BER expressions for differentially detected π/4 DQPSK modulation

Closed form bit-error rate (BER) expressions for differentially detected π/4-shifted differentially encoded quadrature phase-shift keying (QPSK) modulation (π/4 DQPSK) are derived for both additive white Gaussian noise (AWGN) and Rayleigh-fading channels. The derivations are carried out in an exact and most general manner in that in-phase (I) and quadrature (Q) channel bit-error probabilities P_cl and P_cQ are separately obtained in terms of the same-quadrature and cross-quadrature noise Correlation functions, including a measure of noise nonstationarity. We then specialize the general expressions for uncorrelated noise and equal noise powers in successive symbol periods to obtain a useful bit-error probability expression for the AWGN channel in the form P_e≈Q(√(1.1716·E_b/N₀)) where Q(·) is the Gaussian distribution Q-function and E_b /N₀ is the bit energy-to-noise density ratio. Exact BER expressions for the Rayleigh-fading channel that involve the noise parameters are also given and are extended to the case of L-fold diversity combining