**Speaker:** Stephen Casey (American University)

**Title: ** Sampling and Frame Expansions for Ultra-Wideband and Adaptive-Band Signals

**Abstract: **
Our talk develops a new approach to signal sampling, designed to deal with
ultra-wide band (UWB) and adaptive frequency band (AFB) communication systems.
These systems require either very high sampling or rapidly changing sampling rates.
From a signal processing perspective, we have approached this problem
by implementing an appropriate signal decomposition in the analog portion
that provides parallel outputs for integrated digital conversion and processing.
This naturally leads to an architecture with windowed time segmentation and
parallel analog basis expansion. 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 families are key, and we develop families that have variable partitioning
length, variable roll-off and variable smoothness. We then show how these windowing
families preserve orthogonality of any orthonormal systems between adjacent blocks,
and use these to create bases in which do signal expansions in lapped transforms.
We compute error bounds, demonstrating how to decrease error systematically by
constructing more sophisticated basis systems. We also develop the method with
a modified Gegenbauer system designed specifically for UWB signals.

The overarching goal of the theory developed in this talk is to develop a
computable atomic decomposition of time-frequency space. The idea is to come up
with a way of non-uniformly tiling time and frequency so that if the signal has
a burst of high-frequency information, we tile quickly and efficiently in time
and broadly in frequency, whereas if the signal has a relatively low-frequency
segment, we can tile broadly in time and efficiently in frequency. Computability
is key; systems are designed so that they can be implemented in circuitry.

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