The error probability of M-ary PSK block coded modulation schemes

A tight upper bound on the decoding error probability is derived for block-coded modulation structures where an M-ary phase shift keying (M-PSK) signal constellation is employed. This bound, called a tangential sphere bound, is tight for very low (as well as for high) signal-to-noise ratios (SNRs). Berlekamp´s tangential union bound, previously derived for binary codes, can be derived for an M-PSK block coded modulation structure as well. However, it is proven that our tangential sphere bound is tighter than Berlekamp´s (1980) tangential bound. For particular schemes, it is shown that for low SNRs our bound is considerably tighter than the tangential bound. As one of the examples, a multistage decoder is considered