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 and are derived and expressed as a bounded, weighted sum of 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.

Time–Frequency Localization Measures for Packets of Orthogonally Multiplexed Signals

“…Proposition 3. The GHPξ of N -dimensional packet x, defined in (12), can be expressed directly in terms of a GHP of the interference potential function as…”

“…and(11) two averagings are taken into account: a statistical expectation over symbol randomness in s k , and a time/frequency integration over the signal as in(8) and(9). Note that normalization of time/frequency averaging will be explicitly handled in(12).…”

Hermite-based communications

“…As a consequence, it is preferred over the rectangular lattice especially for cases where η → 1. Where work by the author published in [111] treats signal design for the rectangular lattice, upcoming sections exclusively deal with waveform design for the hexagonal lattice. A study underlining the advantage of hexagonal-lattice based communications can be found in [110].…”

“…The orthogonalization procedure -shown in this section -is an extension to the orthogonalization procedure outlined in our work in [111]. The procedure in [111] is limited to the orthogonalization of only neighboring lattice points whereas the procedure in this section includes an arbitrary number of lattice points.…”

“…The procedure in [111] is limited to the orthogonalization of only neighboring lattice points whereas the procedure in this section includes an arbitrary number of lattice points. To analyze the correlation, which is a means to analyze the interference between two waveforms, the AF is used.…”