Closed-Form Expressions for Time-Frequency Operations Involving Hermite Functions

• Published in: IEEE transactions on signal processing Vol. 64; no. 6; pp. 1383 - 1390
• Main Authors: Korevaar, C. Willem, Oude Alink, Mark S., de Boer, Pieter-Tjerk, Kokkeler, Andre B. J., Smit, Gerard J. M.
• Format: Journal Article
• Language: English
• Published: New York IEEE 15.03.2016; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: ambiguity function; Closed-form solutions; Convolution; Correlation; correlation functions; Design analysis; Eigenvalues and eigenfunctions; Exact solutions; Fourier transforms; Functions (mathematics); Gaussian; Hermite functions; Mathematical analysis; Mathematical functions; Polynomials; signal analysis; signal detection; Time-frequency analysis; Wigner distribution; Wigner distribution function
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2015.2488580

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.