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This paper describes a transformation technique aimed at solving the peak-to-average power ratio (PAPR) problem associated with OFDM (orthogonal frequency-division multiplexing). Constant envelope OFDM (CE-OFDM) transforms the OFDM signal, by way of phase modulation, to a signal designed for efficient power amplification. At the receiver, the inverse transformation - phase demodulation - is applied prior to the conventional OFDM demodulator. The performance of CE-OFDM is analyzed in additive white Gaussian noise (AWGN) and fading channels. CE-OFDM is shown to achieve good performance in dense multipath with the use of cyclic prefix transmission in conjunction with a frequency- domain equalizer (FDE). By way of computer simulation and hardware realization, CE-OFDM is shown to compare favorably to conventional OFDM.