M.Sc Student | Hagai Kirshner |
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Subject | On Sampling-Invariant Characteristics in Signal Representation |

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 (*l*_{2}
and the Sobolev space), intertwining relations of *L*_{2}-, *l*_{2}-
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.