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

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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Texas-El Paso, El Paso, TX, USA

    Indranil SenGupta & Maria C. Mariani

  2. College of Mathematics and Computers, Changsha University of Science and Technology, Changsha, Hu’nan, China

    Bo Sun

  3. College of Mathematics and Statistics, Chongqing University, Chongqing, China

    Weisheng Jiang

  4. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Goong Chen

Authors
  1. Indranil SenGupta
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  2. Bo Sun
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  3. Weisheng Jiang
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  4. Goong Chen
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  5. Maria C. Mariani
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Corresponding author

Correspondence to Indranil SenGupta.

Additional information

Communicated by Hans G. Feichtinger.

G. Chen supported in part by Texas ARP grant 010366-0149-2009 and QNRF grants NPRP 09-462-1-074 and 4-1162-1-181.

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SenGupta, I., Sun, B., Jiang, W. et al. Concentration Problems for Bandpass Filters in Communication Theory over Disjoint Frequency Intervals and Numerical Solutions. J Fourier Anal Appl 18, 182–210 (2012). https://doi.org/10.1007/s00041-011-9197-y

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  • DOI: https://doi.org/10.1007/s00041-011-9197-y

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