On Differentially Demodulated CPFSK Anthony Griffin and Desmond P. Taylor griffiab@elec.canterbury.ac.nz and taylor@elec.canterbury.ac.nz Electrical and Electronic Engineering University of Canterbury New Zealand Abstract This paper develops a differential encoder for differentially de- modulated continuous phase frequency shift keying (CPFSK). CPFSK schemes with modulation index , where and are relatively prime positive integers, can be represented by a decomposed model consisting of a continuous phase encoder (CPE) and a memoryless modulator (MM). The differential en- coder is shown to fit well with the CPE and form a decomposed model of differentially encoded CPFSK (DCPFSK). A basic re- ceiver structure for differentially demodulating DCPFSK is pre- sented along with simulation results. An exact formula for the minimum squared Euclidean distance (MSED) of differentially demodulated DCPFSK is also given. 1 Introduction Continuous Phase Modulation (CPM) is an attractive communi- cation scheme as it is a true constant envelope modulation, and can be amplifed by amplifiers working in their non-linear regions. Continuous phase frequency shift keying (CPFSK) is a simple CPM scheme that is very useful due to its narrowband RF require- ments. CPFSK can be decomposed into a two part model [1] that isolates the coding and modulation inherent within CPFSK. This model has allowed codes to be designed specifically for CPFSK with improved performance compared to previous schemes [2] [3]. CPM schemes are sensitive to phase jitter, and therefore require an accurate phase reference for coherent demodulation. Recover- ing the carrier accurately enough can be difficult in more extreme channels. A solution that avoids the need for a phase reference is differential demodulation, where the previous symbol is used to demodulate the current one. A differential CPFSK (DCPFSK) scheme has been developed by Yuan and Taylor [4], but their dif- ferential encoder was designed for bipolar binary CPFSK, and does not lend itself to coding. An -ary differential encoder that interfaces well with the decomposition model would allow the de- velopment of a decomposed differential model, and codes to be designed especially for DCPFSK. In Section 2 we describe CPFSK and its decomposition model with a modulation index of . In Section 3 we de- velop a differential encoder for CPFSK and present the decompo- sition of DCPFSK. We discuss coding DCPFSK, the differential phase trellis, and the squared Euclidean distance and spectrum of DCPFSK in Section 4. The results of some simulations of DCPFSK on the additive white Gaussian noise (AWGN) channel are presented in Section 5, and finally in Section 6 we draw some conclusions. Memoryless Modulator (MM) Continuous Phase Encoder (CPE) Figure 1: Decomposition of CPFSK 2 The Decomposition of CPFSK A CPFSK signal can be described [1] by (1) where is the symbol energy, is the symbol period, is the intial phase offset, and the asymmetric carrier frequency, which is related to the symmetric carrier frequency by . is called the tilted (information- carrying) phase, and is given by (2) which is assumed to be 0 at , and the -ary data sequence is given by (3) The parameter in (2) is called the modulation index. We con- sider only rational modulation indexes of the form where and are relatively prime postive integers. The phase response, , for CPFSK is (4) Using these definitions, CPFSK can be decomposed [1] into a continuous phase encoder (CPE) and a memoryless modulator (MM) as shown in Figure 1.