstituent odeword sequenes. Due to the observation above, the onatenation interleaver is hosen to inlude three sub-interleavers (π,π 2 and π 3 ), one for the systemati bit sequene and one for eah of the parity bit sequenes. The three interleaved bit sequenes are then multiplexed together as the input to the inner ode. The best multiplexing turns out to be the one whih starts with the first parity bit sequene, followed by the systemati bit sequene, and then by the seond parity bit sequene. Obviously, the two parity bit sequenes an be swapped. The sub-interleavers are hosen to be S-random. III. DECODING We employ iterative soft deision deoding in this paper. The basi idea of deoding follows from [7], [9]. The unique feature of our deoding problem is that there are effetively three onstituent enoders in the system: two from the PCCC and the third one being the inner ode. It turns out that the passing of extrinsi information needs to be arefully managed in order to yield a good deoding result. The deoding proedure is depited in Fig. 4. In the diagram, r is the reeived signal, P i ( ) and P o ( ) represent the input and output probability vetors of a deoder, respetively, and P e ( ) represents an extrinsi probability vetor. The a priori probability P i (C I ) is omputed using r. The initial value for P i (U I ) is a vetor of /2. These two piees of information are fed into the CPM APP deoder whih produes the extrinsi probability Po e (U I ). This information is deinterleaved to produe the probability P i (C o ). The probability P i (U o ) is set to be a vetor of /2 (unless U o is known to be biased). The two probabilities, P i (C o ) and P i (U o ),now at as the aprioriinformation for the PCCC APP Deoder whih produes two outputs, Po e (C o ) and P o (U o ). The former is interleaved to give the probability P i (U I ) so that iterative deoding an ontinue. The PCCC APP Deoder, whih is inside the dashed box in Fig. 4, is detailed in Fig. 5. There are two onstituent deoders, DEC and DEC2, whih orrespond to the two onstituent enoders. The extrinsi information from the CPM APP Deoder, i.e., the probability vetor Po e (U I ), is first deinterleaved to separate the systemati and parity omponents, i.e., P i (Cs o ), P i (Cp) o and P i (Cp2). o The first probability vetor, P i (Cs o ), is then multiplied by the deinterleaved probability vetor Po2(C e s o ), whih is the extrinsi information from DEC2 r P i(c I ) P i(u I ) CPM APP DEC Po e (U I ) Û o HARD Po(U o ) DECISION Po e (U I ) P i(cs o ) π P i(c o p2) Fig. 4. P i(c o p) πo Fig. 5. π P i(c o ) PCCC APP DEC π P i(u o ) Iterative soft deision deoding DEC π o DEC2 Po(C e s o ) Po e (Cp) o P e o2(c o s ) P e o (C o p2) P e o (C o ) π Iterative soft deision deoding: The PCCC part P o(u o ) P i(u I ) in the previous iteration. This result and the probability vetor P i (C o p), ating as the aprioriinformation for DEC, are fed into DEC to give the extrinsi information, i.e, the two probability vetors P e o(c o s ) and P e o (C o p). The deoder DEC2 works in a similar way. Then, P e o(c o s ) and the deinterleaved P e o2(c o s ) are multiplied together. The result is ombined together with P e o (C o p) and P e o (C o p2) and they are interleaved to produe P i (U I ) whih is the aprioriinformation about U I for the CPM APP Deoder in the next iteration. By one iteration we mean that the extrinsi information passes through the three onstituent deoders without repeat. IV. SIMULATIONS AND COMPARISONS The proposed sheme in Fig. 2 has been simulated for the AWGN hannel. The onstituent enoder of the PCCC is a 2- state non-reursive enoder with generator polynomial [, + D] or (2, 3) in otal representation while the inner enoder is a 8-state multi-alphabet binary CPM as desribed earlier. The parity bit sequenes of the PCCC are puntured to give the oding rate R o =/2. Coherent demodulations are assumed. The input blok length is 24 and iterations are used (further iterations will only give negligible improvements). The simulation result is shown in Fig. 6 (the lowest urve). For omparison purposes, we have also simulated shemes A, B, E, F. The simulation result of sheme D is shown in Fig. 7. It is lear from Fig. 6 and Fig. 7 that the proposed sheme has the best performane. This is in line with our earlier EXIT hart analysis. Fig. 7 also ompares the simulation result of the proposed sheme with those published in [5], [8] and []. The sheme IEEE Globeom /5/$2. 25 Crown Copyright

5 This full text paper was peer reviewed at the diretion of IEEE Communiations Soiety subjet matter experts for publiation in the IEEE GLOBECOM 25 proeedings. in [5] is same with sheme D. Note that both [5] (Moqvist) and [8] (Narayanan) use the (7, 5) (otal) ode as the outer ode and as the inner ode, whereas [] ( two hannel approah) uses two onvolutionally oded CPM hannels. All the shemes onsume omparable bandwidths with the same oding rate and input blok length. We observe that the proposed sheme outperforms the shemes in [5] and [8] by about.5 db at BER of 5. When ompared with [], our sheme also gives db improvement at BER of 5. However, we reall that the result of [] is ahieved using two CPM hannels whereas others use only one CPM hannel. We may also analyze the deoding omplexity by omparing the numbers of overall deoding states in different shemes. The sheme in [] has overall 6 states (8 states for eah hannel) in deoding omplexity and those in [5] and [8] have 6 states (4 in the outer deoder and 2 in the inner deoder). The proposed sheme has an overall 2 states for deoding omplexity. Therefore, the proposed sheme ahieves the performane improvement at a somewhat higher deoding omplexity than [5], [8] (but lower than the sheme in []). V. CONCLUSION In this paper, we have shown the advantages of using a parallel onatenated onvolutional ode (PCCC) and a multialphabet binary CPM as an outer ode and an inner ode, respetively, for serial onatenated binary CPM systems. Our simulations show that, in the onatenation where the modulator ontains a reursive enoder suh as CPM, using a generi PCCC as an outer ode may be inadequate. Instead, areful design of the individual omponents is required to make the onatenation work effetively. The key design rules are summarized as follows. Firstly, in the serial onatenation with binary CPM as the inner ode, non-reursive onstituent odes for the PCCC perform better than reursive odes at low signal-to-noise ratio. Seondly, the multi-alphabet binary CPM proposed in this paper gives an improved performane than in the onatenation. Thirdly, the onatenation should avoid orrelating the outer odewords from the two onstituent enoders. REFERENCES [] J. B. Anderson, T. Aulin and C. E. Sundberg, Digital Phase Modulation, Plenum, New York, 986. [2] J. G. Proakis, Digital Communiations, 4th ed., MGraw Hill, 2. [3] B. E. Rimoldi, A deomposition approah to CPM, IEEE Trans. Inform. Theory, vol. 34, no. 3, pp , 988. [4] B. E. Rimoldi and Q. Li, Coded ontinuous phase modulation using ring onvolutional odes, IEEE Trans. Commun., vol. 43, no., pp , 995. [5] P. Moqvist and T. Aulin, Serially onatnated ontinuous phase modulation with iterative deoding, IEEE Trans. Commun., vol. 49, no., pp. 9-95, 2. [6] S. Benedetto, D. Divsalar, G. Montorsi and F. Pollara, Serial onatenation of interleaved odes: performane analysis, design, and iterative deoding, IEEE Trans. Inform. Theory, vol. 44, pp , 998. [7] S. Benedetto, D. Divsalar, G. Montorsi and F. Pollara, A soft-input soft-output APP module for iterative deoding of onatenated odes, IEEE Commun. Lett., vol., pp , 997. [8] K. R. Narayanan and G. L. Stüber, Performane of trellis-oded CPM with iterative demodulation and deoding, IEEE Trans. Commun., vol. 49, no. 4, pp , 2. BER Fig. 6. BER Sheme F Sheme E Sheme B Sheme A Proposed Eb/N(dB) Performane for different ombinations of inner and outer enoders 5 SC (Moqvist) SC (Narayanan) Turbo oded (2 hannels) Proposed Eb/N(dB) Fig. 7. Performane for the proposed sheme and shemes in [5], [8], [] [9] C. Berrou and A. Glavieux, Near optimum error orreting oding and deoding: Turbo odes, IEEE Trans. Comm., vol. COM-44, no., pp , 996. [] M. R. Shane and R. D. Wesel, Parallel onatenated turbo odes for ontinuous phase modulation, in Pro. IEEE Wireless Commun. and Networking Conf., WCNC, vol., pp , Sept. 2. [] Stephen G. Wilson and Rihard C. Gaus, Power Spetra of Multi-h phase odes, IEEE Trans. Commun., vol. 29, no. 3, pp , 98. [2] A. Z. Yilmaz and W. E. Stark, Turbo oded ontinuous phase modulation, in Pro. Military Commun. Conf., vol. 2, pp , Ot. 2. [3] B. A. Dave and R. K. Rao, Generalized asymmetri multi-h phaseoded modulation for M-ary data transmission, in Pro. Canadian Conf. Elet. and Comp. Engr., 24, vol., pp , May 24. [4] M. Xiao and T. M. Aulin, Serially onatenated ontinuous phase modulation with symbol interleavers: performane, properties and design priniples, in Pro. IEEE GlobeCom, vol., pp , De. 24. [5] S. ten Brink, Convergene behavior of iteratively deoded parallel onatenated odes, IEEE Trans. Commun., vol. 49, no., pp , Ot. 2. [6] S. ten Brink, Designing iterative deoding shemes with the extrinsi information transfer hart, in Pro. AEU Int. J. Eletron. Commun., vol. 54, pp , Feb. 2. [7] B. Sanavino, G. Montorsi and S. Benedetto, Convergene properties of iterative deoders working at bit and symbol level, in Pro. IEEE GlobeCom, vol. 2, pp. 37-4, Nov. 2. IEEE Globeom /5/$2. 25 Crown Copyright