On linear structure and phase rotation invariant properties of block M-PSK modulation codes

In this correspondence, we investigate two important structural properties of block 2tary PSK modulation codes, namely: linear structure and phase symmetry. For an AWGN channel, the error performance of a modulation code depends on its squared Euclidean distance distribution. Linear structure of a code makes the error performance analysis much easier. Phase symmetry of a code is important in resolving carrier-phase ambiguity and ensuring rapid carrier-phase resynchronization after temporary loss of synchronization. It is desirable for a modulation code to have as many phase symmetries as possible. In this paper, we first represent a 2t-ary modulation code as a code with symbols from the integer group, S2,.ps K = {0, 1,2,...,2 t1}, under the modulo-2 t addition. Then we define the linear structure of block 2t-ary PSK modulation codes over S2_.PSK with respect to the modulo-2 t vector addition, and derive conditions under which a block 2t-ary PSK modulation code is linear. Once the linear structure is developed, we study phase symmetry of a block 2t-ary PSK modulation code. In particular, we derive a necessary and sufficient condition for a block 2t-ary PSK modulation code, which is linear as a binary code, to be invariant under 180"/2 t-h phase rotation, for 1 < h < L Finally, a list of short 8-PSK and 16-PSK modulation codes is given together with their linear structure and the smallest phase rotation for which a code is invariant. ON LINEAR STRUCTURE AND PHASE ROTATION INVARIANT PROPERTIES OF BLOCK 2t-PSK MODULATION CODES