A linearizing approach to coding for generalized MSK

Abstract

Minimum shift keying as an instance of continuous-phase frequency shift keying is a nonlinear modulation which makes designing error correcting codes for it difficult as Euclidean distances between signals and the Hamming distances between symbol sequences do not linearly correspond. We show that the nonlinearity disappears if we employ a sequence mapping we term precoding arising from the equivalence of this modulation to a form of offset quadrature phase shift keying. When we consider the modified forms of this modulation (generalized MSK) to achieve improved spectral efficiency, the nonlinearity and memory make the coding design problem more difficult, and precoding does not resolve the issue. We show that the nonlinearity can be removed in the cases involving pulse shapes of two and three symbol period duration using two stages of coding, the last being a double or quadruple repetition code. Using the Laurent representation of continuous-phase modulation as amplitude modulated pulses, a simple inphase-quadrature modulator can be implemented to generate such coded generalized MSK signals. The inphase and quadrature pulses are shown to be linear combinations of the Laurent pulses. It is shown that the coded signals can be demodulated using a simple inphase-quadrature receiver without sacrificing the performance. It is shown that no intersymbol interference is introduced at the sampled output of the matched filter and in additive white Gaussian noise environment, the noise samples are independent. Therefore, the detection problem for the recovery of the symbols sequence from the decision variable sequence is one corresponding to memoryless linear modulation, avoiding the use of the maximum likelihood sequence detection which would otherwise be needed for optimal performance. Furthermore, it is shown that by applying repetition code to these modulation schemes, the Euclidean distance between different signals is directly related to the Hamming distance between corresponding coded sequences. This may simplify the task of search for good codes applied to the considered case of generalized MSK signals.