|M.Sc Student||Hagai Kirshner|
|Subject||On Sampling-Invariant Characteristics in Signal|
|Department||Department of Electrical Engineering||Supervisor||Professor Porat Moshe|
Most signal processing systems are based on discrete-time signals although the origin of many sources of information is analog. In this work we consider the task of signal representation by a set of basis functions. Presently, without prior knowledge of the signal beyond its samples, no bound on the potential representation error is available. The question raised in this paper is to what extent the sampling process keeps algebraic relations, such as inner product, intact. By interpreting the sampling process as a linear bounded operator of the two Hilbert spaces (l2 and the Sobolev space), intertwining relations of L2-, l2- and Sobolev inner products are derived. This, in turn, gives rise to an upper bound on the representation error, which is demonstrated for finite energy signals and images. No constraints of sampling bandlimited functions are assumed. Based on our theorems, one can then determine the maximum potential representation error induced by the sampling process. We further propose a new discrete approximation scheme for the calculation of the inner product, which is optimal in the sense of minimizing the maximum potential representation error, where several special cases such as sampling in shift-invariant spaces and sampling over finite dimensional spaces are considered. Our approach and results are applicable to signal and image processing systems where analog signals are represented by their sampled versions.