A New Approach to Deriving Closed-Form Bit Error Probability Expressions of MPSK Signals

In conventional approach, the bit error probability (BEP) expressions of $M$ -ary phase-shift keying (MPSK) signals over additive white Gaussian noise (AWGN) channel are derived by averaging error probabilities for all bits of MPSK symbol. The closed-form BEP expressions of MPSK signals over AWGN channel can also be derived by using the weighted conditional symbol error probability (SEP) expressions, which has not been reported. In this paper, we derive the conditional SEP expressions of MPSK signals in terms of Gaussian $Q$ and Owen's $T$ functions by changing the domain of integration from a plane bounded by two rays into a quadrant. By weighting the conditional SEP expressions according to average distance spectrum, the closed-form BEP expressions of MPSK signals over AWGN channel are obtained. Only two summations involving two parameters are required. The closed-form BEP expressions of MPSK signals over Nakagami- $m$ fading channel are derived by using the BEP expressions over AWGN channel and the moment generating function-based approach. The approximate BEP expressions of MPSK signals over Nakagami- $m$ fading channel are obtained by using elementary functions-based approximations of Gaussian $Q$ and Owen’s $T$ functions. The conciseness and computational complexity of our BEP expressions are verified by the comparison with existing results.