Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1354129, 5 pages

https://doi.org/10.1155/2017/1354129

## Poisson Summation Formulae Associated with the Special Affine Fourier Transform and Offset Hilbert Transform

Beijing Information Science & Technology University, Beijing, China

Correspondence should be addressed to Zhi-Hai Zhuo; nc.ude.utsib@iahihzouhz

Received 7 May 2017; Accepted 16 July 2017; Published 15 August 2017

Academic Editor: Alessandro Lo Schiavo

Copyright © 2017 Zhi-Hai Zhuo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates the generalized pattern of Poisson summation formulae from the special affine Fourier transform (SAFT) and offset Hilbert transform (OHT) points of view. Several novel summation formulae are derived accordingly. Firstly, the relationship between SAFT (or OHT) and Fourier transform (FT) is obtained. Then, the generalized Poisson sum formulae are obtained based on above relationships. The novel results can be regarded as the generalizations of the classical results in several transform domains such as FT, fractional Fourier transform, and the linear canonical transform.

#### 1. Introduction

The classical Poisson summation formula, which demonstrates that the sum of infinite samples in the time domain of a signal is equivalent to the sum of infinite samples of in the Fourier domain, is of importance in theories and applications of signal processing [1]. The traditional Poisson sum formula can be represented as follows [2]: or where denotes the Fourier transform (FT) of a signal and stands for the sampling interval. Not only does Poisson summation formula play a key role in various branches of the mathematics, but also it finds numerous applications in lots of fields, for example, mechanics, signal processing community, and many other scientific fields. The Poisson summation formula is related to the Fourier transform, and, with the development of modern signal processing technologies, there are many other kinds of transforms that have been proposed, it is therefore worthwhile and interesting to investigate the Poisson sum formula in deep associated with these kinds of new integral transforms.

The special affine Fourier transform (SAFT) [3, 4], also known as the offset linear canonical transform [5, 6] or the inhomogeneous canonical transform [5], is a six-parameter class of linear integral transform. Many well-known transforms in signal processing and optic systems are its special cases such as Fourier transform (FT), fractional Fourier transform (FRFT), the linear canonical transform (LCT), time shifting and scaling, frequency modulation, pulse chirping, and others [7, 8]. SAFT can be interpreted as a time shifting and frequency modulated version of LCT [9–11], that is much more flexible because of its extra parameters . Recently, it has been widely noticed in many practical applications along with the rapid development of LCT [12–14]. Thus, developing relevant theorems for SAFT is of importance and necessary in optical systems and many signal processing applications as well.

In addition, the generalized Hilbert transform closely related to SAFT, called offset Hilbert transform (OHT), is another powerful tool in the fields of optics and signal processing community [15]. It has been presented recently and widely used for image processing, especially for edge detection and enhancement, because it can emphasize the derivatives of the image [16, 17]. In recent decades, many essential theories and useful applications of SAFT and OHT have been derived from in-depth researching on it [8, 15, 18, 19]. To the best of our knowledge, Poisson sum formula has been generalized with many transforms such as FRFT, LCT, fractional Laplace transform and fractional Hilbert transform [1, 2, 20, 21]. However, none of the research papers throw light on the study of the traditional Poisson sum formula associated with the SAFT and OHT yet. Based on the existing results, the motivation of this paper is to generalize the above-mentioned Poisson sum formula into SAFT and OHT domains.

The rest of this paper is organized as follows. Section 2 gives some fundamental knowledge of SAFT and OHT. In Section 3, we give the relationships between SAFT/OHT and FT in detail. Some novel Poisson summation formulae associated with SAFT are presented in Section 4. Section 5 concludes the paper.

#### 2. Preliminaries

##### 2.1. The Special Affine Fourier Transform

The special affine Fourier transform (SAFT) with real parameters of a signal is defined by the following [5, 22]: where and . Note that, for , the SAFT of a signal is essentially a chirp multiplication and it is of no particular interest for our objective in this work. Hence, without loss of generality, we set in the following section unless stated otherwise. The inverse of an SAFT with parameter is given by an SAFT with parameter , which is where . This can be verified by the definition of SAFT. Most of important transforms can be its special cases when parameter is replaced with specific parameters. For example, when , SAFT coincides with FT; when , SAFT is FRFT; when , SAFT equals LCT. Furthermore, many important theories on SAFT have been investigated [8, 15, 23, 24].

##### 2.2. Offset Hilbert Transform

The offset Hilbert transform (OHT) of a signal is defined as follows [15]: It should be noted that the above definition uses the Cauchy principal value of the integral (denoted here by p.v.). To obtain the relationship between the stand and HT and OHT, we can rewrite (6) as Notice that computing the OHT of a signal is equivalent to multiplying it by a chirp, , then passing the product through a standard Hilbert filter and finally multiplying the output by the chirp, . This relationship between OHT and the classical HT can be shown in Figure 1.