Concentration Problems for Bandpass Filters in Communication Theory over Disjoint Frequency Intervals and Numerical Solutions

Abstract

The concentration problem of maximizing signal strength of bandlimited and timelimited nature is important in communication theory. In this paper we consider two types of concentration problems for the signals which are bandlimited in disjoint frequency-intervals, which constitute a band-pass filter. For the first type the problem is to determine which members of L 2 (−∞, ∞) lose the smallest fraction of their energy when first timelimited and then bandlimited. For the second type the problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted to a given time interval. For both types of problems, basic theoretical properties and numerical algorithms for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L 2 (−∞, ∞) are also proved. Numerical computations are carried out which corroborate the theory. Relationship between eigenvalues of these two types of problems is also established. Several properties of eigenvalues of both types of problems are proved.

An Empirical Bayes Approach to Frequency Estimation

“…The independence assumption for the cross sections may seem quite strong. A more natural assumption could be to require that the strings (26) are sample observations of length N from an exchangeable sequence of N -dimensional random vectors, {y k } L k=1 . This means that the joint probability distribution of the sequence {y k } L k=1 is independent of the order in which the random elements y k are listed [27].…”

“…Therefore, the Law of Large Numbers applies also to our exchangeable sequence of observations {y k } and leads via a second averaging with respect to k, to a probabilistic average (expectation) with respect to the measure governing the ω k 's. In particular a consistent covariance estimate can result by first averaging each N -dimensional sample (26) with respect to t and successively computing a second cross-sectional average with respect to k.…”

A Review of Prolate Spheroidal Wave Functions from the Perspective of Spectral Methods

“…We also refer to the recent works [21,109] along this line. SenGupta et al [75] considered the concentration problem over disjoint frequency intervals. Another very important inspiration of the Slepian's seminal papers is the investigation of differential operators that commute with appropriate integral operators, see for example, the insightful analysis by Grünbaum et al [28][29][30].…”

The finite Hartley new convolutions and solvability of the integral equations with Toeplitz plus Hankel kernels

“…Eq. (1.1) with a Toeplitz p(x − u) or Hankel q(x + u) kernel attracts attention of many authors as they have practical applications in such diverse fields as scattering theory, fluid dynamics, linear filtering theory, and inverse scattering problems in quantum-mechanics, problems in radiative wave transmission, and further applications in Medicine and Biology (see [1][2][3][4][5][6][7][8][9][10][11]). The historical development concerning Eq.…”