A digital pre-distortion system can inversely model a power amplifier of a system to linearize the transmitter. A complex baseband model for digital pre-distortion based on a narrowband signal assumption is unworkable for an ultra wide band Cable television application. Predistortion can use a true wide band model including real-valued basis terms, obtained from a real-valued signal. When raised to a power, both even and odd harmonics or both odd or even other non-linear terms are represented and negative frequency fold-over can be accounted for. A Hilbert transform can be applied. Compressed sensing can be used to reduce the number of basis terms in the true real wide band model to generate a sparse model. Sparse equalization can be added to improve the stability of the digital pre-distortion system.