In this work, we build on the classical time-frequency analysis tools developed by Landau, Slepian and Pollack. We focus on prolate spheroidal wave functions as a tool to analyze time-frequency localized signals. Recent developments in periodic nonuniform sampling by Venkataramani and Bresler allow us to develop several sampling formulas and discrete methods for applying time-frequency localization concepts to multiband signals. Using concepts introduced by Landau, we explore the behavior of the eigenvalues for time-frequency localization operators. By extending the work of Khare and George, we are able to give several alternatives for calculating these eigenvalues. We also include an error analysis for these calculations.;Our presentation includes a brief history and overview of several concepts related to time-frequency analysis and sampling. We develop distribution theory and functional theoretic foundations in order to provide a rigorous basis for our use of the Poisson summation formula, impulse sampling, synthesis/analysis operators and Bessel sequences.;Our overview emphasizes possible applications in digital signal processing and includes several examples. We also include a small code library to demonstrate how to modularize code based on theory.
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