Theory and operational rules for the discrete Hankel transform

Abstract

Previous definitions of a discrete Hankel transform (DHT) have focused on methods to approximate the continuous Hankel integral transform. In this paper, we propose and evaluate the theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier-Bessel expansions. The proposed transform also possesses requisite orthogonality properties which lead to invertibility of the transform. The standard set of shift, modulation, multiplication, and convolution rules are derived. In addition to the theory of the actual manipulated quantities which stand in their own right, this DHT can be used to approximate the continuous forward and inverse Hankel transform in the same manner that the discrete Fourier transform is known to be able to approximate the continuous Fourier transform.

Matlab code for the Discrete Hankel Transform

“…The nth order discrete Hankel transform (DHT) proposed in [1] is defined as the transformation of the discrete vector f to vector F given by…”

“…where nk j is the th k zero of the Bessel function of the first kind of order n [1]. Properties of the DHT as defined in equation (3) are shown in [1].…”

“…The choice of discretization points nk r and nm ρ is also discussed below. The full theory of the discrete Hankel transform is given in [1].…”

“…The conditions and corresponding scaling factors are listed in Table 1. The detailed derivation of these scaling factors was shown in [1]. It can be observed that the scaling factors for the space-limited or frequency limited cases can be derived from each other via …”

“…An additional feature of a carefully derived DFT is that it can be used to approximate the continuous Fourier transform, with relevant sampling and interpolation theories that can be used. Recently, a DHT was proposed as a complete and orthogonal transform that possesses its own mathematical theory, including the standard set of shift, modulation, multiplication and convolution rules [1]. In addition, this DHT can be used to approximate the continuous Hankel transform in the same manner that the Discrete Fourier transform is known to be able to approximate the continuous Fourier transform.…”