Signal representation and recovery under partial information, redundancy, and generalized finite extent constraints

Abstract

We study a number of fundamental issues and problems associated with linear canonical transforms (LCTs) and fractional Fourier transforms (FRTs). First, we study signal representation under generalized finite extent constraints. Then we turn our attention to signal recovery problems under partial and redundant information in multiple transform domains. In the signal representation part, we focus on sampling issues, the number of degrees of freedom, and the timefrequency support of the set of signals which are confined to finite intervals in two arbitrary linear canonical domains. We develop the notion of bicanonical width product, which is the generalization of the ordinary time-bandwidth product, to refer to the number of degrees of freedom of this set of signals. The bicanonical width product is shown to be the area of the time-frequency support of this set of signals, which is simply given by a parallelogram. Furthermore, these signals can be represented in these two LCT domains with the minimum number of samples given by the bicanonical width product. We prove that with these samples the discrete LCT provides a good approximation to the continuous LCT due to the underlying exact relation between them. In addition, the problem of finding the minimum number of samples to represent arbitrary signals is addressed based on the LCT sampling theorem. We show that this problem reduces to a simple geometrical problem, which aims to find the smallest parallelogram enclosing a given time-frequency support. By using this equivalence, we see that the bicanonical width product provides a better fit to the actual number of degrees of freedom of a signal as compared to the time-bandwidth product. We give theoretical bounds on the representational efficiency of this approach. In the process, we accomplish to relate LCT domains to the time-frequency plane. We show that each LCT domain is essentially a scaled FRT domain, and thus any LCT domain can be labeled by the associated fractional order, instead of its three parameters. We apply these concepts knowledge to the analysis of optical systems with arbitrary numbers of apertures. We propose a method to find the largest number of degrees of freedom that can pass through the system. Besides, we investigate the minimum number of samples to represent the wave at any plane in the system. In the signal recovery part of this thesis, we study a class of signal recovery problems where partial information in two or more fractional Fourier domains are available. We propose a novel linear algebraic approach to these problems and use the condition number as a measure of redundant information in given samples. By analyzing the effect of the number of known samples and their distributions on the condition number, we explore the redundancy and information relations between the given data under different partial information conditions.