Methods and apparatus for reducing the average-to-minimum power ratio (AMPR) of communications signals in communications transmitters. An AMPR-reducing method includes detecting a sample of a baseband signal having a magnitude less than a predetermined low-magnitude threshold. The magnitude and/or angle of the baseband signal is modified in the temporal vicinity of the detected low-magnitude sample, to form a modified baseband signal having a reduced AMPR. The baseband signal is modified by scaling an insertion pulse by a complex pulse insertion vector and combining the resulting scaled complex insertion pulse with the baseband signal in the temporal vicinity of the detected low-magnitude sample. The pulse insertion angle may be set to any angle within a pulse insertion vector range determined by a vector defining the detected low-magnitude sample and a vector that is orthogonal to the trajectory of the baseband signal.