This paper presents a method of analysing the performance ofM-ary coherent phase-shift-keyed (c.p.s.k.) signals transmitted through a bandpass nonlinearity, exhibiting a.m./a.m. and a.m./p.m. conversions, and in the presence of additive Gaussian noise both preceding and following the nonlinearity. The method is computationally superior to those previously obtained by other workers, especially for high-level (M> 4) c.p.s.k. systems. In addition, it is capable of extension to include other signal corruptions, such as cochannel interference, both preceding and following the nonlinearity.