Adaptive frequency band (AFB) and ultra-wide-band (UWB) systems require either rapidly changing or very high sampling rates. Conventional analog-to-digital devices have non-adaptive and limited dynamic range. We investigate AFB and UWB sampling via a basis projection method. The method decomposes the signal into a basis over time segments via a continuous-time inner product operation and then samples the basis coefficients in parallel. The signal may then be reconstructed from the basis coefficients to recover the signal in the time domain. We develop the procedure of this method, analyze various methods for signal segmentation and close by creating systems designed for binary signals.