Relation between bit and symbol error probabilities for DPSK and FSK with differential phase detection

In M-ary, Gray coded DPSK, and FSK with differential phase detection the bit error probability is usually approximated by the symbol error probability divided by the number of bits in a symbol. This approximation is known to be excellent for PSK with large signal-to-noise ratios in a Gaussian channel. In other cases this approximation may be questionable. Here we compute the exact values of bit error probability for DPSK and FSK in Gaussian, Rayleigh, and Rician channels. We also compute the relative error involved in this approximation. The conclusion is that except for very low values of signal-to-noise ratio the approximation is good for all channels