Discrete Wavelet Transform E-Book

D. Sundararajan

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 348

Veröffentlichungsjahr: 2015

Beschreibung

Provides easy learning and understanding of DWT from a signal processing point of view * Presents DWT from a digital signal processing point of view, in contrast to the usual mathematical approach, making it highly accessible * Offers a comprehensive coverage of related topics, including convolution and correlation, Fourier transform, FIR filter, orthogonal and biorthogonal filters * Organized systematically, starting from the fundamentals of signal processing to the more advanced topics of DWT and Discrete Wavelet Packet Transform. * Written in a clear and concise manner with abundant examples, figures and detailed explanations * Features a companion website that has several MATLAB programs for the implementation of the DWT with commonly used filters "This well-written textbook is an introduction to the theory of discrete wavelet transform (DWT) and its applications in digital signal and image processing." -- Prof. Dr. Manfred Tasche - Institut für Mathematik, Uni Rostock Full review at https://zbmath.org/?q=an:06492561

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Preface

List of Abbreviations

Chapter 1: Introduction

1.1 The Organization of This Book

Chapter 2: Signals

2.1 Signal Classifications

2.2 Basic Signals

2.3 The Sampling Theorem and the Aliasing Effect

2.4 Signal Operations

2.5 Summary

Exercises

Chapter 3: Convolution and Correlation

3.1 Convolution

3.2 Correlation

3.3 Summary

Exercises

Chapter 4: Fourier Analysis of Discrete Signals

4.1 Transform Analysis

4.2 The Discrete Fourier Transform

4.3 The Discrete-Time Fourier Transform

4.4 Approximation of the DTFT

4.5 The Fourier Transform

4.6 Summary

Exercises

Chapter 5: The z-Transform

5.1 The -Transform

5.2 Properties of the -Transform

5.3 Summary

Exercises

Chapter 6: Finite Impulse Response Filters

6.1 Characterization

6.2 Linear Phase Response

6.3 Summary

Exercises

Chapter 7: Multirate Digital Signal Processing

7.1 Decimation

7.2 Interpolation

7.3 Two-Channel Filter Bank

7.4 Polyphase Form of the Two-Channel Filter Bank

7.5 Summary

Exercises

Chapter 8: The Haar Discrete Wavelet Transform

8.1 Introduction

8.2 The Haar Discrete Wavelet Transform

8.3 The Time-Frequency Plane

8.4 Wavelets from the Filter Coefficients

8.5 The 2-D Haar Discrete Wavelet Transform

8.6 Discontinuity Detection

8.7 Summary

Exercises

Chapter 9: Orthogonal Filter Banks

9.1 Haar Filter

9.2 Daubechies Filter

9.3 Orthogonality Conditions

9.4 Coiflet Filter

9.5 Summary

Exercises

Chapter 10: Biorthogonal Filter Banks

10.1 Biorthogonal Filters

10.2 5/3 Spline Filter

10.3 4/4 Spline Filter

10.4 CDF 9/7 Filter

10.5 Summary

Exercises

Chapter 11: Implementation of the Discrete Wavelet Transform

11.1 Implementation of the DWT with Haar Filters

11.2 Symmetrical Extension of the Data

11.3 Implementation of the DWT with the D4 Filter

11.4 Implementation of the DWT with Symmetrical Filters

11.5 Implementation of the DWT using Factorized Polyphase Matrix

11.6 Summary

Exercises

Chapter 12: The Discrete Wavelet Packet Transform

12.1 The Discrete Wavelet Packet Transform

12.2 Best Representation

12.3 Summary

Exercises

Chapter 13: The Discrete Stationary Wavelet Transform

13.1 The Discrete Stationary Wavelet Transform

13.2 Summary

Exercises

Chapter 14: The Dual-Tree Discrete Wavelet Transform

14.1 The Dual-Tree Discrete Wavelet Transform

14.2 The Scaling and Wavelet Functions

14.3 Computation of the DTDWT

14.4 Summary

Exercises

Chapter 15: Image Compression

15.1 Lossy Image Compression

15.2 Lossless Image Compression

15.3 Recent Trends in Image Compression

15.4 Summary

Exercises

Chapter 16: Denoising

16.1 Denoising

16.2 VisuShrink Denoising Algorithm

16.3 Summary

Exercises

Bibliography

Answers to Selected Exercises

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

Chapter 2: Signals

Figure 2.1 (a) An arbitrary discrete signal; (b) the discrete sinusoidal signal

Figure 2.2 (a) The even component of the signal ; (b) the odd component

Figure 2.3 (a) Cumulative energy of an arbitrary discrete signal; (b) cumulative energy of its transformed version

Figure 2.4 (a) The unit-impulse signal, ; (b) the unit-step signal

Figure 2.5 Discrete sinusoids (dots) and (crosses)

Figure 2.6 Signal (dots) and its shifted versions, (crosses) and (unfilled circles)

Figure 2.7 Signal (dots) and its time-reversed version, (crosses)

Figure 2.8 (a) Symbolic representation of compressing a signal by a factor of 2; (b) signal ; (c) compressed signal,

Figure 2.9 (a) Symbolic representation of expanding a signal by a factor of 2; (b) signal ; (c) expanded signal,

Figure 2.10 Cosine signal and its expanded version

Chapter 3: Convolution and Correlation

Figure 3.1 The linear convolution operation

Figure 3.2 The linear convolution operation as an ordinary multiplication

Figure 3.3 The linear correlation operation

Figure 3.4 The linear correlation operation as a convolution

Chapter 4: Fourier Analysis of Discrete Signals

Figure 4.1 (a) A periodic waveform, , with period four samples and (b) its frequency-domain representation; (c) the frequency components of the waveform in (a); (d) the square error in approximating the waveform in (a) using only the DC component with different amplitudes

Figure 4.2 (a) The unit-impulse signal and (b) its DFT spectrum; (c) the DC signal and (d) its DFT spectrum; (e) the sinusoid and (f) its DFT spectrum

Figure 4.3

Figure 4.4 The magnitude of the frequency response of a highpass filter

Figure 4.5 The magnitude of the frequency response

Figure 4.6 (a) The rectangular signal and (b) its spectrum; (c) the triangular signal, which is the convolution of the signal in (a) with itself, and (d) its spectrum

Figure 4.7 (a) Signal and (b) its DTFT spectrum; (c) expanded version and (d) its DTFT spectrum

Figure 4.8 (a) The sinc function ; (b) its rectangular FT spectrum; (c) samples of (a) with sampling interval seconds; (d) its DTFT spectrum; (e) samples of the sinc function with sampling interval seconds; (f) its DTFT spectrum

Chapter 6: Finite Impulse Response Filters

Figure x.1 Magnitude of the periodic frequency response of an ideal digital lowpass filter

Figure 6.2 Magnitude of the periodic frequency response of an ideal digital highpass filter

Figure 6.3 Magnitude of the periodic frequency response of an ideal digital bandpass filter

Figure 6.4 Typical FIR filter with even-symmetric impulse response and odd number of coefficients

Figure 6.5

Figure 6.6 Typical FIR filter with even-symmetric impulse response and even number of coefficients

Chapter 7: Multirate Digital Signal Processing

Figure 7.1 Decimation of a signal

x

(

n

) by a factor of 2

Figure 7.2 Convolution of two sequences (a), and computation of the DWT coefficients (b)

Figure 7.3 (a) Signal ; (b) its DTFT ; (c) signal ; (d) its DTFT ; (e) signal ; (f) its DTFT ; (g) downsampled signal ; (h) its DTFT

Figure 7.4 Decimation of a signal by a factor of 2 with downsampling followed by filtering

Figure 7.5 Interpolation of a signal by a factor of 2

Figure 7.6 Convolution of an upsampled sequence and (a), and convolution of alternately with and (b)

Figure 7.7 Interpolation of a signal by a factor of 2 with filtering followed by upsampling

Figure 7.8 A frequency-domain representation of a two-channel analysis and synthesis filter bank

Figure 7.9 Polyphase form of an FIR decimation filter

Figure 7.10 Polyphase form of an FIR interpolation filter

Figure 7.11 A two-channel filter bank with no filtering

Figure 7.12 Polyphase form of a two-channel analysis and synthesis filter bank

Figure 7.13 Polyphase form of a two-channel analysis and synthesis filter bank, in detail

Figure 7.14

Figure 7.15 A two-channel filter bank

Chapter 8: The Haar Discrete Wavelet Transform

Figure 8.1 Typical basis functions for Fourier and wavelet transforms; (a) one cycle of a sinusoid; (b) a wavelet

Figure 8.2 Haar DWT basis functions with . (a) ; (b)

Figure 8.3 Time-frequency plane of the DWT, with

Figure 8.4 The 2-level decomposition of a 512-point signal by the DWT

Figure 8.5

Figure 8.6 Haar scaling function from lowpass synthesis filter coefficients.

Figure 8.7 Haar wavelet function from highpass synthesis filter coefficients.

Figure 8.8 (a) Haar scaling function; (b) Haar wavelet function

Figure 8.9 (a) The Haar 2-D scaled scaling filter ; (b) the 2-D scaled wavelet filter ; (c) the 2-D scaled wavelet filter ; (d) the 2-D scaled wavelet filter

Figure 8.10 scaled Haar 2-D DWT filters , , , and

Figure 8.11

Figure 8.12 A image with 256 gray levels

Figure 8.13 The 1-level Haar DWT of the image in Figure 8.12

Figure 8.14 (a) The histogram of the image shown in Figure 8.12; (b) the histogram of the 1-level DWT of the image shown in Figure 8.13; (c) the histogram of the approximation part of the DWT; (d) the histogram of the horizontal component of the detail part of the DWT; (e) the histogram of the vertical component; (f) the histogram of the diagonal component

Figure 8.15 The 2-level Haar DWT of the image in Figure 8.12

Figure 8.16 (a) A signal with a discontinuity; (b) the 1-level Haar DWT of the signal showing the occurrence of the discontinuity in the detail part

Figure 8.17 (a) A signal with two discontinuities of its difference; (b) the 1-level Haar DWT of the signal showing the occurrence of its discontinuities in the detail part

Chapter 9: Orthogonal Filter Banks

Figure 9.1 A two-channel analysis and synthesis filter bank with Haar filters

Figure 9.2 A two-channel analysis and synthesis filter bank with D4 filters

Figure 9.3 (a) The magnitude of the frequency responses of the Haar analysis filters; (b) the phase response of the lowpass filter

Figure 9.4 The magnitude of the frequency responses of the D4 analysis filters; (b) the phase response of the lowpass filter

Figure 9.5 Approximations of the D4 scaling function derived from the synthesis lowpass filter. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after 10 cascade stages

Figure 9.6 Approximations of the D4 wavelet function derived from the synthesis highpass filter. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after 10 cascade stages

Figure 9.7 (a) The D4 2-D scaling filter; (b) the 2-D wavelet filter ; (c) the 2-D wavelet filter ; (d) the 2-D wavelet filter

Figure 9.8 The magnitude of the frequency responses of the C6 analysis filter (solid line) and that of the Daubechies filter (dashed line) of length six

Figure 9.9 C6 scaling function derived from the synthesis lowpass filter. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after 10 cascade stages

Figure 9.10 C6 wavelet function derived from the synthesis highpass filter. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after 10 cascade stages

Chapter 10: Biorthogonal Filter Banks

Figure 10.1 A two-channel analysis and synthesis filter bank with 5/3 spline filters

Figure 10.2 (a) The magnitude of the frequency responses of the 5/3 spline analysis filters; (b) the phase response of the lowpass filter

Figure 10.3 Approximation of the 5/3 spline scaling function derived from the synthesis lowpass filter. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after 10 cascade stages

Figure 10.4 Approximation of the 5/3 spline wavelet function derived from the synthesis highpass filter. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after 10 cascade stages

Figure 10.5 A two-channel analysis and synthesis filter bank with 4/4 spline filters

Figure 10.6 (a) The magnitude of the frequency responses of the 4/4 spline analysis filters; (b) the phase response of the lowpass filter

Figure 10.7 Approximation of the 4/4 spline scaling function derived from the synthesis lowpass filter. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after 10 cascade stages

Figure 10.8 Approximation of the 4/4 spline wavelet function derived from the synthesis highpass filter. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after 10 cascade stages

Figure 10.9 The magnitude of the frequency responses of the CDF 9/7 analysis filters

Figure 10.10 Approximation of the CDF 9/7 scaling function derived from the synthesis lowpass filter. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after 10 cascade stages

Figure 10.11 Approximation of the CDF 9/7 wavelet function derived from the synthesis highpass filter. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after 10 cascade stages

Chapter 11: Implementation of the Discrete Wavelet Transform

Figure 11.1 The 1-level Haar DWT approximation coefficients using the convolution operation

Figure 11.2 The 1-level Haar DWT detail coefficients using the convolution operation

Figure 11.3 The 1-level Haar IDWT using the convolution operation without upsampling

Figure 11.4 Computation of a 2-level 4-point DWT using a two-stage two-channel Haar analysis filter bank

Figure 11.5 Computation of a 2-level 4-point IDWT using a two-stage two-channel Haar synthesis filter bank

Figure 11.6 Computation of a 1-level 2-D Haar DWT using a two-stage filter bank

Figure 11.7 Computation of a 1-level 2-D IDWT using a two-stage filter bank

Figure 11.8 The signal-flow graph of the 1-D Haar 3-level DWT algorithm, with

Figure 11.9 The trace of the 1-D Haar 3-level DWT algorithm, with

Figure 11.10 The signal-flow graph of the 1-D 3-level Haar IDWT algorithm, with

Figure 11.11 The trace of the 1-D 3-level Haar IDWT algorithm, with

Figure 11.12 The trace of the 1-D 1-level Haar DWT algorithm with (a) and without (b) auxiliary memory

Figure 11.13 (a) A signal with half-point extension; (b) a signal with whole-point extension. The dots represent the given signal values and the unfilled circles are the extended signal values

Figure 11.14

Chapter 12: The Discrete Wavelet Packet Transform

Figure 12.1 2-level Haar DWPT basis functions with . (a) ; (b) ; (c) ; (d)

Figure 12.2

Chapter 13: The Discrete Stationary Wavelet Transform

Figure 13.1 Computation of a 2-level 4-point SWT using a two-stage two-channel analysis filter bank

Figure 13.2 Computation of a partial result of a 2-level 4-point ISWT using a two-stage two-channel synthesis filter bank

Figure 13.3 Computation of a partial result of a 2-level 4-point ISWT using a two-stage two-channel synthesis filter bank. Letter C indicates right circular shift by one position

Figure 13.5 Computation of a partial result of a 2-level 4-point ISWT using a two-stage two-channel synthesis filter bank. Letter C indicates right circular shift by one position

Chapter 14: The Dual-Tree Discrete Wavelet Transform

Figure 14.1 DTDWT analysis filter bank

Figure 14.2 The magnitude of the frequency response of the analysis filters

Figure 14.3 DTDWT synthesis filter bank

Figure 14.4 Approximations of the scaling function derived from the second-stage synthesis lowpass filter of Tree R. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after five cascade stages

Figure 14.5 Approximations of the wavelet function derived from the second-stage synthesis highpass filter of Tree R. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after five cascade stages

Figure 14.6 Approximations of the scaling function derived from the second-stage synthesis lowpass filter of Tree I. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after five cascade stages

Figure 14.7 Approximations of the wavelet function derived from the second-stage synthesis highpass filter of Tree I. (a) Filter coefficients; (b) after one stage; (c) after two cascade stages; (d) after three cascade stages; (e) after four cascade stages; (f) after five cascade stages

Figure 14.8 The magnitude of the DFT of the approximation of the complex signal , the real part being the Tree R wavelet function and the imaginary part being the Tree I wavelet function

Figure 14.9

Figure 14.10

Figure 14.11

Chapter 15: Image Compression

Figure 15.1 (a) An arbitrary signal; (b) the 3-level Haar DWT decomposition of the signal

Figure 15.2 (a) Image compression using DWT/DWPT; (b) reconstructing a compressed image

Figure 15.3 Labeling of the DWT coefficients of 3-level decomposition of a image

Figure 15.4 Uniform scalar quantization

Figure 15.5 (a) Hard thresholding; (b) soft thresholding

Figure 15.6 Zigzag scan order of the coefficients of the 1-level DWT decomposition of a 2-D 8 × 8 image

Figure 15.7 Huffman coding tree

Figure 15.9 A reconstructed image after a 2-level DWT decomposition using 5/3 spline filters with the quantization step 16.2287 and the threshold value 16.2287

Figure 15.8 A reconstructed image after a 1-level DWT decomposition using 5/3 spline filters with the quantization step 4.2068 and the threshold value 0

Figure 15.10 A reconstructed image after a 1-level DWT decomposition using 5/3 spline filters with the quantization step 8.5494 and the threshold value 0

Figure 15.11 A reconstructed image after a 1-level DWT decomposition using 5/3 spline filters with the quantization step 17.6688 and the threshold value 0

Figure 15.12 A reconstructed image after a 1-level DWT decomposition using 5/3 spline filters with the quantization step 37.8616 and the threshold value 0

Figure 15.13 A reconstructed image after a 1-level DWT decomposition using 5/3 spline filters with the quantization step 4.2068 and the threshold value 8.4137

Figure 15.14 A reconstructed image after a 1-level DWT decomposition using 5/3 spline filters with the quantization step 4.2068 and the threshold value 12.6205

Figure 15.15 A reconstructed image after a 1-level DWT decomposition using 5/3 spline filters with the quantization step 4.2068 and the threshold value 16.8274

Figure 15.16 A reconstructed image after a 1-level DWT decomposition using 5/3 spline filters with the quantization step 4.2068 and the threshold value 21.0342

Figure 15.17 A reconstructed image after a 2-level DWT decomposition using 5/3 spline filters with the quantization step 7.9856 and the threshold value 0

Figure 15.18 Coefficient trees spanning across different scales of a 2-D image

Chapter 16: Denoising

Figure 16.1 (a) A sinusoidal signal; (b) the signal in (a) soft thresholded with

Figure 15.2 (a) A sinusoidal signal; (b) the signal in (a) with noise; (c) the denoised signal

Figure 15.3 (a) The DWT of the sinusoid; (b) the DWT of the noise; (c) the DWT of the noisy signal

Figure 15.4 (a) The soft thresholded 1-level DWT of the signal; (b) the 4-level DWT of the signal; (c) the soft thresholded 4-level DWT of the signal

Figure 15.5

List of Tables

Chapter 3: Convolution and Correlation

Table 3.1 Three types of data extensions at the borders

Table 3.2 Outputs of convolving the data in Table 3.1 with {3, 1}

Chapter 7: Multirate Digital Signal Processing

Table 7.1 The transformation of a sequence through a two-channel filter bank with no filtering

Table 7.2

Chapter 12: The Discrete Wavelet Packet Transform

Table 12.1

Table 12.2

Table 12.3

Table 12.4

Table 12.5

Table 12.6

Table 12.7

Table 12.8

Table 12.10

Table 12.11

Table 12.12

Table 12.14

Table 12.15

Table 12.16

Table 12.18

Chapter 13: The Discrete Stationary Wavelet Transform

Table 13.1 1-level Haar SWT of {2, 3, − 4, 5}

Table 13.2 2-level Haar SWT of {2, 3, − 4, 5}

Chapter 14: The Dual-Tree Discrete Wavelet Transform

Table 14.1

Table 14.4 Second and subsequent stages filter coefficients – Tree I

Table 14.3 Second and subsequent stages filter coefficients – Tree R

Table 14.2 First-stage filter coefficients – Tree I

Chapter 15: Image Compression

Table 15.1 Assignment of Huffman codes

Table 15.2 Example image

Table 15.3 Level-shifted image

Table 15.4 The row DWT on the left and the 2-D Haar DWT of the level-shifted image on the right

Table 15.5 Quantized image

Table 15.6 2-D Haar DWT of the reconstructed level-shifted image

Table 15.7 Level-shifted reconstructed image

Table 15.8 Reconstructed image

Table 15.9 Example image

Table 15.10 Level-shifted image

Table 15.11 2-D DWT of the image using 5/3 spline filter, assuming whole-point symmetry extension

Table 15.12 Quantized image

Table 15.13 Zigzag pattern

Table 15.14 Reconstructed image

Table 15.15 Input image 15.6.1

Table 15.16 Input image 15.6.2

Table 15.17 Input image 15.6.3

Discrete Wavelet Transform

A Signal Processing Approach

D. Sundararajan

Adhiyamaan College of Engineering, India

This edition first published 2015

© 2015 John Wiley & Sons Singapore Pte. Ltd.

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Library of Congress Cataloging-in-Publication Data applied for.

ISBN: 9781119046066

Preface

The discrete wavelet transform, a generalization of the Fourier analysis, is widely used in many applications of science and engineering. The primary objective of writing this book is to present the essentials of the discrete wavelet transform – theory, implementation, and applications – from a practical viewpoint. The discrete wavelet transform is presented from a digital signal processing point of view. Physical explanations, numerous examples, plenty of figures, tables, and programs enable the reader to understand the theory and algorithms of this relatively difficult transform with minimal effort.

This book is intended to be a textbook for senior-undergraduate-level and graduate-level discrete wavelet transform courses or a supplementary textbook for digital signal/image processing courses in engineering disciplines. For signal and image processing professionals, this book will be useful for self-study. In addition, this book will be a reference for anyone, student or professional, specializing in signal and image processing. The prerequisite for reading this book is a good knowledge of calculus, linear algebra, signals and systems, and digital signal processing at the undergraduate level. The last two of these topics are adequately covered in the first few chapters of this book.

MATLAB® programs are available at the website of the book, www.wiley.com/go/sundararajan/wavelet. Programming is an important component in learning this subject. Answers to selected exercises marked with * are given at the end of the book. A Solutions Manual and slides are available for instructors at the website of the book.

I assume the responsibility for all the errors in this book and would very much appreciate receiving readers' suggestions at [email protected]. I am grateful to my Editor and his team at Wiley for their help and encouragement in completing this project. I thank my family for their support during this endeavor.

D. Sundararajan

List of Abbreviations

bpp

bits per pixel

DFT

discrete Fourier transform

DTDWT

dual-tree discrete wavelet transform

DTFT

discrete-time Fourier transform

DWPT

discrete wavelet packet transform

DWT

discrete wavelet transform

FIR

finite impulse response

FS

Fourier series

FT

Fourier transform

IDFT

inverse discrete Fourier transform

IDTDWT

inverse dual-tree discrete wavelet transform

IDWPT

inverse discrete wavelet packet transform

IDWT

inverse discrete wavelet transform

ISWT

inverse discrete stationary wavelet transform

PR

perfect reconstruction

SWT

discrete stationary wavelet transform

1-D

one-dimensional

2-D

two-dimensional

Chapter 1Introduction

A signal conveys some information. Most of the naturally occurring signals are continuous in nature. More often than not, they are converted to digital form and processed. In digital signal processing, the information is extracted using digital devices. It is the availability of fast digital devices and numerical algorithms that has made digital signal processing important for many applications of science and engineering. Often, the information in a signal is obscured by the presence of noise. Some type of filtering or processing of signals is usually required. To transmit and store a signal, we would like to compress it as much as possible with the required fidelity. These tasks have to be carried out in an efficient manner.

In general, a straightforward solution from the definition of a problem is not the most efficient way of solving it. Therefore, we look for more efficient methods. Typically, the problem is redefined in a new setting by some transformation. We often use -substitutions or integration by parts to simplify the problem in finding the integral of a function. By using logarithms, the more difficult multiplication operation is reduced to addition.