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Digital filtering and prolate functions

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A class of trigonometric polynomials p(x) = \sum _{n=-N}^{N} a_{n} e^{j n \pi x} of unit energy is introduced such that their energy concentration \alpha = \int_{-e}^{e} p^{2}(x) dx in a specified interval (- \epsilon, \epsilon) is maximum. It is shown that the coefficients a_{n} must be the eigenvectors of the system \sum _{m=-N}^{N} frac{\sin (n - m)\pi \epsilon}{(n - m)\epsilon} a_{m} = \lambda a_{n} . corresponding to the maximum eigenvalue X. These polynomials are determined for N = 1, \cdots , 10 and \epsilon = 0.025, \cdots , 0.5 . The resulting family of periodic functions forms the discrete version of the familiar prolate spheroidal wave functions.

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Circuit Theory, IEEE Transactions on  (Volume:19 ,  Issue: 6 )