IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 9, SEPTEMBER 2007 1661 Woven Coded CPFSK With Hierarchical Code Structure Stefan Kempf, Member, IEEE, Sergo Shavgulidze, and Martin Bossert, Senior Member, IEEE Abstract—We introduce hierarchical woven coded continuous phase frequency-shift keying (hierarchical WCCPFSK) as the se- rial concatenation of different outer convolutional codes and inner CPFSK. We compare it to WCCPFSK with identical outer con- volutional codes. With the proposed code combinations, hierarchi- cal WCCPFSK achieves superior decoding capability. Simulations show that it performs better at medium SNRs. Index Terms—Continuous phase modulation (CPM), iterative decoding, woven codes. I. INTRODUCTION C ONTINUOUS PHASE modulation (CPM) comprises a multitude of modulation techniques with constant enve- lope and good bandwidth efficiency [1]. One family of CPM is continuous phase frequency-shift keying (CPFSK), which is especially suited for theoretical analysis of its distance prop- erties [2], [3]. Serially concatenated CPM (SCCPM) is a con- catenation of an outer convolutional encoder, a random bit in- terleaver, and an inner CPM modulator [4]. With this concate- nation technique, it is possible to construct very long codes that achieve large coding gains, while decoding can be done effi- ciently in an iterative manner. The iterative receiver of SCCPM was analyzed in [5] by means of mutual information transfer be- tween the inner demodulator and the outer decoder. SCCPM was generalized in [3] to woven coded CPM with several identical outer convolutional encoders, where inner modulation CPFSK was employed. It was shown that woven coded CPFSK (WC- CPFSK) has a larger lower bound on the free distance than does serially concatenated CPFSK (SCCPFSK), and therefore, achieves superior performance at high SNR. In this paper, we investigate woven coded CPM with inner CPFSK. We show that, in CPFSK, the transmitted bits can be divided into different bit classes, which possess unequal error probabilities under maximum likelihood sequence detection (MLSD). We protect every bit class of the inner CPFSK with different outer codes and obtain a new WCCPFSK system with hierarchical outer code structure. We call this system the hierarchical WCCPFSK. For comparison, we also consider WCCPFSK with identical outer codes as in [3]. In this paper, this system is called classical WCCPFSK. The decoding capa- Paper approved by A. K. Khandani, the Editor for Coding of the IEEE Communications Society. Manuscript received March 15, 2004; revised August 4, 2005 and August 8, 2006. S. Kempf is with the Nokia Siemens Networks, 89081 Ulm, Germany (e-mail: stefan.kempf@nsn.com). M. Bossert is with the Institute of Telecommunications and Applied Information Theory, University of Ulm, 89081 Ulm, Germany (e-mail: martin.bossert@uni-ulm.de). S. Shavgulidze is with the Department of Digital Communication Theory, Georgian Technical University, Tbilisi 0175, Georgia (e-mail: sergo_130@hotmail.com). Digital Object Identifier 10.1109/TCOMM.2007.904353 bility of hierarchical and classical WCCPFSK is investigated by means of mutual information transfer between the demodulator and the decoders, under the assumption of infinitely long in- terleavers. This analysis shows that, with a proper combination of the outer codes, hierarchical WCCPFSK needs up to 0.7 dB less E b /N 0 than does classical WCCPFSK to achieve reliable communication. Moreover, we compare lower bounds on the free distance of hierarchical and classical WCCPFSK. With those code combinations that achieve the best decoding capa- bility, hierarchical WCCPFSK has a smaller lower bound on the free distance than that of the classical WCCPFSK. Finally, we carry out simulations with interleavers of small and medium size, and we observe that hierarchical WCCPFSK gains up to 0.6 dB over classical WCCPFSK at small and medium SNRs. This is caused by the improved decoding capability of hierarchical WCCPFSK. At high SNRs, classical WCCPFSK is better due to its larger lower bound on the free distance. II. DEFINITION OF CPFSK AND SOME PROPERTIES CPFSK has modulation alphabet size of M =2 q for some integer q. Consider q binary information sequences b m , m = 1,...,q. Every sequence b m =(b m (1),b m (2),...) has ele- ments b m (n) ∈{0, 1}, where n denotes the symbol interval number. In every symbol interval, the tuple (b 1 (n),...,b q (n)) is mapped to an M -ary symbol u(n) ∈{0,...,M − 1}. We con- sider natural binary mapping and Gray mapping (see [4]). With natural binary mapping, b 1 (n) is the least significant bit. The CPFSK information sequence is u =(u(1),u(2),...). With tilted phase representation [6], the modulated signal is s(t)=  2E T cos ( 2πf 1 t + ¯ Ψ(t, u)+ ¯ Ψ 0 ) (1) where E is the symbol energy and T is the symbol duration. The modified frequency is defined as f 1 = f c − h(M − 1)/2T , where f c is the carrier frequency, and h is the modulation index. We consider only rational modulation indexes h = K/P with K, P =1, 2,... and gcd(K, P )=1 because then all possible information sequences u and their corresponding modulated signals s(t) can be described by a trellis [1]. With tilted phase representation, the number of states in the trellis is P . ¯ Ψ 0 is an arbitrary phase offset, which is assumed to be zero. The information sequence modulates the tilted phase ¯ Ψ(t, u). In the interval t ∈ [nT ;(n + 1)T ), the tilted phase is ¯ Ψ(t, u)=  2πh  n −1  i =0 u(i)  mod P +4πhu(n)q(t − nT )  mod 2π (2) 0090-6778/$25.00 © 2007 IEEE