  Using an auxiliary random variable technique, we prove that binary
differential phase-shift keying and binary phase-shift keying have the same
asymptotic bit-error rate performance in lognormal fading channels. We also
show that differential quaternary phase-shift keying is exactly 2.32 dB worse
than quaternary phase-shift keying over the lognormal fading channels in high
signal-to-noise ratio regimes.
