Preface

*"The fact is that Gutenberg led to books being put in shelves, and digitisation is taking books off shelves. If you start taking books off shelves then you are only going to find what you are looking for, which does not help those who do not know what they are looking for."*

*- Jeanette Winterson*

Modern technologies surround all of us and they are our most reliable partners for the future. The 21st century ushered in a new era of technology that has been reshaping everyday life, simplifying outdated processes, and even giving rise to entirely new business sectors. On the other hand, contemporary customers and users of products and services expect more and more personalized products and services that can meet their unique needs. Through good-quality work and determination, clients will share with you their business needs and requirements, certain that you will find the right solutions for them. Therefore, companies must continuously improve their competitiveness and explore modern ways and techniques for the full experience of clients. For this, it is necessary to further develop existing methods, adapt them to new applications, or the discovering of new methods.

Methods that have an increasing impact on humanity today and can solve different types of problems even in specific industries are: artificial intelligence, neural networks, machine learning, digital signal processing, spectroscopy, process optimization, methods for analyzing and predicting financial markets, time series analysis etc. Upgrading with Fourier Transformation gives a different meaning to the method. Methods based on Fourier Transformation have great application in all areas of science and engineering. New technologies support their development and have good projected acceleration in the future. The future has five faces: Innovation, Digitalization, Urbanization, Community, and Humanity. The scientific sector should develop each of these faces, but one that occupies a leading position is definitely Digitalization. Digitalization is transforming our industry and making it more efficient and customer-centric than ever before. It strives for the future every day and is struggling to overcome professional challenges.

This book provides a detailed overview of some Fourier Transformation based methods that have an increasing impact on humanity and can solve different types of problems.

The first chapter presents a mathematical theory of a Discrete Hankel Transform (DHT) that is shown to arise from a discretization scheme based on the theory of Fourier–Bessel expansions. The Hankel Transform (HT), also known as the Fourier-Bessel Transform, is a significant integral transform that has been applied in many areas of science and engineering. HT is closely related to the Fourier Transform (HT of 0th order is a 2D FT of a rotationally symmetric function, HT appears in defining the 2D FT in polar coordinates and the spherical, as well as in the definition of the 3D FT in spherical polar coordinates). The scientific contribution of this chapter is to present the theory of DHT as a 'stand-alone' transformation. The author has

systematically demonstrated the standard operating rules for multiplication, modulation, shift, and convolution. Sampling and interpolation theorems were shown, as well as the theory and numerical steps to use the presented discrete theory for the purpose of approximating the continuous HT. Links to the publicly available, open-source numerical code were also included.

The application of the Fourier Transform for obtaining matrix equations (inhomogeneous and homogeneous generalized Fredholm equations) is described in the second chapter. The authors have systematically expanded on the scope of Fourier Transforms by application to a relatively new class (a vector generalization) of integral equations, named Generalized Fredholm Equations (GFE). They included the mathematical proofs for properties of the integral equations, the relationship between homogeneous and inhomogeneous equations, and the mechanism for release of the evanescent waves converting them to travelling ones. The scientific contribution of this chapter is the discovery of a strong relationship between the resonant solutions of the Generalized Homogeneous Fredholm Equations for the electromagnetic field and the resonances observed in scattering in nuclear physics. The physical interpretation of the new class of resonances makes it possible to discern completely new applications in different subjects such as electromagnetic wave propagation or the understanding of meta-materials.

The development of a new technique for spectral analysis for unevenly sampled data, called Non-Uniform Discrete Fourier Transform (NUDFT), is the subject of the third chapter. The new method of dealing with unevenly sampled data was developed and it has interesting anti-aliasing properties. Namely, unlike in electronic devices, it is very difficult to devise procedures to detect aliasing in humans. In electronic devices, aliasing can be easily detected by changing the sampling rate. In humans, fluctuations of heart rate are of the same order as the required changes in the sampling rates. It is therefore very important to develop a proper procedure for detecting aliasing in humans. One of the main points in this chapter is devoted to the problem of how to detect aliasing in the heart rate spectral analysis. The authors conducted an experiment that gave a clear insight into the mechanism of aliasing in the heart rate (R-R) interval spectrum. They discussed the relationship between the R-R interval spectral analysis and the spectral analysis of the corresponding electrocardiography (ECG) signal from which the R-R intervals were evaluated. The spectral analysis of the ECG signal is more sensitive and accurate compared to the R-R interval spectral analysis and is free from aliasing.

A method based on the principle of Fourier spectrum cloning for the denoising of images is proposed in the fourth chapter. This method improves the peak signal to noise ratio (PSNR) and the structural similarity (SSIM) ratio in comparison with spectrum masking denoising. Much refinement can be implemented in the future in order to improve these results obtained with the simplistic application of the cloning principle. The construction of the synthetic replacement part of the spectrum could be synthesized considering different parameters such as border effects or statistical measures on the spectrum, which encourages further research.

The fifth chapter discusses the development of new methods for forecasting time series and application of existing techniques in different areas, which is a permanent concern for both researchers and companies that are interested in gaining competitive advantages. Financial market analysis is important for investors who invest money on the market and want security in multiplying their investment.

**V**

copy (fTCDS).

Between the existing techniques, artificial neural networks have proven to be very good in predicting financial market performance. In this chapter, for time series analysis and forecasting of specific values, the nonlinear autoregressive exogenous (NARX) neural network is used. As an input to the network, both data in the time domain and those in the frequency domain obtained using the Fourier Transform are used. After the experiment was performed, the results were compared to determine the potentially best time series for predicting, as well as the convenience of the domain in which better results are obtained. In addition, the fifth chapter presents valuable information on the various computational intelligence methods in finance. Authors reported proposals for improving the neural network, which is of

The sixth chapter illustrates the importance of Fourier Transform (FT) and the central role that FT plays in optical spectroscopy. FT is especially important in the field of ultrafast spectroscopy because it enables new types of molecular dynamic investigations. With its conceptual approach, without too much mathematical formalism and general technical capabilities, this chapter illustrates how FT helps conceptualize light and helps characterize laser pulses. These pulses can be used to learn about the molecules with which they interact. Consequently, pulsed laser spectroscopy has become an important tool for investigating and characterizing electronic and nuclear structure of protein complexes. In particular, this chapter focuses on femto-second spectroscopy because such systems are now commercially

available and are becoming an essential tool to study molecular dynamics.

and accurate for determining the geographical origin of a sample.

In the eighth chapter, the application of Fourier analysis of cerebral glucose metabolism in color induced long-term potentiation is demonstrated. The study in mice focuses on the implementation of the mechanistic strategies for brain function in color processing using the Fourier analysis of the time series of the standardized uptake values (SUV) as a surrogate marker of cerebral metabolism of glucose. The main aim of the evolutionary trend is to optimize perception of the 'whole' environment by functional coupling of the genes for complementarity of brain hemispheres within self, and between genders. The potential use of these findings in animal models of memory deficits is of great interest to researchers in degenerative brain diseases. This new approach can be useful in resolving the binding problem of conscious experience. In addition, it can have a wide range of applications in several areas, including neuroscience and artificial intelligence. The authors suggested that the latter approach be called functional positron emission tomography spectroscopy (fPETS), analogous to the already known functional transcranial Doppler spectros-

Chapter seven provides valuable information that can be used to develop a reference database of herbs in order to provide basic information on relevant medicinal products for the purpose of authenticity, since the product spectrum can be rapidly matched to validate its geographical origin and predict the anthocyanin content that has been reported as the key component in therapeutic studies. In this sense, the study focused on the database establishment for the authentication of Roselle (*Hibiscus sabdariffa*) raw materials collected from seven selected locations on the western coastline in Peninsular Malaysia. The contribution of this chapter is the developed method of Assured ID software of Roselle, which can be used as a reference database for a sample from an unknown geographical location. The model was based on FTIR spectrum and it showed that this method is rapid, non-destructive,

particular importance.

Between the existing techniques, artificial neural networks have proven to be very good in predicting financial market performance. In this chapter, for time series analysis and forecasting of specific values, the nonlinear autoregressive exogenous (NARX) neural network is used. As an input to the network, both data in the time domain and those in the frequency domain obtained using the Fourier Transform are used. After the experiment was performed, the results were compared to determine the potentially best time series for predicting, as well as the convenience of the domain in which better results are obtained. In addition, the fifth chapter presents valuable information on the various computational intelligence methods in finance. Authors reported proposals for improving the neural network, which is of particular importance.

The sixth chapter illustrates the importance of Fourier Transform (FT) and the central role that FT plays in optical spectroscopy. FT is especially important in the field of ultrafast spectroscopy because it enables new types of molecular dynamic investigations. With its conceptual approach, without too much mathematical formalism and general technical capabilities, this chapter illustrates how FT helps conceptualize light and helps characterize laser pulses. These pulses can be used to learn about the molecules with which they interact. Consequently, pulsed laser spectroscopy has become an important tool for investigating and characterizing electronic and nuclear structure of protein complexes. In particular, this chapter focuses on femto-second spectroscopy because such systems are now commercially available and are becoming an essential tool to study molecular dynamics.

Chapter seven provides valuable information that can be used to develop a reference database of herbs in order to provide basic information on relevant medicinal products for the purpose of authenticity, since the product spectrum can be rapidly matched to validate its geographical origin and predict the anthocyanin content that has been reported as the key component in therapeutic studies. In this sense, the study focused on the database establishment for the authentication of Roselle (*Hibiscus sabdariffa*) raw materials collected from seven selected locations on the western coastline in Peninsular Malaysia. The contribution of this chapter is the developed method of Assured ID software of Roselle, which can be used as a reference database for a sample from an unknown geographical location. The model was based on FTIR spectrum and it showed that this method is rapid, non-destructive, and accurate for determining the geographical origin of a sample.

In the eighth chapter, the application of Fourier analysis of cerebral glucose metabolism in color induced long-term potentiation is demonstrated. The study in mice focuses on the implementation of the mechanistic strategies for brain function in color processing using the Fourier analysis of the time series of the standardized uptake values (SUV) as a surrogate marker of cerebral metabolism of glucose. The main aim of the evolutionary trend is to optimize perception of the 'whole' environment by functional coupling of the genes for complementarity of brain hemispheres within self, and between genders. The potential use of these findings in animal models of memory deficits is of great interest to researchers in degenerative brain diseases. This new approach can be useful in resolving the binding problem of conscious experience. In addition, it can have a wide range of applications in several areas, including neuroscience and artificial intelligence. The authors suggested that the latter approach be called functional positron emission tomography spectroscopy (fPETS), analogous to the already known functional transcranial Doppler spectroscopy (fTCDS).

**IV**

systematically demonstrated the standard operating rules for multiplication, modulation, shift, and convolution. Sampling and interpolation theorems were shown, as well as the theory and numerical steps to use the presented discrete theory for the purpose of approximating the continuous HT. Links to the publicly available,

The application of the Fourier Transform for obtaining matrix equations (inhomogeneous and homogeneous generalized Fredholm equations) is described in the second chapter. The authors have systematically expanded on the scope of Fourier Transforms by application to a relatively new class (a vector generalization) of integral equations, named Generalized Fredholm Equations (GFE). They included the mathematical proofs for properties of the integral equations, the relationship between homogeneous and inhomogeneous equations, and the mechanism for release of the evanescent waves converting them to travelling ones. The scientific contribution of this chapter is the discovery of a strong relationship between the resonant solutions of the Generalized Homogeneous Fredholm Equations for the electromagnetic field and the resonances observed in scattering in nuclear physics. The physical interpretation of the new class of resonances makes it possible to discern completely new applications in different subjects such as electromagnetic

The development of a new technique for spectral analysis for unevenly sampled data, called Non-Uniform Discrete Fourier Transform (NUDFT), is the subject of the third chapter. The new method of dealing with unevenly sampled data was developed and it has interesting anti-aliasing properties. Namely, unlike in electronic devices, it is very difficult to devise procedures to detect aliasing in humans. In electronic devices, aliasing can be easily detected by changing the sampling rate. In humans, fluctuations of heart rate are of the same order as the required changes in the sampling rates. It is therefore very important to develop a proper procedure for detecting aliasing in humans. One of the main points in this chapter is devoted to the problem of how to detect aliasing in the heart rate spectral analysis. The authors conducted an experiment that gave a clear insight into the mechanism of aliasing in the heart rate (R-R) interval spectrum. They discussed the relationship between the R-R interval spectral analysis and the spectral analysis of the corresponding electrocardiography (ECG) signal from which the R-R intervals were evaluated. The spectral analysis of the ECG signal is more sensitive and accurate

compared to the R-R interval spectral analysis and is free from aliasing.

A method based on the principle of Fourier spectrum cloning for the denoising of images is proposed in the fourth chapter. This method improves the peak signal to noise ratio (PSNR) and the structural similarity (SSIM) ratio in comparison with spectrum masking denoising. Much refinement can be implemented in the future in order to improve these results obtained with the simplistic application of the cloning principle. The construction of the synthetic replacement part of the spectrum could be synthesized considering different parameters such as border effects or statistical measures on the spectrum, which encourages further research.

The fifth chapter discusses the development of new methods for forecasting time series and application of existing techniques in different areas, which is a permanent concern for both researchers and companies that are interested in gaining competitive advantages. Financial market analysis is important for investors who invest money on the market and want security in multiplying their investment.

open-source numerical code were also included.

wave propagation or the understanding of meta-materials.

In addition, the reference list included in each chapter contains both historical and extensive analysis, which work together with the articles that describe several key breakthroughs in the mentioned areas of interest.

> **Goran S. Nikolić** Professor, University of Niš, Faculty of Technology, Leskovac, Serbia

Chapter 1

Abstract

polar coordinates

1. Introduction

[1, 2].

1

Natalie Baddour

forward and inverse Hankel transform.

discrete transform in its own right.

The Discrete Hankel Transform

The Hankel transform is an integral transform and is also known as the Fourier-Bessel transform. Until recently, there was no established discrete version of the transform that observed the same sort of relationship to its continuous counterpart as the discrete Fourier transform does to the continuous Fourier transform. Previous definitions of a discrete Hankel transform (DHT) only focused on methods to approximate the integrals of the continuous Hankel integral transform. Recently published work has remedied this and this chapter presents this theory. Specifically, this chapter presents a theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier-Bessel expansions. The standard set of shift, modulation, multiplication, and convolution rules are shown. In addition to being a discrete transform in its own right, this DHT can approximate the continuous

Keywords: Fourier-Bessel, Hankel transform, transform rules, discrete transform,

The Hankel transform has seen applications in many areas of science and engineering. For example, there are applications in propagation of beams and waves, the generation of diffusion profiles and diffraction patterns, imaging and tomographic reconstructions, designs of beams, boundary value problems, etc. The Hankel transform also has a natural relationship to the Fourier transform since the Hankel transform of zeroth order is a 2D Fourier transform of a rotationally symmetric function. Furthermore, the Hankel transform also appears naturally in defining the 2D Fourier transform in polar coordinates and the spherical Hankel transform also appears in the definition of the 3D Fourier transform in spherical polar coordinates

As useful as the Hankel transform may be, there is no discrete Hankel transform (DHT) that exists that has the same relationship to the continuous Hankel transform in the same way that the discrete Fourier transform (DFT) exists alongside the continuous Fourier transform. By this, we mean that the discrete transform exists as a transform in its own right, has its own mathematical theory of the manipulated quantities, and finally as an added bonus, can be used to approximate the continuous version of the transform, with relevant sampling and interpolation theories. Until recently, a discrete Hankel transform merely implied an attempt to discretize the integral(s) of the continuous Hankel transform, rather than an independent

Such a theory of a DHT was recently proposed [3]. Thus, goal of this chapter is to outline the mathematical theory for the DHT. The mathematical standard set of

## **Dragana Z. Marković-Nikolić, PhD** High Technologically Artistic Professional School, Leskovac, Serbia

## Chapter 1 The Discrete Hankel Transform

Natalie Baddour

## Abstract

The Hankel transform is an integral transform and is also known as the Fourier-Bessel transform. Until recently, there was no established discrete version of the transform that observed the same sort of relationship to its continuous counterpart as the discrete Fourier transform does to the continuous Fourier transform. Previous definitions of a discrete Hankel transform (DHT) only focused on methods to approximate the integrals of the continuous Hankel integral transform. Recently published work has remedied this and this chapter presents this theory. Specifically, this chapter presents a theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier-Bessel expansions. The standard set of shift, modulation, multiplication, and convolution rules are shown. In addition to being a discrete transform in its own right, this DHT can approximate the continuous forward and inverse Hankel transform.

Keywords: Fourier-Bessel, Hankel transform, transform rules, discrete transform, polar coordinates

## 1. Introduction

The Hankel transform has seen applications in many areas of science and engineering. For example, there are applications in propagation of beams and waves, the generation of diffusion profiles and diffraction patterns, imaging and tomographic reconstructions, designs of beams, boundary value problems, etc. The Hankel transform also has a natural relationship to the Fourier transform since the Hankel transform of zeroth order is a 2D Fourier transform of a rotationally symmetric function. Furthermore, the Hankel transform also appears naturally in defining the 2D Fourier transform in polar coordinates and the spherical Hankel transform also appears in the definition of the 3D Fourier transform in spherical polar coordinates [1, 2].

As useful as the Hankel transform may be, there is no discrete Hankel transform (DHT) that exists that has the same relationship to the continuous Hankel transform in the same way that the discrete Fourier transform (DFT) exists alongside the continuous Fourier transform. By this, we mean that the discrete transform exists as a transform in its own right, has its own mathematical theory of the manipulated quantities, and finally as an added bonus, can be used to approximate the continuous version of the transform, with relevant sampling and interpolation theories. Until recently, a discrete Hankel transform merely implied an attempt to discretize the integral(s) of the continuous Hankel transform, rather than an independent discrete transform in its own right.

Such a theory of a DHT was recently proposed [3]. Thus, goal of this chapter is to outline the mathematical theory for the DHT. The mathematical standard set of "DFT-like" rules of shift, modulation, multiplication and convolution rules are derived and presented. A Parseval-like theorem is presented, as are sampling and interpolation theorems. The manner in which this DHT can be used to approximate the continuous Hankel transform is also explained.

## 2. Hankel transforms and Bessel series

To start, we define the Hankel transform and Fourier-Bessel series as used in this work.

#### 2.1 Hankel transform

The nth-order Hankel transform Fð Þρ of the function f rð Þ of a real variable, r≥ 0, is defined by the integral [4]

$$F(\rho) = \mathbb{H}\_n(f(r)) = \bigcap\_{n=1}^{\infty} f(r)I\_n(\rho r)r dr,\tag{1}$$

3. Sampling and interpolation theorems for band-limited and

Sampling and interpolation theorems supply the answers to some important questions. For example, given a bandlimited function in frequency space, a sampling theorem answers the question of which samples of the original function are required in order to determine the function completely. The interpolation theorem shows how to interpolate those samples to recover the original function completely. Here, a band-limit means boundedness in frequency. In many applications such as tomography, the notion of a band-limit is not necessarily a property of a function, but rather a limitation of the measurement equipment used to acquire measurements. These measurements are then used to reconstruct some desired function. Thus, the sampling theorem can also answer the question of how band-limits (frequency sensitivities) of measurement equipment determine the resolution of

Given a space-limited function, the sampling theorem answers the question of which samples in frequency space determine the function completely, i.e., those that are required to reconstruct the original function. In other words, the sampling theorem dictates which frequency measurements need to be made. As before, the interpolation theorem will give a formula for interpolating those samples to recons-

We state here the sampling theorem in the same way that Shannon stated it for functions in time and frequency: if a spatial function f rð Þ contains no frequencies higher than W cycles per meter, then it is completely determined by a series of

Proof: suppose that a function f rð Þ is band-limited in the frequency Hankel domain so that its spectrum Fð Þρ is zero outside of an interval 0½ � ; 2πW . The interval is written in this form since W would typically be quoted in units of Hz (cycles per second) if using temporal units or cycles per meter if using spatial units. Therefore, the multiplication by 2π ensures that the final units are in s�<sup>1</sup> or m�1. Let us define W<sup>ρ</sup> ¼ 2πW. Since the Hankel transform Fð Þρ is defined on a finite portion of the

� �, it can be expanded in terms of a Fourier Bessel series as

nk � � <sup>ð</sup><sup>W</sup><sup>ρ</sup> 0

nk � � <sup>f</sup> <sup>j</sup>

In (6), we have used the fact that f rð Þ can be written in terms of its inverse Hankel transform, Eq. (2), in combination with the fact that the function is assumed

Fð Þρ Jn

nk W<sup>ρ</sup>

� � can be written as

j nkρ W<sup>ρ</sup> � �ρd<sup>ρ</sup>

Fð Þ¼ ρ ∑ ∞ k¼1 FkJn j nkρ W<sup>ρ</sup>

where the Fourier Bessel coefficients can be found from Eq. (4) as

<sup>¼</sup> <sup>2</sup> W<sup>2</sup> ρ J 2 <sup>n</sup>þ<sup>1</sup> j

Fk <sup>¼</sup> <sup>2</sup> W<sup>2</sup> ρ J 2 <sup>n</sup>þ<sup>1</sup> j

Hence, a function bandlimited to 0;W<sup>ρ</sup>

nk

<sup>W</sup><sup>ρ</sup> where W<sup>ρ</sup> ¼ 2πW.

� �: (5)

� � (6)

space-limited functions

DOI: http://dx.doi.org/10.5772/intechopen.84399

The Discrete Hankel Transform

those measurements.

real line 0;W<sup>ρ</sup>

band-limited.

3

truct the continuous function completely.

3.1 Sampling theorem for a band-limited function

sampling points given by evaluating f rð Þ at <sup>r</sup> <sup>¼</sup> <sup>j</sup>

where Jnð Þz is the nth-order Bessel function of the first kind. If n is real and n . � 1=2, the transform is self-reciprocating and the inversion formula is given by

$$f(r) = \bigcap\_{\rho}^{\infty} F(\rho) I\_n(\rho r) \rho d\rho. \tag{2}$$

Thus, Hankel transforms take functions in the spatial r domain and transform them to functions in the spatial frequency ρ domain f rð Þ ⇔ Fð Þρ . The notation ⇔ is used to indicate a Hankel transform pair.

#### 2.2 Fourier Bessel series

It is known that functions defined on a finite portion of the real line 0½ � ; R , can be expanded in terms of a Fourier Bessel series [5] given by

$$f(r) = \sum\_{k=1}^{\infty} f\_k J\_n \left(\frac{j\_{nk}r}{R}\right),\tag{3}$$

where the order, n, of the Bessel function is arbitrary and j nk denotes the kth root of the nth Bessel function Jnð Þz . The Fourier Bessel coefficients f <sup>k</sup> of the function f rð Þ are given by

$$f\_k = \frac{2}{R^2 f\_{n+1}^2(j\_{nk})} \int\_0^R f(r) J\_n\left(\frac{j\_{nk}r}{R}\right) r \, dr. \tag{4}$$

Eqs. (3) and (4) can be considered to be a transform pair where the continuous function f rð Þ is forward-transformed to the discrete vector f <sup>k</sup> given in (4). The inverse transform is then the operation which returns f rð Þ if given f <sup>k</sup>, and is given by the summation in Eq. (3). The Fourier Bessel series has the same relationship to the Hankel transform as the Fourier series has to the Fourier transform.

"DFT-like" rules of shift, modulation, multiplication and convolution rules are derived and presented. A Parseval-like theorem is presented, as are sampling and interpolation theorems. The manner in which this DHT can be used to approximate

Fourier Transforms - Century of Digitalization and Increasing Expectations

To start, we define the Hankel transform and Fourier-Bessel series as used in this

The nth-order Hankel transform Fð Þρ of the function f rð Þ of a real variable, r≥ 0,

∞ð

f rð ÞJnð Þ ρr rdr, (1)

Fð Þρ Jnð Þ ρr ρdρ: (2)

, (3)

nk denotes the kth root of

r dr: (4)

0

n . � 1=2, the transform is self-reciprocating and the inversion formula is given by

Thus, Hankel transforms take functions in the spatial r domain and transform them to functions in the spatial frequency ρ domain f rð Þ ⇔ Fð Þρ . The notation ⇔ is used to

It is known that functions defined on a finite portion of the real line 0½ � ; R , can be

the nth Bessel function Jnð Þz . The Fourier Bessel coefficients f <sup>k</sup> of the function f rð Þ

0

function f rð Þ is forward-transformed to the discrete vector f <sup>k</sup> given in (4). The inverse transform is then the operation which returns f rð Þ if given f <sup>k</sup>, and is given by the summation in Eq. (3). The Fourier Bessel series has the same relationship to

the Hankel transform as the Fourier series has to the Fourier transform.

Eqs. (3) and (4) can be considered to be a transform pair where the continuous

f rð ÞJn

j nkr R � �

Fð Þ¼ ρ Hnð Þ¼ f rð Þ

f rð Þ¼

expanded in terms of a Fourier Bessel series [5] given by

where the order, n, of the Bessel function is arbitrary and j

<sup>f</sup> <sup>k</sup> <sup>¼</sup> <sup>2</sup> R2 J 2 <sup>n</sup>þ<sup>1</sup> j nk � � ð R

f rð Þ¼ ∑ ∞ k¼1 f <sup>k</sup> Jn j nkr R � �

where Jnð Þz is the nth-order Bessel function of the first kind. If n is real and

∞ð

0

the continuous Hankel transform is also explained.

2. Hankel transforms and Bessel series

work.

2.1 Hankel transform

is defined by the integral [4]

indicate a Hankel transform pair.

2.2 Fourier Bessel series

are given by

2

## 3. Sampling and interpolation theorems for band-limited and space-limited functions

Sampling and interpolation theorems supply the answers to some important questions. For example, given a bandlimited function in frequency space, a sampling theorem answers the question of which samples of the original function are required in order to determine the function completely. The interpolation theorem shows how to interpolate those samples to recover the original function completely. Here, a band-limit means boundedness in frequency. In many applications such as tomography, the notion of a band-limit is not necessarily a property of a function, but rather a limitation of the measurement equipment used to acquire measurements. These measurements are then used to reconstruct some desired function. Thus, the sampling theorem can also answer the question of how band-limits (frequency sensitivities) of measurement equipment determine the resolution of those measurements.

Given a space-limited function, the sampling theorem answers the question of which samples in frequency space determine the function completely, i.e., those that are required to reconstruct the original function. In other words, the sampling theorem dictates which frequency measurements need to be made. As before, the interpolation theorem will give a formula for interpolating those samples to reconstruct the continuous function completely.

#### 3.1 Sampling theorem for a band-limited function

We state here the sampling theorem in the same way that Shannon stated it for functions in time and frequency: if a spatial function f rð Þ contains no frequencies higher than W cycles per meter, then it is completely determined by a series of sampling points given by evaluating f rð Þ at <sup>r</sup> <sup>¼</sup> <sup>j</sup> nk <sup>W</sup><sup>ρ</sup> where W<sup>ρ</sup> ¼ 2πW.

Proof: suppose that a function f rð Þ is band-limited in the frequency Hankel domain so that its spectrum Fð Þρ is zero outside of an interval 0½ � ; 2πW . The interval is written in this form since W would typically be quoted in units of Hz (cycles per second) if using temporal units or cycles per meter if using spatial units. Therefore, the multiplication by 2π ensures that the final units are in s�<sup>1</sup> or m�1. Let us define W<sup>ρ</sup> ¼ 2πW. Since the Hankel transform Fð Þρ is defined on a finite portion of the real line 0;W<sup>ρ</sup> � �, it can be expanded in terms of a Fourier Bessel series as

$$F(\rho) = \sum\_{k=1}^{\infty} F\_k J\_n \left( \frac{j\_{nk}\rho}{W\_\rho} \right). \tag{5}$$

where the Fourier Bessel coefficients can be found from Eq. (4) as

$$\begin{split} F\_k &= \frac{2}{W\_\rho^2 f\_{n+1}^2(j\_{nk})} \int\_0^{W\_\rho} F(\rho) f\_n \left( \frac{j\_{nk} \rho}{W\_\rho} \right) \rho d\rho \\ &= \frac{2}{W\_\rho^2 f\_{n+1}^2(j\_{nk})} f \left( \frac{j\_{nk}}{W\_\rho} \right) \end{split} \tag{6}$$

In (6), we have used the fact that f rð Þ can be written in terms of its inverse Hankel transform, Eq. (2), in combination with the fact that the function is assumed band-limited.

Hence, a function bandlimited to 0;W<sup>ρ</sup> � � can be written as Fourier Transforms - Century of Digitalization and Increasing Expectations

$$F(\rho) = \begin{cases} \sum\_{k=1}^{\infty} \frac{2}{W\_{\rho}^{2} J\_{n+1}^{2} (j\_{nk})} f\left(\frac{j\_{nk}}{W\_{\rho}}\right) J\_{n}\left(\frac{j\_{nk}\rho}{W\_{\rho}}\right) & \rho < W\_{\rho} \\\\ 0 & \rho \ge W\_{\rho} \end{cases} \tag{7}$$

3.3 Interpretation in terms of a jinc

DOI: http://dx.doi.org/10.5772/intechopen.84399

f rð Þ¼ ∑ ∞ k¼1

� �Jnð Þ <sup>ρ</sup><sup>r</sup> <sup>ρ</sup>d<sup>ρ</sup> <sup>¼</sup> <sup>R</sup>

publication [9], or the classic paper reprint [6]).

∞ð

0

Hankel transform of the Boxcar function is given by

Π<sup>W</sup><sup>ρ</sup> ð Þρ J0ð Þ ρr ρdρ ¼

2

nk � � <sup>f</sup> <sup>j</sup>

∞ð

0

j nk Wρ |{z} generalized shift of <sup>j</sup>

> nkW<sup>2</sup> ρ

<sup>¼</sup> <sup>j</sup>

j 2 nk � rW<sup>ρ</sup>

Π<sup>W</sup><sup>ρ</sup> ð Þ¼ ρ

With this definition of a generalized shift operator, we recognize the integral in Eq. (12) as the inverse Hankel transform of the Boxcar function shifted by <sup>j</sup>

> nk W<sup>ρ</sup>

� �<sup>2</sup> Jnþ<sup>1</sup> <sup>j</sup>

∞ð

8 < :

0

nk � �Jn rW<sup>ρ</sup> � �

1 0≤ ρ≤W<sup>ρ</sup> 0 otherwise �

The boxcar function is a generalized version of the standard Rect function. The Rect function is usually defined as the function which has value 1 over the interval ½ � �1=2; �1=2 and is zero otherwise. Now this is interesting specifically because of the interpretation of Eq. (14). Had we been working in the standard Fourier domain instead of the Hankel domain, the Boxcar function would be replaced with the Rect function and the Hankel transform would be replaced with a standard Fourier transform. Proceeding with this line of thinking, the inverse Fourier transform of the Rect function would be a sinc function, which is the standard interpolation function of the classical Shannon interpolation formula. Hence, the Fourier equivalent interpretation of Eq. (14) is a shifted sinc function. Thus, the formulation above follows exactly the standard Shannon Interpolation formula (see the original

For the relatively simple case of the zeroth-order Hankel transform, the inverse

J0ð Þ ρr ρdρ

<sup>J</sup><sup>1</sup> <sup>W</sup>ρ<sup>r</sup> � � <sup>¼</sup> <sup>W</sup><sup>2</sup>

ρ

J<sup>1</sup> Wρr � � <sup>W</sup>ρ<sup>r</sup> :

W ðρ

0

<sup>¼</sup> <sup>W</sup><sup>ρ</sup> r

nk W<sup>ρ</sup> � � <sup>W</sup>ð<sup>ρ</sup>

In other research work [8], the generalized shift operator Rr<sup>0</sup> indicating a shift of r<sup>0</sup>

0 Jn j nkρ W<sup>ρ</sup>

� �Jnð Þ <sup>ρ</sup><sup>r</sup> <sup>ρ</sup>d<sup>ρ</sup> (12)

nk W<sup>ρ</sup> � �.

(14)

(15)

(16)

Fð Þρ Jnð Þ ρr<sup>0</sup> Jnð Þ ρr ρdρ: (13)

9 = ;

ΠW<sup>ρ</sup> ð Þρ Jnð Þ ρr ρdρ


W<sup>2</sup> ρ J 2 <sup>n</sup>þ<sup>1</sup> j

acting on the function f rð Þ has been defined by the formula

<sup>R</sup><sup>r</sup><sup>0</sup> f rð Þ¼

Eq. (8) states

The Discrete Hankel Transform

More explicitly,

Wðρ

0 Jn j nkρ W<sup>ρ</sup>

where

5

Eq. (7) states that the samples f <sup>j</sup> nk W<sup>ρ</sup> � � determine the function f rð Þ completely since (i) Fð Þρ is determined by Eq. (7), and (ii) f rð Þ is known if Fð Þρ is known. Another way of looking at this is that band-limiting a function to 0;W<sup>ρ</sup> � � results in information about the original function at samples rnk <sup>¼</sup> <sup>j</sup> nk W<sup>ρ</sup> . So, Eq. (7) is the statement of the sampling theorem.

To verify that this sampling theorem is consistent with expectations, we recognize that the zeros of Jnð Þz are almost evenly spaced at intervals of π and that the spacing becomes exactly π in the limit as z ! ∞. To determine the (bandlimited) function f rð Þ completely, we need to sample the function at <sup>f</sup> <sup>j</sup> nk W<sup>ρ</sup> � � <sup>¼</sup> <sup>f</sup> <sup>j</sup> nk 2πW � � and these samples are (eventually) multiples of <sup>π</sup> <sup>ð</sup>2πW<sup>Þ</sup> ¼ <sup>1</sup> <sup>ð</sup>2W<sup>Þ</sup> � � apart, which is consistent with the standard Shannon sampling theorem which requires samples at multiples of <sup>1</sup> <sup>ð</sup>2W<sup>Þ</sup> � [6].

#### 3.2 Interpolation theorem for a band-limited function

It follows from Eq. (7) that f rð Þ can be found by inverse Hankel transformation to give

$$f(r) = \int\_0^{W\_\rho} \left\{ \sum\_{k=1}^\infty \frac{2}{W\_\rho^2 f\_{n+1}^2(j\_{nk})} f\left(\frac{j\_{nk}}{W\_\rho}\right) J\_n\left(\frac{j\_{nk}\rho}{W\_\rho}\right) \right\} I\_n(\rho r) \rho d\rho$$

$$= \sum\_{k=1}^\infty \frac{2}{W\_\rho^2 f\_{n+1}^2(j\_{nk})} f\left(\frac{j\_{nk}}{W\_\rho}\right) \int\_0^{W\_\rho} \left(\frac{j\_{nk}\rho}{W\_\rho}\right) I\_n(\rho r) \rho d\rho.$$

From Watson ([7], p. 134), we have the following result

$$\int\_{\mathbb{T}} f\_n(\alpha \mathbf{z}) \overline{f}\_n(\beta \mathbf{z}) \mathbf{z} d\mathbf{z} = \frac{\mathbf{z} \{a \mathbf{J}\_{n+1}(\alpha \mathbf{z}) \overline{\mathbf{J}}\_n(\beta \mathbf{z}) - \beta \mathbf{J}\_n(\alpha \mathbf{z}) \overline{\mathbf{J}}\_{n+1}(\beta \mathbf{z})\}}{a^2 - \beta^2} \tag{9}$$

Eq. (9) can be used to simplify (8) to give

$$f(r) = \sum\_{k=1}^{\infty} f\left(\frac{j\_{nk}}{W\_{\rho}}\right) \frac{2j\_{nk}}{J\_{n+1}(j\_{nk})} \cdot \frac{J\_n(rW\_{\rho})}{j\_{nk}^2 - r^2W\_{\rho}^2} \tag{10}$$

Eq. (10) gives the formula for interpolating the samples f <sup>j</sup> nk W<sup>ρ</sup> � � to reconstruct the continuous band-limited function f rð Þ. Each term used to reconstruct the original function f rð Þ consists of the samples of the function f rð Þ at <sup>r</sup> <sup>¼</sup> <sup>j</sup> nk W<sup>ρ</sup> � � multiplied by a reconstructing function of the form

$$\frac{2j\_{nk}}{J\_{n+1}(j\_{nk})} \begin{array}{c} J\_n \left( rW\_\rho \right) \\ j\_{nk}^2 - r^2 W\_\rho^2 \end{array} . \tag{11}$$

## 3.3 Interpretation in terms of a jinc

Eq. (8) states

Fð Þ¼ ρ

statement of the sampling theorem.

tiples of <sup>1</sup> <sup>ð</sup>2W<sup>Þ</sup>

to give

4

� [6].

f rð Þ¼

ð

W ðρ

0

¼ ∑ ∞ k¼1

Eq. (9) can be used to simplify (8) to give

reconstructing function of the form

Eq. (7) states that the samples f <sup>j</sup>

∑ ∞ k¼1

8 >><

>>:

W<sup>2</sup> ρ J 2 <sup>n</sup>þ<sup>1</sup> j nk � � <sup>f</sup> <sup>j</sup>

information about the original function at samples rnk <sup>¼</sup> <sup>j</sup>

function f rð Þ completely, we need to sample the function at <sup>f</sup> <sup>j</sup>

these samples are (eventually) multiples of <sup>π</sup> <sup>ð</sup>2πW<sup>Þ</sup> ¼ <sup>1</sup> <sup>ð</sup>2W<sup>Þ</sup>

3.2 Interpolation theorem for a band-limited function

∑ ∞ k¼1

W<sup>2</sup> ρ J 2 <sup>n</sup>þ<sup>1</sup> j nk � � <sup>f</sup> <sup>j</sup>

f rð Þ¼ ∑ ∞ k¼1

Eq. (10) gives the formula for interpolating the samples f <sup>j</sup>

function f rð Þ consists of the samples of the function f rð Þ at <sup>r</sup> <sup>¼</sup> <sup>j</sup>

2j nk Jnþ<sup>1</sup> j nk � �

W<sup>2</sup> ρ J 2 <sup>n</sup>þ<sup>1</sup> j nk � � <sup>f</sup> <sup>j</sup>

2

From Watson ([7], p. 134), we have the following result

f j nk W<sup>ρ</sup> � � 2j

2

2

Fourier Transforms - Century of Digitalization and Increasing Expectations

nk W<sup>ρ</sup> � �

since (i) Fð Þρ is determined by Eq. (7), and (ii) f rð Þ is known if Fð Þρ is known. Another way of looking at this is that band-limiting a function to 0;W<sup>ρ</sup>

To verify that this sampling theorem is consistent with expectations, we recognize that the zeros of Jnð Þz are almost evenly spaced at intervals of π and that the spacing becomes exactly π in the limit as z ! ∞. To determine the (bandlimited)

tent with the standard Shannon sampling theorem which requires samples at mul-

It follows from Eq. (7) that f rð Þ can be found by inverse Hankel transformation

( ) � �

nk W<sup>ρ</sup> � �

Jnð Þ <sup>α</sup><sup>z</sup> Jnð Þ <sup>β</sup><sup>z</sup> zdz <sup>¼</sup> <sup>z</sup> <sup>α</sup>Jnþ<sup>1</sup>ð Þ <sup>α</sup><sup>z</sup> Jnð Þ� <sup>β</sup><sup>z</sup> <sup>β</sup>Jnð Þ <sup>α</sup><sup>z</sup> Jnþ<sup>1</sup>ð Þ <sup>β</sup><sup>z</sup> � �

nk Jnþ<sup>1</sup> j nk � �

> Jn rW<sup>ρ</sup> � �

continuous band-limited function f rð Þ. Each term used to reconstruct the original

j 2 nk � <sup>r</sup><sup>2</sup>W<sup>2</sup> ρ Jn rW<sup>ρ</sup> � �

> nk W<sup>ρ</sup> � �

j 2 nk � <sup>r</sup><sup>2</sup>W<sup>2</sup> ρ

nk W<sup>ρ</sup> � �

> W ðρ

0 Jn j nkρ W<sup>ρ</sup> � �

Jn j nkρ W<sup>ρ</sup>

nk W<sup>ρ</sup> � �

0 ρ≥W<sup>ρ</sup>

Jn j nkρ W<sup>ρ</sup> � �

ρ ,W<sup>ρ</sup>

determine the function f rð Þ completely

nk W<sup>ρ</sup> (7)

and

(8)

(10)

to reconstruct the

multiplied by a

� � results in

. So, Eq. (7) is the

<sup>¼</sup> <sup>f</sup> <sup>j</sup> nk 2πW � �

nk W<sup>ρ</sup> � �

Jnð Þ ρr ρdρ

Jnð Þ ρr ρdρ:

<sup>α</sup><sup>2</sup> � <sup>β</sup><sup>2</sup> (9)

nk W<sup>ρ</sup> � �

: (11)

� � apart, which is consis-

$$f(r) = \sum\_{k=1}^{\infty} \frac{2}{\mathcal{W}\_{\rho}^{2} f\_{n+1}^{2}(j\_{nk})} f\left(\frac{j\_{nk}}{\mathcal{W}\_{\rho}}\right) \quad \int\_{0}^{W\_{\rho}} J\_{n}\left(\frac{j\_{nk}\rho}{\mathcal{W}\_{\rho}}\right) J\_{n}(\rho r) \rho d\rho \tag{12}$$

In other research work [8], the generalized shift operator Rr<sup>0</sup> indicating a shift of r<sup>0</sup> acting on the function f rð Þ has been defined by the formula

$$\mathcal{R}^{r\_0} f(r) = \bigcap\_{0}^{\infty} F(\rho) J\_n(\rho r\_0) J\_n(\rho r) \rho d\rho. \tag{13}$$

With this definition of a generalized shift operator, we recognize the integral in Eq. (12) as the inverse Hankel transform of the Boxcar function shifted by <sup>j</sup> nk W<sup>ρ</sup> � �. More explicitly,

$$\begin{aligned} \left\{ \int\_{\boldsymbol{0}} I\_{n} \left( \frac{j\_{\rm k} \rho}{\boldsymbol{W}\_{\rho}} \right) I\_{n}(\rho r) \rho d\rho = \underbrace{\boldsymbol{\mathcal{R}}\_{\boldsymbol{\nu}\_{\rho}}^{\boldsymbol{j}\_{\rm k}}}\_{\text{generalized}} \underbrace{\left\{ \int\_{0}^{\boldsymbol{\alpha}} \Pi\_{\boldsymbol{W}\_{\boldsymbol{\nu}}}(\rho) I\_{n}(\rho r) \rho d\rho}\_{\text{shift of}} \right\}}\_{\text{shift of}} \tag{14} \\ = \frac{j\_{\rm nk} \boldsymbol{W}\_{\rho}^{2}}{j\_{\rm nk}^{2} - (r \boldsymbol{W}\_{\rho})^{2}} I\_{n+1}(\boldsymbol{j}\_{\rm k}) I\_{n}(r \boldsymbol{W}\_{\rho}) \end{aligned} \tag{14}$$

where

$$\Pi\_{W\_{\rho}}(\rho) = \begin{cases} 1 & 0 \le \rho \le W\_{\rho} \\ 0 & \text{otherwise} \end{cases} \tag{15}$$

The boxcar function is a generalized version of the standard Rect function. The Rect function is usually defined as the function which has value 1 over the interval ½ � �1=2; �1=2 and is zero otherwise. Now this is interesting specifically because of the interpretation of Eq. (14). Had we been working in the standard Fourier domain instead of the Hankel domain, the Boxcar function would be replaced with the Rect function and the Hankel transform would be replaced with a standard Fourier transform. Proceeding with this line of thinking, the inverse Fourier transform of the Rect function would be a sinc function, which is the standard interpolation function of the classical Shannon interpolation formula. Hence, the Fourier equivalent interpretation of Eq. (14) is a shifted sinc function. Thus, the formulation above follows exactly the standard Shannon Interpolation formula (see the original publication [9], or the classic paper reprint [6]).

For the relatively simple case of the zeroth-order Hankel transform, the inverse Hankel transform of the Boxcar function is given by

$$\begin{aligned} \int\_0^\infty \Pi\_{W\_\rho}(\rho) J\_0(\rho r) \rho d\rho &= \int\_0^{W\_\rho} I\_0(\rho r) \rho d\rho\\ &= \frac{W\_\rho}{r} J\_1(W\_\rho r) = W\_\rho^2 \frac{J\_1(W\_\rho r)}{W\_\rho r} .\end{aligned} \tag{16}$$

The function <sup>2</sup>J1ð Þ<sup>r</sup> =<sup>r</sup> is often termed the jinc or sombrero function and is the polar coordinate analog of the sinc function, so Eq. (16) is a scaled version of a jinc function.

In fact, from Eqs. (13), (14) and (16), it follows that the generalized shifted version of the jinc function is given by

$$R^{\frac{j\_{0k}}{W\_{\rho}}} \left\{ \frac{2j\_1 \left( W\_{\rho} r \right)}{W\_{\rho} r} \right\} = \frac{2j\_{0k} J\_1 \left( j\_{0k} \right)}{j\_{0k}^2 - \left( r W\_{\rho} \right)^2} J\_0 \left( r W\_{\rho} \right). \tag{17}$$

Fð Þ¼ ρ

DOI: http://dx.doi.org/10.5772/intechopen.84399

The Discrete Hankel Transform

∞ð

f rð ÞJnð Þ ρr rdr

2

F j nk R � � 2j

4. Intuitive discretization scheme for the Hankel transform

nk � � <sup>F</sup> <sup>j</sup>

nk R � �ð R

0 Jn j nkr R

nk Jnþ<sup>1</sup> j nk � �

Based on the sampling theorems above, in this section we explore how assuming that a function can be simultaneously band-limited and space-limited will naturally lead to a discrete Hankel transform. Although it is known that it is not possible to fulfill both of these conditions exactly, it is possible to keep the spectrum within a given frequency band, and to have the space function very small outside some specified spatial interval (or vice-versa). Hence, it is possible for functions to be

We demonstrated above that a band-limited function, with ρ ,W<sup>ρ</sup> ¼ 2πW can

nk � � <sup>f</sup> <sup>j</sup>

nk W<sup>ρ</sup> � �Jn

Now, suppose that in addition to being band-limited, the function is also effectively space limited. As mentioned above, it is known that a function cannot be finite in both space and spatial frequency (equivalently it cannot be finite in both time and frequency if using the Fourier transform). However, if a function is effectively space limited, this means that there exists an integer N for which

� � <sup>≈</sup>0 for <sup>k</sup> . <sup>N</sup>. In other words, we can find an interval beyond which the

space function is very small. In fact, since the Fourier-Bessel series in (24) is known

nk W<sup>ρ</sup> � �Jn

> j nk W<sup>ρ</sup> j nmW<sup>ρ</sup> j nN

j nkρ W<sup>ρ</sup>

,Wρ, and Eq. (25), summing over infinite k, is exact.

≥W<sup>ρ</sup> and by the assumption of the bandlimited nature

� �: (24)

(for any integer

nmW<sup>ρ</sup> j nN

� � <sup>m</sup> , <sup>N</sup>: (25)

2

nk � � <sup>f</sup> <sup>j</sup>

W<sup>2</sup> ρ J 2 <sup>n</sup>þ<sup>1</sup> j

Evaluating the previous Eq. (24) at the sampling points <sup>ρ</sup>nm <sup>¼</sup> <sup>j</sup>

2

j 2

� �Jnð Þ <sup>ρ</sup><sup>r</sup> rdr

Jnð Þ ρR

nk R

nk � ð Þ <sup>ρ</sup><sup>R</sup> <sup>2</sup> (23)

� � to give the continu-

(22)

0

¼ ∑ ∞ k¼1

Using Eq. (9), Eq. (22) can be simplified to give

"effectively" space and band-limited.

Fð Þ¼ ρ ∑ ∞ k¼1

> W<sup>2</sup> ρ J 2 <sup>n</sup>þ<sup>1</sup> j

nmW<sup>ρ</sup> j nN

nmW<sup>ρ</sup> j nN

4.1 Forward transform

be written as

N) gives for m , N

F j

f <sup>j</sup> nk W<sup>ρ</sup>

7

nmW<sup>ρ</sup> j nN � �

For <sup>m</sup> , <sup>N</sup>, then <sup>ρ</sup>nm <sup>¼</sup> <sup>j</sup>

For <sup>m</sup> <sup>≥</sup> <sup>N</sup>, then <sup>ρ</sup>nm <sup>¼</sup> <sup>j</sup>

of the function, Fð Þ¼ ρnm 0.

¼ ∑ ∞ k¼1

ous function Fð Þρ .

R2 J 2 <sup>n</sup>þ<sup>1</sup> j

Fð Þ¼ ρ ∑ ∞ k¼1

Eq. (23) gives the formula for interpolating the samples F <sup>j</sup>

Hence, for a zeroth-order Fourier Bessel transform, Eq. (12), the expansion for f rð Þ reads

$$f(r) \quad = \sum\_{k=1}^{\infty} f\left(\frac{j\_{0k}}{W\_{\rho}}\right) \frac{1}{f\_1^2(j\_{0k})} \underbrace{\frac{2j\_{0k}J\_1(j\_{0k})J\_0(rW\_{\rho})}{j\_{0k}^2 - \left(rW\_{\rho}\right)^2}}\_{=: \mathbb{R}^{\frac{j\_{0k}}{W\_{\rho}}} \left\{\frac{2j\_1(\mathcal{W}\_{\rho}r)}{W\_{\rho}r}\right\}}\tag{18}$$

Eq. (18) says that the interpolating function is a shifted jinc function, in analogy with a scaled sinc being the interpolating function for the sampling theorem used for Fourier transforms.

#### 3.4 Sampling theorem for a space-limited function

Now consider a space-limited function f rð Þ so that f rð Þ is zero outside of an interval 0½ � ; R . It then follows that it can be expanded on 0½ � ; R in terms of a Fourier Bessel series so that

$$f(r) = \sum\_{k=1}^{\infty} f\_k J\_n \left(\frac{j\_{nk}r}{R}\right),\tag{19}$$

where the Fourier Bessel coefficients can be found from

$$f\_n f = \frac{2}{R^2 j\_{n+1}^2(j\_{nk})} \int\_0^R f(r) f\_n \left(\frac{j\_{nk}r}{R}\right) r dr = \frac{2}{R^2 j\_{n+1}^2(j\_{nk})} F\left(\frac{j\_{nk}}{R}\right). \tag{20}$$

Here, we have used the definition of the Hankel transform Fð Þρ , Eq. (1), in the right hand side of Eq. (20). Hence, the function can be written as

$$f(r) = \begin{cases} \sum\_{k=1}^{\infty} \frac{2}{R^2 f\_{n+1}^2(j\_{nk})} F\left(\frac{j\_{nk}}{R}\right) J\_n\left(\frac{j\_{nk}r}{R}\right) & r \le R\\ 0 & r \ge R \end{cases} \tag{21}$$

From Eq. (21), it is evident that the samples F <sup>j</sup> nk R � � determine the function f rð Þ and hence its transform Fð Þρ completely. Another way of looking at this is that space limiting a function to 0½ � ; R implies discretization in spatial frequency space, at frequencies <sup>ρ</sup>nk <sup>¼</sup> <sup>j</sup> nk R .

#### 3.5 Interpolation theorem for a space-limited function

The Hankel transform of the function can then be found from a forward Hankel transformation as

The Discrete Hankel Transform DOI: http://dx.doi.org/10.5772/intechopen.84399

The function <sup>2</sup>J1ð Þ<sup>r</sup> =<sup>r</sup> is often termed the jinc or sombrero function and is the polar coordinate analog of the sinc function, so Eq. (16) is a scaled version of a jinc

In fact, from Eqs. (13), (14) and (16), it follows that the generalized shifted

<sup>¼</sup> <sup>2</sup><sup>j</sup>

j 2

Eq. (18) says that the interpolating function is a shifted jinc function, in analogy with a scaled sinc being the interpolating function for the sampling theorem used

Now consider a space-limited function f rð Þ so that f rð Þ is zero outside of an interval 0½ � ; R . It then follows that it can be expanded on 0½ � ; R in terms of a Fourier

> f rð Þ¼ ∑ ∞ k¼1 f <sup>k</sup>Jn j nkr R � �

f rð ÞJn

nk R � � Jn j nkr R � �

j nkr R � �

Here, we have used the definition of the Hankel transform Fð Þρ , Eq. (1), in the right

0 r≥R

The Hankel transform of the function can then be found from a forward Hankel

hence its transform Fð Þρ completely. Another way of looking at this is that space limiting a function to 0½ � ; R implies discretization in spatial frequency space, at

nk R � �

Hence, for a zeroth-order Fourier Bessel transform, Eq. (12), the expansion for

<sup>0</sup>kJ<sup>1</sup> j 0k � �

� �<sup>2</sup> <sup>J</sup><sup>0</sup> rW<sup>ρ</sup>

<sup>0</sup><sup>k</sup> � rW<sup>ρ</sup> � �<sup>2</sup> |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

� �: (17)

, (19)

nk R � �

determine the function f rð Þ and

: (20)

(21)

(18)

<sup>0</sup><sup>k</sup> � rW<sup>ρ</sup>

2j <sup>0</sup>kJ<sup>1</sup> j 0k � �J<sup>0</sup> rW<sup>ρ</sup> � �

> j 2

¼R j <sup>0</sup><sup>k</sup> <sup>W</sup><sup>ρ</sup> <sup>2</sup>J1ð Þ <sup>W</sup>ρ<sup>r</sup> Wρr n o

r dr <sup>¼</sup> <sup>2</sup> R2 J 2 <sup>n</sup>þ<sup>1</sup> j nk � � <sup>F</sup> <sup>j</sup>

r , R

function.

f rð Þ reads

for Fourier transforms.

Bessel series so that

version of the jinc function is given by

R j 0k Wρ

f rðÞ ¼ ∑

∞ k¼1

3.4 Sampling theorem for a space-limited function

where the Fourier Bessel coefficients can be found from

0

hand side of Eq. (20). Hence, the function can be written as

2

3.5 Interpolation theorem for a space-limited function

<sup>f</sup> <sup>k</sup> <sup>¼</sup> <sup>2</sup> R2 J 2 <sup>n</sup>þ<sup>1</sup> j nk � � ð R

> ∞ k¼1

> > nk R .

R2 J 2 <sup>n</sup>þ<sup>1</sup> j nk � � <sup>F</sup> <sup>j</sup>

From Eq. (21), it is evident that the samples F <sup>j</sup>

f rð Þ¼ <sup>∑</sup>

frequencies <sup>ρ</sup>nk <sup>¼</sup> <sup>j</sup>

transformation as

6

8 < : f j 0k W<sup>ρ</sup> � � 1 J 2 1 j 0k � �

2J<sup>1</sup> Wρr � � Wρr � �

Fourier Transforms - Century of Digitalization and Increasing Expectations

$$\begin{aligned} F(\rho) &= \int\_0^\infty f(r) f\_n(\rho r) r dr \\ &= \sum\_{k=1}^\infty \frac{2}{R^2 f\_{n+1}^2(j\_{nk})} F\left(\frac{j\_{nk}}{R}\right) \Big|\_0^R \left(\frac{j\_{nk} r}{R}\right) f\_n(\rho r) r dr \end{aligned} \tag{22}$$

Using Eq. (9), Eq. (22) can be simplified to give

$$F(\rho) = \sum\_{k=1}^{\infty} F\left(\frac{j\_{nk}}{R}\right) \frac{2j\_{nk}}{J\_{n+1}\left(j\_{nk}\right)} \frac{J\_n(\rho R)}{j\_{nk}^2 - \left(\rho R\right)^2} \tag{23}$$

Eq. (23) gives the formula for interpolating the samples F <sup>j</sup> nk R � � to give the continuous function Fð Þρ .

## 4. Intuitive discretization scheme for the Hankel transform

Based on the sampling theorems above, in this section we explore how assuming that a function can be simultaneously band-limited and space-limited will naturally lead to a discrete Hankel transform. Although it is known that it is not possible to fulfill both of these conditions exactly, it is possible to keep the spectrum within a given frequency band, and to have the space function very small outside some specified spatial interval (or vice-versa). Hence, it is possible for functions to be "effectively" space and band-limited.

#### 4.1 Forward transform

We demonstrated above that a band-limited function, with ρ ,W<sup>ρ</sup> ¼ 2πW can be written as

$$F(\rho) = \sum\_{k=1}^{\infty} \frac{2}{\mathcal{W}\_{\rho}^{2} J\_{n+1}^{2} (j\_{nk})} f\left(\frac{j\_{nk}}{\mathcal{W}\_{\rho}}\right) J\_{n}\left(\frac{j\_{nk}\rho}{\mathcal{W}\_{\rho}}\right). \tag{24}$$

Evaluating the previous Eq. (24) at the sampling points <sup>ρ</sup>nm <sup>¼</sup> <sup>j</sup> nmW<sup>ρ</sup> j nN (for any integer N) gives for m , N

$$F\left(\frac{j\_{nm}W\_{\rho}}{j\_{nN}}\right) = \sum\_{k=1}^{\infty} \frac{2}{W\_{\rho}^{2}f\_{n+1}^{2}(j\_{nk})} f\left(\frac{j\_{nk}}{W\_{\rho}}\right) J\_{n}\left(\frac{j\_{nk}}{W\_{\rho}}\frac{j\_{nm}W\_{\rho}}{j\_{nN}}\right) \quad m \le N. \tag{25}$$

For <sup>m</sup> , <sup>N</sup>, then <sup>ρ</sup>nm <sup>¼</sup> <sup>j</sup> nmW<sup>ρ</sup> j nN ,Wρ, and Eq. (25), summing over infinite k, is exact. For <sup>m</sup> <sup>≥</sup> <sup>N</sup>, then <sup>ρ</sup>nm <sup>¼</sup> <sup>j</sup> nmW<sup>ρ</sup> j nN ≥W<sup>ρ</sup> and by the assumption of the bandlimited nature of the function, Fð Þ¼ ρnm 0.

Now, suppose that in addition to being band-limited, the function is also effectively space limited. As mentioned above, it is known that a function cannot be finite in both space and spatial frequency (equivalently it cannot be finite in both time and frequency if using the Fourier transform). However, if a function is effectively space limited, this means that there exists an integer N for which f <sup>j</sup> nk W<sup>ρ</sup> � � <sup>≈</sup>0 for <sup>k</sup> . <sup>N</sup>. In other words, we can find an interval beyond which the space function is very small. In fact, since the Fourier-Bessel series in (24) is known to converge, it is known that limk!<sup>∞</sup> <sup>f</sup> <sup>j</sup> nk W<sup>ρ</sup> <sup>¼</sup> 0, which means that for any arbitrarily small ε, there must exist an integer N for which f <sup>j</sup> nk W<sup>ρ</sup> , <sup>ε</sup> for <sup>k</sup> . <sup>N</sup>.

Hence, using the argument of "effectively space limited" in the preceding paragraph, we can terminate the series in Eq. (25) at a suitably chosen N that ensures the effective space limit. Terminating the series at k ¼ N is the same as assuming that f rð Þ≈0 for <sup>r</sup> . <sup>R</sup> <sup>¼</sup> <sup>j</sup> nN W<sup>ρ</sup> . Noting that at k ¼ N, the last term in (25) is Jn j nNj nm <sup>j</sup> nN <sup>¼</sup> Jn <sup>j</sup> nm <sup>¼</sup> 0, then after terminating at <sup>N</sup>, Eq. (25) becomes for m ¼ 1::N � 1

$$F\left(\frac{j\_{nm}W\_{\rho}}{j\_{nN}}\right) = \sum\_{k=1}^{N-1} \frac{2}{W\_{\rho}^2 f\_{n+1}^2(j\_{nk})} f\left(\frac{j\_{nk}}{W\_{\rho}}\right) f\_n\left(\frac{j\_{nk}j\_{nm}}{j\_{nN}}\right). \tag{26}$$

Eqs. (29) and (30) are the fundamental relations used for the discrete Hankel

The preceding development shows that a "natural," N-dimensional discretization scheme in finite space 0½ � ; R and finite frequency space 0;W<sup>ρ</sup>

to a finite space domain and vice-versa. The size of the transform N, can be

ffiffiffiffiffi 2 πz r

It is pointed out in [10] that the zeros of Jnð Þz are almost evenly spaced at intervals of π and that the spacing becomes exactly π in the limit as z ! ∞. In fact, it is shown in [10] that a simple asymptotic form for the Bessel function is given by

nkW<sup>ρ</sup> j nN

The relationship WρR ¼ j nN can be used to change from finite frequency domain

cos z � n þ

2

2

� � π

n 2

2

Eq. (32) becomes a better approximation to Jnð Þz as z ! ∞. The zeros of the cosine function are at odd multiples of π=2. Therefore, an approximation to the Bessel

nk <sup>≈</sup> <sup>2</sup><sup>k</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup>

<sup>2</sup>πWR <sup>¼</sup> <sup>j</sup> nN <sup>≈</sup> <sup>2</sup><sup>N</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup>

2WR≈ N þ

This is exactly analogous to the corresponding expression for Fourier transforms. Specifically, for temporal Fourier transforms Shannon [6] argued that "If the function is limited to the time interval T and the samples are spaced 1/(2 W) seconds apart (where W is the bandwidth), there will be a total of 2WT samples in the interval. All samples outside will be substantially zero. To be more precise, we can define a function to be limited to the time interval T if, and only if, all the samples outside this interval are exactly zero. Then we can say that any function limited to the bandwidth W and the time interval T can be specified by giving N ¼ 2WT numbers". Following this line of thinking, Eq. (35) states that for an nth-order Hankel transform, any function limited to the bandwidth W and the space interval R can be specified by giving N ¼ ð Þ 2WR � n=2 numbers and it will certainly be true that specifying N ¼ 2WR numbers will specify the function, in exact analogy to

� � π

For larger values of N as would typically be used in a discretization scheme, then

1 2 � � π

2

� � (32)

� � (35)

: (33)

<sup>2</sup> (34)

<sup>ρ</sup>nk <sup>¼</sup> <sup>j</sup> nk <sup>R</sup> <sup>¼</sup> <sup>j</sup>

Jnð Þz ≈

j

Using this approximation, then WρR ¼ j nN becomes

� � is

k ¼ 1…N � 1: (31)

transform proposed in the following sections.

DOI: http://dx.doi.org/10.5772/intechopen.84399

given by

zero, j

rnk <sup>¼</sup> <sup>j</sup> nk W<sup>ρ</sup> ¼ j nkR j nN

The Discrete Hankel Transform

determined from WρR ¼ j nN.

nk is given by

we can write approximately

Shannon's result.

9

4.3 Intuitive discretization scheme and kernel

In this case, the truncated sum in Eq. (26) does not represent Fð Þ ρnm exactly due to the truncation at N terms, but should provide a reasonably good approximation since the Fourier-Bessel series is known to converge and we are assuming that we have terminated the series at the point where additional f <sup>j</sup> nk W<sup>ρ</sup> terms do not contribute significantly.

#### 4.2 Inverse transform

Concomitantly, we know that for any space-limited function then for r , R, we can write

$$f(r) = \sum\_{m=1}^{\infty} \frac{2}{R^2 j\_{n+1}^2(j\_{nm})} F\left(\frac{j\_{nm}}{R}\right) J\_n\left(\frac{j\_{nm}r}{R}\right). \tag{27}$$

More specifically, suppose that we follow the logic from the previous section that the function f rð Þ that was bandlimited but also "effectively space-limited" due the truncation of the series in Eq. (25) at <sup>N</sup>. In that case then <sup>R</sup> <sup>¼</sup> <sup>j</sup> nN <sup>W</sup><sup>ρ</sup> and the band-limit implies that <sup>F</sup>ð Þ¼ <sup>ρ</sup> 0 for <sup>ρ</sup> .Wρ. Following this logic and using <sup>R</sup> <sup>¼</sup> <sup>j</sup> nN W<sup>ρ</sup> , then Eq. (27) becomes

$$f(r) = \sum\_{m=1}^{N-1} \frac{2\mathcal{W}\_{\rho}^{2}}{j\_{nN}^{2} j\_{n+1}^{2} (j\_{nm})} F\left(\frac{j\_{nm}\mathcal{W}\_{\rho}}{j\_{nN}}\right) I\_{n}\left(\frac{j\_{nm}\mathcal{W}\_{\rho}}{j\_{nN}}r\right) \tag{28}$$

where we truncated the series in Eq. (28) at N by using the fact that Fð Þ¼ ρ 0 for ρ≥W<sup>ρ</sup> to deduce that F <sup>j</sup> nmW<sup>ρ</sup> <sup>j</sup> nN <sup>¼</sup> 0 for <sup>m</sup> <sup>≥</sup> <sup>N</sup>. Evaluating (28) at rnk <sup>¼</sup> <sup>j</sup> nkR j nN ¼ j nk W<sup>ρ</sup> gives for k ¼ 1::N � 1

$$f\left(\frac{j\_{nk}}{W\_{\rho}}\right) = \sum\_{m=1}^{N-1} \frac{2W\_{\rho}^2}{j\_{nN}^2 J\_{n+1}^2(j\_{nm})} F\left(\frac{j\_{nm}W\_{\rho}}{j\_{nN}}\right) J\_n\left(\frac{j\_{nm}j\_{nk}}{j\_{nN}}\right). \tag{29}$$

Compare Eq. (29) to the "forward" transform from Eq. (26), repeated here for convenience, where we found that for m ¼ 1::N � 1

$$F\left(\frac{\dot{j}\_{nm}W\_{\rho}}{\dot{j}\_{nN}}\right) = \sum\_{k=1}^{N-1} \frac{2}{W\_{\rho}^{2}J\_{n+1}^{2}(\dot{j}\_{nk})} f\left(\frac{\dot{j}\_{nk}}{W\_{\rho}}\right) J\_{n}\left(\frac{\dot{j}\_{nk}\dot{j}\_{nm}}{\dot{j}\_{nN}}\right). \tag{30}$$

to converge, it is known that limk!<sup>∞</sup> <sup>f</sup> <sup>j</sup>

W<sup>ρ</sup>

nmW<sup>ρ</sup> j nN 

F j

f rð Þ≈0 for <sup>r</sup> . <sup>R</sup> <sup>¼</sup> <sup>j</sup> nN

tribute significantly.

4.2 Inverse transform

can write

Eq. (27) becomes

ρ≥W<sup>ρ</sup> to deduce that F <sup>j</sup>

f j nk W<sup>ρ</sup> 

F j

gives for k ¼ 1::N � 1

8

m ¼ 1::N � 1

¼ Jn j nm

Jn j nNj nm j nN 

trarily small ε, there must exist an integer N for which f <sup>j</sup>

Fourier Transforms - Century of Digitalization and Increasing Expectations

¼ ∑ N�1 k¼1

have terminated the series at the point where additional f <sup>j</sup>

f rð Þ¼ ∑ ∞ m¼1

f rð Þ¼ ∑ N�1 m¼1 W<sup>2</sup> ρ J 2 <sup>n</sup>þ<sup>1</sup> j nk <sup>f</sup> <sup>j</sup>

nk W<sup>ρ</sup> 

Hence, using the argument of "effectively space limited" in the preceding paragraph, we can terminate the series in Eq. (25) at a suitably chosen N that ensures the effective space limit. Terminating the series at k ¼ N is the same as assuming that

. Noting that at k ¼ N, the last term in (25) is

nk W<sup>ρ</sup> 

Jn j nkj nm j nN 

> nk W<sup>ρ</sup>

<sup>¼</sup> 0, then after terminating at <sup>N</sup>, Eq. (25) becomes for

2

In this case, the truncated sum in Eq. (26) does not represent Fð Þ ρnm exactly due to the truncation at N terms, but should provide a reasonably good approximation since the Fourier-Bessel series is known to converge and we are assuming that we

Concomitantly, we know that for any space-limited function then for r , R, we

nm R Jn j nmr R 

nmW<sup>ρ</sup> j nN 

Jn j nmW<sup>ρ</sup> j nN r 

<sup>¼</sup> 0 for <sup>m</sup> <sup>≥</sup> <sup>N</sup>. Evaluating (28) at rnk <sup>¼</sup> <sup>j</sup>

nmW<sup>ρ</sup> j nN 

> nk W<sup>ρ</sup>

Jn j nmj nk j nN 

Jn j nk j nm j nN 

2

More specifically, suppose that we follow the logic from the previous section that the function f rð Þ that was bandlimited but also "effectively space-limited" due the

where we truncated the series in Eq. (28) at N by using the fact that Fð Þ¼ ρ 0 for

2W<sup>2</sup> ρ

Compare Eq. (29) to the "forward" transform from Eq. (26), repeated here for

2

W<sup>2</sup> ρ J 2 <sup>n</sup>þ<sup>1</sup> j nk <sup>f</sup> <sup>j</sup>

R2 J 2 <sup>n</sup>þ<sup>1</sup> j nm <sup>F</sup> <sup>j</sup>

truncation of the series in Eq. (25) at <sup>N</sup>. In that case then <sup>R</sup> <sup>¼</sup> <sup>j</sup> nN

j 2 nNJ 2 <sup>n</sup>þ<sup>1</sup> j nm <sup>F</sup> <sup>j</sup>

nmW<sup>ρ</sup> j nN 

> ¼ ∑ N�1 m¼1

convenience, where we found that for m ¼ 1::N � 1

nmW<sup>ρ</sup> j nN  j 2 nNJ 2 <sup>n</sup>þ<sup>1</sup> j nm <sup>F</sup> <sup>j</sup>

¼ ∑ N�1 k¼1

implies that <sup>F</sup>ð Þ¼ <sup>ρ</sup> 0 for <sup>ρ</sup> .Wρ. Following this logic and using <sup>R</sup> <sup>¼</sup> <sup>j</sup> nN

2W<sup>2</sup> ρ ¼ 0, which means that for any arbi-

, ε for k . N.

: (26)

terms do not con-

: (27)

<sup>W</sup><sup>ρ</sup> and the band-limit

nkR j nN ¼ j nk W<sup>ρ</sup>

: (29)

: (30)

(28)

W<sup>ρ</sup> , then

nk W<sup>ρ</sup>  Eqs. (29) and (30) are the fundamental relations used for the discrete Hankel transform proposed in the following sections.

## 4.3 Intuitive discretization scheme and kernel

The preceding development shows that a "natural," N-dimensional discretization scheme in finite space 0½ � ; R and finite frequency space 0;W<sup>ρ</sup> � � is given by

$$
\sigma\_{nk} = \frac{j\_{nk}}{W\_{\rho}} = \frac{j\_{nk}R}{j\_{nN}} \qquad \qquad \rho\_{nk} = \frac{j\_{nk}}{R} = \frac{j\_{nk}W\_{\rho}}{j\_{nN}} \qquad \qquad k = 1...N-1. \tag{31}
$$

The relationship WρR ¼ j nN can be used to change from finite frequency domain to a finite space domain and vice-versa. The size of the transform N, can be determined from WρR ¼ j nN.

It is pointed out in [10] that the zeros of Jnð Þz are almost evenly spaced at intervals of π and that the spacing becomes exactly π in the limit as z ! ∞. In fact, it is shown in [10] that a simple asymptotic form for the Bessel function is given by

$$J\_n(z) \approx \sqrt{\frac{2}{\pi z}} \cos \left[ z - \left( n + \frac{1}{2} \right) \frac{\pi}{2} \right] \tag{32}$$

Eq. (32) becomes a better approximation to Jnð Þz as z ! ∞. The zeros of the cosine function are at odd multiples of π=2. Therefore, an approximation to the Bessel zero, j nk is given by

$$j\_{nk} \approx \left(2k + n - \frac{1}{2}\right)\frac{\pi}{2}.\tag{33}$$

Using this approximation, then WρR ¼ j nN becomes

$$2\pi \text{WR} = j\_{nN} \approx \left(2N + n - \frac{1}{2}\right)\frac{\pi}{2} \tag{34}$$

For larger values of N as would typically be used in a discretization scheme, then we can write approximately

$$2\text{W}\mathbf{R} \approx \left(\mathbf{N} + \frac{n}{2}\right) \tag{35}$$

This is exactly analogous to the corresponding expression for Fourier transforms. Specifically, for temporal Fourier transforms Shannon [6] argued that "If the function is limited to the time interval T and the samples are spaced 1/(2 W) seconds apart (where W is the bandwidth), there will be a total of 2WT samples in the interval. All samples outside will be substantially zero. To be more precise, we can define a function to be limited to the time interval T if, and only if, all the samples outside this interval are exactly zero. Then we can say that any function limited to the bandwidth W and the time interval T can be specified by giving N ¼ 2WT numbers". Following this line of thinking, Eq. (35) states that for an nth-order Hankel transform, any function limited to the bandwidth W and the space interval R can be specified by giving N ¼ ð Þ 2WR � n=2 numbers and it will certainly be true that specifying N ¼ 2WR numbers will specify the function, in exact analogy to Shannon's result.

## 4.4 Proposed kernel for the discrete transform

The preceding sections show that both forward and inverse discrete versions of the transforms contain an expression of the form

$$\frac{2}{J\_{n+1}^2(j\_{nk})} J\_n \left(\frac{j\_{nk} j\_{nm}}{j\_{nN}}\right). \tag{36}$$

<sup>Y</sup> nNY nN <sup>¼</sup> <sup>Ι</sup>, (41)

m,kf rð Þ nk : (42)

k,mFð Þ ρnm : (43)

1≤ m, k≤ N � 1: (44)

m,k: (45)

k,i ¼ δmi: (46)

where I is the N � 1 dimensional identity matrix and we have written the N � 1

transforms given in Eqs. (26) and (29) can be expressed in terms of Y nN. The

j nN W<sup>2</sup> ρ ∑ N�1 k¼1

W<sup>2</sup> ρ j nN ∑ N�1 m¼1

Fð Þ¼ ρnm

Similarly, the inverse transform, Eq. (29), can be written as

f rð Þ¼ nk

m,k as Y nN. The forward and inverse truncated and discretized

Y nN

Y nN

Following the notation in [12], we can also define a different ð Þ� N � 1 ð Þ N � 1

nk <sup>j</sup> nN

In Eq. (44), the superscripts n and N refer to the order of the Bessel function and the dimension of the space that are being considered, respectively. The subscripts m and k refer to the (m,k)th entry of the matrix. From (39), it can be seen that

k,m so that TnN is a real, symmetric matrix. The relationship between the

nk <sup>¼</sup> <sup>Y</sup> nN

¼ ∑ N�1 k¼1

TnN m,kTnN

<sup>T</sup>nNT nN <sup>¼</sup> <sup>T</sup>nN <sup>T</sup> nN <sup>T</sup> <sup>¼</sup> <sup>I</sup>: (47)

square matrix Y nN

The Discrete Hankel Transform

forward transform, Eq. (26), can be written as

DOI: http://dx.doi.org/10.5772/intechopen.84399

5.2 Another choice of transformation matrix

TnN m,k ¼ 2

TnN m,k <sup>¼</sup> <sup>T</sup>nN

TnN

m,k and Y nN

transformation matrix with the (m,k)th entry given by

Jnþ<sup>1</sup> j

m,k matrices is given by

J 2 <sup>n</sup>þ<sup>1</sup> j nm <sup>J</sup> 2 <sup>n</sup>þ<sup>1</sup> j nk <sup>j</sup> 2 nN

transform from Eq. (26) can be written in as

∑ N�1 k¼1 4 Jn <sup>j</sup> nmj nk <sup>=</sup><sup>j</sup> nN Jn <sup>j</sup>

written in matrix form as

are real.

11

Jn <sup>j</sup> nmj nk <sup>=</sup><sup>j</sup> nN

nm Jnþ<sup>1</sup> <sup>j</sup>

T nN m,k

The orthogonality relationship, Eq. (37), can be written as

Jnþ<sup>1</sup> j nm

Jnþ<sup>1</sup> j

nkj ni=<sup>j</sup> nN

Eq. (40) states that the rows and columns of the matrix TnN are orthonormal so that TnN is an orthogonal matrix. Furthermore, T nN is also symmetric. Eq. (46) can be

Therefore, the TnN matrix is unitary and furthermore orthogonal since the entries

Using the symmetric, orthogonal transformation matrix TnN, the forward

This leads to a natural choice of kernel for the discrete transform, as shall be outlined in the next section. To aid in in the choice of kernel for the discrete transform, we present a useful discrete orthogonality relationship shown in [11] that for 1≤ m, i ≤ N � 1

$$\sum\_{k=1}^{N-1} \frac{4J\_n\left(\frac{j\_{nm}j\_{nk}}{j\_{nN}}\right)J\_n\left(\frac{j\_{nk}j\_{ni}}{j\_{nN}}\right)}{J\_{n+1}^2(j\_{nk})} = j\_{nN}^2 J\_{n+1}^2(j\_{nm})\delta\_{mi} \tag{37}$$

where j nm represents the mth zero of the nth-order Bessel function Jnð Þ x , and δmi is the Kronecker delta function, defined as

$$
\delta\_{mn} = \begin{cases} 1 & \text{if } m = n \\ 0 & \text{otherwise} \end{cases} \tag{38}
$$

If written in matrix notation, then the Kronecker delta of Eq. (38) is the identity matrix.

Fisk-Johnson discusses the analytical derivation of Eq. (37) in the appendix of [11]. Eq. (37) is exactly true in the limit as N ! ∞ and is true for N . 30 within the limits of computational error �10�<sup>7</sup> . For smaller values of <sup>N</sup>, Eq. (37) holds with the worst case for the smallest value of <sup>N</sup> giving �10�3.

## 5. Transformation matrices

### 5.1 Transformation matrix

With inspiration from the notation in [11], and an additional scaling factor of 1=j nN, we define an ð Þ� N � 1 ð Þ N � 1 transformation matrix with the (m,k)th entry given by

$$Y\_{m,k}^{nN} = \frac{2}{j\_{nN} J\_{n+1}^2(j\_{nk})} J\_n \left(\frac{j\_{nm} j\_{nk}}{j\_{nN}}\right) \qquad \qquad 1 \le m, k \le N - 1. \tag{39}$$

In Eq. (39), the superscripts n and N refer to the order of the Bessel function and the dimension of the space that are being considered, respectively. The subscripts m and k refer to the (m,k)th entry of the transformation matrix.

Furthermore, the orthogonality relationship, Eq. (37), states that

$$\sum\_{k=1}^{N-1} Y\_{i,k}^{nN} Y\_{k,m}^{nN} = \sum\_{k=1}^{N-1} 4 \frac{J\_n \left( {}^{j\_{n\bar{j}n}}/{}\_{j\_{nN}} \right) J\_n \left( {}^{j\_{n\bar{j}m}}/{}\_{j\_{nN}} \right)}{j\_{nN}^2 j\_{n+1}^2 (j\_{n\bar{k}}) f\_{n+1}^2 (j\_{nm})} = \delta\_{\bar{i}m}.\tag{40}$$

Eq. (40) states that the rows and columns of the matrix Y nN m,k are orthonormal and can be written in matrix form as

The Discrete Hankel Transform DOI: http://dx.doi.org/10.5772/intechopen.84399

4.4 Proposed kernel for the discrete transform

the transforms contain an expression of the form

∑ N�1 k¼1

the Kronecker delta function, defined as

5. Transformation matrices

Y nN

∑<sup>N</sup>�<sup>1</sup> <sup>k</sup>¼<sup>1</sup> <sup>Y</sup> nN i,k Y nN

can be written in matrix form as

m,k <sup>¼</sup> <sup>2</sup> j nNJ 2 <sup>n</sup>þ<sup>1</sup> j nk Jn

5.1 Transformation matrix

entry given by

10

4Jn j nmj nk j nN Jn j nkj ni j nN 

the worst case for the smallest value of <sup>N</sup> giving �10�3.

J 2 <sup>n</sup>þ<sup>1</sup> j nk <sup>¼</sup> <sup>j</sup>

that for 1≤ m, i ≤ N � 1

where j

matrix.

The preceding sections show that both forward and inverse discrete versions of

j nkj nm j nN 

> 2 nNJ 2 <sup>n</sup>þ<sup>1</sup> j nm

nm represents the mth zero of the nth-order Bessel function Jnð Þ x , and δmi is

0 otherwise :

<sup>δ</sup>mn <sup>¼</sup> 1 if <sup>m</sup> <sup>¼</sup> <sup>n</sup>

If written in matrix notation, then the Kronecker delta of Eq. (38) is the identity

Fisk-Johnson discusses the analytical derivation of Eq. (37) in the appendix of [11]. Eq. (37) is exactly true in the limit as N ! ∞ and is true for N . 30 within the limits of computational error �10�<sup>7</sup> . For smaller values of <sup>N</sup>, Eq. (37) holds with

With inspiration from the notation in [11], and an additional scaling factor of 1=j nN, we define an ð Þ� N � 1 ð Þ N � 1 transformation matrix with the (m,k)th

In Eq. (39), the superscripts n and N refer to the order of the Bessel function and the dimension of the space that are being considered, respectively. The subscripts m

> Jn <sup>j</sup> nij nk =<sup>j</sup> nN

> > j 2 nNJ 2 <sup>n</sup>þ<sup>1</sup> j nk J 2 <sup>n</sup>þ<sup>1</sup> j nm

Jn <sup>j</sup> nkj nm =<sup>j</sup> nN 

j nmj nk j nN 

and k refer to the (m,k)th entry of the transformation matrix.

k,m <sup>¼</sup> <sup>∑</sup><sup>N</sup>�<sup>1</sup>

Eq. (40) states that the rows and columns of the matrix Y nN

Furthermore, the orthogonality relationship, Eq. (37), states that

<sup>k</sup>¼<sup>1</sup> <sup>4</sup>

: (36)

δmi (37)

1≤ m, k≤ N � 1: (39)

<sup>¼</sup> <sup>δ</sup>im: (40)

m,k are orthonormal and

(38)

2

This leads to a natural choice of kernel for the discrete transform, as shall be outlined in the next section. To aid in in the choice of kernel for the discrete transform, we present a useful discrete orthogonality relationship shown in [11]

J 2 <sup>n</sup>þ<sup>1</sup> j nk Jn

Fourier Transforms - Century of Digitalization and Increasing Expectations

$$Y^{nN}Y^{nN} = \mathbf{I},\tag{41}$$

where I is the N � 1 dimensional identity matrix and we have written the N � 1 square matrix Y nN m,k as Y nN. The forward and inverse truncated and discretized transforms given in Eqs. (26) and (29) can be expressed in terms of Y nN. The forward transform, Eq. (26), can be written as

$$F(\rho\_{nm}) = \frac{j\_{nN}}{\mathcal{W}\_{\rho}^2} \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} f(r\_{nk}). \tag{42}$$

Similarly, the inverse transform, Eq. (29), can be written as

$$f(r\_{nk}) = \frac{\mathcal{W}\_{\rho}^{2}}{j\_{nN}} \sum\_{m=1}^{N-1} Y\_{k,m}^{nN} F(\rho\_{nm}). \tag{43}$$

#### 5.2 Another choice of transformation matrix

Following the notation in [12], we can also define a different ð Þ� N � 1 ð Þ N � 1 transformation matrix with the (m,k)th entry given by

$$T\_{m,k}^{nN} = 2 \frac{J\_n\left(j\_{m\bar{j}nk} / \bar{j}\_{nN}\right)}{J\_{n+1}\left(j\_{nm}\right)J\_{n+1}\left(j\_{nk}\right)j\_{nN}} \qquad \qquad 1 \le m, k \le N-1. \tag{44}$$

In Eq. (44), the superscripts n and N refer to the order of the Bessel function and the dimension of the space that are being considered, respectively. The subscripts m and k refer to the (m,k)th entry of the matrix. From (39), it can be seen that TnN m,k <sup>¼</sup> <sup>T</sup>nN k,m so that TnN is a real, symmetric matrix. The relationship between the TnN m,k and Y nN m,k matrices is given by

$$T\_{m,k}^{nN} \frac{J\_{n+1}(j\_{nm})}{J\_{n+1}(j\_{nk})} = Y\_{m,k}^{nN}. \tag{45}$$

The orthogonality relationship, Eq. (37), can be written as

$$\sum\_{k=1}^{N-1} \mathbf{4} \frac{\int\_{\mathbf{n}} \left( {}\_{m}\boldsymbol{j}\_{n\mathbf{n}} / {}\_{j}\boldsymbol{j}\_{n\mathbf{N}} \right) \mathbf{J}\_{n} \left( {}\_{n}\boldsymbol{j}\_{n\mathbf{i}} / {}\_{j}\boldsymbol{j}\_{n\mathbf{N}} \right)}{\int\_{n+1}^{2} {}\_{n+1} \left( {}\_{m}\boldsymbol{j}\_{m\mathbf{n}} \right) \mathbf{J}\_{n\mathbf{k}}^{2} \left( {}\_{n}\boldsymbol{j}\_{n\mathbf{k}} \right) \mathbf{j}\_{n\mathbf{N}}^{2}} = \sum\_{k=1}^{N-1} T\_{m,k}^{nN} T\_{k,i}^{nN} = \delta\_{mi}. \tag{46}$$

Eq. (40) states that the rows and columns of the matrix TnN are orthonormal so that TnN is an orthogonal matrix. Furthermore, T nN is also symmetric. Eq. (46) can be written in matrix form as

$$T^{nN}T^{nN} = T^{nN} \left(T^{nN}\right)^T = \mathbf{I}.\tag{47}$$

Therefore, the TnN matrix is unitary and furthermore orthogonal since the entries are real.

Using the symmetric, orthogonal transformation matrix TnN, the forward transform from Eq. (26) can be written in as

Fourier Transforms - Century of Digitalization and Increasing Expectations

$$\begin{split} F(\rho\_{nm}) &= \frac{R^2}{j\_{nN}} \sum\_{k=1}^{N-1} T\_{m,k}^{nN} \frac{J\_{n+1}(j\_{nm})}{J\_{n+1}(j\_{nk})} f(r\_{nk}) \\ &= \frac{j\_{nN}}{\mathcal{W}\_{\rho}^2} \sum\_{k=1}^{N-1} T\_{m,k}^{nN} \frac{J\_{n+1}(j\_{nm})}{J\_{n+1}(j\_{nk})} f(r\_{nk}) \end{split} \tag{48}$$

∑ N�1 p¼1 ∑ N�1 k¼1

DOI: http://dx.doi.org/10.5772/intechopen.84399

be inverted by (52).

The Discrete Hankel Transform

7. Generalized Parseval theorem

<sup>g</sup><sup>T</sup><sup>h</sup> <sup>¼</sup> <sup>T</sup>nN<sup>G</sup> � �<sup>T</sup>

<sup>g</sup><sup>T</sup><sup>h</sup> <sup>¼</sup> <sup>Y</sup> nN<sup>G</sup> � �<sup>T</sup>

N�1 k¼1

¼ ∑ N�1 p¼1 ∑ N�1 q¼1

Jnþ<sup>1</sup> j

J 2 <sup>n</sup>þ<sup>1</sup> j nk � � <sup>∑</sup> N�1 p¼1 Y nN k,pGp ∑ N�1 q¼1 Y nN k,qHq

<sup>g</sup> <sup>¼</sup> <sup>T</sup>nNG, <sup>h</sup> <sup>¼</sup> <sup>T</sup> nN<sup>H</sup> then

hk Jnþ<sup>1</sup> <sup>j</sup> ð Þ nk

> ∑ N�1 k¼1

gk Jnþ<sup>1</sup> j nk � �

> ∑ N�1 k¼1

transpose.

13

hk Jnþ<sup>1</sup> j

modified Parseval relationship

gk Jnþ<sup>1</sup> j

energy preserving, meaning that

nk � � ! hk

nk � � <sup>¼</sup> <sup>∑</sup>

Y nN m,kY nN k,p Fp ¼ ∑ N�1 p¼1

The inside summations as indicated in Eq. (55) are recognized as yielding the Diracdelta function, the orthogonality property of Eq. (40) (or Eq. (46) if using T nN), which in turn yields the desired result. This proves that the DHT given by (50) can

Inner products are preserved and thus energies are preserved under the TnN matrix formulation. To see this, consider any two vectors given by the transform

<sup>T</sup>nN<sup>H</sup> <sup>¼</sup> <sup>G</sup><sup>T</sup> <sup>T</sup> nN � �<sup>T</sup>

The Y nN matrix formulation does not directly preserve inner products:

1

J 2 <sup>n</sup>þ<sup>1</sup> j

nk � � ! <sup>¼</sup> <sup>∑</sup>

However, under the Y nN formulation, the inner product between gk

1

np � � <sup>∑</sup> N�1 k¼1

Making use of the now-present Dirac-delta function, Eq. (58) simplifies to give a

N�1 p¼1

In other words, under a DHT using the Y nN matrix, inner products of the scaled functions are preserved but not the inner products of the functions themselves. As a consequence of the orthogonality property of TnN, the TnN based DHT is

where the overbar indicates a conjugate transpose and the superscript T indicates a

F T F ¼ f T

0 @ TnN


<sup>Y</sup> nN<sup>H</sup> <sup>¼</sup> <sup>G</sup><sup>T</sup> <sup>Y</sup> nN � �<sup>T</sup>

4 <sup>j</sup> nkj np <sup>=</sup><sup>j</sup> nN � �Jn <sup>j</sup>

j 2 nNJ 2 <sup>n</sup>þ<sup>1</sup> j nk � �<sup>J</sup> 2 <sup>n</sup>þ<sup>1</sup> j nq � �

Hp

Jnþ<sup>1</sup> j np � �

is preserved. To see this, we calculate the inner product between them as

δmpFp ¼ Fm (55)

<sup>H</sup> <sup>¼</sup> <sup>G</sup><sup>T</sup>H: (56)

Y nNH: (57)

nkj nq <sup>=</sup><sup>j</sup> nN � �

Gp

f: (60)

1

A: (59)

Jnþ<sup>1</sup> j np � �


> 1 A

0 @ Jnþ<sup>1</sup> <sup>j</sup> ð Þ nk

HqGp

and

(58)


Similarly, the inverse discrete transform of Eq. (29) can be written as

$$\begin{split} f(r\_{nk}) &= \frac{j\_{nN}}{R^2} \sum\_{m=1}^{N-1} T\_{k,m}^{nN} \frac{J\_{n+1}(j\_{nk})}{J\_{n+1}(j\_{nm})} F(\rho\_{nm}) \\ &= \frac{W\_{\rho}^2}{j\_{nN}} \sum\_{m=1}^{N-1} T\_{k,m}^{nN} \frac{J\_{n+1}(j\_{nk})}{J\_{n+1}(j\_{nm})} F(\rho\_{nm}). \end{split} \tag{49}$$

## 6. Discrete forward and inverse Hankel transform

From the previous section is it clear that the two natural choices of kernel for a DHT are either Y nN m,k or TnN m,k. Y nN m,k is closer to the discretized version of the continuous Hankel transform that we hope the discrete version emulates. However, TnN m,k is an orthogonal and symmetric matrix, therefore it is energy preserving and will be shown to lead to a Parseval-type relationship if chosen as the kernel for the DHT. Thus, to define a discrete Hankel transform (DHT), we can use either formulation:

$$F\_m = \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} f\_k \quad \text{or} \quad F\_m = \sum\_{k=1}^{N-1} T\_{m,k}^{nN} f\_k. \tag{50}$$

Here, the transform is of any N � 1 dimensional vector f <sup>k</sup> to any N � 1 dimensional vector Fm for the integers m, k where 1≤ m, k , N � 1. This can be written in matrix form as

$$\mathbf{F} = Y^{nN} \mathbf{f} \qquad \text{or} \qquad \mathbf{F} = T^{nN} \mathbf{f} \tag{51}$$

where F is any N � 1 dimensional column vector and f is also any column vector, defined in the same manner.

The inverse discrete Hankel transform (IDHT) is then given by

$$f\_k = \sum\_{m=1}^{N-1} Y\_{k,m}^{nN} F\_m \qquad \text{or} \qquad f\_k = \sum\_{m=1}^{N-1} T\_{k,m}^{nN} F\_m. \tag{52}$$

This can also be written in matrix form as

$$\mathbf{f} = Y^{nN} \mathbf{F} \qquad \text{or} \qquad \mathbf{f} = T^{nN} \mathbf{F}. \tag{53}$$

We note that the forward and inverse transforms are the same. Proof

We show the proof for the Y nN formulation, but it proceeds similarly if Y nN is replaced with TnN. Substituting Eq. (52) into the right hand side of (50) gives

$$\sum\_{k=1}^{N-1} Y\_{m,k}^{nN} f\_k = \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} \left[ \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} F\_p \right]. \tag{54}$$

Switching the order of the summation in Eq. (54) gives

The Discrete Hankel Transform DOI: http://dx.doi.org/10.5772/intechopen.84399

Fð Þ¼ ρnm

f rð Þ¼ nk

R2 j nN ∑ N�1 k¼1 T nN m,k

Fourier Transforms - Century of Digitalization and Increasing Expectations

<sup>¼</sup> <sup>j</sup> nN W<sup>2</sup> ρ ∑ N�1 k¼1 T nN m,k

Similarly, the inverse discrete transform of Eq. (29) can be written as

j nN <sup>R</sup><sup>2</sup> <sup>∑</sup> N�1 m¼1 TnN k,m

<sup>¼</sup> <sup>W</sup><sup>2</sup> ρ j nN ∑ N�1 m¼1 TnN k,m

6. Discrete forward and inverse Hankel transform

are either Y nN

form as

Proof

12

m,k or TnN

defined in the same manner.

f <sup>k</sup> ¼ ∑ N�1 m¼1

This can also be written in matrix form as

Y nN

We note that the forward and inverse transforms are the same.

∑ N�1 k¼1

Y nN

Switching the order of the summation in Eq. (54) gives

m,k f <sup>k</sup> ¼ ∑

m,k. Y nN

Fm ¼ ∑ N�1 k¼1

Y nN

Jnþ<sup>1</sup> j nm � �

Jnþ<sup>1</sup> j nk � � f rð Þ nk

(48)

(49)

m,k is an

Jnþ<sup>1</sup> j nm � �

Jnþ<sup>1</sup> j nk � � f rð Þ nk

Jnþ<sup>1</sup> j nk � �

Jnþ<sup>1</sup> j nm � � <sup>F</sup>ð Þ <sup>ρ</sup>nm

Jnþ<sup>1</sup> j nk � �

Jnþ<sup>1</sup> j nm � � <sup>F</sup>ð Þ <sup>ρ</sup>nm :

m,k is closer to the discretized version of the continuous

N�1 k¼1

<sup>F</sup> <sup>¼</sup> <sup>Y</sup> nN<sup>f</sup> or <sup>F</sup> <sup>¼</sup> <sup>T</sup> nN<sup>f</sup> (51)

N�1 m¼1

<sup>f</sup> <sup>¼</sup> <sup>Y</sup> nN<sup>F</sup> or <sup>f</sup> <sup>¼</sup> <sup>T</sup> nNF: (53)

Y nN k,pFp " #

TnN

TnN

m,k f <sup>k</sup>: (50)

k,mFm: (52)

: (54)

From the previous section is it clear that the two natural choices of kernel for a DHT

m,k f <sup>k</sup> or Fm ¼ ∑

Here, the transform is of any N � 1 dimensional vector f <sup>k</sup> to any N � 1 dimensional vector Fm for the integers m, k where 1≤ m, k , N � 1. This can be written in matrix

where F is any N � 1 dimensional column vector and f is also any column vector,

k,mFm or f <sup>k</sup> ¼ ∑

We show the proof for the Y nN formulation, but it proceeds similarly if Y nN is replaced with TnN. Substituting Eq. (52) into the right hand side of (50) gives

> N�1 k¼1

Y nN m,k ∑ N�1 p¼1

The inverse discrete Hankel transform (IDHT) is then given by

Hankel transform that we hope the discrete version emulates. However, TnN

orthogonal and symmetric matrix, therefore it is energy preserving and will be shown to lead to a Parseval-type relationship if chosen as the kernel for the DHT. Thus, to define a discrete Hankel transform (DHT), we can use either formulation:

$$\sum\_{p=1}^{N-1} \underbrace{\sum\_{k=1}^{N-1} Y\_{m,k}^{nN} Y\_{k,p}^{nN}}\_{\delta\_{mp}} F\_p = \sum\_{p=1}^{N-1} \delta\_{mp} F\_p = F\_m \tag{55}$$

The inside summations as indicated in Eq. (55) are recognized as yielding the Diracdelta function, the orthogonality property of Eq. (40) (or Eq. (46) if using T nN), which in turn yields the desired result. This proves that the DHT given by (50) can be inverted by (52).

## 7. Generalized Parseval theorem

Inner products are preserved and thus energies are preserved under the TnN matrix formulation. To see this, consider any two vectors given by the transform <sup>g</sup> <sup>¼</sup> <sup>T</sup>nNG, <sup>h</sup> <sup>¼</sup> <sup>T</sup> nN<sup>H</sup> then

$$\mathbf{g}^T \mathbf{h} = \left(T^{nN} \mathbf{G}\right)^T T^{nN} \mathbf{H} = \mathbf{G}^T \underbrace{\left(T^{nN}\right)^T T^{nN}}\_{=I} \mathbf{H} = \mathbf{G}^T \mathbf{H}.\tag{56}$$

The Y nN matrix formulation does not directly preserve inner products:

$$\mathbf{g}^{T}\mathbf{h} = \left(Y^{nN}\mathbf{G}\right)^{T}Y^{nN}\mathbf{H} = \mathbf{G}^{T}\left(Y^{nN}\right)^{T}Y^{nN}\mathbf{H}.\tag{57}$$

However, under the Y nN formulation, the inner product between gk Jnþ<sup>1</sup> <sup>j</sup> ð Þ nk and hk Jnþ<sup>1</sup> <sup>j</sup> ð Þ nk is preserved. To see this, we calculate the inner product between them as

$$\sum\_{k=1}^{N-1} \frac{\mathcal{g}\_k}{f\_{n+1}(j\_{nk}) f\_{n+1}(j\_{nk})} = \sum\_{k=1}^{N-1} \frac{1}{f\_{n+1}^2(j\_{nk})} \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} G\_p \sum\_{q=1}^{N-1} Y\_{k,q}^{nN} H\_q$$

$$= \sum\_{p=1}^{N-1} \sum\_{q=1}^{N-1} \frac{1}{f\_{n+1}^2(j\_{np})} \underbrace{\sum\_{k=1}^{N-1} \frac{4 \left(j\_{nk} \gamma\_{j\_{nk}} \right) f\_n \left(j\_{nk} j\_{nj} \right)\_{nN}}\_{f\_{nN}^2 \mathcal{J}\_{n+1}^2 \left(j\_{nk} \right) f\_{nN}^2}}\_{\delta\_{pq}} H\_q G\_p$$

Making use of the now-present Dirac-delta function, Eq. (58) simplifies to give a modified Parseval relationship

$$\sum\_{k=1}^{N-1} \left( \frac{\mathcal{G}\_k}{J\_{n+1}(j\_{nk})} \right) \left( \frac{h\_k}{J\_{n+1}(j\_{nk})} \right)^{-1} = \sum\_{p=1}^{N-1} \left( \frac{H\_p}{J\_{n+1}(j\_{np})} \right) \left( \frac{G\_p}{J\_{n+1}(j\_{np})} \right). \tag{59}$$

In other words, under a DHT using the Y nN matrix, inner products of the scaled functions are preserved but not the inner products of the functions themselves.

As a consequence of the orthogonality property of TnN, the TnN based DHT is energy preserving, meaning that

$$
\overline{\mathbf{F}}^T \mathbf{F} = \overline{\mathbf{f}}^T \mathbf{f}.\tag{60}
$$

where the overbar indicates a conjugate transpose and the superscript T indicates a transpose.

For the formulation with Y nN as the transformation kernel, the equivalent expression is

$$\overline{\mathbf{F}}^T \mathbf{F} = \left( Y^{nN} \overline{\mathbf{f}} \right)^T Y^{nN} \mathbf{f} = \overline{\mathbf{f}}^T \left( Y^{nN} \right)^T Y^{nN} \mathbf{f}. \tag{61}$$

As before, Y nN

The Discrete Hankel Transform

DOI: http://dx.doi.org/10.5772/intechopen.84399

given by

pair:

k,m<sup>0</sup> represents the m0th column of the transformation matrix <sup>Y</sup> nN.

Y nN m,kY nN k,m<sup>0</sup> is

k,m<sup>0</sup> ¼ δmm<sup>0</sup> , (67)

iωt

, (70)

, which inevitably encounters difficulty

k,ko is a N � 1 square matrix (in other words,

k,ko is another N � 1

dω: (69)

k,m<sup>0</sup> ⇔ δmmo : (68)

From the forward definition of the transform, Eq. (50), the transform of Yn,N

N�1 k¼1

where we have used the orthogonality relationship (40). This gives us another DHT

For a one-dimensional Fourier transform, one of the known transform rules is

In Eq. (69), ^fð Þ <sup>ω</sup> is the Fourier transform of f xð Þ, <sup>F</sup>�<sup>1</sup> denotes an inverse Fourier transform and e�ia<sup>ω</sup> is the kernel of the Fourier transform operator. Motivated by this result, we define a generalized-shift operator by finding the inverse DHT of the DHT of the function multiplied by the DHT kernel. This is a discretized version of the definition of a generalized shift operator as proposed by Levitan [8] (he suggested the complex conjugate of the Fourier operator, which for Fourier transforms is the inverse transform operator). We thus propose the definition of the

> Y nN k,p Y nN p,ko Fp n o

when the subscript k � ko falls outside of the range 1½ � ; N � 1 . We note that if all

a two dimensional structure), whereas the original un-shifted f <sup>k</sup> is an N � 1 vector. For the discrete Fourier transform, when the shifted subscript k � ko falls outside the range of the indices, is it usually interpreted modulo the size of the DFT. However, the kernel of the Fourier transform is periodic so this does not create difficulties for the DFT. The Bessel functions are not periodic so the same trick cannot be used with the Hankel transform. In fact, this lack of periodicity and lack of simple relationship between Jnð Þ x � y and Jnð Þ x is the reason that the continuous Hankel transform does not have a convolution-multiplication rule [13]. Thus, the notation <sup>f</sup> <sup>k</sup>�ko would not make mathematical sense when used with the DHT. With the definition given by Eq. (70), no such confusion arises since the definition is

2π

∞ð

e �ia<sup>ω</sup>^fð Þ <sup>ω</sup> n o <sup>e</sup>

�∞


k,ko implying a k0-shifted version of f <sup>k</sup>. This generalizes

m,k f <sup>k</sup> ¼ ∑

Yn,N

�ia<sup>ω</sup>^fð Þ <sup>ω</sup> n o <sup>¼</sup> <sup>1</sup>

Fm ¼ ∑ N�1 k¼1

8.3 The generalized shift operator

f xð Þ¼ � <sup>a</sup> <sup>F</sup>�<sup>1</sup> <sup>e</sup>

generalized-shifted function to be given by

the notion of the shift, usually denoted <sup>f</sup> <sup>k</sup>�ko

possible shifts ko are considered, then f shift

unambiguous for all allowable values of k and ko.

vector, with the notation f shift

15

f shift k,ko ¼ ∑ N�1 p¼1

where 1≤k, ko <sup>≤</sup> <sup>N</sup> � 1 . For a single, fixed value of ko, then <sup>f</sup> shift

the shift rule, which states that

Y nN

It is obvious from Eq. (59) that if we define the "scaled" vector

$$f\_k^{\text{Scaled}} = \frac{f\_k}{J\_{n+1}(j\_{nk})} \qquad \qquad \text{and} \quad F\_p^{\text{Scaled}} = \frac{F\_p}{J\_{n+1}(j\_{np})},\tag{62}$$

then by straighforward substitution of scaled vectors and their conjugates, it follows that

$$
\overline{\left(\mathbf{F}^{\rmScaled}\right)^{T}\mathbf{F}^{\rmScaled}} = \overline{\left(\mathbf{f}^{\rmScaled}\right)^{T}\mathbf{f}^{\rmScaled}}.\tag{63}
$$

## 8. Transform rules

In keeping with the development of the well-known discrete Fourier transform, we develop the standard toolkit of rules for the discrete Hankel transform. In the following, Y nN is used but all expressions apply equally if Y nN is replaced with TnN.

#### 8.1 Transform of Kronecker-Delta function

The discrete counterpart of the Dirac-delta function is the Kronecker-delta function, δkk<sup>0</sup> . We recall that the DHT as defined above is a matrix transform from a N � 1 dimensional vector to another. The vector δkko is interpreted as the vector as having zero entries everywhere except at position k ¼ k<sup>0</sup> (k<sup>0</sup> fixed so δkk<sup>0</sup> is a vector), or in other words, the k0th column of the N � 1 sized identity matrix. The DHT of the Kronecker-delta can be found from the definition of the forward transform via

$$\mathbb{H}(\delta\_{kk\_\circ}) = \sum\_{k=1}^{N-1} Y\_{m,k} \delta\_{kk\_\circ} = Y\_{m,k\_0}^{nN} \tag{64}$$

The symbol Hð Þ� is used to denote the operation of taking the discrete Hankel transform. This gives us our first DHT transform pair of order n dimension N � 1, and we denote this relationship as

$$
\delta\_{kk\_o} \Leftrightarrow Y\_{m,k\_0}^{n,N} \tag{65}
$$

Here, f <sup>k</sup> ⇔ Fm denotes a transform pair and Y nN m,k<sup>0</sup> is k0th column of the matrix Y nN.

#### 8.2 Inverse transform of the Kronecker Delta function

From Eq. (65), we can deduce the vector f <sup>k</sup> that transforms to the Kroneckerdelta vector δmmo function. Namely, we take the forward transform of

$$f\_k = Y\_{k,m\_0}^{n,N}.\tag{66}$$

For the formulation with Y nN as the transformation kernel, the equivalent

<sup>Y</sup> nN<sup>f</sup> <sup>¼</sup> <sup>f</sup>

and <sup>F</sup>Scaled

then by straighforward substitution of scaled vectors and their conjugates, it follows

<sup>F</sup>Scaled <sup>¼</sup> <sup>f</sup>Scaled <sup>T</sup>

In keeping with the development of the well-known discrete Fourier transform, we develop the standard toolkit of rules for the discrete Hankel transform. In the following, Y nN is used but all expressions apply equally if Y nN is replaced with TnN.

The discrete counterpart of the Dirac-delta function is the Kronecker-delta function, δkk<sup>0</sup> . We recall that the DHT as defined above is a matrix transform from a N � 1 dimensional vector to another. The vector δkko is interpreted as the vector as having zero entries everywhere except at position k ¼ k<sup>0</sup> (k<sup>0</sup> fixed so δkk<sup>0</sup> is a vector), or in other words, the k0th column of the N � 1 sized identity matrix. The DHT of the Kronecker-delta can be found from the definition of the forward

H δkko ð Þ¼ ∑

N�1 k¼1

The symbol Hð Þ� is used to denote the operation of taking the discrete Hankel transform. This gives us our first DHT transform pair of order n dimension N � 1,

δkko ⇔ Yn,N

From Eq. (65), we can deduce the vector f <sup>k</sup> that transforms to the Kronecker-

<sup>f</sup> <sup>k</sup> <sup>¼</sup> <sup>Y</sup>n,N k,m<sup>0</sup>

delta vector δmmo function. Namely, we take the forward transform of

Ym,kδkko <sup>¼</sup> <sup>Y</sup> nN

<sup>T</sup> <sup>Y</sup> nN <sup>T</sup>

<sup>p</sup> <sup>¼</sup> Fp

Jnþ<sup>1</sup> j np

Y nNf: (61)

, (62)

fScaled: (63)

m,k<sup>0</sup> (64)

m,k<sup>0</sup> (65)

m,k<sup>0</sup> is k0th column of the matrix

: (66)

expression is

that

F T

f Scaled <sup>k</sup> <sup>¼</sup> <sup>f</sup> <sup>k</sup>

8. Transform rules

transform via

Y nN.

14

and we denote this relationship as

Here, f <sup>k</sup> ⇔ Fm denotes a transform pair and Y nN

8.2 Inverse transform of the Kronecker Delta function

<sup>F</sup> <sup>¼</sup> <sup>Y</sup> nN<sup>f</sup> <sup>T</sup>

> Jnþ<sup>1</sup> j nk

<sup>F</sup>Scaled <sup>T</sup>

8.1 Transform of Kronecker-Delta function

It is obvious from Eq. (59) that if we define the "scaled" vector

Fourier Transforms - Century of Digitalization and Increasing Expectations

As before, Y nN k,m<sup>0</sup> represents the m0th column of the transformation matrix <sup>Y</sup> nN. From the forward definition of the transform, Eq. (50), the transform of Yn,N k,m<sup>0</sup> is given by

$$F\_m = \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} f\_k = \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} Y\_{k,m\_0}^{nN} = \delta\_{mm\_0},\tag{67}$$

where we have used the orthogonality relationship (40). This gives us another DHT pair:

$$Y\_{k,m\_0}^{n,N} \Leftrightarrow \delta\_{mm\_0}.\tag{68}$$

## 8.3 The generalized shift operator

For a one-dimensional Fourier transform, one of the known transform rules is the shift rule, which states that

$$f(\mathbf{x} - a) = \mathbb{F}^{-1}\left\{ e^{-i a \boldsymbol{\alpha}} \hat{f}(\boldsymbol{\alpha}) \right\} = \frac{1}{2\pi} \int\_{-\infty}^{\infty} \left\{ e^{-i a \boldsymbol{\alpha}} \hat{f}(\boldsymbol{\alpha}) \right\} \, e^{i \boldsymbol{\alpha} t} d\boldsymbol{\alpha}.\tag{69}$$

In Eq. (69), ^fð Þ <sup>ω</sup> is the Fourier transform of f xð Þ, <sup>F</sup>�<sup>1</sup> denotes an inverse Fourier transform and e�ia<sup>ω</sup> is the kernel of the Fourier transform operator. Motivated by this result, we define a generalized-shift operator by finding the inverse DHT of the DHT of the function multiplied by the DHT kernel. This is a discretized version of the definition of a generalized shift operator as proposed by Levitan [8] (he suggested the complex conjugate of the Fourier operator, which for Fourier transforms is the inverse transform operator). We thus propose the definition of the generalized-shifted function to be given by

$$f\_{k,k\_o}^{shift} = \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} \underbrace{\left\{ Y\_{p,k\_o}^{nN} F\_p \right\}}\_{\text{shift in Hankel}},\tag{70}$$

where 1≤k, ko <sup>≤</sup> <sup>N</sup> � 1 . For a single, fixed value of ko, then <sup>f</sup> shift k,ko is another N � 1 vector, with the notation f shift k,ko implying a k0-shifted version of f <sup>k</sup>. This generalizes the notion of the shift, usually denoted <sup>f</sup> <sup>k</sup>�ko , which inevitably encounters difficulty when the subscript k � ko falls outside of the range 1½ � ; N � 1 . We note that if all possible shifts ko are considered, then f shift k,ko is a N � 1 square matrix (in other words, a two dimensional structure), whereas the original un-shifted f <sup>k</sup> is an N � 1 vector. For the discrete Fourier transform, when the shifted subscript k � ko falls outside the range of the indices, is it usually interpreted modulo the size of the DFT. However, the kernel of the Fourier transform is periodic so this does not create difficulties for the DFT. The Bessel functions are not periodic so the same trick cannot be used with the Hankel transform. In fact, this lack of periodicity and lack of simple relationship between Jnð Þ x � y and Jnð Þ x is the reason that the continuous Hankel transform does not have a convolution-multiplication rule [13]. Thus, the notation <sup>f</sup> <sup>k</sup>�ko would not make mathematical sense when used with the DHT. With the definition given by Eq. (70), no such confusion arises since the definition is unambiguous for all allowable values of k and ko.

The shifted function f shift k,ko can also be expressed in terms of the original unshifted function f <sup>k</sup> . Using the definition of Fm from Eq. (50) and a dummy change of variable, then Eq. (70) becomes

$$f\_{k,k\_s}^{shift} = \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} Y\_{p,k\_s}^{nN} F\_p = \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} Y\_{p,k\_s}^{nN} \sum\_{m=1}^{N-1} Y\_{p,m}^{nN} f\_m. \tag{71}$$

8.5 Modulation

The Discrete Hankel Transform

DOI: http://dx.doi.org/10.5772/intechopen.84399

Then Eq. (78) becomes

∑ N�1 k¼1

transform.

8.6 Convolution

over all possible shifts.

17

Y nN

m,k f <sup>k</sup> ¼ ∑

Interchanging the order of summation gives

N�1 k¼1

Y nN m,kY nN k,ko

∑ N�1 p¼1 ∑ N�1 k¼1

the frequency domain. This yields another transform pair:

defined. The convolution of two functions is defined as

<sup>f</sup> <sup>k</sup> <sup>¼</sup> <sup>Y</sup> nN k,ko

We consider the forward DHT of a function "modulated" in the space domain

N�1 k¼1

Y nN m,kY nN

Y nN

Y nN m,kY nN k,ko ∑ N�1 p¼1

> Gp <sup>¼</sup> <sup>G</sup>shift m,ko

vector <sup>g</sup> is multiplied by the ð Þ <sup>k</sup>; ko th entry of <sup>Y</sup> nN for a fixed value of ko. No

m,k f <sup>k</sup> ¼ ∑

gk ¼ ∑ N�1 p¼1

Y nN m,kY nN k,ko Y nN k,p

Y nN k,ko


By comparing Eq. (81) with Eqs. (72) and (73), we recognize the shift operator as indicated in (81). This produces a modulation-shift rule as would be expected so that the forward DHT of a modulated function is equivalent to a generalized shift in

> gk <sup>⇔</sup> <sup>G</sup>shift m,ko

In other words, Eq. (82) says that modulation in the space domain is equivalent to shift in the frequency domain, as would be expected for a (generalized) Fourier

We consider the convolution using the generalized shifted function previously

N�1 k0¼1 gko hshift k,ko

<sup>f</sup> <sup>k</sup> <sup>¼</sup> <sup>g</sup><sup>∗</sup> ð Þ <sup>h</sup> <sup>k</sup> <sup>¼</sup> <sup>∑</sup>

The meaning of Eq. (83) follows from the traditional definition of a convolution: multiply one of the functions by a shifted version of a second function and then sum

Subsequently, from the definition of the inverse transforms, we obtain

gk ¼ ∑ N�1 k¼1

k,ko gk is that the kth entry of the

k,ko gk, (78)

k,pGp: (79)

Y nN

k,ko gk are N � 1

k,pGp: (80)

: (81)

: (82)

: (83)

gk. Here, the interpretation of <sup>f</sup> <sup>k</sup> <sup>¼</sup> <sup>Y</sup> nN

∑ N�1 k¼1

and write gk in terms of its inverse transform

summation is implied so this is not a dot product; both f <sup>k</sup> and Y nN

Y nN

vectors. Again, we implement the definition of the forward transform

Changing the order of summation gives

$$f\_{k,k\_o}^{shift} = \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} Y\_{p,k\_o}^{nN} F\_p = \sum\_{m=1}^{N-1} \underbrace{\sum\_{p=1}^{N-1} Y\_{k,p}^{nN} Y\_{p,k\_o}^{nN} Y\_{p,m}^{nN}}\_{\text{shift operator}} \,\tag{72}$$

As indicated in Eq. (72), the quantity in brackets can be considered to be a type of shift operator acting on the original unshifted function. We can define this as

$$\mathbf{S}\_{k,k\_o,m}^{nN} = \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} Y\_{p,k\_o}^{nN} Y\_{p,m}^{nN}.\tag{73}$$

It then follows that Eq. (72) can be written as

$$f\_{k,k\_o}^{sh\text{jf}} = \sum\_{m=1}^{N-1} \mathbb{S}\_{k,k\_o,m}^{nN} f\_m. \tag{74}$$

This triple-product shift operator is similar to previous definitions of shift operators for multidimensional Fourier transforms that rely on Hankel transforms [1, 2] and of generalized Hankel convolutions [14–16].

#### 8.4 Transform of the generalized shift operator

We now consider the forward DHT transform of the shifted function f shift k,ko . From the definition, the DHT of the shifted function can be found from

$$\sum\_{k=1}^{N-1} Y\_{m,k}^{nN} f\_{k,k\_o}^{shjt} = \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} Y\_{p,k\_o}^{nN} F\_p. \tag{75}$$

Changing the order of summation gives

$$\underbrace{\sum\_{p=1}^{N-1} \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} Y\_{k,p}^{nN}}\_{=\delta\_{mp}} Y\_{p,k\_{\bullet}}^{nN} F\_p = \sum\_{p=1}^{N-1} \delta\_{mp} Y\_{p,k\_{\bullet}}^{nN} F\_p = Y\_{m,k\_{\bullet}}^{nN} F\_m. \tag{76}$$

This yields another transform pair and is the shift-modulation rule. This rule analogous to the shift-modulation rule for regular Fourier transforms whereby a shift in the spatial domain is equivalent to modulation in the frequency domain:

$$f\_{k,k\_o}^{shj\dagger} \Leftrightarrow Y\_{m,k\_o}^{nN} F\_m. \tag{77}$$

Note that Eq. (77) does not imply a summation over the m index. For a fixed value of ko on the left hand side, the corresponding transformed value of Fm is multiplied by the ð Þ <sup>m</sup>; ko th entry of the <sup>Y</sup> nN matrix.

## 8.5 Modulation

The shifted function f shift

of variable, then Eq. (70) becomes

f shift k,ko ¼ ∑ N�1 p¼1

Changing the order of summation gives

f shift k,ko ¼ ∑ N�1 p¼1

Y nN k,pY nN p,ko

Fourier Transforms - Century of Digitalization and Increasing Expectations

Y nN k,pY nN p,ko

S nN

It then follows that Eq. (72) can be written as

of generalized Hankel convolutions [14–16].

8.4 Transform of the generalized shift operator

∑ N�1 k¼1

> Y nN m,kY nN k,p


Changing the order of summation gives

by the ð Þ <sup>m</sup>; ko th entry of the <sup>Y</sup> nN matrix.

16

∑ N�1 p¼1 ∑ N�1 k¼1

Y nN m,k <sup>f</sup> shift

k,ko,m ¼ ∑

f shift k,ko ¼ ∑ N�1 m¼1

the definition, the DHT of the shifted function can be found from

k,ko ¼ ∑ N�1 k¼1

Y nN p,ko

> f shift k,ko <sup>⇔</sup> <sup>Y</sup> nN m,ko

k,ko can also be expressed in terms of the original un-

Y nN k,pY nN p,ko Y nN p,m


Y nN

p,mf <sup>m</sup>: (71)

f <sup>m</sup>: (72)

k,ko

Fp: (75)

Fm: (76)

. From

p,m: (73)

k,ko,m f <sup>m</sup>: (74)

Y nN k,pY nN p,ko ∑ N�1 m¼1

shifted function f <sup>k</sup> . Using the definition of Fm from Eq. (50) and a dummy change

Fp ¼ ∑ N�1 p¼1

Fp ¼ ∑ N�1 m¼1 ∑ N�1 p¼1

As indicated in Eq. (72), the quantity in brackets can be considered to be a type of shift operator acting on the original unshifted function. We can define this as

> Y nN k,pY nN p,ko Y nN

S nN

This triple-product shift operator is similar to previous definitions of shift operators for multidimensional Fourier transforms that rely on Hankel transforms [1, 2] and

We now consider the forward DHT transform of the shifted function f shift

Y nN m,k ∑ N�1 p¼1

Fp ¼ ∑ N�1 p¼1

This yields another transform pair and is the shift-modulation rule. This rule analogous to the shift-modulation rule for regular Fourier transforms whereby a shift in the spatial domain is equivalent to modulation in the frequency domain:

Note that Eq. (77) does not imply a summation over the m index. For a fixed value of ko on the left hand side, the corresponding transformed value of Fm is multiplied

Y nN k,pY nN p,ko

δmpY nN p,ko Fp <sup>¼</sup> <sup>Y</sup> nN m,ko

Fm: (77)

N�1 p¼1

We consider the forward DHT of a function "modulated" in the space domain <sup>f</sup> <sup>k</sup> <sup>¼</sup> <sup>Y</sup> nN k,ko gk. Here, the interpretation of <sup>f</sup> <sup>k</sup> <sup>¼</sup> <sup>Y</sup> nN k,ko gk is that the kth entry of the vector <sup>g</sup> is multiplied by the ð Þ <sup>k</sup>; ko th entry of <sup>Y</sup> nN for a fixed value of ko. No summation is implied so this is not a dot product; both f <sup>k</sup> and Y nN k,ko gk are N � 1 vectors. Again, we implement the definition of the forward transform

$$\sum\_{k=1}^{N-1} Y\_{m,k}^{nN} f\_k = \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} Y\_{k,k\_o}^{nN} \mathbf{g}\_{k^o} \tag{78}$$

and write gk in terms of its inverse transform

$$\mathbf{g}\_k = \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} \mathbf{G}\_p. \tag{79}$$

Then Eq. (78) becomes

$$\sum\_{k=1}^{N-1} Y\_{m,k}^{nN} f\_k = \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} Y\_{k,k\_0}^{nN} g\_k = \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} Y\_{k,k\_0}^{nN} \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} G\_p. \tag{80}$$

Interchanging the order of summation gives

$$\underbrace{\sum\_{p=1}^{N-1} \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} Y\_{k,k\_o}^{nN} Y\_{k,p}^{nN}}\_{\text{shift operator}} \mathbf{G}\_p = \mathbf{G}\_{m,k\_o}^{\text{shift}}.\tag{81}$$

By comparing Eq. (81) with Eqs. (72) and (73), we recognize the shift operator as indicated in (81). This produces a modulation-shift rule as would be expected so that the forward DHT of a modulated function is equivalent to a generalized shift in the frequency domain. This yields another transform pair:

$$Y\_{k,k\_o}^{nN} \underset{k}{\Leftrightarrow} G\_{m,k\_o}^{shj\hat{t}}.\tag{82}$$

In other words, Eq. (82) says that modulation in the space domain is equivalent to shift in the frequency domain, as would be expected for a (generalized) Fourier transform.

### 8.6 Convolution

We consider the convolution using the generalized shifted function previously defined. The convolution of two functions is defined as

$$f\_k = (\mathbf{g}^\* h)\_k = \sum\_{k\_0=1}^{N-1} \mathbf{g}\_{k\_\sigma} h\_{k\_\cdot, k\_\sigma}^{shift}. \tag{83}$$

The meaning of Eq. (83) follows from the traditional definition of a convolution: multiply one of the functions by a shifted version of a second function and then sum over all possible shifts.

Subsequently, from the definition of the inverse transforms, we obtain

Fourier Transforms - Century of Digitalization and Increasing Expectations

$$\begin{split} f\_{k} &= \sum\_{k\_{0}=1}^{N-1} \mathbf{g}\_{k\_{\circ}} h\_{k\_{\circ}k\_{\circ}}^{shift} = \sum\_{k\_{0}=1}^{N-1} \underbrace{\sum\_{q=1}^{N-1} Y\_{k\_{\circ},q}^{nN} \mathbf{G}\_{q}}\_{\mathbf{g}\_{bs}} \underbrace{\sum\_{p=1}^{N-1} Y\_{k,p}^{nN} Y\_{p,k\_{\circ}}^{nN} H\_{p}}\_{h\_{k,k\_{\circ}}} \\ &= \sum\_{q=1}^{N-1} \sum\_{p=1}^{N-1} \underbrace{\sum\_{k\_{0}=1}^{N-1} Y\_{p,k\_{\circ}}^{nN} Y\_{k,q}^{nN}}\_{=\mathbf{s}\_{pq}} Y\_{k,p}^{nN} H\_{p} \mathbf{G}\_{q}. \end{split} \tag{84}$$

This gives us yet another transform pair that says that multiplication in the

GqHshift

Interchanging the roles of G and H in Eq. (91) demonstrates that convolution in the

m, <sup>q</sup> ¼ ∑ N�1 q¼1

Eqs. (26) and (29) show how the DHT can be used to calculate the continuous Hankel transform at finite points. From Eqs. (26) and (29), it is clear that given a continuous function f rð Þ evaluated at the discrete points rnk (given by Eq. (31)) in the space domain (1≤k≤ N � 1), its nth-order Hankel-transform function Fð Þρ evaluated at the discrete points ρnm (given in Eq. (31)) in the frequency domain

Similarly, given a continuous function Fð Þρ evaluated at the discrete points ρnm in the frequency domain (1≤ m ≤ N � 1), its nth-order inverse Hankel transform f rð Þ evaluated at the discrete points rnk (1≤ k≤ N � 1), can be approximately

m,kF m½� ) <sup>f</sup> <sup>¼</sup> <sup>1</sup>

). The scaling factor α chosen for using the DHT to approximate the

, where R is the effective space limit and W<sup>ρ</sup> is the effective band

For both the forward and inverse transforms, <sup>α</sup> is a scaling factor and <sup>α</sup> <sup>¼</sup> <sup>R</sup><sup>2</sup>

CHT depends on whether information is known about the band-limit or space-limit. We already introduced the idea of an effective limit in the previous sections, where a function was defined as being "effectively limited in space by R" means that if r . R, then f rð Þ≈0 for all r . R. In other words, the function can be made as close to zero as desired by selecting an R that is large enough. The same idea can be applied to the

The relationship WρR ¼ j nN, derived in the previous sections, holds between the ranges in space and frequency. Choosing N determines the dimension (size) of the DHT and determines j nN. The determination of j nN (via choosing N) determines the range in one domain once the range in the other domain is chosen. In fact, any two of R, Wρ, j nN can be chosen but the third must follow from WρR ¼ j nN. A similar relationship applies when using the discrete Fourier transform, any two of the range

spatial frequency domain, where the effective domain is denoted by Wρ.

Gshift

m, <sup>q</sup> <sup>¼</sup> <sup>G</sup><sup>∗</sup> ð Þ <sup>H</sup> <sup>m</sup>: (91)

m, <sup>q</sup> Hq <sup>¼</sup> <sup>H</sup><sup>∗</sup> ð Þ <sup>G</sup> <sup>m</sup>: (92)

m,k f k½� ) <sup>F</sup> <sup>¼</sup> <sup>α</sup><sup>Y</sup> nN<sup>f</sup> (93)

<sup>α</sup> <sup>Y</sup> nN<sup>F</sup> (94)

j nN or

spatial domain is equivalent to convolution in the transform domain:

gkhk ⇔ ∑ N�1 q¼1

> N�1 q¼1

GqHshift

9. Using the DHT to approximate the continuous transform

transform domain also commutes:

The Discrete Hankel Transform

DOI: http://dx.doi.org/10.5772/intechopen.84399

<sup>G</sup><sup>∗</sup> ð Þ <sup>H</sup> <sup>m</sup> <sup>¼</sup> <sup>∑</sup>

9.1 Approximation to the continuous transform

(1≤ m ≤ N � 1), can be approximately given by

f k½ �¼ <sup>1</sup>

W<sup>2</sup> ρ <sup>α</sup> <sup>∑</sup> N�1 m¼1

f k½ �¼ f rð Þ nk .

given by

equivalently <sup>α</sup> <sup>¼</sup> <sup>j</sup> nN

limit (in m�<sup>1</sup>

19

F m½ �¼ α ∑

N�1 k¼1

Y nN

where α is a scaling factor to be discussed below, and F m½ �¼ Fð Þ ρnm ,

Y nN

But from the orthogonality relationship (40), the summation over k<sup>0</sup> gives the Kronecker delta function, so that Eq. (84) becomes

$$\begin{aligned} (\mathbf{g}^\* h)\_k &= \sum\_{k\_0=1}^{N-1} \mathbf{g}\_{k\_\sigma} h\_{k,k\_\sigma}^{shift} = \sum\_{q=1}^{N-1} \sum\_{p=1}^{N-1} \delta\_{pq} Y\_{k,p}^{nN} H\_P G\_q \\ &= \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} (H\_p G\_p) \end{aligned} \tag{85}$$

The right hand side of Eq. (85) is clearly the inverse transform of the product of the transforms HpFp. This gives us another transform pair

$$(\mathbf{g}^\* h)\_k = \sum\_{k\_0=1}^{N-1} \mathbf{g}\_{k\_\flat} h\_{k\_\flat, k\_\flat}^{\text{shift}} \Leftrightarrow H\_m \mathbf{G}\_m. \tag{86}$$

It follows from Eq. (85) that interchanging the roles of g and h will yield the same result, meaning

$$\sum\_{k\_0=1}^{N-1} \mathbf{g}\_{k,k\_0}^{shift} h\_{k\_0} \;= \sum\_{p=1}^{N-1} Y\_{k,p}^{nN} \mathbf{G}\_p H\_p. \tag{87}$$

Therefore, it follows that

$$(h^\* \mathbf{g})\_k = \sum\_{k\_0=1}^{N-1} \mathbf{g}\_{k\_\* k\_0}^{sh \text{jft}} h\_{k\_o} = \sum\_{k\_0=1}^{N-1} \mathbf{g}\_{k\_o} h\_{k\_\* k\_o}^{sh \text{jft}} = (\mathbf{g}^\* h)\_k. \tag{88}$$

#### 8.7 Multiplication

We now consider the forward transform of a product in the space domain f <sup>k</sup> ¼ gkhk so that

$$\sum\_{k=1}^{N-1} Y\_{m,k}^{nN} g\_k h\_k = \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} \underbrace{\sum\_{q=1}^{N-1} Y\_{k,q}^{nN} G\_q}\_{g\_k} \underbrace{\sum\_{p=1}^{N-1} Y\_{k,p}^{nN} H\_p}\_{h\_k} \tag{89}$$

Rearranging gives

$$\sum\_{k=1}^{N-1} Y\_{m,k}^{nN} g\_k h\_k = \sum\_{q=1}^{N-1} G\_q \sum\_{p=1}^{N-1} \underbrace{\sum\_{k=1}^{N-1} Y\_{m,k}^{nN} Y\_{k,q}^{nN} Y\_{k,p}^{nN}}\_{\text{shift operator}} H\_p$$

$$= \sum\_{q=1}^{N-1} G\_q H\_{m,q}^{shft} = (G^\* H)\_m.$$

f <sup>k</sup> ¼ ∑ N�1 k0¼1 gko hshift k,ko ¼ ∑ N�1 k0¼1 ∑ N�1 q¼1 Y nN ko, qGq

Kronecker delta function, so that Eq. (84) becomes

<sup>g</sup><sup>∗</sup> ð Þ <sup>h</sup> <sup>k</sup> <sup>¼</sup> <sup>∑</sup>

transforms HpFp. This gives us another transform pair

∑ N�1 k0¼1

> N�1 k0¼1

m,k gkhk ¼ ∑

m,kgkhk ¼ ∑

g shift k,ko

> N�1 k¼1

> > N�1 q¼1

¼ ∑ N�1 q¼1

Gq ∑ N�1 p¼1 ∑ N�1 k¼1 Y nN m,kY nN k, qY nN k,p

GqHshift

<sup>h</sup><sup>∗</sup> ð Þ<sup>g</sup> <sup>k</sup> <sup>¼</sup> <sup>∑</sup>

∑ N�1 k¼1

> ∑ N�1 k¼1 Y nN

Y nN

result, meaning

Therefore, it follows that

8.7 Multiplication

f <sup>k</sup> ¼ gkhk so that

Rearranging gives

18

N�1 k0¼1 gko hshift k,ko ¼ ∑ N�1 q¼1 ∑ N�1 p¼1

¼ ∑ N�1 p¼1 Y nN k,p HpGp � �

<sup>g</sup><sup>∗</sup> ð Þ <sup>h</sup> <sup>k</sup> <sup>¼</sup> <sup>∑</sup>

g shift k,ko

N�1 k0¼1 gko hshift


But from the orthogonality relationship (40), the summation over k<sup>0</sup> gives the

The right hand side of Eq. (85) is clearly the inverse transform of the product of the

It follows from Eq. (85) that interchanging the roles of g and h will yield the same

hko ¼ ∑ N�1 p¼1

> hko ¼ ∑ N�1 k0¼1 gko hshift

We now consider the forward transform of a product in the space domain

Y nN m,k ∑ N�1 q¼1

Y nN

Y nN k, qGq

∑ N�1 p¼1


m, <sup>q</sup> <sup>¼</sup> <sup>G</sup><sup>∗</sup> ð Þ <sup>H</sup> <sup>m</sup>:

Y nN k,pHp


Hp


Y nN p,ko Y nN ko,q


δpqY nN

¼ ∑ N�1 q¼1 ∑ N�1 p¼1 ∑ N�1 k0¼1

Fourier Transforms - Century of Digitalization and Increasing Expectations

∑ N�1 p¼1 Y nN k,pY nN p,ko Hp


(84)

(85)

Y nN k,pHpGq:

k,pHpGq

k,ko ⇔ HmGm: (86)

k,pGpHp: (87)

k,ko <sup>¼</sup> <sup>g</sup><sup>∗</sup> ð Þ <sup>h</sup> <sup>k</sup>: (88)

: (89)

(90)

This gives us yet another transform pair that says that multiplication in the spatial domain is equivalent to convolution in the transform domain:

$$\mathbf{g}\_k h\_k \Leftrightarrow \sum\_{q=1}^{N-1} \mathbf{G}\_q H\_{m,q}^{sh\hat{\mathbf{y}}\hat{\mathbf{t}}} = (\mathbf{G}^\* H)\_m. \tag{91}$$

Interchanging the roles of G and H in Eq. (91) demonstrates that convolution in the transform domain also commutes:

$$(G^\*H)\_m = \sum\_{q=1}^{N-1} G\_q H\_{m,q}^{sh\text{fit}} = \sum\_{q=1}^{N-1} G\_{m,q}^{sh\text{fit}} H\_q = (H^\*G)\_m. \tag{92}$$

## 9. Using the DHT to approximate the continuous transform

### 9.1 Approximation to the continuous transform

Eqs. (26) and (29) show how the DHT can be used to calculate the continuous Hankel transform at finite points. From Eqs. (26) and (29), it is clear that given a continuous function f rð Þ evaluated at the discrete points rnk (given by Eq. (31)) in the space domain (1≤k≤ N � 1), its nth-order Hankel-transform function Fð Þρ evaluated at the discrete points ρnm (given in Eq. (31)) in the frequency domain (1≤ m ≤ N � 1), can be approximately given by

$$F[m] = a \sum\_{k=1}^{N-1} Y\_{m,k}^{nN} f[k] \qquad \Rightarrow \qquad \mathbf{F} = aY^{nN} \mathbf{f} \tag{93}$$

where α is a scaling factor to be discussed below, and F m½ �¼ Fð Þ ρnm , f k½ �¼ f rð Þ nk .

Similarly, given a continuous function Fð Þρ evaluated at the discrete points ρnm in the frequency domain (1≤ m ≤ N � 1), its nth-order inverse Hankel transform f rð Þ evaluated at the discrete points rnk (1≤ k≤ N � 1), can be approximately given by

$$f[k] = \frac{1}{a} \sum\_{m=1}^{N-1} Y\_{m,k}^{nN} F[m] \quad \Rightarrow \qquad \mathbf{f} = \frac{1}{a} Y^{nN} \mathbf{F} \tag{94}$$

For both the forward and inverse transforms, <sup>α</sup> is a scaling factor and <sup>α</sup> <sup>¼</sup> <sup>R</sup><sup>2</sup> j nN or equivalently <sup>α</sup> <sup>¼</sup> <sup>j</sup> nN W<sup>2</sup> ρ , where R is the effective space limit and W<sup>ρ</sup> is the effective band limit (in m�<sup>1</sup> ). The scaling factor α chosen for using the DHT to approximate the CHT depends on whether information is known about the band-limit or space-limit. We already introduced the idea of an effective limit in the previous sections, where a function was defined as being "effectively limited in space by R" means that if r . R, then f rð Þ≈0 for all r . R. In other words, the function can be made as close to zero as desired by selecting an R that is large enough. The same idea can be applied to the spatial frequency domain, where the effective domain is denoted by Wρ.

The relationship WρR ¼ j nN, derived in the previous sections, holds between the ranges in space and frequency. Choosing N determines the dimension (size) of the DHT and determines j nN. The determination of j nN (via choosing N) determines the range in one domain once the range in the other domain is chosen. In fact, any two of R, Wρ, j nN can be chosen but the third must follow from WρR ¼ j nN. A similar relationship applies when using the discrete Fourier transform, any two of the range in each domain and the size of the DFT can be chosen independently. In previous sections, we showed that the size of the DHT required can be quickly approximated from 2WR <sup>¼</sup> <sup>W</sup>ρ<sup>R</sup> <sup>π</sup> <sup>≈</sup> <sup>N</sup> <sup>þ</sup> <sup>n</sup> 2 .

and inverse transform. The second and third steps in the list above are only needed if the function (vector) to be transformed is not already given as a set of discrete points. In the case of a continuous function, it is important to evaluate the function at the sampling points in Eq. (95). Failing to do so results in the function not being properly transformed since there is a necessary relationship between the sampling points and the transformation matrix Y nN. In order to perform the steps listed above, several Matlab functions have been developed. These functions are shown in

Additionally, the matlab script GuidetoDHT.m is included to illustrate the

The software was tested by using the DHT to approximate the computation of both the continuous Hankel forward and inverse transforms and comparing the results with known (continuous) forward and inverse Hankel transform pairs.

For the purpose of testing the accuracy of the DHT and IDHT, the dynamic error

This error function compares the difference between the exact function values f vð Þ (evaluated from the continuous function) and the function values estimated via the

used. The methodology of the testing is given in further detail in [18] and also in the

(Eq. (95))

of kind—developed in [17]

In this chapter, the theory of the discrete Hankel transform as a "standalone"

transform was motivated and presented. The standard operating rules for

mated samples. The dynamic error uses the ratio of the absolute error to the maximum amplitude of the function on a log scale. Therefore, negative decibel errors imply an accurate discrete estimation of the true transform value. The transform was also tested for accuracy on itself by performing consecutive forward and then inverse transformation. This is done to verify that the transforms themselves

f vð Þ� f <sup>∗</sup> j j ð Þ<sup>ν</sup> max f <sup>∗</sup> j j ð Þ<sup>v</sup>

ð Þν , scaled with the maximum value of the discretely esti-

(96)

<sup>N</sup> <sup>∑</sup><sup>N</sup>

Calculation of k Bessel zeros of the nth-order Bessel function

Creation of sample points in the frequency domain

Creation of Y nN matrix from the zeros

Creation of sample points in the space domain (Eq. (95))

<sup>i</sup>¼<sup>1</sup> fi � fi <sup>∗</sup> 

was

e vð Þ¼ 20 log <sup>10</sup>

do not add errors. For this evaluation, the average absolute error <sup>1</sup>

execution of the necessary computational steps.

Different transform orders n were evaluated.

∗

Function name Calling sequence Description

(n,k,kind)

(R,zeros)

(R,zeros)

(n,N,zeros)

10. Summary and conclusions

9.4 Verification of the software

The Discrete Hankel Transform

DOI: http://dx.doi.org/10.5772/intechopen.84399

was used, defined as [12]

discrete transform, f

theory paper [3].

Table 1.

21

Set of available functions.

besselzero besselzero

freqSampler freqSampler

spaceSampler spaceSampler

YmatrixAssembly YmatrixAssembly

Table 1.

## 9.2 Sampling points

In order to properly use the discrete transform to approximate the continuous transform, a function has to be sampled at specific discretization points. For a finite spatial range 0½ � ; R and a Hankel transform of order n, these sampling points are given in the space domain as rnk and frequency domain by ρnm, given in Eq. (31) and repeated here for convenience

$$r\_{nk} = \frac{j\_{nk}}{W\_{\rho}} = \frac{j\_{nk}R}{j\_{nN}} \qquad \qquad \rho\_{nm} = \frac{j\_{nm}}{R} = \frac{j\_{nm}W\_{\rho}}{j\_{nN}} \qquad \qquad k, m = 1...N-1 \tag{95}$$

It is important to note that as in the case of the computation of the transformation matrix Y nN, the first Bessel zero j <sup>n</sup><sup>1</sup> used in computing the discretization points is the first non-zero value. Eq. (95) demonstrates that some of the ideas known for the DFT also apply to the DHT. That is, making the spatial domain larger (larger R) implies making the sampling density tighter in frequency (the ρnm get closer together). Similarly, making the frequency domain larger (larger Wρ) implies a tighter sampling density (smaller step size) in the spatial domain. Although j nm are not equispaced, they are nearly so for higher values of m and for purposes of developing quick intuitions on ideas such as sampling density, if is convenient to approximately think of j nk <sup>≈</sup> <sup>k</sup> <sup>þ</sup> <sup>n</sup> 2 π.

#### 9.3 Implementation and availability of the software

The software used to calculate the DHT is based on the MATLAB programming language. The software can be downloaded from


The implementation of the discrete Hankel transform is decomposed into distinct functions. These functions consist of various steps that have to be performed in order to properly execute the transform. These steps are as follows:


The steps are the same regardless if the function is in the space or frequency domain. Furthermore, the Y nN transformation matrix is used for both the forward

### The Discrete Hankel Transform DOI: http://dx.doi.org/10.5772/intechopen.84399

in each domain and the size of the DFT can be chosen independently. In previous sections, we showed that the size of the DHT required can be quickly approximated

In order to properly use the discrete transform to approximate the continuous transform, a function has to be sampled at specific discretization points. For a finite spatial range 0½ � ; R and a Hankel transform of order n, these sampling points are given in the space domain as rnk and frequency domain by ρnm, given in Eq. (31) and

> nmW<sup>ρ</sup> j nN

It is important to note that as in the case of the computation of the transforma-

The software used to calculate the DHT is based on the MATLAB programming

The implementation of the discrete Hankel transform is decomposed into distinct functions. These functions consist of various steps that have to be performed in

• Generate of N sample points (if using the DHT to approximate the continuous

The steps are the same regardless if the function is in the space or frequency domain. Furthermore, the Y nN transformation matrix is used for both the forward

is the first non-zero value. Eq. (95) demonstrates that some of the ideas known for the DFT also apply to the DHT. That is, making the spatial domain larger (larger R) implies making the sampling density tighter in frequency (the ρnm get closer together). Similarly, making the frequency domain larger (larger Wρ) implies a tighter sampling density (smaller step size) in the spatial domain. Although j

not equispaced, they are nearly so for higher values of m and for purposes of developing quick intuitions on ideas such as sampling density, if is convenient to

k, m ¼ 1…N � 1 (95)

nm are

<sup>n</sup><sup>1</sup> used in computing the discretization points

from 2WR <sup>¼</sup> <sup>W</sup>ρ<sup>R</sup>

9.2 Sampling points

rnk <sup>¼</sup> <sup>j</sup> nk W<sup>ρ</sup> ¼ j nkR j nN

approximately think of j

transform)

20

• Sample the function (if needed)

• Create the Y nN transformation matrix

• Perform the matrix-function multiplication

repeated here for convenience

tion matrix Y nN, the first Bessel zero j

<sup>π</sup> <sup>≈</sup> <sup>N</sup> <sup>þ</sup> <sup>n</sup>

2 .

<sup>ρ</sup>nm <sup>¼</sup> <sup>j</sup>

Fourier Transforms - Century of Digitalization and Increasing Expectations

nk <sup>≈</sup> <sup>k</sup> <sup>þ</sup> <sup>n</sup>

9.3 Implementation and availability of the software

• http://dx.doi.org/10.6084/m9.figshare.1453205

language. The software can be downloaded from

2 π.

• https://github.com/uchouinard/DiscreteHankelTransform

order to properly execute the transform. These steps are as follows:

• Calculate N Bessel zeros of the Bessel function of order n

nm <sup>R</sup> <sup>¼</sup> <sup>j</sup> and inverse transform. The second and third steps in the list above are only needed if the function (vector) to be transformed is not already given as a set of discrete points. In the case of a continuous function, it is important to evaluate the function at the sampling points in Eq. (95). Failing to do so results in the function not being properly transformed since there is a necessary relationship between the sampling points and the transformation matrix Y nN. In order to perform the steps listed above, several Matlab functions have been developed. These functions are shown in Table 1.

Additionally, the matlab script GuidetoDHT.m is included to illustrate the execution of the necessary computational steps.

## 9.4 Verification of the software

The software was tested by using the DHT to approximate the computation of both the continuous Hankel forward and inverse transforms and comparing the results with known (continuous) forward and inverse Hankel transform pairs. Different transform orders n were evaluated.

For the purpose of testing the accuracy of the DHT and IDHT, the dynamic error was used, defined as [12]

$$e(\nu) = 20 \log\_{10} \left[ \frac{|f(\nu) - f^\*(\nu)|}{\max |f^\*(\nu)|} \right] \tag{96}$$

This error function compares the difference between the exact function values f vð Þ (evaluated from the continuous function) and the function values estimated via the discrete transform, f ∗ ð Þν , scaled with the maximum value of the discretely estimated samples. The dynamic error uses the ratio of the absolute error to the maximum amplitude of the function on a log scale. Therefore, negative decibel errors imply an accurate discrete estimation of the true transform value. The transform was also tested for accuracy on itself by performing consecutive forward and then inverse transformation. This is done to verify that the transforms themselves do not add errors. For this evaluation, the average absolute error <sup>1</sup> <sup>N</sup> <sup>∑</sup><sup>N</sup> <sup>i</sup>¼<sup>1</sup> fi � fi <sup>∗</sup> was used. The methodology of the testing is given in further detail in [18] and also in the theory paper [3].


#### Table 1.

Set of available functions.

## 10. Summary and conclusions

In this chapter, the theory of the discrete Hankel transform as a "standalone" transform was motivated and presented. The standard operating rules for

multiplication, modulation, shift and convolution were also demonstrated. Sampling and interpolation theorems were shown. The theory and numerical steps to use the presented discrete theory for the purpose of approximating the continuous Hankel transform was also shown. Links to the publicly available, open-source numerical code were also included.

References

1767-1777

27(10):2144-2155

Press; 2000. pp. 9.1-9.30

IEEE. 1998;86(2):447-457

[8] Levitan BM. Generalized displacement operators. In: Encyclopedia of Mathematics. Heidelberg: Springer. p. 2002

IRE. 1949;37(1):10-21

2005

23

[7] Watson GN. A Treatise on the Theory of Bessel Functions. Cambridge, UK: Cambridge University Press; 1995

611-622

[1] Baddour N. Operational and convolution properties of two-

The Discrete Hankel Transform

[2] Baddour N. Operational and convolution properties of threedimensional Fourier transforms in spherical polar coordinates. Journal of the Optical Society of America. A. 2010;

[3] Baddour N, Chouinard U. Theory and operational rules for the discrete Hankel transform. JOSA A. 2015;32(4):

[4] Piessens R. The Hankel transform. In: The Transforms and Applications Handbook. Vol. 2. Boca Raton: CRC

[5] Schroeder J. Signal processing via Fourier-Bessel series expansion. Digital Signal Processing. 1993;3(2):112-124

[6] Shannon CE. Communication in the presence of noise. Proceedings of the

[9] Shannon CE. Communication in the presence of noise. Proceedings of the

[10] Arfken GB. Mathematical Methods for Physicists. 6th ed. Boston: Elsevier;

[11] Johnson HF. An improved method for computing a discrete Hankel

dimensional Fourier transforms in polar coordinates. Journal of the Optical Society of America. A. 2009;26(8):

DOI: http://dx.doi.org/10.5772/intechopen.84399

transform. Computer Physics

2004;21(1):53-58

Communications. 1987;43(2):181-202

[12] Guizar-Sicairos M, Gutiérrez-Vega JC. Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields. Journal of the Optical Society of America. A.

[13] Baddour N. Application of the generalized shift operator to the Hankel transform. Springerplus. 2014;3(1):246

[14] Belhadj M, Betancor JJ. Hankel convolution operators on entire functions and distributions. Journal of

[15] de Sousa Pinto J. A generalised Hankel convolution. SIAM Journal on Mathematical Analysis. 1985;16(6):

[16] Malgonde SP, Gaikawad GS. On a generalized Hankel type convolution of generalized functions. Proceedings of the Indian Academy of Sciences– Mathematical Sciences. 2001;111(4):

[17] G. von Winckel, "Bessel Function Zeros—File Exchange—MATLAB Central." [Online]. Available from: http://www.mathworks.com/ matlabcentral/fileexchange/6794 bessel-function-zeros. [Accessed:

Mathematical Analysis and Applications. 2002;276(1):40-63

1335-1346

471-487

06-Jun-2015]

[18] Chouinard U. Numerical simulations for the discrete Hankel transform [B.A.Sc. thesis]. Ottawa, Canada: University of Ottawa; 2015

## Acknowledgements

The author acknowledges the contributions of Mr. Ugo Chouinard, who developed and tested the numerical code to which links are provided in this chapter. This work was financially supported by the Natural Sciences and Engineering Research Council of Canada.

## Conflict of interest

The author declares that there are no conflicting interests.

## Author details

Natalie Baddour Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario, Canada

\*Address all correspondence to: nbaddour@uottawa.ca

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Discrete Hankel Transform DOI: http://dx.doi.org/10.5772/intechopen.84399

## References

multiplication, modulation, shift and convolution were also demonstrated. Sampling and interpolation theorems were shown. The theory and numerical steps to use the presented discrete theory for the purpose of approximating the continuous Hankel transform was also shown. Links to the publicly available, open-source

Fourier Transforms - Century of Digitalization and Increasing Expectations

The author acknowledges the contributions of Mr. Ugo Chouinard, who developed and tested the numerical code to which links are provided in this chapter. This work was financially supported by the Natural Sciences and Engineering Research

Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: nbaddour@uottawa.ca

provided the original work is properly cited.

The author declares that there are no conflicting interests.

numerical code were also included.

Acknowledgements

Council of Canada.

Conflict of interest

Author details

Natalie Baddour

Canada

22

[1] Baddour N. Operational and convolution properties of twodimensional Fourier transforms in polar coordinates. Journal of the Optical Society of America. A. 2009;26(8): 1767-1777

[2] Baddour N. Operational and convolution properties of threedimensional Fourier transforms in spherical polar coordinates. Journal of the Optical Society of America. A. 2010; 27(10):2144-2155

[3] Baddour N, Chouinard U. Theory and operational rules for the discrete Hankel transform. JOSA A. 2015;32(4): 611-622

[4] Piessens R. The Hankel transform. In: The Transforms and Applications Handbook. Vol. 2. Boca Raton: CRC Press; 2000. pp. 9.1-9.30

[5] Schroeder J. Signal processing via Fourier-Bessel series expansion. Digital Signal Processing. 1993;3(2):112-124

[6] Shannon CE. Communication in the presence of noise. Proceedings of the IEEE. 1998;86(2):447-457

[7] Watson GN. A Treatise on the Theory of Bessel Functions. Cambridge, UK: Cambridge University Press; 1995

[8] Levitan BM. Generalized displacement operators. In: Encyclopedia of Mathematics. Heidelberg: Springer. p. 2002

[9] Shannon CE. Communication in the presence of noise. Proceedings of the IRE. 1949;37(1):10-21

[10] Arfken GB. Mathematical Methods for Physicists. 6th ed. Boston: Elsevier; 2005

[11] Johnson HF. An improved method for computing a discrete Hankel

transform. Computer Physics Communications. 1987;43(2):181-202

[12] Guizar-Sicairos M, Gutiérrez-Vega JC. Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields. Journal of the Optical Society of America. A. 2004;21(1):53-58

[13] Baddour N. Application of the generalized shift operator to the Hankel transform. Springerplus. 2014;3(1):246

[14] Belhadj M, Betancor JJ. Hankel convolution operators on entire functions and distributions. Journal of Mathematical Analysis and Applications. 2002;276(1):40-63

[15] de Sousa Pinto J. A generalised Hankel convolution. SIAM Journal on Mathematical Analysis. 1985;16(6): 1335-1346

[16] Malgonde SP, Gaikawad GS. On a generalized Hankel type convolution of generalized functions. Proceedings of the Indian Academy of Sciences– Mathematical Sciences. 2001;111(4): 471-487

[17] G. von Winckel, "Bessel Function Zeros—File Exchange—MATLAB Central." [Online]. Available from: http://www.mathworks.com/ matlabcentral/fileexchange/6794 bessel-function-zeros. [Accessed: 06-Jun-2015]

[18] Chouinard U. Numerical simulations for the discrete Hankel transform [B.A.Sc. thesis]. Ottawa, Canada: University of Ottawa; 2015

Chapter 2

Abstract

Fourier Transforms for

Juan Manuel Velazquez Arcos,

and Jaime Granados Samaniego

to give a sound support of the Fourier transforms.

Fourier transforms, vector-matrix equations

1. Introduction

25

Keywords: Fredholm equations, electromagnetic resonances,

electromagnetic confinement, evanescent waves, left-hand materials,

There is a very broad class of problems on physics that requires a tool that not only serves to handle the mathematical problem related to the solution of some differential equation describing the behavior of a system but that gives us an alternative description of them from a distinct point of view in a manner that allows us to discover some hidden physical properties, that is, we need to generalize the application of the Fourier transform from the conventional task to achieve a set of algebraic equations to a complete alternative formulation in terms of the Fourier transform of the integral Fredholm equations [1–5, 13, 17]. Many of the problems we want to consider are those related with vector fields like the electromagnetic. For this situation we dedicate the present chapter first to the integral equation formulation of the electromagnetic traveling waves, and then, by the application of

Generalized Fredholm Equations

Ricardo Teodoro Paez Hernandez, Alejandro Perez Ricardez

In this chapter we take the conventional Fredholm integral equations as a guideline to define a broad class of equations we name generalized Fredholm equations with a larger scope of applications. We show first that these new kind of equations are really vector-integral equations with the same properties but with redefined and also enlarged elements in its structure replacing the old traditional concepts like in the case of the source or inhomogeneous term with the generalized source useful for describing the electromagnetic wave propagation. Then we can apply a Fourier transform to the new equations in order to obtain matrix equations to both types, inhomogeneous and homogeneous generalized Fredholm equations. Meanwhile, we discover new properties of the field we can describe with this new technology, that is, mean; we recognize that the old concept of nuclear resonances is present in the new equations and reinterpreted as the brake of the confinement of the electromagnetic field. It is important to say that some segments involving mathematical details of our present work were published somewhere by us, as part of independent researches with different specific goals, and we recall them as a tool

## Chapter 2
