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Spectral Analysis and Singular Value Computations of the Noncompact Frequency Response and Compression Operators in Sampled-Data Systems

This paper is motivated by the problem of computing the frequency response gain of general sampled-data systems with noncompact frequency response operators. We first show that, with the J-unitary transformation, the computation in the noncompact operator case can be reduced, in principle, to that in the compact operator case, to which an existing efficient and reliable bisection method can be applied. At the same time, however, we point out that there arise some critical problems in this reduction to the compact case which could be serious enough to invalidate the apparent success in the reduction. Through some spectral analysis of operators involving or related to the frequency response operators, we eventually prove that these critical problems can be circumvented after all, and we give an explicit result that shows how to compute the frequency response gain with a bisection method dealing only with finite-dimensional matrices. Extending the arguments, we also give a bisection method to compute the singular values of the frequency response operators and the associated compression operators.