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Supply modulation for RF power amplifiers according to the modulated signal's envelope (i.e. envelope tracking and polar modulation) is an effective approach for achieving high efficiency in transmitters. Various current and emerging wireless standards produce signal envelopes with wide bandwidths and high peak-to-average ratio (PAR), which makes efficient supply modulation very challenging. In this paper, a reduced bandwidth class H supply modulation scheme is proposed, which provides efficient supply modulation with high PAR and wide bandwidth. It leverages a novel bandwidth reduction DSP algorithm to effectively exploit an efficient narrowband switching modulator without increasing its switching losses. Analyses and simulation results for LTE and WiMAX signals indicate significant modulator efficiency improvements (up to 15%) over similar existing techniques.