Codes for Combined Phase and Amplitude Modulated Signals in a Four-Dimensional Space

Data transmission with combined amplitude and phase modulation is considered with signal design carried out in a four-dimensional space of two amplitude and two phase variables. The energy for each signal is fixed. Quaternions are used to represent signal vectors and an algebra for quaternions is introduced which allows distance between signal vectors and equivalence between signal codes to be defined. It is also used to characterize the structure of codes. Code construction is based on regular four-dimensional polytopes and particular interest is focused on group codes with 8, 24, 48 and 120 elements. Two classes of phase modulated signals have interest as reference codes. Code evaluation is carried out by calculation of the distance distribution and the error probability assuming opitmum coherent detection. The group codes are found to be between 1.3 and 3.4 dB better than the best phase modulated codes. By using a suitable representation for a signal code it may be realized by using a few amplitude and several phase values.