AFRL-AFOSR-VA-TR-2016-0103 New Techniques in Time-Frequency Analysis

The project New Techniques in Time-Frequency Analysis: Adaptive Band, Ultra-Wide Band and Multi-Rate Signal Processing led to the development of new techniques and theories in the analysis of signals. These techniques and theories were extensions of known techniques – sampling, Fourier, Gabor and wavelet analysis, and new approaches to analysis – using combinations of analysis, geometry, group theory, and number theory. Every target item in the original statement of objectives was achieved, and several new areas of research were open up. There were four main areas of study. Techniques were developed to deal with classes of signals for which the known techniques have limitations, e.g., adaptive frequency band (AFB) and ultra-wide band (UWB) signals. These signals are of interest to Air Force communications systems, but the theory and techniques we have developed and propose to continue developing fit in the context of expanding the general theory. Our techniques involve signal segmentation, projection, expansion in bases, transform analysis, and multi-rate methods. 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 develop a procedure to preserve orthonormality between blocks. This involves adapting an developing windowing systems for time-frequency analysis. We then demonstrate the connection of these techniques with Gabor and wavelet analysis. We also gave a technique for multi-rate sub-Nyquist sampling. This then allows for processing of wider bandwidth signals at smaller bandwidth rates. These techniques are based on our work on multichannel deconvolution. The “tricks” for the multi-rate theory come from clever uses of number theory and complex analysis. We also developed a program of study connecting sampling theory with the geometry of the signal and its domain. It is relatively easy to demonstrate this connection in Euclidean spaces, but one quickly gets into open problems when the underlying space is not Euclidean. In Euclidean space, the minimal sampling rate for Paley-Wiener functions on Rd, the Nyquist rate, is a function of the band-width. No such rate has yet been determined for hyperbolic or spherical spaces. We look to develop a structure for the tiling of frequency spaces in both Euclidean and non-Euclidean domains. In particular, we establish Nyquist tiles and sampling groups in Euclidean geometry, and discuss the extension of these concepts to hyperbolic and spherical geometry and general orientable surfaces. We close by discussing our work in the analysis of point processes. We have developed a collection of algorithms that analyze periodic phenomena generated by either a single or multiple generators. These work even when the data is extremely sparse and noisy. The algorithms use number theory in novel ways to extract the underlying period(s) by modifying the Euclidean algorithm to determine the period from a sparse set of noisy measurements. We divide our analysis into two cases periodic processes created by a single source, and those processes created by several sources. We wish to extract the fundamental period of the generators, and, in the second case, to deinterleave the processes. DISTRIBUTION A: Distribution approved for public release. New Techniques in Time-Frequency Analysis 1