It is known that, under certain conditions, the logarithm of the envelope and the instantaneous phase deviation of a singlesideband amplitude-modulated (SSB-AM) channel signal form Hilbert pairs. This minimum phase relationship has previously been employed to demodulate SSB-AM asynchronously and without distortion. The demodulator consisted of an envelope detector followed by nonlinear post-detection processing to reconstruct the phase deviation and, finally, the transmitted baseband signal. We now establish a function theoretic basis to study asynchronously detected SSB-AM signals having nonminimum phase. The received baseband signal from the envelope-to-phase demodulator is obtained in terms of the transmitted baseband signal, carrier level, and a function determined by the upper half-plane zeros of the baseband analytic signal. It is shown that the received signal exhibits nonlinear distortion. Furthermore, if the transmitted baseband signal is band-limited, we show that the baseband distortion is limited to the same band as the transmitted signal. Finally, bounds are obtained on the difference between the received baseband signal and the transmitted baseband signal.