Aliasing error in muliti-channel sampling

Abstract

In this dissertation we treat the aliasing error in multi-channel sampling. The aliasing error is the difference between a non-bandlimited function and its representation in a series of the Riesz basis of the Paley-Wiener space. It is a well-known fact that sampling functions in multi-channel sampling comprise the Riesz basis for the Paley-Wiener space even though they are not a translation of one as in ordinary sampling. Thus, we define a series representation by this Riesz basis for a non-bandlimited function and the aliasing error in multi-channel sampling in the same way as in the ordinary sampling. The Poisson summation formula plays a key role in deriving the upper bound for the aliasing error. Contrary to the ordinary sampling, we can derive the upper bound for the aliasing error in multi-channel sampling by having as many Poisson summation formulas as transfer functions. Also, we need some conditions on transfer functions in order to use the Poisson summation formula because the formula does not mean the pointwise convergence in general. We show that the aliasing error in multi-channel sampling is bounded by a constant and a non-bandlimited function``s representation in a series of the Riesz basis of the Paley-Wiener space in multi-channel sampling is an approximate representation to the original function in a sense that we can obtain a series representation with as small aliasing error as we want. We also extend this argument to the multi-dimensional case.