A constant amplitude Radio Frequency (RF) signal is created by ordering complex information bearing symbols in the frequency domain along with their complex conjugates, and performing an inverse Fourier transform. This produces an analytic real-only transformed baseband signal. The real-only baseband signal is used to linearly vary the phase angle of a carrier wave while its amplitude remains constant. After reception, multi-path distortion is canceled. A time series of recovered phase angle is un-transformed with a FFT (fast Fourier transform) to produce I (in-phase) and Q (quadrature) samples. Demodulation occurs in a receiver by recovering carrier's phase angle vs. time. Forward error correction may be applied to data if desired. This may be called PM-OFDM (Phase Modulated-orthogonal frequency division multiplexing). This modulation method produces a relatively high amplitude Continuous Wave (CW) component to assist demodulation, and subcarriers near the CW can be zeroed-out to improve the CW recovery and tracking of frequency error and phase noise. The PM-OFDM signal can also be linearly precoded, creating PM-LP-OFDM. PM-LP-OFDM has a characteristic of a lower phase angle crest factor, enabling a lower peak phase rotation. Outer frequency subcarriers can be nulled to reduce occupied bandwidth, or to allow simultaneous reception of multi-user transmissions by a receiver. This can be called PM-LP-OFMDA (phase modulated-linear precoded-orthogonal frequency division multiple access).