Closed-form expressions for time-frequency operations involving Hermite functions

C. Willem Korevaar, Mark S. Oude Alink, Pieter-Tjerk de Boer, André B.J. Kokkeler, Gerard J.M. Smit

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    3 Citations (Scopus)
    168 Downloads (Pure)


    The product, convolution, correlation, Wigner distribution function (WDF) and ambiguity function (AF) of two Hermite functions of arbitrary order n and m are derived and expressed as a bounded, weighted sum of n+m Hermite functions. It was already known that these mathematical operations performed on Gaussians (Hermite functions of the zeroth-order) lead to a result which can be expressed as a Gaussian function again. We generalize this reciprocity to Hermite functions of arbitrary order. The product, convolution, correlation, WDF, and AF operations performed on two Hermite functions of arbitrary order lead to remarkably similar closed-form expressions, where the difference between the operations is primarily determined by distinct phase changes of the weights of the Hermite functions in the result. The closed-form expressions are generalized to the class of square-integrable functions. A key insight from the closed-form expressions is applied to the design of orthogonal, time-frequency localized communication signals which are characterized by an AF with rotational symmetry. In addition to this application, the theoretical expressions may prove useful for signal analysis in fields ranging from communications, radar and image processing to quantum mechanics.
    Original languageEnglish
    Pages (from-to)1383 -1390
    Number of pages8
    JournalIEEE transactions on signal processing
    Issue number6
    Publication statusPublished - 16 Mar 2016


    • Correlation
    • Correlation functions
    • Hermite functions
    • Wigner distribution function
    • Signal analysis
    • Signal detection
    • Eigenvalues and eigenfunctions
    • Polynomials
    • Closed-form solutions
    • Ambiguity function
    • Fourier transforms
    • Convolution
    • Time-frequency analysis


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