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