Multi-channel and irregular sampling for signals and linear time-varying systems

Abstract

This dissertation consists of two parts: sampling of functions and sampling of operators. Sampling theory for functions/signals has been developed in order to reconstruct continuous signals from their discrete values. Operator sampling is introduced to prove operator identification for linear time-varying systems. For sampling of functions, we derive multi-channel sampling formula considering multiple parallel filter banks and suitable transfer functions. The classical sampling theorem is extended to the Kramer`s Lemma by replacing the Fourier kernel with any integral kernel. An irregular sampling formula is also provided in Kramer`s Lemma. Combining the concept of Kramer`s Lemma and multi-channel sampling, we derive a multi-channel, irregular sampling formula in a general abstract Hilbert space. This abstract Hilbert space setting gives us more flexibility to choose signals which are not necessarily band-limited, and to distribute sampling rates arbitrarily to each channel. Oversampling is considered in a single-channel sampling in the general abstract Hilbert space, and recovery of missing samples is discussed. We also provide periodic nonuniform sampling formulas for multi-band signals, where multi-band structure can be arbitrary. Periodic nonuniform sampling is known for an efficient scheme for analysis of multi-band signals, and interpreted as a special case of multi-channel sampling. Furthermore, we discuss the conditions on the coefficient matrices for the reconstruction formula to be a Riesz basis or frame expansion so that the expansion is stable under the error which might occur during sampling process. We consider an interesting situation that the estimation of the spectrum, but only partial, is required for multi-band signals. Sampling of operators is a special case of the operator identification. Operator identification, or channel measurement, is one of the fundamental questions in communications engineering. It is desirable to identify...