A communications transmitter configured to reduce the average-to-minimum magnitude ratio (AMR) of a communications signal includes a symbol mapper, a pulse-shaping filter, an AMR reduction circuit, and a modulator. The symbol mapper operates to generate a sequence of symbols from a binary-source data stream containing a message to be transmitted, and the pulse-shaping filter generates a baseband signal based on the sequence of symbols. The AMR reduction circuit is configured to compare a magnitude of a local minimum of samples of the baseband signal to various magnitude threshold levels, and to modify the baseband signal in one of two manners depending on the relationship of the magnitude of the local minimum and the various threshold levels. Finally, the modulator operates to modulate a carrier signal based on the modulation information contained in the modified baseband signal.