Sampling Architectures for Ultra-Wideband Signals Stephen D. Casey Department of Mathematics and Statistics, American University, Washington, DC 20016 USA Email: scasey@american.edu Howard S. Cohl Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8910 USA Email: howard.cohl@nist.gov Abstract—Ultra-wideband (UWB) signal processing is a tech- nology that has tremendous potential to develop advances in communication and information technology. However, it also presents challenges to the signal processing community, and, in particular, to sampling theory. This article outlines a UWB signal processing system via a basis projection and a basis system designed specifically 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 windows are key, and we develop windows that have variable partitioning length, variable roll-off and variable smoothness. They preserve orthogonality of any orthonormal system between adjacent blocks. In this paper, we develop new windows, and give an outline for a new architecture for the projection. We then use this projection with a basis system designed to work with UWB signals, implementing modified Gegenbauer functions designed specifically for these signals. I. I NTRODUCTION Ultra-wideband (UWB) signal processing is a technology with many features that promise potential advances in wireless communications, networking, radar, imaging, and positioning systems. This article outlines a UWB signal processing system via a basis projection and a basis system designed especially for UWB signals. The method first windows the signal and then decomposes the signal into an orthonormal basis via a continuous-time inner product operation, computing the basis coefficients in parallel. We call this procedure the Projection Method. The windows are key, and we develop windows that have variable partitioning length, variable roll-off and variable smoothness. They are designed to preserve orthogonality of any orthonormal system between adjacent blocks. We then use the Projection Method with a basis system designed to work with UWB signals. This system is a modified Gegenbauer system designed specifically for UWB signals. This system minimizes the Gibbs phenomenon, giving the point values of a piecewise smooth signal with essentially the same accuracy as a smooth approximation, making it the ideal system to use for the Projection Method as applied to these signals. The use of the Gegenbauer system for UWB signals is known in the engineering community. Justification for this system is given by numerical simulation [13], [16] (and references therein). In this article, we provide an outline of an analytic justification, and we give new methods for creating windows and new outlines for the system architecture that advance our previous work [4]. Mathematical definitions and computations for the paper follow those given in Benedetto [1]. A UWB communication system is a large bandwidth system based on the transmission of very short pulses with relatively low energy. These systems operate by running as signaling waveforms, baseband pulses of very short duration, rather than the traditional method using a sinusoidal carrier. The UWB technique has a fine time resolution which makes it a technology appropriate for accurate ranging. The large bandwidth of a UWB system is dominated by its pulse shape and duration. This large system bandwidth relative to the information bandwidth allows UWB systems to operate with a low power spectral density. Such a low power spectral density implies that the UWB signal may be kept near or below the noise floor of detection devices. For these reasons, UWB technology has many potential advantages, such as high data rate, low probability of interception and detection, system simplicity, low cost, reduced average power consumption, weak sensitivity to the near-far problem and immunity to interference. However, UWB systems present challenges to current meth- ods of signal processing. From a signal processing perspective, we can approach this problem by implementing an appropriate signal decomposition in the analog portion that provides par- allel outputs for integrated digital conversion and processing [3]. This naturally leads to an architecture with windowed time segmentation and parallel analog basis expansion. The method represents a change of view in sampling, from that of a stationary view of a signal used in classical sampling to an “short-time windowed stationary” view. This viewpoint gives that the time and frequency space “tile” occupied by the signal is processed quickly. The windows give us the tools to partition time-frequency so that the UWB signal can be partitioned uniformly but also quickly and efficiently. With the blocks, the signal can be sampled in parallel [3]. II. THE PROJECTION METHOD Classical sampling theory applies to functions that are square integrable and band-limited. A function in L 2 (R) whose Fourier transform  f (ω)=  R f (t)e −2πitω dt is com- pactly supported and has several smoothness and growth properties given in the Paley-Wiener Theorem. The choice to have 2π in the exponent simplifies certain expressions, e.g., for f,g ∈ L 1 ∩L 2 (R),  f,  g ∈ L 1 ∩L 2 (  R), we have Plancherel- Parseval – ‖f ‖ L 2 (R) = ‖  f ‖ L 2 ( b R) 〈f,g〉 = 〈  f,  g〉. The Paley- Wiener Space PW Ω is defined as PW Ω = {f continuous : 978-1-5386-1565-2/17/$31.00 c 2017 IEEE 2017 I NTERNATIONAL CONFERENCE ON SAMPLING THEORY AND APPLICATIONS (SAMPTA) 246