Preface

Chapter 7 **Neural Network Principles and Applications 115** Amer Zayegh and Nizar Al Bassam

**Dynamic Systems 133**

**VI** Contents

and Mahardhika Pratama

Chapter 8 **Applications of General Regression Neural Networks in**

Ahmad Jobran Al-Mahasneh, Sreenatha Anavatti, Matthew Garratt

Digital system design requires unembellished simulation that eliminates potential risks and harm to users and manufacturers. Most of modern design and analysis tools are targeted at custom integrated circuits that are costly and time prohibitive to implement at high level designs. The purpose of this book is to provide a review on advanced digital system design and simulation through computer aided design (CAD) and machine learning tools. We present the practical applications of CAD and machine learning modeling and synthesis in digital system design to construct a basis for effective design and provide a tutorial of digi‐ tal systems functionality. We review theoretical principles, discrete mathematical models, computer simulations and machine learning methods in related areas. In this book, imple‐ mentation of frequency analysis methods is presented at software and hardware levels. Var‐ iable Digital Filter (VDF) is presented as one of the vastly used hardware processing tools in the field of digital systems. A detailed description is provided on the advanced design meth‐ ods of VDFs. Practical application of field-programmable gate array (FPGA) in a control sys‐ tem is presented in this book and an evolutionary algorithm is presented for functional verification of FPGA design. Deep learning has been used as an efficient tool for compres‐ sion of digital networks and increasing the processing speed. Some of the useful deep learn‐ ing methods have been introduced and the applications of them is presented in digital system design and verification. Several architectures are introduced and evaluated includ‐ ing general regression neural networks and convolutional neural networks.

As the editor of the book, I would like to acknowledge the contribution of the authors. These efforts allowed the updated materials in the field of digital systems to be available for the researchers and users in this field.

> **Dr. Vahid Asadpour** Sadjad University of Technology Research Scientist at University of California Los Angeles (UCLA)

**Section 1**

**Digital Systems and Firmware Algorithms**

**Digital Systems and Firmware Algorithms**

**Chapter 1**

Provisional chapter

**Fourier Transform Profilometry in LabVIEW**

Fourier transform profilometry (FTP) is an established non-contact method for 3D sensing in many scientific and industrial applications, such as quality control and biomedical imaging. This phase-based technique has the advantages of high resolution and noise robustness compared to intensity-based approaches. In FTP, a sinusoidal grating is projected onto the surface of an object, the shape information is encoded into a deformed fringe pattern recorded by a camera. The object shape is decoded by calculating the Fourier transform, filtering in the spatial frequency domain, and calculating the inverse Fourier transform; afterward, a conversion of the measured phase to object height is carried out. FTP has been extensively studied and extended for achieving better slope measurement, better separation of height information from noise, and robustness to discontinuities in the fringe pattern. Most of the literature on FTP disregards the software implementation aspects. In this chapter, we return to the basics of FTP and explain in detail the software implementation in LabVIEW, one of the most used data acquisition platforms in engineering. We show results on three applications for FTP in 3D metrology.

DOI: 10.5772/intechopen.78548

Keywords: 3D reconstruction, Fourier transform profilometry, FTP, LabVIEW

Three-dimensional (3D) shape measurement techniques are widely used in many different fields such as mechanical engineering, industry monitoring, robotics, biomedicine, dressmaking, among others [1]. These techniques can be classified as passive, like in stereo vision in which two or more cameras are used to obtain the 3D reconstruction of a scene, or as active, like in fringe projection profilometry (FPP) in which a projection device is used to project a pattern onto the object to be reconstructed. When compared with other 3D measurement techniques, FPP has the advantages of high measurement accuracy and high density. There are two types of FPP

> © 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited.

© 2018 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.

Fourier Transform Profilometry in LabVIEW

Andrés G. Marrugo, Jesús Pineda, Lenny A. Romero,

Andrés G. Marrugo, Jesús Pineda, Lenny A. Romero,

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Raúl Vargas and Jaime Meneses

Raúl Vargas and Jaime Meneses

http://dx.doi.org/10.5772/intechopen.78548

Abstract

1. Introduction

#### **Chapter 1** Provisional chapter

#### **Fourier Transform Profilometry in LabVIEW** Fourier Transform Profilometry in LabVIEW

DOI: 10.5772/intechopen.78548

Andrés G. Marrugo, Jesús Pineda, Lenny A. Romero, Raúl Vargas and Jaime Meneses Andrés G. Marrugo, Jesús Pineda, Lenny A. Romero, Raúl Vargas and Jaime Meneses

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.78548

#### Abstract

Fourier transform profilometry (FTP) is an established non-contact method for 3D sensing in many scientific and industrial applications, such as quality control and biomedical imaging. This phase-based technique has the advantages of high resolution and noise robustness compared to intensity-based approaches. In FTP, a sinusoidal grating is projected onto the surface of an object, the shape information is encoded into a deformed fringe pattern recorded by a camera. The object shape is decoded by calculating the Fourier transform, filtering in the spatial frequency domain, and calculating the inverse Fourier transform; afterward, a conversion of the measured phase to object height is carried out. FTP has been extensively studied and extended for achieving better slope measurement, better separation of height information from noise, and robustness to discontinuities in the fringe pattern. Most of the literature on FTP disregards the software implementation aspects. In this chapter, we return to the basics of FTP and explain in detail the software implementation in LabVIEW, one of the most used data acquisition platforms in engineering. We show results on three applications for FTP in 3D metrology.

Keywords: 3D reconstruction, Fourier transform profilometry, FTP, LabVIEW

#### 1. Introduction

Three-dimensional (3D) shape measurement techniques are widely used in many different fields such as mechanical engineering, industry monitoring, robotics, biomedicine, dressmaking, among others [1]. These techniques can be classified as passive, like in stereo vision in which two or more cameras are used to obtain the 3D reconstruction of a scene, or as active, like in fringe projection profilometry (FPP) in which a projection device is used to project a pattern onto the object to be reconstructed. When compared with other 3D measurement techniques, FPP has the advantages of high measurement accuracy and high density. There are two types of FPP

© 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited. © 2018 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.

methods: phase shifting and Fourier-transform profilometry (FTP). Phase-shifting methods offer high-resolution measurement at the expense of projecting several patterns onto the object [2–4], whereas FTP is popular because only one deformed fringe pattern image is needed [5]. For this reason, FTP has been used in many dynamic applications [6] such as vibration measurement of micromechanical devices [7] and measurement of real-time deformation fields [8].

2. FTP fundamentals

ved through the camera is given by

than the spatial carrier frequency f <sup>0</sup>

following form

There are many implementations of FPP. However, all share the same underlying principle. A typical FPP setup consists of a projection device and a camera as shown in Figure 1. A fringe pattern is projected onto a test object, and the resulting image is acquired by the camera from a different direction. The acquired fringe pattern image is distorted according to the object shape. In terms of information theory, it is said that the object shape is encoded into a deformed fringe pattern acquired by the camera. The object shape is recovered/decoded by comparison to the original (undeformed) fringe pattern image. Therefore, the phase shift between the reference and

By projecting a fringe pattern onto the reference plane, the fringe pattern (with period p<sup>0</sup> ¼1=f <sup>0</sup>)

Likewise, when the object is placed on the reference plane, the deformed fringe pattern obser-

where a0ð Þ x; y and a xð Þ ; y represent the non-uniform background illumination, b0ð Þ x; y and b xð Þ ; y the contrast of the fringe pattern. f <sup>0</sup> is the fundamental frequency of the observed fringe pattern (also called carrier frequency). ϕ0ð Þ x; y and ϕð Þ x; y are the original phase modulation on the reference plane R where z xð Þ¼ ; y 0 and the phase modulations resulting from the object height distribution, respectively. a xð Þ ; y , b xð Þ ; y and ϕð Þ x; y are assumed to vary much slower

The input fringe pattern from Eqs. (1) and (2) can be rewritten using Euler's formula in the

<sup>g</sup>0ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>a</sup>0ð Þþ <sup>x</sup>; <sup>y</sup> <sup>b</sup>0ð Þ <sup>x</sup>; <sup>y</sup> cos 2π<sup>f</sup> <sup>0</sup><sup>x</sup> <sup>þ</sup> <sup>ϕ</sup>0ð Þ <sup>x</sup>; <sup>y</sup> : (1)

Fourier Transform Profilometry in LabVIEW http://dx.doi.org/10.5772/intechopen.78548 5

g xð Þ¼ ; <sup>y</sup> a xð Þþ ; <sup>y</sup> b xð Þ ; <sup>y</sup> cos 2π<sup>f</sup> <sup>0</sup><sup>x</sup> <sup>þ</sup> <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> , (2)

. The principle of FTP is shown schematically in Figure 2.

the deformed image contains the information of the object shape.

on the reference plane observed through the camera can be modeled as

Figure 2. Principle of the filtering via Fourier transform (FT) method. IFT, inverse FT.

FTP was proposed by Takeda et al. [5, 9] in 1982 and has since become one of the most used methods [3, 10]. Its main advantages are full-field analysis, high precision, noise-robustness [11], among others. In FTP, a Ronchi grating, or a sinusoidal grating, or a fringe pattern from a digital projector is projected onto an object, and the depth information of the object is encoded into the deformed fringe pattern recorded by an image acquisition device as shown in Figure 1. The surface shape can be decoded by calculating the Fourier transform, filtering in the spatial frequency domain, and calculating the inverse Fourier transform. Compared with other fringe analysis methods, FTP can accomplish a fully automatic distinction between a depression and an elevation of the object shape. It requires no fringe order assignments or fringe center determination, and it needs no interpolation between fringes because it gives height distribution at each pixel over the entire field. Since FTP requires only one or two images of the deformed fringe pattern, it has become one of the most popular methods for real-time 3D reconstruction of dynamic scenes.

Although FTP has been extensively studied and used in many applications, to the best of our knowledge a complete reference in which the implementation details are fully described is nonexistent. In this chapter, we describe the FTP fundamentals and the implementation of an FTP system in LabVIEW one of the most used engineering development platforms for data acquisition and laboratory automation. The chapter is organized as follows. In Section 2 we describe the FTP fundamentals and a general calibration method, in Section 3 we describe how FTP is implemented in LabVIEW, and finally in Section 4 we show three applications of FTP for 3D reconstruction.

Figure 1. Fringe projection system.

### 2. FTP fundamentals

methods: phase shifting and Fourier-transform profilometry (FTP). Phase-shifting methods offer high-resolution measurement at the expense of projecting several patterns onto the object [2–4], whereas FTP is popular because only one deformed fringe pattern image is needed [5]. For this reason, FTP has been used in many dynamic applications [6] such as vibration measurement of

FTP was proposed by Takeda et al. [5, 9] in 1982 and has since become one of the most used methods [3, 10]. Its main advantages are full-field analysis, high precision, noise-robustness [11], among others. In FTP, a Ronchi grating, or a sinusoidal grating, or a fringe pattern from a digital projector is projected onto an object, and the depth information of the object is encoded into the deformed fringe pattern recorded by an image acquisition device as shown in Figure 1. The surface shape can be decoded by calculating the Fourier transform, filtering in the spatial frequency domain, and calculating the inverse Fourier transform. Compared with other fringe analysis methods, FTP can accomplish a fully automatic distinction between a depression and an elevation of the object shape. It requires no fringe order assignments or fringe center determination, and it needs no interpolation between fringes because it gives height distribution at each pixel over the entire field. Since FTP requires only one or two images of the deformed fringe pattern, it has become one of the most popular methods for real-time 3D

Although FTP has been extensively studied and used in many applications, to the best of our knowledge a complete reference in which the implementation details are fully described is nonexistent. In this chapter, we describe the FTP fundamentals and the implementation of an FTP system in LabVIEW one of the most used engineering development platforms for data acquisition and laboratory automation. The chapter is organized as follows. In Section 2 we describe the FTP fundamentals and a general calibration method, in Section 3 we describe how FTP is implemented in LabVIEW, and finally in Section 4 we show three applications of FTP

micromechanical devices [7] and measurement of real-time deformation fields [8].

reconstruction of dynamic scenes.

for 3D reconstruction.

4 Digital Systems

Figure 1. Fringe projection system.

There are many implementations of FPP. However, all share the same underlying principle. A typical FPP setup consists of a projection device and a camera as shown in Figure 1. A fringe pattern is projected onto a test object, and the resulting image is acquired by the camera from a different direction. The acquired fringe pattern image is distorted according to the object shape. In terms of information theory, it is said that the object shape is encoded into a deformed fringe pattern acquired by the camera. The object shape is recovered/decoded by comparison to the original (undeformed) fringe pattern image. Therefore, the phase shift between the reference and the deformed image contains the information of the object shape.

By projecting a fringe pattern onto the reference plane, the fringe pattern (with period p<sup>0</sup> ¼1=f <sup>0</sup>) on the reference plane observed through the camera can be modeled as

$$g\_0(\mathbf{x}, y) = a\_0(\mathbf{x}, y) + b\_0(\mathbf{x}, y) \cos \left[ 2\pi f\_0 \mathbf{x} + \phi\_0(\mathbf{x}, y) \right]. \tag{1}$$

Likewise, when the object is placed on the reference plane, the deformed fringe pattern observed through the camera is given by

$$\mathbf{g}(\mathbf{x}, y) = a(\mathbf{x}, y) + b(\mathbf{x}, y)\cos\left[2\pi f\_0 \mathbf{x} + \phi(\mathbf{x}, y)\right],\tag{2}$$

where a0ð Þ x; y and a xð Þ ; y represent the non-uniform background illumination, b0ð Þ x; y and b xð Þ ; y the contrast of the fringe pattern. f <sup>0</sup> is the fundamental frequency of the observed fringe pattern (also called carrier frequency). ϕ0ð Þ x; y and ϕð Þ x; y are the original phase modulation on the reference plane R where z xð Þ¼ ; y 0 and the phase modulations resulting from the object height distribution, respectively. a xð Þ ; y , b xð Þ ; y and ϕð Þ x; y are assumed to vary much slower than the spatial carrier frequency f <sup>0</sup> . The principle of FTP is shown schematically in Figure 2. The input fringe pattern from Eqs. (1) and (2) can be rewritten using Euler's formula in the following form

Figure 2. Principle of the filtering via Fourier transform (FT) method. IFT, inverse FT.

$$\mathbf{g}(\mathbf{x},y) = a(\mathbf{x},y) + c(\mathbf{x},y)\exp\{2\pi i f\_0 \mathbf{x}\} + c^\*(\mathbf{x},y)\exp\{-2\pi i f\_0 \mathbf{x}\},\tag{3}$$

with

$$c(\mathbf{x}, y) = \frac{1}{2} b(\mathbf{x}, y) \exp\{i\phi(\mathbf{x}, y)\},\tag{4}$$

unwrapped by using a suitable phase unwrapping algorithm [12] that gives the desired phase map as shown in Figure 2. The phase map Δϕð Þ x; y is proportional to the height of the object

The calibration of FPP systems plays an essential role in the accuracy of the 3D reconstructions. Here we describe a simple yet extensively used calibration called the reference-plane-based

The optical axis geometry of the FTP measurement system is depicted in Figure 3. The optical

reference to measure the height of the object z xð Þ ; y . d is the distance between the projector and the camera, l<sup>0</sup> is the distance between the camera and the reference plane. The fringe pattern image (with period p) is formed by the projector lens on plane I through point O. p is related to the carrier frequency by f <sup>0</sup> ¼ 1=p<sup>0</sup> ¼ cosθ=p. The height of the object surface is measured relative to R. From the point of view of the projector, point A on the object surface has the

denotes a point on the reference plane. On the camera sensor, point A on the object surface and point D on the reference plane are imaged on the same pixel. By subtracting the reference phase map from the object phase map, we obtain the phase difference at this specific pixel

0

<sup>c</sup> � Ec of a camera lens at a point O on a

Fourier Transform Profilometry in LabVIEW http://dx.doi.org/10.5772/intechopen.78548 7

<sup>c</sup> � Ec and serves as a

<sup>C</sup>, where the superindex R

0

technique, i.e., to convert the unwrapped phase map Δϕð Þ x; y to height z.

reference plane R. This reference plane is normal to the optical axis E

same phase value as point <sup>C</sup> on the reference plane <sup>R</sup>, <sup>Φ</sup><sup>A</sup> <sup>¼</sup> <sup>Φ</sup><sup>R</sup>

<sup>p</sup> � Ep of a projector lens crosses the optical axis E

surface.

axis E 0

2.1. System calibration

Figure 3. Fringe projection system.

where ∗ denotes a complex conjugate.

Next, the phase of the fringe patterns is recovered using the Fourier Transform method. Using one-dimensional notation for simplicity, when we compute the Fourier transform of Eqs. (1) and (2) the Fourier spectrum of the fringe signals splits intro three spectrum components separated from each other, which gives

$$\mathcal{G}(f\_x, y) = A(f\_x, y) + \mathcal{C}(f\_x - f\_0, y) + \mathcal{C}^\*(f\_x + f\_0, y),\tag{5}$$

as shown in two dimensions in Figure 2. With an appropriate filter function, for instance, a Hanning filter, the spectra are filtered to let only the fundamental component C f <sup>x</sup> � <sup>f</sup> <sup>0</sup>; <sup>y</sup> � �. A Hanning window is given by [11],

$$H(f\_x) = 0.50 \left[ 1 + \cos \left( \beta \pi \frac{f\_x - f\_0}{f\_c} \right) \right] \tag{6}$$

where f <sup>c</sup> is the cutoff frequency at a 50% attenuation ratio, β ¼ 1=2 and f <sup>x</sup> varies from f <sup>0</sup> � f <sup>c</sup>=β to f <sup>0</sup> þ f <sup>c</sup>=β. The inverse Fourier Transform is applied to the filtered component, and a complex signal is obtained

$$
\widehat{g}\_0(\mathbf{x}, y) = \frac{1}{2} b(\mathbf{x}, y) \exp\{i(2\pi f\_0 \mathbf{x} + \phi\_0(\mathbf{x}, y))\},
\tag{7}
$$

$$
\widehat{g}(\mathbf{x}, y) = \frac{1}{2} b(\mathbf{x}, y) \exp\{i(2\pi f\_0 \mathbf{x} + \phi(\mathbf{x}, y))\}.\tag{8}
$$

The variable related to height distribution is the phase change Δϕð Þ x; y [9]:

$$
\Delta\phi(\mathbf{x},y) = \Phi(\mathbf{x},y) - \Phi\_0(\mathbf{x},y) = \phi(\mathbf{x},y) - \phi\_0(\mathbf{x},y),
\tag{9}
$$

with

$$\Phi\_0(\mathbf{x}, y) = \tan^{-1} \left( \frac{\mathfrak{T}[\widehat{\mathcal{G}}\_0(\mathbf{x}, y)]}{\mathfrak{R}[\widehat{\mathcal{G}}\_0(\mathbf{x}, y)]} \right), \tag{10}$$

$$\Phi(\mathbf{x}, y) = \tan^{-1} \left( \frac{\mathfrak{T}[\widehat{\mathcal{G}}(\mathbf{x}, y)]}{\mathfrak{R}[\widehat{\mathcal{G}}(\mathbf{x}, y)]} \right), \tag{11}$$

where ℑ½ �: and ℜ½ �: denote the imaginary and the real part, respectively. The phases obtained from Eqs. (10) and (11) are wrapped into the principal value ½ � �π; π . The wrapped phase is unwrapped by using a suitable phase unwrapping algorithm [12] that gives the desired phase map as shown in Figure 2. The phase map Δϕð Þ x; y is proportional to the height of the object surface.

#### 2.1. System calibration

g xð Þ¼ ; <sup>y</sup> a xð Þþ ; <sup>y</sup> c xð Þ ; <sup>y</sup> exp 2πif <sup>0</sup><sup>x</sup> � � <sup>þ</sup> <sup>c</sup>

c xð Þ¼ ; y

H f <sup>x</sup>

<sup>b</sup>g0ð Þ¼ <sup>x</sup>; <sup>y</sup>

<sup>b</sup>g xð Þ¼ ; <sup>y</sup>

1 2

> 1 2

The variable related to height distribution is the phase change Δϕð Þ x; y [9]:

where ∗ denotes a complex conjugate.

separated from each other, which gives

Hanning window is given by [11],

signal is obtained

with

1 2

Next, the phase of the fringe patterns is recovered using the Fourier Transform method. Using one-dimensional notation for simplicity, when we compute the Fourier transform of Eqs. (1) and (2) the Fourier spectrum of the fringe signals splits intro three spectrum components

as shown in two dimensions in Figure 2. With an appropriate filter function, for instance, a Hanning filter, the spectra are filtered to let only the fundamental component C f <sup>x</sup> � <sup>f</sup> <sup>0</sup>; <sup>y</sup> � �. A

� � <sup>¼</sup> <sup>0</sup>:50 1 <sup>þ</sup> cos βπ <sup>f</sup> <sup>x</sup> � <sup>f</sup> <sup>0</sup>

where f <sup>c</sup> is the cutoff frequency at a 50% attenuation ratio, β ¼ 1=2 and f <sup>x</sup> varies from f <sup>0</sup> � f <sup>c</sup>=β to f <sup>0</sup> þ f <sup>c</sup>=β. The inverse Fourier Transform is applied to the filtered component, and a complex

<sup>Φ</sup>0ð Þ¼ <sup>x</sup>; <sup>y</sup> tan�<sup>1</sup> <sup>ℑ</sup>½ � <sup>b</sup>g0ð Þ <sup>x</sup>; <sup>y</sup>

<sup>Φ</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> tan�<sup>1</sup> <sup>ℑ</sup>½ � <sup>b</sup>g xð Þ ; <sup>y</sup>

where ℑ½ �: and ℜ½ �: denote the imaginary and the real part, respectively. The phases obtained from Eqs. (10) and (11) are wrapped into the principal value ½ � �π; π . The wrapped phase is

G f <sup>x</sup>; <sup>y</sup> � � <sup>¼</sup> A f <sup>x</sup>; <sup>y</sup> � � <sup>þ</sup> C f <sup>x</sup> � <sup>f</sup> <sup>0</sup>; <sup>y</sup> � � <sup>þ</sup> <sup>C</sup><sup>∗</sup> <sup>f</sup> <sup>x</sup> <sup>þ</sup> <sup>f</sup> <sup>0</sup>; <sup>y</sup> � �, (5)

f c

Δϕð Þ¼ x; y Φð Þ� x; y Φ0ð Þ¼ x; y ϕð Þ� x; y ϕ0ð Þ x; y , (9)

<sup>ℜ</sup>½ � <sup>b</sup>g0ð Þ <sup>x</sup>; <sup>y</sup> � �

<sup>ℜ</sup>½ � <sup>b</sup>g xð Þ ; <sup>y</sup> � �

b xð Þ ; <sup>y</sup> exp <sup>i</sup> <sup>2</sup>π<sup>f</sup> <sup>0</sup><sup>x</sup> <sup>þ</sup> <sup>ϕ</sup>0ð Þ <sup>x</sup>; <sup>y</sup> � � � � , (7)

b xð Þ ; <sup>y</sup> exp <sup>i</sup> <sup>2</sup>π<sup>f</sup> <sup>0</sup><sup>x</sup> <sup>þ</sup> <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � � � � : (8)

� � � �

with

6 Digital Systems

<sup>∗</sup>ð Þ <sup>x</sup>; <sup>y</sup> exp �2πif <sup>0</sup><sup>x</sup> � �, (3)

, (6)

, (10)

, (11)

b xð Þ ; <sup>y</sup> exp <sup>i</sup>ϕð Þ <sup>x</sup>; <sup>y</sup> � �, (4)

The calibration of FPP systems plays an essential role in the accuracy of the 3D reconstructions. Here we describe a simple yet extensively used calibration called the reference-plane-based technique, i.e., to convert the unwrapped phase map Δϕð Þ x; y to height z.

The optical axis geometry of the FTP measurement system is depicted in Figure 3. The optical axis E 0 <sup>p</sup> � Ep of a projector lens crosses the optical axis E 0 <sup>c</sup> � Ec of a camera lens at a point O on a reference plane R. This reference plane is normal to the optical axis E 0 <sup>c</sup> � Ec and serves as a reference to measure the height of the object z xð Þ ; y . d is the distance between the projector and the camera, l<sup>0</sup> is the distance between the camera and the reference plane. The fringe pattern image (with period p) is formed by the projector lens on plane I through point O. p is related to the carrier frequency by f <sup>0</sup> ¼ 1=p<sup>0</sup> ¼ cosθ=p. The height of the object surface is measured relative to R. From the point of view of the projector, point A on the object surface has the same phase value as point <sup>C</sup> on the reference plane <sup>R</sup>, <sup>Φ</sup><sup>A</sup> <sup>¼</sup> <sup>Φ</sup><sup>R</sup> <sup>C</sup>, where the superindex R denotes a point on the reference plane. On the camera sensor, point A on the object surface and point D on the reference plane are imaged on the same pixel. By subtracting the reference phase map from the object phase map, we obtain the phase difference at this specific pixel

Figure 3. Fringe projection system.

$$
\Delta\Phi\_{AD} = \Phi\_A - \Phi\_D^R = \Phi\_\mathbb{C}^R - \Phi\_D^R = \Phi\_{\text{CD}}^R. \tag{12}
$$

model, and the best data obtained in step II. Finally in step IV, with the complete model, we

Fourier Transform Profilometry in LabVIEW http://dx.doi.org/10.5772/intechopen.78548 9

In this section, we explain the details of the FTP software implementation in LabVIEW. LabVIEW stands for Laboratory Virtual Instrument Engineering Workbench and is a systemdesign platform and development environment for a visual programming language from National Instruments [21]. It allows integrating hardware, acquiring and analyzing data, and sharing results. Because it is a visual programming language based on function blocks, it is a highly intuitive integrated development environment (IDE) for engineers and scientists familiar with block diagrams and flowcharts. Every LabVIEW block diagram also has an associated

The acquisition and processing strategies described in this section require the installation of the

• NI vision acquisition software, which installs NI-IMAQdx. This software driver allows the integration of cameras with different control protocols such as USB3 Vision, GigE Vision devices, IEEE 1394 cameras compatible with IIDC, IP (Ethernet) and DirectShow compatible USB devices (e.g., cameras, webcams, microscopes, scanners). NI vision acquisition software also includes the driver NI-IMAQ for acquiring from analog cameras, digital parallel and Camera Link, as well as NI Smart Cameras. This hardware compatibility is the main advantage of using LabVIEW for vision systems. This compatibility greatly facilitates the development of applications for different types of cameras and

• NI vision development module (VDM). This package provides machine vision and image processing functions. It includes IMAQ Vision, a library of powerful functions for vision processing. In this library, there is a group of VIs that analyze and process images in the frequency domain. We will make use of these functions throughout the entire chapter.

NI VDM and Vision Acquisition Software are supported on the following operating systems: • Windows 10; Windows 8.1; Windows 7 (SP1) 32-bit; Windows 7 (SP1) 64-bit; Windows Embedded Standard 7 (SP1); Windows Server 2012 R2 64-bit; Windows Server 2008 R2 (SP1)

There are two primary ways to obtain images in LabVIEW: loading an image file or acquiring directly from a camera. The wiring diagram in Figure 5(a) illustrates how to perform a continuous (grab) acquisition in LabVIEW using Vision Acquisition Software. A Grab acquisition begins by initializing the camera specified by the Camera Name Control and configuring the driver for acquiring images continuously. Using IMAQ Create, we create a temporary

can find mathematical expressions that convert phase maps to XYZ-coordinates.

3. LabVIEW implementation

following software components:

busses.

64-bit.

3.1. Image acquisition

front panel, which is the user interface of the application.

The triangles ΔEpEcA and ΔCDA are similar, and the height AB of point A on the object surface relative to the reference plane can be related to the distance between points C and D

$$
\Delta z(x, y) = \overline{AB} \approx \frac{l\_0}{d} \overline{CD} \approx \Delta \Phi\_{\text{CD}}^R = \Phi\_A - \Phi\_D^R. \tag{13}
$$

Combining Eqs. (12) and (13) a proportional relation between the phase map and the surface height can be obtained for any point ð Þ x; y

$$
\Delta z(\mathbf{x}, y) \propto \Delta \phi(\mathbf{x}, y) = \Phi(\mathbf{x}, y) - \Phi\_0(\mathbf{x}, y) \tag{14}
$$

where Φð Þ x; y is the object phase map and Φ0ð Þ x; y is the reference plane phase map. Assuming the reference plane has a depth of z0, the depth value for each measured point can be represented as

$$z(\mathbf{x}, y) = z\_0 + k\_0 \times [\Phi(\mathbf{x}, y) - \Phi\_0]. \tag{15}$$

where k<sup>0</sup> is a constant determined through calibration and z<sup>0</sup> is usually set to 0.

We have shown how the object surface height is related to the recovered phase through FTP. The model described by Eq. (15) has many underlying assumptions and is often extended to cover more degrees of freedom. Moreover, a general calibration process in FPP can be carried out employing the methodology shown in Figure 4. First, we propose a model that best describes the system, while also considering metrological requirements such as speed, robustness, accuracy, flexibility and reconstruction scale. Some authors have proposed to use several calibration models based on polynomial or fractional fitting functions [13, 14], bilinear interpolation by look-up table (LUT) [15] and stereo triangulation [16–18]. These calibration models require different strategies or techniques that allow relating metric coordinates with phase values. In step II, we select or design a strategy that fits the proposed calibration model and characteristics of the elements to a given experimental setup, such as the type of projector (i.e., analog or digital projection) and camera (i.e., monochrome or color). These strategies consist in projecting and capturing fringe patterns onto 3D-objects [19] or 2D-targets [16, 20] with highly accurate known measurements. In some cases, the calibration consists in displacing the targets along the z axis using a linear translation stage [19]. The purpose is to obtain a correspondence between a metric coordinate system and the phase images captured with the camera. In step III, the correspondences are used to calculate the parameters that are part of the proposed

Figure 4. General calibration methodology.

model, and the best data obtained in step II. Finally in step IV, with the complete model, we can find mathematical expressions that convert phase maps to XYZ-coordinates.

### 3. LabVIEW implementation

ΔΦAD <sup>¼</sup> <sup>Φ</sup><sup>A</sup> � <sup>Φ</sup><sup>R</sup>

<sup>Δ</sup>z xð Þ¼ ; <sup>y</sup> AB <sup>≈</sup> <sup>l</sup><sup>0</sup>

height can be obtained for any point ð Þ x; y

Figure 4. General calibration methodology.

represented as

8 Digital Systems

<sup>D</sup> <sup>¼</sup> <sup>Φ</sup><sup>R</sup>

The triangles ΔEpEcA and ΔCDA are similar, and the height AB of point A on the object surface

<sup>d</sup> CD <sup>∝</sup> ΔΦ<sup>R</sup>

Combining Eqs. (12) and (13) a proportional relation between the phase map and the surface

where Φð Þ x; y is the object phase map and Φ0ð Þ x; y is the reference plane phase map. Assuming the reference plane has a depth of z0, the depth value for each measured point can be

We have shown how the object surface height is related to the recovered phase through FTP. The model described by Eq. (15) has many underlying assumptions and is often extended to cover more degrees of freedom. Moreover, a general calibration process in FPP can be carried out employing the methodology shown in Figure 4. First, we propose a model that best describes the system, while also considering metrological requirements such as speed, robustness, accuracy, flexibility and reconstruction scale. Some authors have proposed to use several calibration models based on polynomial or fractional fitting functions [13, 14], bilinear interpolation by look-up table (LUT) [15] and stereo triangulation [16–18]. These calibration models require different strategies or techniques that allow relating metric coordinates with phase values. In step II, we select or design a strategy that fits the proposed calibration model and characteristics of the elements to a given experimental setup, such as the type of projector (i.e., analog or digital projection) and camera (i.e., monochrome or color). These strategies consist in projecting and capturing fringe patterns onto 3D-objects [19] or 2D-targets [16, 20] with highly accurate known measurements. In some cases, the calibration consists in displacing the targets along the z axis using a linear translation stage [19]. The purpose is to obtain a correspondence between a metric coordinate system and the phase images captured with the camera. In step III, the correspondences are used to calculate the parameters that are part of the proposed

where k<sup>0</sup> is a constant determined through calibration and z<sup>0</sup> is usually set to 0.

relative to the reference plane can be related to the distance between points C and D

<sup>C</sup> � <sup>Φ</sup><sup>R</sup>

<sup>D</sup> <sup>¼</sup> <sup>Φ</sup><sup>R</sup>

CD <sup>¼</sup> <sup>Φ</sup><sup>A</sup> � <sup>Φ</sup><sup>R</sup>

Δz xð Þ ; y ∝Δϕð Þ¼ x; y Φð Þ� x; y Φ0ð Þ x; y , (14)

z xð Þ¼ ; y z<sup>0</sup> þ k<sup>0</sup> � ½ � Φð Þ� x; y Φ<sup>0</sup> , (15)

CD: (12)

<sup>D</sup>: (13)

In this section, we explain the details of the FTP software implementation in LabVIEW. LabVIEW stands for Laboratory Virtual Instrument Engineering Workbench and is a systemdesign platform and development environment for a visual programming language from National Instruments [21]. It allows integrating hardware, acquiring and analyzing data, and sharing results. Because it is a visual programming language based on function blocks, it is a highly intuitive integrated development environment (IDE) for engineers and scientists familiar with block diagrams and flowcharts. Every LabVIEW block diagram also has an associated front panel, which is the user interface of the application.

The acquisition and processing strategies described in this section require the installation of the following software components:


NI VDM and Vision Acquisition Software are supported on the following operating systems:

• Windows 10; Windows 8.1; Windows 7 (SP1) 32-bit; Windows 7 (SP1) 64-bit; Windows Embedded Standard 7 (SP1); Windows Server 2012 R2 64-bit; Windows Server 2008 R2 (SP1) 64-bit.

#### 3.1. Image acquisition

There are two primary ways to obtain images in LabVIEW: loading an image file or acquiring directly from a camera. The wiring diagram in Figure 5(a) illustrates how to perform a continuous (grab) acquisition in LabVIEW using Vision Acquisition Software. A Grab acquisition begins by initializing the camera specified by the Camera Name Control and configuring the driver for acquiring images continuously. Using IMAQ Create, we create a temporary

Figure 5. Grab acquisition in LabVIEW. (a) Block diagram. (b) Image indicator in front panel.

memory location for the acquired image. This function returns an IMAQ image reference to the buffer in memory where the image is stored. The reference is the input to the IMAQ Grab VI for starting the acquisition. The grabbed image is displayed on the LabVIEW front panel using an Image Indicator (see Figure 5(b)), which points to the location in memory referenced by the IMAQ image reference. A while loop statement allows adding each grabbed image to the image indicator as a single frame. Finally, the image acquisition is finished by calling the IMAQ close VI that releases resources associated with the camera and the interface.

acquisition settings is one of the most relevant processes during configuration and allows the simultaneous manipulation of camera attributes like Exposure Time, Trigger Mode, Gain, Gamma Factor, among others. For this example, we configured the acquisition for working in a continuous acquisition with inline processing mode, which continuously acquires images until an event stops the acquisition. Additionally, the Exposure Time attribute can be modified during the acquisition process by using a Numeric Control. As with the example in Figure 5, the captured image is displayed in a secondary image indicator during the Case Structure execution. In Fringe Projection systems, the manipulation of certain camera attributes (e.g., the Exposure Time attribute) is required to capture high-quality images and to enable to work under different lighting environments with different constraints. In the example above, we introduced the possibility of manipulating camera attributes during acquisition using the Vision Acquisition Express. This manipulation of attributes is also possible by programming a simple snap, grab, or sequence operation based on low-level VIs (as in the example in Figure 5) using IMAQdx property nodes. The attribute manipulation requires providing the property node with the name of the attribute we want to modify and identifying the attribute representation, which can be an integer, float, Boolean, enumeration, string or command. In general, cameras share many attributes; however, they often have specific attributes depending on the manufacturer. These attributes should be known beforehand to ensure good acquisition control. At the development stage, LabVIEW does not know or display the name of the attributes or representations. Furthermore, if the documentation is not available, we suggest using the Measurement and Automation Explorer (MAX). MAX is a tool that allows the configuration of different acquisition parameters and is useful when it is required to manipulate attributes of a device with a specific interface within the LabVIEW programming environment. For example, suppose we want to modify the exposure time of our camera (Basler Aca 1600-60gm), but we do not have information about supported attributes. Here is where MAX becomes a

Figure 6. Continuous acquisition using IMAQ vision acquisition express. (a) Block diagram. (b) Image indicator in front

Fourier Transform Profilometry in LabVIEW http://dx.doi.org/10.5772/intechopen.78548 11

panel.

The acquired image is written to a file in a specified format by using the IMAQ Write File 2 VI. The graphics file formats supported by this function are BMP (windows bitmap), JPEG, PNG (portable network graphics), and TIFF (tagged image file format). However, note that lossy compression formats, such as JPEG, introduce image artifacts and should be avoided to ensure accurate image-based measurements. The saved image can be displayed in a secondary image indicator by enabling the Snapshot option. When enabling the Snapshot Mode, the Image Display control continues to display the image as it was when the image was saved during the Case Structure execution, even when the inspection image has changed. To configure the Image Display control for working in Snapshot Mode, right-click on the control on the front panel and enable the Snapshot option.

Another way to acquire an image using a camera is presented in the Figure 6. This example uses the NI Vision Acquisition Express to perform the acquisition stage. The Vision Acquisition Express VI is located in the Vision Express palette in LabVIEW, and it is commonly used to quickly develop image acquisition applications due to its versatility and intuitive development environment. Double-clicking on the Vision Acquisition Express VI makes a configuration window appear which allows choosing a device from the list of available acquisition sources, selecting an acquisition type, and configuring the acquisition settings. Concerning the acquisition types, there are four main modes: single acquisition with processing, continuous acquisition with inline processing, finite acquisition with inline processing and finite acquisition with postprocessing. The last two acquisition types are similar, except that for a finite acquisition with post-processing the images are only available after they are all acquired. The configuration of the

Figure 6. Continuous acquisition using IMAQ vision acquisition express. (a) Block diagram. (b) Image indicator in front panel.

memory location for the acquired image. This function returns an IMAQ image reference to the buffer in memory where the image is stored. The reference is the input to the IMAQ Grab VI for starting the acquisition. The grabbed image is displayed on the LabVIEW front panel using an Image Indicator (see Figure 5(b)), which points to the location in memory referenced by the IMAQ image reference. A while loop statement allows adding each grabbed image to the image indicator as a single frame. Finally, the image acquisition is finished by calling the

The acquired image is written to a file in a specified format by using the IMAQ Write File 2 VI. The graphics file formats supported by this function are BMP (windows bitmap), JPEG, PNG (portable network graphics), and TIFF (tagged image file format). However, note that lossy compression formats, such as JPEG, introduce image artifacts and should be avoided to ensure accurate image-based measurements. The saved image can be displayed in a secondary image indicator by enabling the Snapshot option. When enabling the Snapshot Mode, the Image Display control continues to display the image as it was when the image was saved during the Case Structure execution, even when the inspection image has changed. To configure the Image Display control for working in Snapshot Mode, right-click on the control on the front

Another way to acquire an image using a camera is presented in the Figure 6. This example uses the NI Vision Acquisition Express to perform the acquisition stage. The Vision Acquisition Express VI is located in the Vision Express palette in LabVIEW, and it is commonly used to quickly develop image acquisition applications due to its versatility and intuitive development environment. Double-clicking on the Vision Acquisition Express VI makes a configuration window appear which allows choosing a device from the list of available acquisition sources, selecting an acquisition type, and configuring the acquisition settings. Concerning the acquisition types, there are four main modes: single acquisition with processing, continuous acquisition with inline processing, finite acquisition with inline processing and finite acquisition with postprocessing. The last two acquisition types are similar, except that for a finite acquisition with post-processing the images are only available after they are all acquired. The configuration of the

IMAQ close VI that releases resources associated with the camera and the interface.

Figure 5. Grab acquisition in LabVIEW. (a) Block diagram. (b) Image indicator in front panel.

panel and enable the Snapshot option.

10 Digital Systems

acquisition settings is one of the most relevant processes during configuration and allows the simultaneous manipulation of camera attributes like Exposure Time, Trigger Mode, Gain, Gamma Factor, among others. For this example, we configured the acquisition for working in a continuous acquisition with inline processing mode, which continuously acquires images until an event stops the acquisition. Additionally, the Exposure Time attribute can be modified during the acquisition process by using a Numeric Control. As with the example in Figure 5, the captured image is displayed in a secondary image indicator during the Case Structure execution.

In Fringe Projection systems, the manipulation of certain camera attributes (e.g., the Exposure Time attribute) is required to capture high-quality images and to enable to work under different lighting environments with different constraints. In the example above, we introduced the possibility of manipulating camera attributes during acquisition using the Vision Acquisition Express. This manipulation of attributes is also possible by programming a simple snap, grab, or sequence operation based on low-level VIs (as in the example in Figure 5) using IMAQdx property nodes. The attribute manipulation requires providing the property node with the name of the attribute we want to modify and identifying the attribute representation, which can be an integer, float, Boolean, enumeration, string or command. In general, cameras share many attributes; however, they often have specific attributes depending on the manufacturer. These attributes should be known beforehand to ensure good acquisition control. At the development stage, LabVIEW does not know or display the name of the attributes or representations. Furthermore, if the documentation is not available, we suggest using the Measurement and Automation Explorer (MAX). MAX is a tool that allows the configuration of different acquisition parameters and is useful when it is required to manipulate attributes of a device with a specific interface within the LabVIEW programming environment. For example, suppose we want to modify the exposure time of our camera (Basler Aca 1600-60gm), but we do not have information about supported attributes. Here is where MAX becomes a powerful tool for vision system developers. This attribute verification is done by selecting the desired attribute from the Camera Attributes tab in the Measurement and Automation Explorer and identifying its name (i.e., ExposureTimeAbs) and representation (i.e., floatingpoint format). Therefore, the section of the block diagram inside a red box in Figure 5 can be modified in order to allow setting the ExposureTimeAbs attribute value using a Property Node as shown in Figure 7.

Both acquisition methods have their advantages and disadvantages concerning their implementation in vision systems. On the one hand, the use of the NI Vision Acquisition Express allows to quickly and easily develop acquisition applications, even without having a high knowledge of the tools for image acquisition offered by LabVIEW. However, this could be a disadvantage if our purpose is to have complete control over the acquisition. On the other hand, the low-level VIs provide greater control and versatility over the application development, but the implementation of vision systems based on low-level VIs can be a complicated task for novice users of NI Vision Acquisition Software and LabVIEW.

Fringe Projection systems can also take advantage of a computer to generate sinusoidal fringe patterns that are projected using a digital projector. The key to a successful 3D reconstruction system based on digital fringe projection focuses on generating high-quality fringes to meet the metrological requirements. Ideally, assuming the projector is linear in that it projects grayscale values ranging from 0 to 255 (0 black, and 255 white), the computer-generated fringe patterns

<sup>2</sup> <sup>1</sup> <sup>þ</sup> cos

where pd represents the number of pixels per fringe period, φ refers to the phase shift, and ð Þ i; j are the pixel indices. Eq. (16) is implemented using the numeric functions provided by the NI LabVIEW Base Package. An example of a pattern generator block diagram is shown in Figure 9. In this program the Numeric Indicators enable the modification of the fringe pitch and the

An alternative to a block diagram implementation of Eq. (16) LabVIEW provides a MathScript RT Module as a scripting language. The module allows the combination of textual and graphical approaches for algorithm development. In Figure 10 we provide an example on how to

Once the fringe images have been generated, they are sent to a digital video projector for projection. A video projector is essentially a second monitor. Therefore the fringe image is displayed by

2πi pd þ φ

, (16)

Fourier Transform Profilometry in LabVIEW http://dx.doi.org/10.5772/intechopen.78548 13

I ið Þ¼ ; <sup>j</sup> <sup>255</sup>

phase shift according to the application requirements.

Figure 9. Block diagram for fringe pattern generation.

use the MathScript RT Module for fringe generation in LabVIEW.

can be described as follows,

Figure 8. Reading an image file in LabVIEW.

Once the acquired fringe image file has been written to disk, it is loaded for processing. The block diagram in Figure 8 illustrates how to perform this procedure in LabVIEW. The IMAQ ReadFile VI opens and reads an image from a file stored on the computer into an image reference. The loaded pixels are converted automatically into the image type supplied by IMAQ Create VI. From now on we refer to the Fringe Image to the loaded fringe image.

#### 3.2. Fringe pattern projection

In the previous section, we described several acquisition methods for capturing images from a camera in LabVIEW. However, in fringe projection systems there are many different fringe pattern projection technologies and choosing the correct one becomes extremely important for an accurate three-dimensional reconstruction. A fringe pattern projector can be considered as an analog device (e.g., LED pattern projector) or as a digital device (e.g., DLP, LCoS, and LCD digital display technologies). LED pattern projectors are ideal for high-resolution threedimensional reconstruction applications. If equipped with an objective lens and a stripe pattern reticle, these projectors offer great versatility for manipulating the optics of the system and obtaining results according to the metrological requirements. The main disadvantage of this type of projection system is the impossibility of manipulating the projected fringe pattern. Therefore, its use is often restricted to techniques in which only a single fringe image is necessary to obtain the 3D information, such as in the case of FTP.

Figure 7. Setting the ExposureTimeAbs attribute value using a property node.

Figure 8. Reading an image file in LabVIEW.

powerful tool for vision system developers. This attribute verification is done by selecting the desired attribute from the Camera Attributes tab in the Measurement and Automation Explorer and identifying its name (i.e., ExposureTimeAbs) and representation (i.e., floatingpoint format). Therefore, the section of the block diagram inside a red box in Figure 5 can be modified in order to allow setting the ExposureTimeAbs attribute value using a Property

Both acquisition methods have their advantages and disadvantages concerning their implementation in vision systems. On the one hand, the use of the NI Vision Acquisition Express allows to quickly and easily develop acquisition applications, even without having a high knowledge of the tools for image acquisition offered by LabVIEW. However, this could be a disadvantage if our purpose is to have complete control over the acquisition. On the other hand, the low-level VIs provide greater control and versatility over the application development, but the implementation of vision systems based on low-level VIs can be a complicated

Once the acquired fringe image file has been written to disk, it is loaded for processing. The block diagram in Figure 8 illustrates how to perform this procedure in LabVIEW. The IMAQ ReadFile VI opens and reads an image from a file stored on the computer into an image reference. The loaded pixels are converted automatically into the image type supplied by IMAQ Create VI. From now on we refer to the Fringe Image to the loaded fringe image.

In the previous section, we described several acquisition methods for capturing images from a camera in LabVIEW. However, in fringe projection systems there are many different fringe pattern projection technologies and choosing the correct one becomes extremely important for an accurate three-dimensional reconstruction. A fringe pattern projector can be considered as an analog device (e.g., LED pattern projector) or as a digital device (e.g., DLP, LCoS, and LCD digital display technologies). LED pattern projectors are ideal for high-resolution threedimensional reconstruction applications. If equipped with an objective lens and a stripe pattern reticle, these projectors offer great versatility for manipulating the optics of the system and obtaining results according to the metrological requirements. The main disadvantage of this type of projection system is the impossibility of manipulating the projected fringe pattern. Therefore, its use is often restricted to techniques in which only a single fringe image is

task for novice users of NI Vision Acquisition Software and LabVIEW.

necessary to obtain the 3D information, such as in the case of FTP.

Figure 7. Setting the ExposureTimeAbs attribute value using a property node.

Node as shown in Figure 7.

12 Digital Systems

3.2. Fringe pattern projection

Fringe Projection systems can also take advantage of a computer to generate sinusoidal fringe patterns that are projected using a digital projector. The key to a successful 3D reconstruction system based on digital fringe projection focuses on generating high-quality fringes to meet the metrological requirements. Ideally, assuming the projector is linear in that it projects grayscale values ranging from 0 to 255 (0 black, and 255 white), the computer-generated fringe patterns can be described as follows,

$$I(i,j) = \frac{255}{2} \left[ 1 + \cos\left(\frac{2\pi i}{p\_d} + \varphi\right) \right],\tag{16}$$

where pd represents the number of pixels per fringe period, φ refers to the phase shift, and ð Þ i; j are the pixel indices. Eq. (16) is implemented using the numeric functions provided by the NI LabVIEW Base Package. An example of a pattern generator block diagram is shown in Figure 9. In this program the Numeric Indicators enable the modification of the fringe pitch and the phase shift according to the application requirements.

An alternative to a block diagram implementation of Eq. (16) LabVIEW provides a MathScript RT Module as a scripting language. The module allows the combination of textual and graphical approaches for algorithm development. In Figure 10 we provide an example on how to use the MathScript RT Module for fringe generation in LabVIEW.

Once the fringe images have been generated, they are sent to a digital video projector for projection. A video projector is essentially a second monitor. Therefore the fringe image is displayed by

Figure 9. Block diagram for fringe pattern generation.


2D array to perform the filtering procedure, thus obtaining the fundamental frequency spectrum in the frequency domain. The following step is to compute the inverse Fourier transform of the fundamental component. The Inverse FFT VI is for computing the inverse discrete Fourier transform (IDFT) of a complex 2D array. By using this function, we calculate the inverse FFT of the fundamental component which contains the 3D information. Finally, we obtain the phase by applying Eq. (11). Here, we use Complex To Re/Im Function to break the complex 2D array into its rectangular components and Inverse Tangent(2 Input) Function for performing the arctangent operation. With the example in Figure 12(a) we illustrate the phase retrieval process in LabVIEW. In this figure, the Fringe Image and Hanning W refer to the fringe pattern image shown in Figure 12(b) and the Hanning window filter array, respectively.

Fourier Transform Profilometry in LabVIEW http://dx.doi.org/10.5772/intechopen.78548 15

In Section 2 we showed that in FTP a filtering procedure is performed to obtain the fundamental frequency spectrum in the frequency domain. Once the Fourier transform is computed, the resultant spectrum is filtered by a 2-D Hanning window defined by Eq. (6). In LabVIEW, the IMAQ Select Rectangle VI is commonly used to specify a rectangular region of interest (ROI) in an image. We use the IMAQ Select Rectangle VI for manually selecting the region in the Fourier spectrum corresponding to the fundamental frequency component. Here, the image is displayed in an external display window and through the use of the rectangle tools, provided by the IMAQ Select Rectangle VI, we estimate the optimal size and location of the filtering

Figure 12. Phase retrieval process in LabVIEW. (a) Block diagram. (b) Fringe pattern image. (c) Wrapped phase map.

The resultant wrapped phase map is shown in Figure 12(c).

3.4. Hanning filter design

Figure 10. Fringe pattern generation example using the LabVIEW MathScript RT module.

using the External Display VIs provided by the NI Vision Development Module. Here, we use IMAQ WindDraw VI to display the image in an external image window. The image window appears automatically when the VI is executed. Having beforehand the information from all the available displays on the computer, including their resolution and bounding rectangles, we set the position of the image window to be displayed on the desired monitor. This setting is done with IMAQ WindMove VI. Additionally, using IMAQ WindSetup VI the appearance and attributes of the window can be modified to hide the title bar. Note that the default value for this attribute is TRUE which shows the title bar. The block diagram in Figure 11 illustrates a projection stage in LabVIEW. Here, we use a Property Node for obtaining the information about all the monitors on the computer. The Disp.AllMonitors property Returns information about their bounding rectangles and bit depths.

#### 3.3. Phase retrieval

Phase retrieval is carried out by Fourier transform profilometry. In LabVIEW, the IMAQ FFT VI computes the discrete Fourier transform of the fringe image. This function creates a complex image in which low frequencies are located at the edges, and high frequencies are grouped at the center of the image. Note that for the IMAQ FFT VI a reference to the destination image must be specified and configured as a Complex(CSG) image. Once the deformed fringe pattern is 2-D Fourier transformed, the resulting spectra are converted into a complex

Figure 11. Second monitor configuration in LabVIEW.

2D array to perform the filtering procedure, thus obtaining the fundamental frequency spectrum in the frequency domain. The following step is to compute the inverse Fourier transform of the fundamental component. The Inverse FFT VI is for computing the inverse discrete Fourier transform (IDFT) of a complex 2D array. By using this function, we calculate the inverse FFT of the fundamental component which contains the 3D information. Finally, we obtain the phase by applying Eq. (11). Here, we use Complex To Re/Im Function to break the complex 2D array into its rectangular components and Inverse Tangent(2 Input) Function for performing the arctangent operation. With the example in Figure 12(a) we illustrate the phase retrieval process in LabVIEW. In this figure, the Fringe Image and Hanning W refer to the fringe pattern image shown in Figure 12(b) and the Hanning window filter array, respectively. The resultant wrapped phase map is shown in Figure 12(c).

### 3.4. Hanning filter design

using the External Display VIs provided by the NI Vision Development Module. Here, we use IMAQ WindDraw VI to display the image in an external image window. The image window appears automatically when the VI is executed. Having beforehand the information from all the available displays on the computer, including their resolution and bounding rectangles, we set the position of the image window to be displayed on the desired monitor. This setting is done with IMAQ WindMove VI. Additionally, using IMAQ WindSetup VI the appearance and attributes of the window can be modified to hide the title bar. Note that the default value for this attribute is TRUE which shows the title bar. The block diagram in Figure 11 illustrates a projection stage in LabVIEW. Here, we use a Property Node for obtaining the information about all the monitors on the computer. The Disp.AllMonitors property Returns information about their bounding rectan-

Figure 10. Fringe pattern generation example using the LabVIEW MathScript RT module.

Phase retrieval is carried out by Fourier transform profilometry. In LabVIEW, the IMAQ FFT VI computes the discrete Fourier transform of the fringe image. This function creates a complex image in which low frequencies are located at the edges, and high frequencies are grouped at the center of the image. Note that for the IMAQ FFT VI a reference to the destination image must be specified and configured as a Complex(CSG) image. Once the deformed fringe pattern is 2-D Fourier transformed, the resulting spectra are converted into a complex

gles and bit depths.

14 Digital Systems

3.3. Phase retrieval

Figure 11. Second monitor configuration in LabVIEW.

In Section 2 we showed that in FTP a filtering procedure is performed to obtain the fundamental frequency spectrum in the frequency domain. Once the Fourier transform is computed, the resultant spectrum is filtered by a 2-D Hanning window defined by Eq. (6). In LabVIEW, the IMAQ Select Rectangle VI is commonly used to specify a rectangular region of interest (ROI) in an image. We use the IMAQ Select Rectangle VI for manually selecting the region in the Fourier spectrum corresponding to the fundamental frequency component. Here, the image is displayed in an external display window and through the use of the rectangle tools, provided by the IMAQ Select Rectangle VI, we estimate the optimal size and location of the filtering

Figure 12. Phase retrieval process in LabVIEW. (a) Block diagram. (b) Fringe pattern image. (c) Wrapped phase map.

discrete Fourier transform of the Fringe Image. The resultant complex spectrum is displayed using an external display window as shown in Figure 13(b). By using the selection tools located on the right side of the window, we can manually select the rectangular area of interest. The IMAQ Select Rectangle VI returns the coordinates (i.e., left, top, right and button) of the chosen rectangle as a cluster. Therefore, it is necessary to access each element from the cluster to extract the window information. For this reason, we add the Unbundle By Name function to the block diagram which unbundles a cluster element by name. Based on this information, we calculate the size and location of the Hanning window filter. Finally, using the Hanning Window VI two 1-D Hanning windows are created whose lengths correspond to the size of x and y of the filtering window, respectively. The two-dimensional Hanning window is obtained by the separable product of these two 1-D Hanning windows [22]. The block diagram in Figure 14(a) illustrates the filtering design stage in LabVIEW. dx and dy, in Figure 14(b), relate to the size in x and y of the selected filtering window, respectively. Finally, the obtained 2D

Fourier Transform Profilometry in LabVIEW http://dx.doi.org/10.5772/intechopen.78548 17

The phase unwrapping process is carried out comparing the wrapped phase at neighborhoods and adding, or subtracting, an integer number of 2π, thus obtaining a continuous phase. This definition is for the one-dimensional phase unwrapping process. However, for two-dimensional (2-D) phase unwrapping this is not readily applicable, and additional steps must be taken to

Figure 15. Bidimensional phase unwrapping in LabVIEW. (a) Wrapped phase map. (b) Unwrapped phase map.

Hanning window is shown in Figure 14(c).

3.5. Phase unwrapping

Figure 13. Manual selection of the filtering window. (a) Block diagram. (b) External display window and rectangle tools.

window that guarantees the separation between the fundamental frequency component and other unwanted contributions. The block diagram shown in Figure 13(a) indicates the IMAQ Select Rectangle VI to manually select the region corresponding to the first order spectrum. The Fringe Image is the fringe pattern image in Figure 12(b). The IMAQ FFT VI computes the

Figure 14. Hanning filter design in LabVIEW. (a) Continuation of the block diagram in Figure 13(a). (b) Fourier transform magnitude spectra displayed by the external window in Figure 13(b). dx and dy relate to the size in x and y of the filtering window, respectively. (c) 2D-hanning window.

discrete Fourier transform of the Fringe Image. The resultant complex spectrum is displayed using an external display window as shown in Figure 13(b). By using the selection tools located on the right side of the window, we can manually select the rectangular area of interest.

The IMAQ Select Rectangle VI returns the coordinates (i.e., left, top, right and button) of the chosen rectangle as a cluster. Therefore, it is necessary to access each element from the cluster to extract the window information. For this reason, we add the Unbundle By Name function to the block diagram which unbundles a cluster element by name. Based on this information, we calculate the size and location of the Hanning window filter. Finally, using the Hanning Window VI two 1-D Hanning windows are created whose lengths correspond to the size of x and y of the filtering window, respectively. The two-dimensional Hanning window is obtained by the separable product of these two 1-D Hanning windows [22]. The block diagram in Figure 14(a) illustrates the filtering design stage in LabVIEW. dx and dy, in Figure 14(b), relate to the size in x and y of the selected filtering window, respectively. Finally, the obtained 2D Hanning window is shown in Figure 14(c).

#### 3.5. Phase unwrapping

window that guarantees the separation between the fundamental frequency component and other unwanted contributions. The block diagram shown in Figure 13(a) indicates the IMAQ Select Rectangle VI to manually select the region corresponding to the first order spectrum. The Fringe Image is the fringe pattern image in Figure 12(b). The IMAQ FFT VI computes the

Figure 14. Hanning filter design in LabVIEW. (a) Continuation of the block diagram in Figure 13(a). (b) Fourier transform magnitude spectra displayed by the external window in Figure 13(b). dx and dy relate to the size in x and y of the filtering

window, respectively. (c) 2D-hanning window.

16 Digital Systems

Figure 13. Manual selection of the filtering window. (a) Block diagram. (b) External display window and rectangle tools.

The phase unwrapping process is carried out comparing the wrapped phase at neighborhoods and adding, or subtracting, an integer number of 2π, thus obtaining a continuous phase. This definition is for the one-dimensional phase unwrapping process. However, for two-dimensional (2-D) phase unwrapping this is not readily applicable, and additional steps must be taken to

Figure 15. Bidimensional phase unwrapping in LabVIEW. (a) Wrapped phase map. (b) Unwrapped phase map.

obtain the unwrapped solution. The conventional approach for 2-D phase unwrapping can be accomplished by applying 1-D phase unwrapping first row-wise followed by 1-D phase unwrapping column-wise in two steps. The block diagram in Figure 15(a) illustrates this process. Here, the Unwrap Phase VI unwraps a 1D-phase array by eliminating discontinuities whose absolute values exceed π. Thus, a for loop is required to compute the continuous phase for each row of the 2-D wrapped phase array. For 1-D phase unwrapping column-wise, we use the Transpose Matrix Function to calculate the conjugate transpose of the resultant array before executing the for loop statement. Figure 15(b) and (c) show a wrapped phase map and its unwrapped counterpart, respectively. In addition to this approach, many 2D phase-unwrapping algorithms have been proposed, especially to address discontinuities and noise [12]. These other methods can also be implemented in LabVIEW either with block diagrams, using math scripts, with precompiled C++ code in .dll files, or via integration of external functions with other environments such as MATLAB. However, an explanation of the details of these other approaches is beyond the scope of this chapter.

> Another application of FPP is in facial metrology, where several patterns are projected onto the face to obtain a 3D digital model. 3D shape measurement of faces plays an important role in several fields like in the biomedical sciences, biometrics, security, and entertainment. Human face models are widely used in medical applications for 3D facial expression recognition [24] and measurement of stretch marks [25]. Usually, the main challenge is the movement of the patient. The movement can produce errors or noise in the 3D reconstruction affecting its accuracy. Hence, 3D scanning techniques that require few images in the reconstruction process, like FTP, are commonly used. In Figure 18 we show an experimental result of reconstructing a live human face. The captured image with the deformed fringe pattern is shown in Figure 18(a). In Figure 18(b) and (c) we show the 3D geometry acquired rendered in shaded mode and with texture mapping, respectively. Note that several facial regions with hairs, like the eyebrows, are reconstructed with high detail. While other areas, under shadows, like the right side of the

Fourier Transform Profilometry in LabVIEW http://dx.doi.org/10.5772/intechopen.78548 19

Figure 17. (a) 3D reconstructed shape. (b) Cross section of the 3D reconstruction.

Finally, another area where FPP has frequently been used is in cultural heritage preservation. The preservation of cultural heritage works requires accurately scanning sculptures, archeological remains, paintings, etc. In Figure 19 we show the 3D reconstruction of a sculpture replica.

Figure 18. (a) Fringe pattern onto face. (b) 3D rendered model in shaded mode. (c) 3D rendered model with color texture

nose, are not correctly reconstructed.

mapping.

### 4. Applications

FPP is often used as a non-contact surface analysis technique in industry inspection. In this section, we show the 3D surface reconstruction of a dented steel pipe. A dent is a permanent plastic deformation of the cross-section of the pipe. In the example shown in Figure 16 the dent was produced penetrating the pipe with a diamond cone indenter. In Figure 16(a) and (b) we show the tested object, and the deformed fringe pattern image, respectively. The goal is to measure the depth of the dent with high accuracy and to obtain the surface shape of the pipe for subsequent deformation analysis. In Figure 16(c) and (d), we show the wrapped, and unwrapped phases obtained by FTP, respectively. The unwrapped phase map is converted to metric coordinates using a calibration model. In Figure 17(a), we show the reconstructed pipe shape with the texture map. A profile across the reconstructed pipe, thought the dent, is shown in Figure 17(b). Analyzing this profile, we can measure the depth of the dent to approximately 4 mm.

Figure 16. FTP analysis of a indented pipe. (a) Texture image. (b) Deformed fringe pattern. (c) Wrapped phase. (d) Unwrapped phase.

Figure 17. (a) 3D reconstructed shape. (b) Cross section of the 3D reconstruction.

obtain the unwrapped solution. The conventional approach for 2-D phase unwrapping can be accomplished by applying 1-D phase unwrapping first row-wise followed by 1-D phase unwrapping column-wise in two steps. The block diagram in Figure 15(a) illustrates this process. Here, the Unwrap Phase VI unwraps a 1D-phase array by eliminating discontinuities whose absolute values exceed π. Thus, a for loop is required to compute the continuous phase for each row of the 2-D wrapped phase array. For 1-D phase unwrapping column-wise, we use the Transpose Matrix Function to calculate the conjugate transpose of the resultant array before executing the for loop statement. Figure 15(b) and (c) show a wrapped phase map and its unwrapped counterpart, respectively. In addition to this approach, many 2D phase-unwrapping algorithms have been proposed, especially to address discontinuities and noise [12]. These other methods can also be implemented in LabVIEW either with block diagrams, using math scripts, with precompiled C++ code in .dll files, or via integration of external functions with other environments such as MATLAB. However, an explanation of the details of these other appro-

FPP is often used as a non-contact surface analysis technique in industry inspection. In this section, we show the 3D surface reconstruction of a dented steel pipe. A dent is a permanent plastic deformation of the cross-section of the pipe. In the example shown in Figure 16 the dent was produced penetrating the pipe with a diamond cone indenter. In Figure 16(a) and (b) we show the tested object, and the deformed fringe pattern image, respectively. The goal is to measure the depth of the dent with high accuracy and to obtain the surface shape of the pipe for subsequent deformation analysis. In Figure 16(c) and (d), we show the wrapped, and unwrapped phases obtained by FTP, respectively. The unwrapped phase map is converted to metric coordinates using a calibration model. In Figure 17(a), we show the reconstructed pipe shape with the texture map. A profile across the reconstructed pipe, thought the dent, is shown in Figure 17(b). Analyzing this profile, we can measure the depth of the dent to

Figure 16. FTP analysis of a indented pipe. (a) Texture image. (b) Deformed fringe pattern. (c) Wrapped phase.

aches is beyond the scope of this chapter.

4. Applications

18 Digital Systems

approximately 4 mm.

(d) Unwrapped phase.

Another application of FPP is in facial metrology, where several patterns are projected onto the face to obtain a 3D digital model. 3D shape measurement of faces plays an important role in several fields like in the biomedical sciences, biometrics, security, and entertainment. Human face models are widely used in medical applications for 3D facial expression recognition [24] and measurement of stretch marks [25]. Usually, the main challenge is the movement of the patient. The movement can produce errors or noise in the 3D reconstruction affecting its accuracy. Hence, 3D scanning techniques that require few images in the reconstruction process, like FTP, are commonly used. In Figure 18 we show an experimental result of reconstructing a live human face. The captured image with the deformed fringe pattern is shown in Figure 18(a). In Figure 18(b) and (c) we show the 3D geometry acquired rendered in shaded mode and with texture mapping, respectively. Note that several facial regions with hairs, like the eyebrows, are reconstructed with high detail. While other areas, under shadows, like the right side of the nose, are not correctly reconstructed.

Finally, another area where FPP has frequently been used is in cultural heritage preservation. The preservation of cultural heritage works requires accurately scanning sculptures, archeological remains, paintings, etc. In Figure 19 we show the 3D reconstruction of a sculpture replica.

Figure 18. (a) Fringe pattern onto face. (b) 3D rendered model in shaded mode. (c) 3D rendered model with color texture mapping.

[5] Takeda M, Ina H, Kobayashi S. Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry. Journal of the Optical Society of America.

Fourier Transform Profilometry in LabVIEW http://dx.doi.org/10.5772/intechopen.78548 21

[6] Su X, Zhang Q. Dynamic 3-D shape measurement method: A review. Optics and Lasers in

[7] Petitgrand S, Yahiaoui R, Danaie K, Bosseboeuf A, Gilles J. 3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope. Optics

[8] Felipe-Sesé L, Siegmann P, Díaz FA, Patterson EA. Integrating fringe projection and digital image correlation for high-quality measurements of shape changes. Optical Engi-

[9] Takeda M, Mutoh K. Fourier transform profilometry for the automatic measurement of

[10] Takeda M. Fourier fringe analysis and its application to metrology of extreme physical

[11] Lin JF, Su X. Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes. Optical Engineering. 1995;34(11):3297-

[12] Ghiglia DC, Pritt MD. Two-Dimensional Phase Unwrapping: Theory, Algorithms, and

[13] Huntley JM, Saldner H. Temporal phase-unwrapping algorithm for automated interfero-

[14] Liu H, Su W-H, Reichard K, Yin S. Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement. Optics Communica-

[15] Merner L, Wang Y, Zhang S. Accurate calibration for 3D shape measurement system using a binary defocusing technique. Optics and Lasers in Engineering. 2013;51(5):514-519

[16] Zhang S, Huang PS. Novel method for structured light system calibration. Optical Engi-

[17] Li K, Bu J, Zhang D. Lens distortion elimination for improving measurement accuracy of fringe projection profilometry. Optics and Lasers in Engineering. 2016;85:53-64

[18] Arciniegas J, González AL, Quintero LA, Contreras CR, Meneses JE. Sistema de reconstrucción tridimensional aplicado a la exploración superficial de fallas y defectos en tuberías con refuerzo no metálico para el transporte de hidrocarburos. Revista Investigaciones Aplicadas.

[19] Hu Q, Huang PS, Fu Q, Chiang F-P. Calibration of a three-dimensional shape measure-

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Figure 19. FTP 3D reconstruction of a sculpture replica of "Figura reclinada 92 - Gertrudis" by Fernando Botero [23]. (a) Texture image. (b) 3D reconstruction.

### Acknowledgements

This work has been partly funded by Colciencias (Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas) project (538871552485) and by the Universidad Tecnológica de Bolívar (Dirección de Investigación, Emprendimiento e Innovación). J. Pineda and R. Vargas thank Universidad Tecnológica de Bolívar for a Master's degree scholarship

### Author details

Andrés G. Marrugo<sup>1</sup> \*, Jesús Pineda<sup>1</sup> , Lenny A. Romero<sup>2</sup> , Raúl Vargas<sup>1</sup> and Jaime Meneses<sup>3</sup>


3 Grupo de Óptica y Tratamiento de Señales, Universidad Industrial de Santander, Bucaramanga, Colombia

### References


Acknowledgements

20 Digital Systems

(a) Texture image. (b) 3D reconstruction.

degree scholarship

Author details

Andrés G. Marrugo<sup>1</sup>

Bucaramanga, Colombia

Engineering. 2010;48(2):133-140

References

pp. 1

\*, Jesús Pineda<sup>1</sup>

\*Address all correspondence to: agmarrugo@utb.edu.co

This work has been partly funded by Colciencias (Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas) project (538871552485) and by the Universidad Tecnológica de Bolívar (Dirección de Investigación, Emprendimiento e Innovación). J. Pineda and R. Vargas thank Universidad Tecnológica de Bolívar for a Master's

Figure 19. FTP 3D reconstruction of a sculpture replica of "Figura reclinada 92 - Gertrudis" by Fernando Botero [23].

, Lenny A. Romero<sup>2</sup>

2 Facultad de Ciencias Básicas, Universidad Tecnológica de Bolívar, Cartagena, Colombia

[1] Zhang S. Handbook of 3D Machine Vision: Optical Metrology and Imaging. CRC Press; 2013.

[2] Gorthi SS, Rastogi P. Fringe projection techniques: Whither we are? Optics and Lasers in

[3] Zappa E, Busca G. Static and dynamic features of Fourier transform profilometry: A

[4] Hariharan P, Oreb BF, Eiju T. Digital phase-shifting interferometry: A simple errorcompensating phase calculation algorithm. Applied Optics. 1987;26(13):2504-2506

review. Optics and Lasers in Engineering. 2012;50(8):1140-1151

1 Facultad de Ingeniería, Universidad Tecnológica de Bolívar, Cartagena, Colombia

3 Grupo de Óptica y Tratamiento de Señales, Universidad Industrial de Santander,

, Raúl Vargas<sup>1</sup> and Jaime Meneses<sup>3</sup>


[20] Huang Z, Xi J, Yu Y, Guo Q. Accurate projector calibration based on a new point-to-point mapping relationship between the camera and projector images. Applied Optics. 2015; 54(3):347-356

**Chapter 2**

Provisional chapter

**Recent Advances in Variable Digital Filters**

Variable digital filters are widely used in a number of applications of signal processing because of their capability of self-tuning frequency characteristics such as the cutoff frequency and the bandwidth. This chapter introduces recent advances on variable digital filters, focusing on the problems of design and realization, and application to adaptive filtering. In the topic on design and realization, we address two major approaches: one is the frequency transformation and the other is the multi-dimensional polynomial approximation of filter coefficients. In the topic on adaptive filtering, we introduce the details of adaptive band-

DOI: 10.5772/intechopen.79198

Keywords: variable digital filter, frequency transformation, polynomial approximation,

Digital filter is well known as one of the essential and fundamental components in signal processing devices. In addition, many signal processing applications such as digital audio equipment and telecommunication systems sometimes require simultaneous realization of digital filtering and real-time control of filter characteristics. Such requirements can be fulfilled by means of variable digital filters (VDFs). Research on VDFs emerged in the 1970s and since then, many results have been reported. Among them, details of the results until the 1990s are

The problems that should be solved in development of VDFs are essentially the same as those in digital filters of fixed characteristics. Hence, research topics on VDFs as well as fixed characteristic filters are broadly classified into three categories [2]: the approximation problem, the realization

> © 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited.

© 2018 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.

pass/band-stop filtering that include the well-known adaptive notch filtering.

adaptive notch filtering, adaptive band-pass/band-stop filtering

Recent Advances in Variable Digital Filters

Shunsuke Koshita, Masahide Abe and

Shunsuke Koshita, Masahide Abe and

http://dx.doi.org/10.5772/intechopen.79198

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Masayuki Kawamata

Masayuki Kawamata

Abstract

1. Introduction

widely reviewed in [1].


#### **Chapter 2** Provisional chapter

#### **Recent Advances in Variable Digital Filters** Recent Advances in Variable Digital Filters

DOI: 10.5772/intechopen.79198

Shunsuke Koshita, Masahide Abe and Masayuki Kawamata Shunsuke Koshita, Masahide Abe and Masayuki Kawamata

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.79198

#### Abstract

[20] Huang Z, Xi J, Yu Y, Guo Q. Accurate projector calibration based on a new point-to-point mapping relationship between the camera and projector images. Applied Optics. 2015;

[21] Travis J, Kring J. LabVIEW for Everyone: Graphical Programming Made Easy and Fun.

[23] IPCC. Monumento Artistico "Figura Reclinada de la Gorda Gertrudis" [Online]. 2016.

[24] Zhang S. Recent progresses on real-time 3d shape measurement using digital fringe

[25] Gómez ALG, Fonseca JEM, Téllez JL. Proyección de franjas en metrología óptica facial.

[22] Easton RL Jr. Fourier Methods in Imaging. John Wiley & Sons; 2010. pp. 567

Available from: http://www.ipcc.gov.co/index.php/noticias/item/260-botero

projection techniques. Optics and Lasers in Engineering. 2010;48(2):149-158

54(3):347-356

22 Digital Systems

Prentice-Hall; 2007. pp. 10

INGE CUC. 2012;8(1):191-206

Variable digital filters are widely used in a number of applications of signal processing because of their capability of self-tuning frequency characteristics such as the cutoff frequency and the bandwidth. This chapter introduces recent advances on variable digital filters, focusing on the problems of design and realization, and application to adaptive filtering. In the topic on design and realization, we address two major approaches: one is the frequency transformation and the other is the multi-dimensional polynomial approximation of filter coefficients. In the topic on adaptive filtering, we introduce the details of adaptive bandpass/band-stop filtering that include the well-known adaptive notch filtering.

Keywords: variable digital filter, frequency transformation, polynomial approximation, adaptive notch filtering, adaptive band-pass/band-stop filtering

### 1. Introduction

Digital filter is well known as one of the essential and fundamental components in signal processing devices. In addition, many signal processing applications such as digital audio equipment and telecommunication systems sometimes require simultaneous realization of digital filtering and real-time control of filter characteristics. Such requirements can be fulfilled by means of variable digital filters (VDFs). Research on VDFs emerged in the 1970s and since then, many results have been reported. Among them, details of the results until the 1990s are widely reviewed in [1].

The problems that should be solved in development of VDFs are essentially the same as those in digital filters of fixed characteristics. Hence, research topics on VDFs as well as fixed characteristic filters are broadly classified into three categories [2]: the approximation problem, the realization

© 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited. © 2018 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.

problem, and the implementation problem. Moreover, in the field of VDFs, application-oriented results have also been actively reported. One of the famous applications is adaptive notch filters, which have been studied since the 1980s and the details will be reviewed in this chapter.

In the sequel, fundamentals of VDFs are first reviewed. Then, recent results on VDFs are introduced and discussed with focus on the approximation problem, the realization problem, and the applications. Such topics include some results proposed by the authors of this chapter.

### 2. Fundamentals of VDFs

### 2.1. Definition

VDFs are defined as the frequency selective digital filters (e.g., low-pass filters and band-pass filters) of which frequency characteristics can be changed in real time by means of controlling some parameters. A popular example of such VDFs is shown in Figure 1, which is the variable low-pass filter (VLPF) of which cutoff frequency can be changed by controlling the single parameter η. Another example shown in Figure 2 is the variable band-pass filter (VBPF), where the bandwidth is fixed and the pass-band center frequency can be changed by the single parameter ξ.

It should be noted that VDFs are different from "filters with variable (adjustable) coefficients" which are used in adaptive filtering. Details of the differences are as follows:

• In the case of general adaptive filtering, all filter coefficients are changed by an adaptive algorithm. On the other hand, most of the coefficients of a VDF are fixed or given as some functions of a few variable parameters. For example, in the VLPF of Figure 1, only the single parameter η can be changed, and the other coefficients are fixed or given as functions of η.

2.2. How to obtain VDFs

Figure 3. Procedure to obtain VDF.

Figure 2. Example of VBPF.

in the form of

describe the transfer function in the form of

This subsection reviews the procedure to obtain VDFs. The required procedure is basically the same as that in the case of fixed characteristic filters, where three important problems must be considered as shown in Figure 3: approximation, realization, and implementation [2]. In this chapter, we pay special attention to the approximation problem and the realization problem. The approximation problem is to obtain an input-output characterization such as transfer function from a prescribed specification of a VDF. The realization problem is to determine a structure (i.e., an appropriate set of adders and multipliers or an appropriate list of primitive

In the approximation problem for VDFs, the required task is to describe an input-output relationship (e.g., transfer function) of the VDF in such a manner that the description includes variable parameters. For example, consider the approximation problem for the VLPF shown in Figure 1. If one wishes to obtain this VLPF as an FIR filter, the approximation problem is to

N

hkð Þ <sup>η</sup> <sup>z</sup>�<sup>k</sup> (1)

Recent Advances in Variable Digital Filters http://dx.doi.org/10.5772/intechopen.79198 25

k¼0

and it is also necessary to describe each coefficient hkð Þ η as a function of η. Therefore, the approximation problem for this VLPF is to determine a set of functions f g hkð Þ η ð Þ 0 ≤ k ≤ N . Similarly, if one wishes to obtain IIR-type VLPF, it is necessary to describe the transfer function

H zð Þ¼ ; <sup>η</sup> <sup>X</sup>

operations for filtering) corresponding to the input-output characterization.

• VDFs are different from general adaptive filters with respect to the mechanism of changing the frequency characteristics. In VDFs, the characteristics are changed but the frequency selectivity such as the low-pass and the band-pass shape is preserved. In other words, VDFs control the frequency characteristics under the constraint of preservation of frequency selectivity. On the other hand, general adaptive filters do not require this constraint. This means that such adaptive filters converge to optimal ones of which characteristics do not necessarily possess frequency selectivity.

Figure 1. Example of VLPF.

Figure 3. Procedure to obtain VDF.

problem, and the implementation problem. Moreover, in the field of VDFs, application-oriented results have also been actively reported. One of the famous applications is adaptive notch filters,

In the sequel, fundamentals of VDFs are first reviewed. Then, recent results on VDFs are introduced and discussed with focus on the approximation problem, the realization problem, and the applications. Such topics include some results proposed by the authors of this chapter.

VDFs are defined as the frequency selective digital filters (e.g., low-pass filters and band-pass filters) of which frequency characteristics can be changed in real time by means of controlling some parameters. A popular example of such VDFs is shown in Figure 1, which is the variable low-pass filter (VLPF) of which cutoff frequency can be changed by controlling the single parameter η. Another example shown in Figure 2 is the variable band-pass filter (VBPF), where the bandwidth is fixed and the pass-band center frequency can be changed by the single parameter ξ. It should be noted that VDFs are different from "filters with variable (adjustable) coefficients"

• In the case of general adaptive filtering, all filter coefficients are changed by an adaptive algorithm. On the other hand, most of the coefficients of a VDF are fixed or given as some functions of a few variable parameters. For example, in the VLPF of Figure 1, only the single parameter η can be changed, and the other coefficients are fixed or given as functions of η.

• VDFs are different from general adaptive filters with respect to the mechanism of changing the frequency characteristics. In VDFs, the characteristics are changed but the frequency selectivity such as the low-pass and the band-pass shape is preserved. In other words, VDFs control the frequency characteristics under the constraint of preservation of frequency selectivity. On the other hand, general adaptive filters do not require this constraint. This means that such adaptive filters converge to optimal ones of which character-

which are used in adaptive filtering. Details of the differences are as follows:

istics do not necessarily possess frequency selectivity.

which have been studied since the 1980s and the details will be reviewed in this chapter.

2. Fundamentals of VDFs

2.1. Definition

24 Digital Systems

Figure 1. Example of VLPF.

#### 2.2. How to obtain VDFs

This subsection reviews the procedure to obtain VDFs. The required procedure is basically the same as that in the case of fixed characteristic filters, where three important problems must be considered as shown in Figure 3: approximation, realization, and implementation [2]. In this chapter, we pay special attention to the approximation problem and the realization problem. The approximation problem is to obtain an input-output characterization such as transfer function from a prescribed specification of a VDF. The realization problem is to determine a structure (i.e., an appropriate set of adders and multipliers or an appropriate list of primitive operations for filtering) corresponding to the input-output characterization.

In the approximation problem for VDFs, the required task is to describe an input-output relationship (e.g., transfer function) of the VDF in such a manner that the description includes variable parameters. For example, consider the approximation problem for the VLPF shown in Figure 1. If one wishes to obtain this VLPF as an FIR filter, the approximation problem is to describe the transfer function in the form of

$$H(z,\eta) = \sum\_{k=0}^{N} h\_k(\eta) z^{-k} \tag{1}$$

and it is also necessary to describe each coefficient hkð Þ η as a function of η. Therefore, the approximation problem for this VLPF is to determine a set of functions f g hkð Þ η ð Þ 0 ≤ k ≤ N . Similarly, if one wishes to obtain IIR-type VLPF, it is necessary to describe the transfer function in the form of

$$H(z,\eta) = \frac{\sum\_{k=0}^{M} b\_k(\eta)z^{-k}}{1 + \sum\_{m=1}^{N} a\_m(\eta)z^{-m}}\tag{2}$$

prototype filter is stable and <sup>∣</sup>η<sup>∣</sup> <sup>&</sup>lt; 1 is satisfied. Also, note that <sup>∣</sup>T ej<sup>ω</sup>; <sup>η</sup> <sup>∣</sup> <sup>¼</sup> 1 holds for any <sup>η</sup>

We next discuss the realization problem for this VLPF. From the realization point of view, Eq. (3) means that a block diagram of this VLPF can be obtained by replacing each delay element z�<sup>1</sup> in the prototype filter with the all-pass filter T zð Þ ; η . However, in most cases, such replacement causes delay-free loops and results in H zð Þ ; η with unrealizable block diagram. To explain this problem, consider a second-order IIR prototype filter with the transfer function given by

and the block diagram given by the direct form as in Figure 4(a). Applying the aforementioned replacement of delay elements with T zð Þ ; η yields the VLPF of which the block diagram corresponds to Figure 4(b). It is now clear that Figure 4(b) includes delay-free loops, and hence it is impossible to implement this block diagram. It is well known that delay-free loops can be avoided by means of mathematical manipulations of transfer function or difference equation. However, such manipulations are not good solutions in the case of VDF realization. For example, applying <sup>z</sup>�<sup>1</sup> T zð Þ ; <sup>η</sup> to <sup>H</sup>pð Þ<sup>z</sup> given by Eq. (4) and then performing mathematical manipulations, we obtain the transfer function of the second-order VLPF as follows:

<sup>b</sup><sup>0</sup> <sup>þ</sup> <sup>b</sup>1z�<sup>1</sup> <sup>þ</sup> <sup>b</sup>2z�<sup>2</sup>

<sup>1</sup>ð Þþ η b<sup>0</sup>

<sup>1</sup>ð Þþ η a<sup>0</sup>

<sup>1</sup>ð Þ¼ <sup>η</sup> �2<sup>η</sup> <sup>þ</sup> <sup>a</sup><sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup> � <sup>2</sup>a2<sup>η</sup> 1 � a1η þ a2η<sup>2</sup>

> <sup>b</sup><sup>0</sup> � <sup>b</sup>1<sup>η</sup> <sup>þ</sup> <sup>b</sup>2η<sup>2</sup> 1 � a1η þ a2η<sup>2</sup>

<sup>b</sup>0η<sup>2</sup> � <sup>b</sup>1<sup>η</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>1</sup> � <sup>a</sup>1<sup>η</sup> <sup>þ</sup> <sup>a</sup>2η<sup>2</sup> :

If we implement the VLPF using this description, the computational cost significantly increases

to the change of η. In particular, the filter coefficients in Eq. (5) are rational polynomials that require divisions for recalculation of filter coefficients, causing very high implementation cost. One of the popular methods to overcome this problem is the Taylor approximation-based description [4]. This method applies the first-order Taylor series approximation to all of the rational polynomials of filter coefficients in VDFs, under the assumption that the absolute

<sup>1</sup>ð Þ η and b<sup>0</sup>

<sup>2</sup>ð Þ η must be recalculated according

<sup>0</sup>ð Þ η , b<sup>0</sup>

<sup>1</sup>ð Þ¼ <sup>η</sup> �2b0<sup>η</sup> <sup>þ</sup> <sup>b</sup><sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup> � <sup>2</sup>b2<sup>η</sup> 1 � a1η þ a2η<sup>2</sup>

<sup>2</sup>ð Þ η

<sup>2</sup>ð Þ η

<sup>1</sup> <sup>þ</sup> <sup>a</sup>1z�<sup>1</sup> <sup>þ</sup> <sup>a</sup>2z�<sup>2</sup> (4)

Recent Advances in Variable Digital Filters http://dx.doi.org/10.5772/intechopen.79198 27

(5)

Hpð Þ¼ z

H zð Þ¼ ; η

a0

a0

b0 <sup>0</sup>ð Þ¼ η

b0

b0 <sup>2</sup>ð Þ¼ η

<sup>1</sup>ð Þ η , a<sup>0</sup>

<sup>2</sup>ð Þ η , b<sup>0</sup>

because the filter coefficients a<sup>0</sup>

b0 <sup>0</sup>ð Þþ η b<sup>0</sup>

1 þ a<sup>0</sup>

<sup>2</sup>ð Þ¼ <sup>η</sup> <sup>η</sup><sup>2</sup> � <sup>a</sup>1<sup>η</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> 1 � a1η þ a2η<sup>2</sup>

and ω because T zð Þ ; η is all-pass.

and to determine the filter coefficients as the functions f g amð Þ η ð Þ 1 ≤ m ≤ N and f g bkð Þ η ð Þ 1 ≤ k ≤ M .

### 3. Research topics on VDFs

This section introduces research topics on VDFs from the viewpoints of the approximation problem and the realization problem. Two methods have been widely used for approximation and realization of VDFs: one is based on the variable transformation of transfer functions and the other is based on the multi-dimensional (M-D) polynomial approximation of filter coefficients. In the sequel details of these two methods are reviewed and some recent results on these two methods are introduced.

#### 3.1. VDFs based on variable transformation of transfer functions

In this method, we first need to design the transfer function of "prototype filter," which is usually low pass, and its coefficients are fixed (i.e., variable parameters are not included in this transfer function). Next, we apply a variable transformation to this prototype transfer function and obtain a desired VDF, where the variable transformation makes use of a function which includes variable parameters that are associated with the components to be changed in frequency characteristics. Many approaches exist for variable transformations, and the most famous approach is the frequency transformation [3]. The frequency transformation makes use of all-pass functions for the variable transformation. Although details of the frequency transformation are well reviewed in [1], this chapter will also review this topic with some additional discussions. This is because many results using the frequency transformation have been still reported in recent years and some of such results include the authors' works.

Now, consider again the VLPF shown in Figure 1. If frequency transformation is used to obtain this VLPF, the first step is to prepare the transfer function of a prototype low-pass filter. Such a transfer function is denoted by Hpð Þz . Then, applying the following frequency transformation to Hpð Þz , we can obtain the desired VLPF with the transfer function H zð Þ ; η :

$$\begin{aligned} H(z,\eta) &= H\_{\mathbb{P}}(z)\big|\_{z^{-1}\leftarrow T(z,\eta)}\\ T(z,\eta) &= \frac{z^{-1}-\eta}{1-\eta z^{-1}} \end{aligned} \tag{3}$$

where T zð Þ ; η is the first-order all-pass function. By changing the value of η in H zð Þ ; η , we can control the cutoff frequency of the VLPF. If η > 0, the cutoff frequency becomes lower than that of the prototype filter. The converse holds if η < 0. Stability of this VLPF is guaranteed if the prototype filter is stable and <sup>∣</sup>η<sup>∣</sup> <sup>&</sup>lt; 1 is satisfied. Also, note that <sup>∣</sup>T ej<sup>ω</sup>; <sup>η</sup> <sup>∣</sup> <sup>¼</sup> 1 holds for any <sup>η</sup> and ω because T zð Þ ; η is all-pass.

H zð Þ¼ ; η

3.1. VDFs based on variable transformation of transfer functions

ð Þ 1 ≤ k ≤ M .

26 Digital Systems

3. Research topics on VDFs

these two methods are introduced.

P<sup>M</sup>

<sup>1</sup> <sup>þ</sup> <sup>P</sup><sup>N</sup>

and to determine the filter coefficients as the functions f g amð Þ η ð Þ 1 ≤ m ≤ N and f g bkð Þ η

This section introduces research topics on VDFs from the viewpoints of the approximation problem and the realization problem. Two methods have been widely used for approximation and realization of VDFs: one is based on the variable transformation of transfer functions and the other is based on the multi-dimensional (M-D) polynomial approximation of filter coefficients. In the sequel details of these two methods are reviewed and some recent results on

In this method, we first need to design the transfer function of "prototype filter," which is usually low pass, and its coefficients are fixed (i.e., variable parameters are not included in this transfer function). Next, we apply a variable transformation to this prototype transfer function and obtain a desired VDF, where the variable transformation makes use of a function which includes variable parameters that are associated with the components to be changed in frequency characteristics. Many approaches exist for variable transformations, and the most famous approach is the frequency transformation [3]. The frequency transformation makes use of all-pass functions for the variable transformation. Although details of the frequency transformation are well reviewed in [1], this chapter will also review this topic with some additional discussions. This is because many results using the frequency transformation have

been still reported in recent years and some of such results include the authors' works.

H zð Þ¼ ; η Hpð Þz

to Hpð Þz , we can obtain the desired VLPF with the transfer function H zð Þ ; η :

T zð Þ¼ ; η

Now, consider again the VLPF shown in Figure 1. If frequency transformation is used to obtain this VLPF, the first step is to prepare the transfer function of a prototype low-pass filter. Such a transfer function is denoted by Hpð Þz . Then, applying the following frequency transformation

> � � z�<sup>1</sup> T zð Þ ;η

> > (3)

<sup>z</sup>�<sup>1</sup> � <sup>η</sup> 1 � ηz�<sup>1</sup>

where T zð Þ ; η is the first-order all-pass function. By changing the value of η in H zð Þ ; η , we can control the cutoff frequency of the VLPF. If η > 0, the cutoff frequency becomes lower than that of the prototype filter. The converse holds if η < 0. Stability of this VLPF is guaranteed if the

<sup>k</sup>¼<sup>0</sup> bkð Þ <sup>η</sup> <sup>z</sup>�<sup>k</sup>

<sup>m</sup>¼<sup>1</sup> amð Þ <sup>η</sup> <sup>z</sup>�<sup>m</sup> (2)

We next discuss the realization problem for this VLPF. From the realization point of view, Eq. (3) means that a block diagram of this VLPF can be obtained by replacing each delay element z�<sup>1</sup> in the prototype filter with the all-pass filter T zð Þ ; η . However, in most cases, such replacement causes delay-free loops and results in H zð Þ ; η with unrealizable block diagram. To explain this problem, consider a second-order IIR prototype filter with the transfer function given by

$$H\_{\mathbb{P}}(z) = \frac{b\_0 + b\_1 z^{-1} + b\_2 z^{-2}}{1 + a\_1 z^{-1} + a\_2 z^{-2}}\tag{4}$$

and the block diagram given by the direct form as in Figure 4(a). Applying the aforementioned replacement of delay elements with T zð Þ ; η yields the VLPF of which the block diagram corresponds to Figure 4(b). It is now clear that Figure 4(b) includes delay-free loops, and hence it is impossible to implement this block diagram. It is well known that delay-free loops can be avoided by means of mathematical manipulations of transfer function or difference equation. However, such manipulations are not good solutions in the case of VDF realization. For example, applying <sup>z</sup>�<sup>1</sup> T zð Þ ; <sup>η</sup> to <sup>H</sup>pð Þ<sup>z</sup> given by Eq. (4) and then performing mathematical manipulations, we obtain the transfer function of the second-order VLPF as follows:

$$\begin{aligned} H(z,\eta) &= \frac{b\_0'(\eta) + b\_1'(\eta) + b\_2'(\eta)}{1 + d\_1(\eta) + d\_2'(\eta)} \\ d\_1'(\eta) &= \frac{-2\eta + a\_1(1 + \eta^2) - 2a\_2\eta}{1 - a\_1\eta + a\_2\eta^2} \\ d\_2'(\eta) &= \frac{\eta^2 - a\_1\eta + a\_2}{1 - a\_1\eta + a\_2\eta^2} \\ b\_0'(\eta) &= \frac{b\_0 - b\_1\eta + b\_2\eta^2}{1 - a\_1\eta + a\_2\eta^2} \\ b\_1'(\eta) &= \frac{-2b\_0\eta + b\_1(1 + \eta^2) - 2b\_2\eta}{1 - a\_1\eta + a\_2\eta^2} \\ b\_2'(\eta) &= \frac{b\_0\eta^2 - b\_1\eta + b\_2}{1 - a\_1\eta + a\_2\eta^2} .\end{aligned} (5)$$

If we implement the VLPF using this description, the computational cost significantly increases because the filter coefficients a<sup>0</sup> <sup>1</sup>ð Þ η , a<sup>0</sup> <sup>2</sup>ð Þ η , b<sup>0</sup> <sup>0</sup>ð Þ η , b<sup>0</sup> <sup>1</sup>ð Þ η and b<sup>0</sup> <sup>2</sup>ð Þ η must be recalculated according to the change of η. In particular, the filter coefficients in Eq. (5) are rational polynomials that require divisions for recalculation of filter coefficients, causing very high implementation cost.

One of the popular methods to overcome this problem is the Taylor approximation-based description [4]. This method applies the first-order Taylor series approximation to all of the rational polynomials of filter coefficients in VDFs, under the assumption that the absolute

Figure 4. Problem in realization of VLPF based on the frequency transformation: (a) second-order prototype filter, and (b) VLPF given by applying <sup>z</sup>�<sup>1</sup> T zð Þ ; <sup>η</sup> to the prototype filter.

values of all variable parameters are small. For example, in the case of Eq. (5), it is assumed that ∣η∣ ≪ 1 and the filter coefficients are approximated to

$$\begin{aligned} d\_1(\eta) &\approx a\_1 + (a\_1^2 - 2 - 2a\_2)\eta \\ d\_2(\eta) &\approx a\_2 + (a\_1 a\_2 - a\_1)\eta \\ b\_0'(\eta) &\approx b\_0 + (a\_1 b\_0 - b\_1)\eta \\ b\_1'(\eta) &\approx b\_1 + (a\_1 b\_1 - 2b\_0 - 2b\_2)\eta \\ b\_2'(\eta) &\approx b\_2 + (a\_1 b\_2 - b\_1)\eta . \end{aligned} \tag{6}$$

require recalculation of filter coefficients even if the value of η is changed. This is because all of

Recent Advances in Variable Digital Filters http://dx.doi.org/10.5772/intechopen.79198 29

Figure 5. Second-order VLPF based on the frequency transformation and first-order Taylor series approximation.

Although the VLPFs based on the Taylor approximation provide an effective realization method, they have a serious drawback that the range of variable cutoff frequency is quite limited. This limitation is due to the assumption of ∣η∣ ≪ 1, which means that the approximation error becomes larger as the cutoff frequency of the VLPFs goes far from that of the prototype filter. In addition, the VLPFs may become unstable if the value of ∣η∣ is inappropriately large. In order to overcome these problems, some alternative methods are proposed [4–6]. All of these methods make use of low sensitivity structures for realization of block diagrams for the prototype filter. Then the replacement <sup>z</sup>�<sup>1</sup> T zð Þ ; <sup>η</sup> and the Taylor approximation are applied to such block diagrams, leading to the desired VDFs. Although the methods given by [4–6] can be applied to the limited classes of transfer functions, the Taylor approximation error becomes smaller than the standard VDFs based on the direct form. This approach is also extended to the

There are some other approaches for the reduction of the Taylor approximation error. In [8], the approach based on wave digital filters is presented. Although this approach requires the knowledge of analog filter theory, very high precision is attained in the resultant VDFs, and hence the variable cutoff frequency can be controlled in relatively wide range. In [9], statespace representation is used for construction of the block diagram of the prototype filter, and series approximations are applied to avoid the significant increase of the implementation cost of frequency transformation-based VDFs. This approach does not need any restriction that appeared in the conventional methods, and hence the method of [9] can be applied to arbitrary transfer functions and arbitrary state-space structures. Furthermore, in [10], the VDFs based on the combination of frequency transformation and coefficient decimation are proposed, and it is

the multipliers except for η in this block diagram are realized as fixed coefficients.

2-D VDFs [7].

These new coefficients do not require divisions, and hence the VLPF can be realized in terms of additions and multiplications, as shown in Figure 5. In addition, this realization does not

Figure 5. Second-order VLPF based on the frequency transformation and first-order Taylor series approximation.

require recalculation of filter coefficients even if the value of η is changed. This is because all of the multipliers except for η in this block diagram are realized as fixed coefficients.

Although the VLPFs based on the Taylor approximation provide an effective realization method, they have a serious drawback that the range of variable cutoff frequency is quite limited. This limitation is due to the assumption of ∣η∣ ≪ 1, which means that the approximation error becomes larger as the cutoff frequency of the VLPFs goes far from that of the prototype filter. In addition, the VLPFs may become unstable if the value of ∣η∣ is inappropriately large. In order to overcome these problems, some alternative methods are proposed [4–6]. All of these methods make use of low sensitivity structures for realization of block diagrams for the prototype filter. Then the replacement <sup>z</sup>�<sup>1</sup> T zð Þ ; <sup>η</sup> and the Taylor approximation are applied to such block diagrams, leading to the desired VDFs. Although the methods given by [4–6] can be applied to the limited classes of transfer functions, the Taylor approximation error becomes smaller than the standard VDFs based on the direct form. This approach is also extended to the 2-D VDFs [7].

values of all variable parameters are small. For example, in the case of Eq. (5), it is assumed

Figure 4. Problem in realization of VLPF based on the frequency transformation: (a) second-order prototype filter, and

<sup>1</sup> � 2 � 2a<sup>2</sup> η

(6)

that ∣η∣ ≪ 1 and the filter coefficients are approximated to

(b) VLPF given by applying <sup>z</sup>�<sup>1</sup> T zð Þ ; <sup>η</sup> to the prototype filter.

28 Digital Systems

a0

a0

b0

b0

b0

<sup>1</sup>ð Þ <sup>η</sup> <sup>≈</sup> <sup>a</sup><sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

<sup>2</sup>ð Þ η ≈ a<sup>2</sup> þ ð Þ a1a<sup>2</sup> � a<sup>1</sup> η

<sup>0</sup>ð Þ η ≈ b<sup>0</sup> þ ð Þ a1b<sup>0</sup> � b<sup>1</sup> η

<sup>2</sup>ð Þ η ≈ b<sup>2</sup> þ ð Þ a1b<sup>2</sup> � b<sup>1</sup> η:

<sup>1</sup>ð Þ η ≈ b<sup>1</sup> þ ð Þ a1b<sup>1</sup> � 2b<sup>0</sup> � 2b<sup>2</sup> η

These new coefficients do not require divisions, and hence the VLPF can be realized in terms of additions and multiplications, as shown in Figure 5. In addition, this realization does not There are some other approaches for the reduction of the Taylor approximation error. In [8], the approach based on wave digital filters is presented. Although this approach requires the knowledge of analog filter theory, very high precision is attained in the resultant VDFs, and hence the variable cutoff frequency can be controlled in relatively wide range. In [9], statespace representation is used for construction of the block diagram of the prototype filter, and series approximations are applied to avoid the significant increase of the implementation cost of frequency transformation-based VDFs. This approach does not need any restriction that appeared in the conventional methods, and hence the method of [9] can be applied to arbitrary transfer functions and arbitrary state-space structures. Furthermore, in [10], the VDFs based on the combination of frequency transformation and coefficient decimation are proposed, and it is shown through FPGA implementation and performance evaluation that the proposed method attains very low cost for hardware implementation.

3.2. VDFs based on M-D polynomial approximation of filter coefficients

attained in VDFs.

ψ1;ψ2; ⋯;ψ<sup>K</sup>

cn m<sup>ψ</sup><sup>1</sup>

; m<sup>ψ</sup><sup>2</sup>

To the authors' best knowledge, the VDFs based on the M-D polynomial approximation of filter coefficients have been most actively studied [11–23] in the field of VDFs. One of the significant benefits of this approach over the frequency transformation-based VDFs is that many kinds of variable characteristics as well as variable cutoff frequencies can be attained. For example, this approach can provide VLPFs with variable transition bandwidth and variable stopband attenuation, as shown in Figure 6. In addition, since this approach is applicable to FIR filters as well as IIR filters, linear-phase characteristics and variable group delay can be

The first step to obtain this type of VDFs is to determine a set of K variable parameters

� � which correspond to the desired variable components of frequency characteristics such as cutoff frequency, transition bandwidth, and stopband attenuation. Such variable parameters are referred to as spectral parameters. After this step, filter coefficients of the desired VDFs are described as M-D polynomials with respect to these variable parameters. For example,

N

n¼0

cn m<sup>ψ</sup><sup>1</sup>

hn ψ1;ψ2; ⋯; ψ<sup>K</sup>

; m<sup>ψ</sup><sup>2</sup>

� � is described in terms of the following M-D polynomial:

; ⋯m<sup>ψ</sup><sup>K</sup> � �ψ<sup>m</sup>ψ<sup>1</sup>

� � � � , the standard approach is based on the

� �z�<sup>n</sup> (7)

<sup>1</sup> <sup>ψ</sup><sup>m</sup>ψ<sup>2</sup>

, M<sup>ψ</sup><sup>2</sup>

<sup>2</sup> <sup>⋯</sup>ψ<sup>m</sup>ψ<sup>K</sup>

Recent Advances in Variable Digital Filters http://dx.doi.org/10.5772/intechopen.79198 31

<sup>K</sup> : (8)

, ⋯, M<sup>ψ</sup><sup>K</sup> denote the

the transfer function of an N-th order VDF with K variable parameters is described by

H z;ψ1; ψ2; ⋯;ψ<sup>K</sup> � � <sup>¼</sup> <sup>X</sup>

Mψ<sup>1</sup>

X Mψ<sup>2</sup>

⋯ M X ψK

; m<sup>ψ</sup><sup>2</sup> ; ⋯m<sup>ψ</sup><sup>K</sup>

mψ<sup>K</sup> ¼0

The approximation problem for this kind of VDFs is to determine the set of coefficients

orders of the M-D polynomials that, respectively, correspond to the variables ψ1,ψ2, ⋯,ψK.

mψ<sup>2</sup> ¼0

mψ<sup>1</sup> ¼0

� � � � for 0 <sup>≤</sup> <sup>n</sup> <sup>≤</sup> <sup>N</sup>. Here, it should be noted that <sup>M</sup><sup>ψ</sup><sup>1</sup>

Figure 6. Example of VLPF based on the M-D polynomial approximation of filter coefficients.

and each filter coefficient hn ψ1;ψ2; ⋯;ψ<sup>K</sup>

� � <sup>¼</sup> <sup>X</sup>

hn ψ1; ψ2; ⋯;ψ<sup>K</sup>

; ⋯m<sup>ψ</sup><sup>K</sup>

In order to obtain the set cn m<sup>ψ</sup><sup>1</sup>

As discussed above, the problem of delay-free loops is an important issue in the approximation/realization of frequency transformation-based VDFs. It should be noted that, however, this problem does not always happen. In general, this problem happens if the all-pass function in the frequency transformation has a nonzero constant term in the numerator. This case corresponds to the VDFs with variable bandwidth. In other words, the problem of delay-free loops does not happen when the VDFs have fixed bandwidth, as shown in Figure 2.

We conclude this subsection with a summary of the merits and the drawbacks of the frequency transformation-based VDFs. The merits are as follows:


Next, the drawbacks are summarized as follows:


#### 3.2. VDFs based on M-D polynomial approximation of filter coefficients

shown through FPGA implementation and performance evaluation that the proposed method

As discussed above, the problem of delay-free loops is an important issue in the approximation/realization of frequency transformation-based VDFs. It should be noted that, however, this problem does not always happen. In general, this problem happens if the all-pass function in the frequency transformation has a nonzero constant term in the numerator. This case corresponds to the VDFs with variable bandwidth. In other words, the problem of delay-free

We conclude this subsection with a summary of the merits and the drawbacks of the frequency

• Variable characteristics can be easily obtained because the theory of controlling cutoff

• If Taylor approximation is not carried out, the frequency transformation preserves many useful properties on the shape of magnitude responses. For example, when a prototype low-pass filter is the Butterworth filter that possesses the monotonic and maximally flat magnitude response, the VDFs given by applying frequency transformations to this pro-

totype filter also possess the monotonic and maximally flat magnitude responses.

• The aforementioned merit facilitates the design of adaptive band-pass or band-stop filters because the cost function for adaptive filtering becomes unimodal, leading to an adaptive algorithm that converges to the globally optimal solution. Details will be discussed in the

• Compared with the VDFs based on the M-D polynomial approximation, the frequency transformation-based VDFs require much less computational cost in the filtering.

• As stated earlier, if the bandwidth needs to be variable in VDFs, the frequency transformation causes delay-free loops and this problem must be appropriately solved.

• If one wishes to obtain VDFs with multiple passbands or stopbands such as VBPFs, VBSFs, and variable multi-band filters, it is necessary to use high-order all-pass functions for the frequency transformation. As a result, the order of VDFs becomes higher than that of the prototype filter. For example, the order of the frequency transformation-based VBPFs and VBSFs becomes doubled as compared with the order of the prototype filter. • Linear-phase VDFs cannot be obtained because the all-pass functions to be used in the frequency transformation are IIR filters. Even if a prototype filter is FIR, applying the

• Realization of variable characteristics is quite limited. To be specific, the frequency transformation can provide only the VDFs with variable cutoff frequencies. In other words, other components such as the transition bandwidth and the stopband attenuation cannot

loops does not happen when the VDFs have fixed bandwidth, as shown in Figure 2.

attains very low cost for hardware implementation.

transformation-based VDFs. The merits are as follows:

Next, the drawbacks are summarized as follows:

next section.

30 Digital Systems

be controlled.

frequency is based on the simple variable transformations.

frequency transformations simply results in IIR-type VDFs.

To the authors' best knowledge, the VDFs based on the M-D polynomial approximation of filter coefficients have been most actively studied [11–23] in the field of VDFs. One of the significant benefits of this approach over the frequency transformation-based VDFs is that many kinds of variable characteristics as well as variable cutoff frequencies can be attained. For example, this approach can provide VLPFs with variable transition bandwidth and variable stopband attenuation, as shown in Figure 6. In addition, since this approach is applicable to FIR filters as well as IIR filters, linear-phase characteristics and variable group delay can be attained in VDFs.

The first step to obtain this type of VDFs is to determine a set of K variable parameters ψ1;ψ2; ⋯;ψ<sup>K</sup> � � which correspond to the desired variable components of frequency characteristics such as cutoff frequency, transition bandwidth, and stopband attenuation. Such variable parameters are referred to as spectral parameters. After this step, filter coefficients of the desired VDFs are described as M-D polynomials with respect to these variable parameters. For example, the transfer function of an N-th order VDF with K variable parameters is described by

$$H(z, \psi\_1, \psi\_2, \dots, \psi\_K) = \sum\_{n=0}^{N} h\_n(\psi\_1, \psi\_2, \dots, \psi\_K) z^{-n} \tag{7}$$

and each filter coefficient hn ψ1;ψ2; ⋯;ψ<sup>K</sup> � � is described in terms of the following M-D polynomial:

$$h\_n(\boldsymbol{\psi}\_1, \boldsymbol{\psi}\_2, \dots, \boldsymbol{\psi}\_K) = \sum\_{m\_{\boldsymbol{\psi}\_1}=0}^{M\_{\boldsymbol{\psi}\_1}} \sum\_{m\_{\boldsymbol{\psi}\_2}=0}^{M\_{\boldsymbol{\psi}\_2}} \cdots \sum\_{m\_{\boldsymbol{\psi}\_K}=0}^{M\_{\boldsymbol{\psi}\_K}} c\_n(m\_{\boldsymbol{\psi}\_1}, m\_{\boldsymbol{\psi}\_2}, \dots, m\_{\boldsymbol{\psi}\_K}) \boldsymbol{\psi}\_1^{m\_{\boldsymbol{\psi}\_1}} \boldsymbol{\psi}\_2^{m\_{\boldsymbol{\psi}\_2}} \cdots \boldsymbol{\psi}\_K^{m\_{\boldsymbol{\psi}\_K}}.\tag{8}$$

The approximation problem for this kind of VDFs is to determine the set of coefficients cn m<sup>ψ</sup><sup>1</sup> ; m<sup>ψ</sup><sup>2</sup> ; ⋯m<sup>ψ</sup><sup>K</sup> � � � � for 0 <sup>≤</sup> <sup>n</sup> <sup>≤</sup> <sup>N</sup>. Here, it should be noted that <sup>M</sup><sup>ψ</sup><sup>1</sup> , M<sup>ψ</sup><sup>2</sup> , ⋯, M<sup>ψ</sup><sup>K</sup> denote the orders of the M-D polynomials that, respectively, correspond to the variables ψ1,ψ2, ⋯,ψK. In order to obtain the set cn m<sup>ψ</sup><sup>1</sup> ; m<sup>ψ</sup><sup>2</sup> ; ⋯m<sup>ψ</sup><sup>K</sup> � � � � , the standard approach is based on the

Figure 6. Example of VLPF based on the M-D polynomial approximation of filter coefficients.

minimization of an error function with respect to approximation of a prescribed ideal characteristic of the desired VDF and a curve fitting method to describe the desired M-D polynomials.

In realization of the VDFs given as above, Farrow structure [24] is widely used. To explain this, consider a simple VDF with a single variable parameter ψ1. The transfer function of this VDF is given by

$$\begin{split} H(\boldsymbol{z}, \boldsymbol{\psi}\_{1}) &= \sum\_{n=0}^{N} h\_{n}(\boldsymbol{\psi}\_{1}) \boldsymbol{z}^{-n} \\ &= \sum\_{n=0}^{N} \sum\_{m\_{\psi\_{1}}=0}^{M\_{\psi\_{1}}} c\_{n}(m\_{\psi\_{1}}) \boldsymbol{\psi}\_{1}^{m\_{\psi\_{1}}} \boldsymbol{z}^{-n} \end{split} \tag{9}$$

filters do not include ψ1, recalculation of their coefficients according to the change of ψ<sup>1</sup> is not required. In this sense, the Farrow structure is suitable for the implementation of M-D polyno-

Recent Advances in Variable Digital Filters http://dx.doi.org/10.5772/intechopen.79198 33

A drawback of the M-D polynomial approximation-based VDFs is the high computational cost in the filtering because the filter coefficients are described by M-D polynomials. In addition, this approach limits the range of variable characteristics. As in the case of frequency transformation with Taylor approximation, this limitation comes from the M-D polynomial approximation. Furthermore, since this approach requires a number of filters with fixed coefficients, their hardware implementation may cause an increase of characteristic degradations that comes from finite wordlength effects such as coefficient sensitivity and roundoff noise. However, such degradations can be suppressed by using high accuracy filter structures, and this

In addition to the aforementioned two approaches, many other methods have also been presented in the literature. In [25], VDFs with variable bandwidth without delay-free loops can be achieved at low cost by means of cascade connection of a single subfilter. In [26–28], by applying the frequency response masking and the fast filterbank to design of VDFs, significant

Also, VDFs for adaptive filtering have been widely studied. One of the famous methods in such VDFs is the variable notch filters with second-order IIR transfer functions. All of these variable notch filters successfully provide the variable characteristics by simple mechanism without delay-free loops or increase of computational cost. Other adaptive-filter-oriented VDFs include notch filters with variable attenuation at the notch frequency, comb filters with variable bandwidth, and variable attenuation. Details of these topics will be addressed in the

In this section, we first pay attention to adaptive notch filters (ANFs) that are the special case of adaptive band-stop or band-pass filters. The ANFs are the most famous application of VDFs to adaptive signal processing, and many results on the ANFs have been reported since the 1980s. In addition to the ANFs, this section also introduces some other types of VDFs that are applied

As shown in Figure 8, an ANF plays a central role in automatic detection and suppression of an unknown sinusoid immersed in a wide-band signal such as white noise. In order to detect and suppress the sinusoid, the ANF is controlled by an adaptive algorithm in such a manner that the notch frequency ω<sup>0</sup> of the ANF converges to the unknown frequency ω<sup>s</sup> of the

reduction of implementation cost over the VDFs with the Farrow structure is attained.

mial approximation-based VDFs.

3.3. VDFs based on other approaches

next section.

to adaptive filtering.

4.1. ANF based on all-pass filter

approach has been recently proposed by the authors [23].

4. Research topics on VDFs for adaptive filtering

which can be rewritten as

$$H(z, \psi\_1) = \sum\_{m\_{\psi\_1} = 0}^{M\_{\psi\_1}} \left( \sum\_{n=0}^{N} c\_n(m\_{\psi\_1}) z^{-n} \right) \psi\_1^{m\_{\psi\_1}}.\tag{10}$$

Now, by using the following definition

$$H\_{m\_{\psi\_1}}(z) = \sum\_{n=0}^{N} c\_n \left( m\_{\psi\_1} \right) z^{-n}, \quad 0 \le m\_{\psi\_1} \le M\_{\psi\_{1'}} \tag{11}$$

the description of the VDF H z;ψ<sup>1</sup> � � becomes

$$H(z, \psi\_1) = \sum\_{m\_{\psi\_1} = 0}^{M\_{\psi\_1}} H\_{m\_{\psi\_1}}(z) \psi\_1^{m\_{\psi\_1}}.\tag{12}$$

Using this description, we can realize H z; ψ<sup>1</sup> � � by means of the Farrow structure as shown in Figure 7. The block diagram of Figure 7 is interpreted as the parallel combination of the set of N-th order FIR filters with fixed coefficients and the weights ψ1. Since these N-th order FIR

Figure 7. Realization of M-D polynomial approximation-based VDF based on the Farrow structure.

filters do not include ψ1, recalculation of their coefficients according to the change of ψ<sup>1</sup> is not required. In this sense, the Farrow structure is suitable for the implementation of M-D polynomial approximation-based VDFs.

A drawback of the M-D polynomial approximation-based VDFs is the high computational cost in the filtering because the filter coefficients are described by M-D polynomials. In addition, this approach limits the range of variable characteristics. As in the case of frequency transformation with Taylor approximation, this limitation comes from the M-D polynomial approximation. Furthermore, since this approach requires a number of filters with fixed coefficients, their hardware implementation may cause an increase of characteristic degradations that comes from finite wordlength effects such as coefficient sensitivity and roundoff noise. However, such degradations can be suppressed by using high accuracy filter structures, and this approach has been recently proposed by the authors [23].

#### 3.3. VDFs based on other approaches

minimization of an error function with respect to approximation of a prescribed ideal characteristic of the desired VDF and a curve fitting method to describe the desired M-D

In realization of the VDFs given as above, Farrow structure [24] is widely used. To explain this, consider a simple VDF with a single variable parameter ψ1. The transfer function of this VDF is

> hn ψ<sup>1</sup> � �z�<sup>n</sup>

X Mψ<sup>1</sup>

cn m<sup>ψ</sup><sup>1</sup> � �ψ<sup>m</sup>ψ<sup>1</sup>

cn m<sup>ψ</sup><sup>1</sup> � �z�<sup>n</sup> !

� �z�n, <sup>0</sup> <sup>≤</sup> <sup>m</sup><sup>ψ</sup><sup>1</sup> <sup>≤</sup> <sup>M</sup><sup>ψ</sup><sup>1</sup>

ð Þ<sup>z</sup> <sup>ψ</sup><sup>m</sup>ψ<sup>1</sup>

<sup>1</sup> z�<sup>n</sup>

ψ<sup>m</sup>ψ<sup>1</sup>

<sup>1</sup> : (10)

, (11)

<sup>1</sup> : (12)

� � by means of the Farrow structure as shown in

(9)

mψ<sup>1</sup> ¼0

X N

n¼0

N

n¼0

n¼0

<sup>¼</sup> <sup>X</sup> N

Mψ<sup>1</sup>

mψ<sup>1</sup> ¼0

cn m<sup>ψ</sup><sup>1</sup>

Mψ<sup>1</sup>

mψ<sup>1</sup> ¼0

Figure 7. The block diagram of Figure 7 is interpreted as the parallel combination of the set of N-th order FIR filters with fixed coefficients and the weights ψ1. Since these N-th order FIR

Hm<sup>ψ</sup><sup>1</sup>

H z;ψ<sup>1</sup> � � <sup>¼</sup> <sup>X</sup>

H z; ψ<sup>1</sup>

Hm<sup>ψ</sup><sup>1</sup>

� � <sup>¼</sup> <sup>X</sup>

ð Þ¼ <sup>z</sup> <sup>X</sup> N

n¼0

� � becomes

H z;ψ<sup>1</sup>

� � <sup>¼</sup> <sup>X</sup>

Figure 7. Realization of M-D polynomial approximation-based VDF based on the Farrow structure.

polynomials.

32 Digital Systems

given by

which can be rewritten as

Now, by using the following definition

the description of the VDF H z;ψ<sup>1</sup>

Using this description, we can realize H z; ψ<sup>1</sup>

In addition to the aforementioned two approaches, many other methods have also been presented in the literature. In [25], VDFs with variable bandwidth without delay-free loops can be achieved at low cost by means of cascade connection of a single subfilter. In [26–28], by applying the frequency response masking and the fast filterbank to design of VDFs, significant reduction of implementation cost over the VDFs with the Farrow structure is attained.

Also, VDFs for adaptive filtering have been widely studied. One of the famous methods in such VDFs is the variable notch filters with second-order IIR transfer functions. All of these variable notch filters successfully provide the variable characteristics by simple mechanism without delay-free loops or increase of computational cost. Other adaptive-filter-oriented VDFs include notch filters with variable attenuation at the notch frequency, comb filters with variable bandwidth, and variable attenuation. Details of these topics will be addressed in the next section.

### 4. Research topics on VDFs for adaptive filtering

In this section, we first pay attention to adaptive notch filters (ANFs) that are the special case of adaptive band-stop or band-pass filters. The ANFs are the most famous application of VDFs to adaptive signal processing, and many results on the ANFs have been reported since the 1980s. In addition to the ANFs, this section also introduces some other types of VDFs that are applied to adaptive filtering.

#### 4.1. ANF based on all-pass filter

As shown in Figure 8, an ANF plays a central role in automatic detection and suppression of an unknown sinusoid immersed in a wide-band signal such as white noise. In order to detect and suppress the sinusoid, the ANF is controlled by an adaptive algorithm in such a manner that the notch frequency ω<sup>0</sup> of the ANF converges to the unknown frequency ω<sup>s</sup> of the sinusoid. Hence, the ANF can be considered as the VDF with variable notch frequency, and the value of ω<sup>0</sup> at the steady state becomes the estimate of the frequency ω<sup>s</sup> of the sinusoid. Therefore, ANFs are used not only for the detection/suppression of a sinusoid, but also for the frequency estimation.

Although the ANF shown in Figure 8 is intended to suppress a sinusoid, the ANF is also capable of enhancement of the sinusoid and suppression of the white noise. This can be achieved by using a peaking filter, which is also called a resonator or an inverse notch filter, as an adaptive filter instead of using a notch filter. Alternatively, the notch filter can also be used: in this case, the sinusoid can be enhanced by subtracting the output of the notch filter from the input signal.<sup>1</sup> Such systems together with the ones shown in Figure 8 are widely used in many practical applications such as radar, sonar, telecommunication system with the suppression of narrowband interference and howling suppressor in speech processing.

In the sequel, we explain the fundamentals of ANFs, that is, their problem statement and the mechanism of control of the notch frequency. As shown in Figure 8, the problem statement of ANFs usually describes the input signal as the sum of a sinusoid and a white noise. Hence, the input signal, denoted by u nð Þ, is given by

$$u(n) = A \sin \left(\omega\_s n + \phi\right) + w(n) \tag{13}$$

There are some methods to describe the transfer function of the notch filter for adaptive filtering. In this chapter we focus on the one based on the second-order all-pass filter [29]. This

2

T zð Þ¼ ; <sup>η</sup>; <sup>ξ</sup> <sup>η</sup> � ð Þ <sup>1</sup> <sup>þ</sup> <sup>η</sup> <sup>ξ</sup>z�<sup>1</sup> <sup>þ</sup> <sup>z</sup>�<sup>2</sup>

In this notch filter, the parameter η determines the 3-dB notch width, and the parameter ξ determines the notch frequency ω0. This means that the notch filter given in this way can control the notch width and the notch frequency independently. Also, it is interesting to note that this notch filter can be interpreted as a VDF given by the frequency transformation [30]: it is clear that this notch filter is obtained by applying the frequency transformation

1

To be more precise, this notch filter has the same transfer function as that of the second-order Butterworth band-stop filter [31]. Therefore this notch filter has unity gain at ω ¼ 0 and ω ¼ π, and zero gain at ω0. In addition, the magnitude response of this notch filter is monotonically

Figure 9 shows the block diagram of ANF based on this notch filter. As stated earlier, when the ANF attains steady state, the component of the sinusoid in the input u nð Þ is suppressed at

<sup>1</sup> � <sup>2</sup>ξz�<sup>1</sup> <sup>þ</sup> <sup>z</sup>�<sup>2</sup>

ð Þ 1 þ T zð Þ ; η; ξ (14)

Recent Advances in Variable Digital Filters http://dx.doi.org/10.5772/intechopen.79198 35

<sup>1</sup> � ð Þ <sup>1</sup> <sup>þ</sup> <sup>η</sup> <sup>ξ</sup>z�<sup>1</sup> <sup>þ</sup> <sup>η</sup>z�<sup>2</sup> : (15)

<sup>1</sup> � ð Þ <sup>1</sup> <sup>þ</sup> <sup>η</sup> <sup>ξ</sup>z�<sup>1</sup> <sup>þ</sup> <sup>η</sup>z�<sup>2</sup> : (16)

<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>z</sup>�<sup>1</sup> : (17)

H zð Þ¼ ; <sup>η</sup>; <sup>ξ</sup> <sup>1</sup>

notch filter is described by the following transfer function

where T zð Þ ; η; ξ is the second-order all-pass filter of the form

<sup>z</sup>�<sup>1</sup> T zð Þ ; <sup>η</sup>; <sup>ξ</sup> to the prototype filter of the form

Figure 9. ANF based on the second-order all-pass filter.

H zð Þ¼ ; <sup>η</sup>; <sup>ξ</sup> <sup>1</sup> <sup>þ</sup> <sup>η</sup>

2

Hpð Þ¼ z

decreasing in 0 < ω < ω<sup>0</sup> and monotonically increasing in ω<sup>0</sup> < ω < π.

Hence Eq. (14) is described as

where A and ω<sup>s</sup> are, respectively, the amplitude and frequency of the unknown sinusoid, and ϕ is the random initial phase uniformly distributed in 0½ Þ ; 2π . The signal w nð Þ is a zero-mean white noise, and it is uncorrerated to ϕ. Based on this setup, let y nð Þ be the output signal of the ANF.

Figure 8. Detection and suppression of sinusoid using ANF.

<sup>1</sup> Note that this approach depends on the characteristic of a notch filter, and hence the use of an inappropriate notch filter may result in failure of enhancement of a sinusoid. The reason of this lies in the fact that the signal which is obtained by subtracting the output of a band-stop filter from the input is not necessarily equivalent to the output of a band-pass filter. However, in the case of the ANF based on a second-order all-pass filter, the frequency characteristic of the notch filter satisfies complementary properties that allow us to successfully obtain a signal equivalent to the band-pass-filtered signal by subtracting the notch-filtered signal from the input.

There are some methods to describe the transfer function of the notch filter for adaptive filtering. In this chapter we focus on the one based on the second-order all-pass filter [29]. This notch filter is described by the following transfer function

$$H(z, \eta, \xi) = \frac{1}{2} (1 + T(z, \eta, \xi)) \tag{14}$$

where T zð Þ ; η; ξ is the second-order all-pass filter of the form

$$T(z,\eta,\xi) = \frac{\eta - (1+\eta)\xi z^{-1} + z^{-2}}{1 - (1+\eta)\xi z^{-1} + \eta z^{-2}}.\tag{15}$$

Hence Eq. (14) is described as

sinusoid. Hence, the ANF can be considered as the VDF with variable notch frequency, and the value of ω<sup>0</sup> at the steady state becomes the estimate of the frequency ω<sup>s</sup> of the sinusoid. Therefore, ANFs are used not only for the detection/suppression of a sinusoid, but also for

Although the ANF shown in Figure 8 is intended to suppress a sinusoid, the ANF is also capable of enhancement of the sinusoid and suppression of the white noise. This can be achieved by using a peaking filter, which is also called a resonator or an inverse notch filter, as an adaptive filter instead of using a notch filter. Alternatively, the notch filter can also be used: in this case, the sinusoid can be enhanced by subtracting the output of the notch filter from the input signal.<sup>1</sup> Such systems together with the ones shown in Figure 8 are widely used in many practical applications such as radar, sonar, telecommunication system with the sup-

In the sequel, we explain the fundamentals of ANFs, that is, their problem statement and the mechanism of control of the notch frequency. As shown in Figure 8, the problem statement of ANFs usually describes the input signal as the sum of a sinusoid and a white noise. Hence, the

where A and ω<sup>s</sup> are, respectively, the amplitude and frequency of the unknown sinusoid, and ϕ is the random initial phase uniformly distributed in 0½ Þ ; 2π . The signal w nð Þ is a zero-mean white noise, and it is uncorrerated to ϕ. Based on this setup, let y nð Þ be the output signal of the ANF.

Note that this approach depends on the characteristic of a notch filter, and hence the use of an inappropriate notch filter may result in failure of enhancement of a sinusoid. The reason of this lies in the fact that the signal which is obtained by subtracting the output of a band-stop filter from the input is not necessarily equivalent to the output of a band-pass filter. However, in the case of the ANF based on a second-order all-pass filter, the frequency characteristic of the notch filter satisfies complementary properties that allow us to successfully obtain a signal equivalent to the band-pass-filtered signal

u nð Þ¼ <sup>A</sup> sin <sup>ω</sup>sn <sup>þ</sup> <sup>ϕ</sup> <sup>þ</sup> w nð Þ (13)

pression of narrowband interference and howling suppressor in speech processing.

the frequency estimation.

34 Digital Systems

input signal, denoted by u nð Þ, is given by

Figure 8. Detection and suppression of sinusoid using ANF.

by subtracting the notch-filtered signal from the input.

1

$$H(z,\eta,\xi) = \frac{1+\eta}{2} \frac{1-2\xi z^{-1}+z^{-2}}{1-(1+\eta)\xi z^{-1}+\eta z^{-2}}.\tag{16}$$

In this notch filter, the parameter η determines the 3-dB notch width, and the parameter ξ determines the notch frequency ω0. This means that the notch filter given in this way can control the notch width and the notch frequency independently. Also, it is interesting to note that this notch filter can be interpreted as a VDF given by the frequency transformation [30]: it is clear that this notch filter is obtained by applying the frequency transformation <sup>z</sup>�<sup>1</sup> T zð Þ ; <sup>η</sup>; <sup>ξ</sup> to the prototype filter of the form

$$H\_{\mathbb{P}}(z) = \frac{1}{2} \left( \mathbb{1} + z^{-1} \right). \tag{17}$$

To be more precise, this notch filter has the same transfer function as that of the second-order Butterworth band-stop filter [31]. Therefore this notch filter has unity gain at ω ¼ 0 and ω ¼ π, and zero gain at ω0. In addition, the magnitude response of this notch filter is monotonically decreasing in 0 < ω < ω<sup>0</sup> and monotonically increasing in ω<sup>0</sup> < ω < π.

Figure 9 shows the block diagram of ANF based on this notch filter. As stated earlier, when the ANF attains steady state, the component of the sinusoid in the input u nð Þ is suppressed at

Figure 9. ANF based on the second-order all-pass filter.

the output signal y nð Þ. Here, it should be noted that many adaptive algorithms assume that the notch width is fixed, and that only the notch frequency ω<sup>0</sup> is controlled to estimate the frequency of the sinusoid. For this reason, we focus on how to control ω0.

4.2. ANFs based on other approaches

notch filter [44] is very well known:

Other types of ANFs have also been well studied. For example, the following second-order

where a and r correspond to the parameters that, respectively, control the notch frequency and the notch width. Hence, in this case, the parameter a is controlled by an adaptive algorithm to estimate ωs. This notch filter is designed by the famous method called the constrained poles

Since the transfer function of this notch filter is different from that of the all-pass-based notch filter, the properties of these notch filters are also somewhat different. For example, the allpass-based notch filter has the unity peak gain, whereas the peak gain of the CPZ-based notch filter depends on the notch width. This also makes the difference with respect to the value of E y<sup>2</sup>ð Þ <sup>n</sup> , see [52] for the details. Another difference between these two notch filters is that the all-pass-based ANFs provide unbiased frequency estimation, whereas the CPZ-based ANFs do not. Although this fact shows a drawback of the CPZ-based ANFs, many adaptive algorithms

In addition to the CPZ-based notch filters, there exist many other types of notch filters. In [53, 54], the specific second-order transfer function is constructed in such a manner that it corresponds to a lattice structure. In [55–58], the bilinear transformation to a second-order analog filter is applied to the notch filter design. In [59], the frequency transformation is used

All of the adaptive filters that were addressed in previous subsections are based on secondorder VDFs. On the other hand, there exist some results on high-order VDFs in adaptive signal processing. Needless to say, second-order ANFs have a drawback that it is difficult to realize sharp cutoff characteristics, causing insufficient frequency selectivity and relatively poor signal-to-noise ratio (SNR) at the output signal. On the other hand, in [31], the authors improve the output SNR by means of higher-order VBPFs or VBSFs instead of using second-order notch filters in the adaptive filtering. As shown in Figure 10, high-order filters can realize sharper

Compared with ANFs, little has been studied on the adaptive filtering based on the highorder VDFs. To the authors' best knowledge, the most significant work is found in [60–63], where fourth-order Butterworth VBPF and VBSF are applied to adaptive filtering, and their center frequencies are controlled by adaptive algorithms. Furthermore, the convergence characteristics are also theoretically analyzed. In this work, it is also claimed that the use of much higher-order VDFs for adaptive filtering is almost impossible because higher-order transfer functions involve mathematically more complicated descriptions, and hence it is conjectured that formulations of filter coefficients with variable characteristics and adaptive

to design a notch filter, but the prototype filter used here is different from Eq. (17).

cutoff characteristics than second-order filters and provide higher output SNR.

<sup>1</sup> <sup>þ</sup> az�<sup>1</sup> <sup>þ</sup> <sup>z</sup>�<sup>2</sup>

<sup>1</sup> <sup>þ</sup> arz�<sup>1</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup>z�<sup>2</sup> (18)

Recent Advances in Variable Digital Filters http://dx.doi.org/10.5772/intechopen.79198 37

H zð Þ¼ ;r; a

and zeros (CPZ), and this notch filter has been most widely used for ANFs [44–51].

to reduce the bias have been proposed for the CPZ-based ANFs.

4.3. Adaptive filtering based on high-order VBPFs/VBSFs

The most standard method to control ω<sup>0</sup> is based on the minimization of a cost function by means of the gradient descent method. Although this is similar to general adaptive filters, ANFs differ from the general adaptive filters in that the cost function to be used in ANFs is the mean square output, that is, E y<sup>2</sup>ð Þ <sup>n</sup> . In other words, ANFs do not usually deal with the error signal between a reference signal and the filter output.<sup>2</sup> Since ANFs control ω0, the cost function E y<sup>2</sup>ð Þ <sup>n</sup> must be formulated as a function of <sup>ω</sup>0. This can be successfully achieved and, in addition, E y<sup>2</sup>ð Þ <sup>n</sup> becomes unimodal if the input signal is given as in Eq. (13) and the ANF has monotonic magnitude response. Therefore, in such a case, the optimal notch frequency that minimizes E y<sup>2</sup>ð Þ <sup>n</sup> can be successfully found by the gradient descent method. In fact, the optimal value of ω<sup>0</sup> coincides with ω<sup>s</sup> if the all-pass-based ANF is used [32–34]. Hence, using the gradient of E y<sup>2</sup>ð Þ <sup>n</sup> with respect to <sup>ω</sup><sup>0</sup> in an adaptive algorithm allows <sup>ω</sup><sup>0</sup> to converge to ωs, leading to detection/suppression of the sinusoid.

Remark 1 If the transfer function of the ANF is not based on the all-pass function, the optimal value of ω<sup>0</sup> may slightly deviate from ωs. In other words, the frequency estimation is biased. This topic will be addressed in the next subsection.

However, the gradient descent method has a serious drawback that the convergence speed becomes very slow when the initial value of ω<sup>0</sup> is distant from ωs. To overcome this problem, many strategies have been proposed. In [32–34], the normalized lattice structure is applied to construct the notch filter, and the adaptive algorithm makes use of the state variable of the normalized lattice structure instead of the information of the gradient. This approach is called the Simplified Lattice Algorithm (SLA) and successfully accelerates the convergence speed at low computational cost. Furthermore, in [35], the authors have extended the SLA and proposed a new algorithm called the Affine Combination Lattice Algorithm (ACLA), and it has been proved that the ACLA achieves faster convergence than the SLA. Other approaches to improve the convergence speed include the methods based on the least square algorithm with forgetting factor [36], parallel combination of multiple notch filters with different notch width [37, 38], and construction of additional monotonically increasing function for the gradient [39, 40].

There are many other important research topics on the ANFs. One of them is the theoretical analysis of the behavior of ANFs at steady state. In [41], a steady-state analysis is presented for ANFs based on the one-multiplier lattice structure. This analysis enables us to evaluate the performance of ANFs such as the accuracy of frequency estimation. Also, in [42], the authors propose a unified method on the steady-state analysis of frequency estimation MSE (mean square error) for the SLA and the ACLA. As another research topic, in [43] fundamental frequency estimation using inverse notch filter is proposed.

<sup>2</sup> Although some literature refers to y nð Þ as the error signal, in the authors' opinion this terminology is incorrect. This is because the error in ANFs should be defined as the difference between the frequency of the sinusoid and its estimate, i.e. ω<sup>s</sup> � ω0. This quantity clearly differs from y nð Þ.

#### 4.2. ANFs based on other approaches

the output signal y nð Þ. Here, it should be noted that many adaptive algorithms assume that the notch width is fixed, and that only the notch frequency ω<sup>0</sup> is controlled to estimate the

The most standard method to control ω<sup>0</sup> is based on the minimization of a cost function by means of the gradient descent method. Although this is similar to general adaptive filters, ANFs differ from the general adaptive filters in that the cost function to be used in ANFs is the mean square output, that is, E y<sup>2</sup>ð Þ <sup>n</sup> . In other words, ANFs do not usually deal with the error signal between a reference signal and the filter output.<sup>2</sup> Since ANFs control ω0, the cost function E y<sup>2</sup>ð Þ <sup>n</sup> must be formulated as a function of <sup>ω</sup>0. This can be successfully achieved and, in addition, E y<sup>2</sup>ð Þ <sup>n</sup> becomes unimodal if the input signal is given as in Eq. (13) and the ANF has monotonic magnitude response. Therefore, in such a case, the optimal notch frequency that minimizes E y<sup>2</sup>ð Þ <sup>n</sup> can be successfully found by the gradient descent method. In fact, the optimal value of ω<sup>0</sup> coincides with ω<sup>s</sup> if the all-pass-based ANF is used [32–34]. Hence, using the gradient of E y<sup>2</sup>ð Þ <sup>n</sup> with respect to <sup>ω</sup><sup>0</sup> in an adaptive algorithm allows <sup>ω</sup><sup>0</sup> to

Remark 1 If the transfer function of the ANF is not based on the all-pass function, the optimal value of ω<sup>0</sup> may slightly deviate from ωs. In other words, the frequency estimation is biased. This topic will be

However, the gradient descent method has a serious drawback that the convergence speed becomes very slow when the initial value of ω<sup>0</sup> is distant from ωs. To overcome this problem, many strategies have been proposed. In [32–34], the normalized lattice structure is applied to construct the notch filter, and the adaptive algorithm makes use of the state variable of the normalized lattice structure instead of the information of the gradient. This approach is called the Simplified Lattice Algorithm (SLA) and successfully accelerates the convergence speed at low computational cost. Furthermore, in [35], the authors have extended the SLA and proposed a new algorithm called the Affine Combination Lattice Algorithm (ACLA), and it has been proved that the ACLA achieves faster convergence than the SLA. Other approaches to improve the convergence speed include the methods based on the least square algorithm with forgetting factor [36], parallel combination of multiple notch filters with different notch width [37, 38], and construction of additional monotonically increasing

There are many other important research topics on the ANFs. One of them is the theoretical analysis of the behavior of ANFs at steady state. In [41], a steady-state analysis is presented for ANFs based on the one-multiplier lattice structure. This analysis enables us to evaluate the performance of ANFs such as the accuracy of frequency estimation. Also, in [42], the authors propose a unified method on the steady-state analysis of frequency estimation MSE (mean square error) for the SLA and the ACLA. As another research topic, in [43] fundamental

Although some literature refers to y nð Þ as the error signal, in the authors' opinion this terminology is incorrect. This is because the error in ANFs should be defined as the difference between the frequency of the sinusoid and its estimate, i.e.

frequency of the sinusoid. For this reason, we focus on how to control ω0.

converge to ωs, leading to detection/suppression of the sinusoid.

addressed in the next subsection.

36 Digital Systems

function for the gradient [39, 40].

ω<sup>s</sup> � ω0. This quantity clearly differs from y nð Þ.

2

frequency estimation using inverse notch filter is proposed.

Other types of ANFs have also been well studied. For example, the following second-order notch filter [44] is very well known:

$$H(z,r,a) = \frac{1+az^{-1}+z^{-2}}{1+arz^{-1}+r^2z^{-2}}\tag{18}$$

where a and r correspond to the parameters that, respectively, control the notch frequency and the notch width. Hence, in this case, the parameter a is controlled by an adaptive algorithm to estimate ωs. This notch filter is designed by the famous method called the constrained poles and zeros (CPZ), and this notch filter has been most widely used for ANFs [44–51].

Since the transfer function of this notch filter is different from that of the all-pass-based notch filter, the properties of these notch filters are also somewhat different. For example, the allpass-based notch filter has the unity peak gain, whereas the peak gain of the CPZ-based notch filter depends on the notch width. This also makes the difference with respect to the value of E y<sup>2</sup>ð Þ <sup>n</sup> , see [52] for the details. Another difference between these two notch filters is that the all-pass-based ANFs provide unbiased frequency estimation, whereas the CPZ-based ANFs do not. Although this fact shows a drawback of the CPZ-based ANFs, many adaptive algorithms to reduce the bias have been proposed for the CPZ-based ANFs.

In addition to the CPZ-based notch filters, there exist many other types of notch filters. In [53, 54], the specific second-order transfer function is constructed in such a manner that it corresponds to a lattice structure. In [55–58], the bilinear transformation to a second-order analog filter is applied to the notch filter design. In [59], the frequency transformation is used to design a notch filter, but the prototype filter used here is different from Eq. (17).

### 4.3. Adaptive filtering based on high-order VBPFs/VBSFs

All of the adaptive filters that were addressed in previous subsections are based on secondorder VDFs. On the other hand, there exist some results on high-order VDFs in adaptive signal processing. Needless to say, second-order ANFs have a drawback that it is difficult to realize sharp cutoff characteristics, causing insufficient frequency selectivity and relatively poor signal-to-noise ratio (SNR) at the output signal. On the other hand, in [31], the authors improve the output SNR by means of higher-order VBPFs or VBSFs instead of using second-order notch filters in the adaptive filtering. As shown in Figure 10, high-order filters can realize sharper cutoff characteristics than second-order filters and provide higher output SNR.

Compared with ANFs, little has been studied on the adaptive filtering based on the highorder VDFs. To the authors' best knowledge, the most significant work is found in [60–63], where fourth-order Butterworth VBPF and VBSF are applied to adaptive filtering, and their center frequencies are controlled by adaptive algorithms. Furthermore, the convergence characteristics are also theoretically analyzed. In this work, it is also claimed that the use of much higher-order VDFs for adaptive filtering is almost impossible because higher-order transfer functions involve mathematically more complicated descriptions, and hence it is conjectured that formulations of filter coefficients with variable characteristics and adaptive

approach for the detection of multiple sinusoids is also proposed in [66–68], where comb filters

Recent Advances in Variable Digital Filters http://dx.doi.org/10.5772/intechopen.79198 39

Furthermore, adaptive filtering based on VLPFs can be found in the literature [69]. It should be noted that, in general, realization of adaptive low-pass filtering is much more difficult than adaptive notch filtering or adaptive band-pass/band-stop filtering. The reason of this lies in the difficulty in the problem setup that can describe a unimodal cost function. However in the work of [69], a unimodal cost function is successfully obtained by considering the detection of passband-edge frequency of a low-pass filtered signal and using the approach of weighted cost

This chapter has reviewed recent research activities on VDFs with focus on the approximation problem, the realization problem, and the applications to adaptive filtering. Since this chapter has paid attention to 1-D VDFs with variable magnitude responses, the introduction of other types of VDFs such as M-D VDFs and variable fractional-delay filters has been omitted. For a similar reason, VDF applications other than adaptive filtering have also been omitted. Although VDFs have been studied for a long time, many elegant results are still being proposed, and hence

Department of Electronic Engineering, Graduate School of Engineering, Tohoku University,

[1] Stoyanov G, Kawamata M. Variable digital filters. Journal of Signal Processing. July 1997;

[3] Constantinides AG. Spectral transformations for digital filters. IEE Proceedings. Aug.

[4] Mitra SK, Neuvo Y, Roivainen H. Design of recursive digital filters with variable characteristics. International Journal of Circuit Theory and Applications. 1990;18:107-119

[2] Roberts RA, Mullis CT. Digital Signal Processing. Boston, USA: Addison-Wesley; 1987

the research on VDFs will continue to be an active area of investigation.

Shunsuke Koshita\*, Masahide Abe and Masayuki Kawamata \*Address all correspondence to: kosita@mk.ecei.tohoku.ac.jp

with variable bandwidth and variable notch gain are applied to adaptive filtering.

function.

5. Conclusion

Author details

Sendai, Japan

References

1(4):275-289

1970;117(8):1585-1590

Figure 10. Example of adaptive band-pass filtering: (a) using second-order VBPF, and (b) using high-order VBPF.

control of them become very complicated. However, in the authors' recent work [31], we have successfully realized adaptive filtering based on higher-order VBPFs/VBSFs, where we have derived a gradient descent method-based adaptive algorithm for arbitrary-order VBPFs/VBSFs in a simple form by means of frequency transformation in terms of the block diagram as well as the mathematical description. As a result, it is demonstrated in [31] that the use of higher-order VBPFs/VBSFs for adaptive filtering leads to higher output SNR than the use of ANFs.

As stated above, adaptive band-pass/band-stop filtering based on high-order VBPFs/VSFs can be realized in a simple manner. However, there are still many open problems such as mathematical discussion of convergence of the adaptive algorithm, improvement of convergence speed, and suppression of large quantization errors that are generated due to the nature of high-order narrowband filters. Although the problem of quantization errors can be solved by means of the state-space-based VBPFs/VBSFs [64], further investigations are need to cope with the other problems.

#### 4.4. Other VDFs for adaptive filtering

In addition to the ANFs and higher-order adaptive band-pass/band-stop filtering, many applications of other VDFs to adaptive filtering have been presented. In [65], adaptive filtering based on the cascade connection of second-order all-pass filters is proposed. This method is shown to be superior to the standard ANF-based methods for the detection of multiple sinusoids. Another approach for the detection of multiple sinusoids is also proposed in [66–68], where comb filters with variable bandwidth and variable notch gain are applied to adaptive filtering.

Furthermore, adaptive filtering based on VLPFs can be found in the literature [69]. It should be noted that, in general, realization of adaptive low-pass filtering is much more difficult than adaptive notch filtering or adaptive band-pass/band-stop filtering. The reason of this lies in the difficulty in the problem setup that can describe a unimodal cost function. However in the work of [69], a unimodal cost function is successfully obtained by considering the detection of passband-edge frequency of a low-pass filtered signal and using the approach of weighted cost function.

### 5. Conclusion

This chapter has reviewed recent research activities on VDFs with focus on the approximation problem, the realization problem, and the applications to adaptive filtering. Since this chapter has paid attention to 1-D VDFs with variable magnitude responses, the introduction of other types of VDFs such as M-D VDFs and variable fractional-delay filters has been omitted. For a similar reason, VDF applications other than adaptive filtering have also been omitted. Although VDFs have been studied for a long time, many elegant results are still being proposed, and hence the research on VDFs will continue to be an active area of investigation.
