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 signal processing via a basis projection method.

The method first windows the signal and then decomposes the signal into a basis via a continuous-time inner product operation, computing the basis coefficients in parallel. The windowing systems are key, and we develop systems that have variable partitioning length, variable roll-off and variable smoothness. These include smooth bounded adaptive partitions of unity (BAPU systems) created using B-splines, systems developed to preserve orthogonality of any orthonormal systems between adjacent blocks, and {\it almost orthogonal} windowing systems that are more computable/constructible than the orthogonality preserving systems. We construct the basis projection method for all three types of windows, analyze various methods for signal segmentation and create systems designed for binary signals. We close by putting the theory into the context of frames.