Discrete fractional Hankel transform based on a nonsymmetric kernel matrix

Abstract

A discrete fractional Hankel transform (DFRHT) is developed based on the eigen decomposition of a nonsymmetric involutory kernel matrix Y of a discrete Hankel transform (DHT). The two eigen spaces pertaining to the two distinct eigenvalues of Y are not orthogonal because Y is not a normal matrix. Explicit expressions are derived for the oblique projection matrices of Y on its eigen spaces by applying a contributed generalized spectral theorem of linear algebra. Expressions are derived for the dimensions of the two eigen spaces in terms of the trace of matrix Y. An analytic form is derived for the inverse of a modal matrix of Y in order to circumvent the need for numerically evaluating it when computing the sophisticated fractional transform.The fact that the Hankel transform (HT) is self-reciprocating implies that taking the HT once can be viewed as a rotation by an angle π radians in the space – Hankel frequency plane. Being a generalization of the HT, the fractional Hankel transform (FRHT) of fractional order a corresponds to a rotation by an arbitrary angle α where α=πa.Initial orthonormal bases are individually generated for the two eigen spaces by the singular value decomposition of the oblique projection matrices on those spaces. In order for the DFRHT to approximate its continuous counterpart, namely the FRHT, it is desirable to define it in terms of eigenvectors which are as close as possible to samples of the eigen functions of the FRHT. After deriving a three term recursion formula for the eigen functions, a sampling scheme is proposed such that a finite number of samples will be representative of the behavior of those functions. Final optimal orthonormal eigenvectors are individually generated for the two eigen spaces.