Ultra-wide band (UWB) systems require either rapidly changing or very high sampling rates. Conventional analog-to-digital devices have limited dynamic range. We investigate UWB signal processing via a basis projection method and a basis system designed for UWB signals. 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 systems developed to preserve orthogonality of any orthonormal systems between adjacent blocks and almost orthogonal windowing systems that are more computable/constructible than the orthogonality preserving systems, built using B-splines. We construct the basis projection method, developing the method with a modified Gegenbauer system designed specifically for UWB signals.
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