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

    Research output: Contribution to journalArticleAcademicpeer-review

    3 Citations (Scopus)
    2 Downloads (Pure)

    Abstract

    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 languageUndefined
    Pages (from-to)1383 -1390
    Number of pages8
    JournalIEEE transactions on signal processing
    Volume64
    Issue number6
    DOIs
    Publication statusPublished - 16 Mar 2016

    Keywords

    • EWI-26994
    • METIS-316916
    • Correlation
    • Correlation functions
    • Hermite functions
    • Wigner distribution function
    • Signal analysis
    • Signal detection
    • Eigenvalues and eigenfunctions
    • Polynomials
    • IR-100339
    • Closed-form solutions
    • Ambiguity function
    • Fourier transforms
    • Convolution
    • Time-frequency analysis

    Cite this

    @article{3c43c314439b48999388fb90d4eb1e9f,
    title = "Closed-form expressions for time-frequency operations involving Hermite functions",
    abstract = "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.",
    keywords = "EWI-26994, METIS-316916, Correlation, Correlation functions, Hermite functions, Wigner distribution function, Signal analysis, Signal detection, Eigenvalues and eigenfunctions, Polynomials, IR-100339, Closed-form solutions, Ambiguity function, Fourier transforms, Convolution, Time-frequency analysis",
    author = "C.W. Korevaar and {Oude Alink}, M.S. and {de Boer}, Pieter-Tjerk and Kokkeler, {Andre B.J.} and Smit, {Gerardus Johannes Maria}",
    note = "eemcs-eprint-26994",
    year = "2016",
    month = "3",
    day = "16",
    doi = "10.1109/TSP.2015.2488580",
    language = "Undefined",
    volume = "64",
    pages = "1383 --1390",
    journal = "IEEE transactions on signal processing",
    issn = "1053-587X",
    publisher = "IEEE",
    number = "6",

    }

    Closed-form expressions for time-frequency operations involving Hermite functions. / Korevaar, C.W.; Oude Alink, M.S.; de Boer, Pieter-Tjerk; Kokkeler, Andre B.J.; Smit, Gerardus Johannes Maria.

    In: IEEE transactions on signal processing, Vol. 64, No. 6, 16.03.2016, p. 1383 -1390.

    Research output: Contribution to journalArticleAcademicpeer-review

    TY - JOUR

    T1 - Closed-form expressions for time-frequency operations involving Hermite functions

    AU - Korevaar, C.W.

    AU - Oude Alink, M.S.

    AU - de Boer, Pieter-Tjerk

    AU - Kokkeler, Andre B.J.

    AU - Smit, Gerardus Johannes Maria

    N1 - eemcs-eprint-26994

    PY - 2016/3/16

    Y1 - 2016/3/16

    N2 - 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.

    AB - 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.

    KW - EWI-26994

    KW - METIS-316916

    KW - Correlation

    KW - Correlation functions

    KW - Hermite functions

    KW - Wigner distribution function

    KW - Signal analysis

    KW - Signal detection

    KW - Eigenvalues and eigenfunctions

    KW - Polynomials

    KW - IR-100339

    KW - Closed-form solutions

    KW - Ambiguity function

    KW - Fourier transforms

    KW - Convolution

    KW - Time-frequency analysis

    U2 - 10.1109/TSP.2015.2488580

    DO - 10.1109/TSP.2015.2488580

    M3 - Article

    VL - 64

    SP - 1383

    EP - 1390

    JO - IEEE transactions on signal processing

    JF - IEEE transactions on signal processing

    SN - 1053-587X

    IS - 6

    ER -