Minimum Euclidean distance for combinations of short rate 1/2 convolutional codes and CPFSK modulation

Abstract

Continuous phase frequency shift keying (CPFSK) is a constant amplitude modulation method with good spectral sidelobe properties. Good error probability properties can be obtained with coherent maximum-likelihood detection. In this paper we study the Euclidean distance properties of signals formed by a conventional rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</tex> convolutional encoder followed by a binary or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</tex> -level CPFSK modulator. The minimum Euclidean distance is calculated for these signal sets as a function of the modulation index and the observation interval length. The optimum detector is discussed for rational modulation index values. The best obtainable codes are found for the case of short rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</tex> codes with binary or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</tex> -level CPFSK modulation. Lists of the best codes are given. Among the results are that the noncatastrophic rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</tex> convolutional codes with optimum free Hamming distance do not in general give the best Euclidean distance with CPFSK.