Digital filters design for signal and image processing

Title:

Series:

Digital signal and image processing series

Publication Information:

London : ISTE, 2006

ISBN:

9781905209453

Subject Term:

Electric filters, Digital

Signal processing -- Digital techniques

Image processing -- Digital techniques

Added Author:

Najim, Mohamed

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010160811	TK7872.F5 D53 2006	Open Access Book	Book	Searching... Unknown

Dealing with digital filtering methods for 1-D and 2-D signals, this book provides the theoretical background in signal processing, covering topics such as the z-transform, Shannon sampling theorem and fast Fourier transform. An entire chapter is devoted to the design of time-continuous filters which provides a useful preliminary step for analog-to-digital filter conversion.
Attention is also given to the main methods of designing finite impulse response (FIR) and infinite impulse response (IIR) filters. Bi-dimensional digital filtering (image filtering) is investigated and a study on stability analysis, a very useful tool when implementing IIR filters, is also carried out. As such, it will provide a practical and useful guide to those engaged in signal processing.

Author Notes

Mohamed Najim has published several books, more than 220 technical papers and has taught courses in digital signal processing for more than 30 years.

Yannick Berthoumieu and Eric Grivel and Mohamed NajimMohamed Najim and Eric GrivelEric Grivel and Yannick BerthoumieuDaniel Bastard and Eric GrivelYannick Berthoumieu and Eric Grivel and Mohamed NajimEric Grivel and Mohamed NajimMohamed Najim and Eric GrivelPhilippe BolonYannick BerthoumieuMichel BarretMichel Barret

Introduction	p. xiii
Chapter 1 Introduction to Signals and Systems	p. 1
1.1 Introduction	p. 1
1.2 Signals: categories, representations and characterizations	p. 1
1.2.1 Definition of continuous-time and discrete-time signals	p. 1
1.2.2 Deterministic and random signals	p. 6
1.2.3 Periodic signals	p. 8
1.2.4 Mean, energy and power	p. 9
1.2.5 Autocorrelation function	p. 12
1.3 Systems	p. 15
1.4 Properties of discrete-time systems	p. 16
1.4.1 Invariant linear systems	p. 16
1.4.2 Impulse responses and convolution products	p. 16
1.4.3 Causality	p. 17
1.4.4 Interconnections of discrete-time systems	p. 18
1.5 Bibliography	p. 19
Chapter 2 Discrete System Analysis	p. 21
2.1 Introduction	p. 21
2.2 The z-transform	p. 21
2.2.1 Representations and summaries	p. 21
2.2.2 Properties of the z-transform	p. 28
2.2.2.1 Linearity	p. 28
2.2.2.2 Advanced and delayed operators	p. 29
2.2.2.3 Convolution	p. 30
2.2.2.4 Changing the z-scale	p. 31
2.2.2.5 Contrasted signal development	p. 31
2.2.2.6 Derivation of the z-transform	p. 31
2.2.2.7 The sum theorem	p. 32
2.2.2.8 The final-value theorem	p. 32
2.2.2.9 Complex conjugation	p. 32
2.2.2.10 Parseval's theorem	p. 33
2.2.3 Table of standard transform	p. 33
2.3 The inverse z-transform	p. 34
2.3.1 Introduction	p. 34
2.3.2 Methods of determining inverse z-transforms	p. 35
2.3.2.1 Cauchy's theorem: a case of complex variables	p. 35
2.3.2.2 Development in rational fractions	p. 37
2.3.2.3 Development by algebraic division of polynomials	p. 38
2.4 Transfer functions and difference equations	p. 39
2.4.1 The transfer function of a continuous system	p. 39
2.4.2 Transfer functions of discrete systems	p. 41
2.5 Z-transforms of the autocorrelation and intercorrelation functions	p. 44
2.6 Stability	p. 45
2.6.1 Bounded input, bounded output (BIBO) stability	p. 46
2.6.2 Regions of convergence	p. 46
2.6.2.1 Routh's criterion	p. 48
2.6.2.2 Jury's criterion	p. 49
Chapter 3 Frequential Characterization of Signals and Filters	p. 51
3.1 Introduction	p. 51
3.2 The Fourier transform of continuous signals	p. 51
3.2.1 Summary of the Fourier series decomposition of continuous signals	p. 51
3.2.1.1 Decomposition of finite energy signals using an orthonormal base	p. 51
3.2.1.2 Fourier series development of periodic signals	p. 52
3.2.2 Fourier transforms and continuous signals	p. 57
3.2.2.1 Representations	p. 57
3.2.2.2 Properties	p. 58
3.2.2.3 The duality theorem	p. 59
3.2.2.4 The quick method of calculating the Fourier transform	p. 59
3.2.2.5 The Wiener-Khintchine theorem	p. 63
3.2.2.6 The Fourier transform of a Dirac comb	p. 63
3.2.2.7 Another method of calculating the Fourier series development of a periodic signal	p. 66
3.2.2.8 The Fourier series development and the Fourier transform	p. 68
3.2.2.9 Applying the Fourier transform: Shannon's sampling theorem	p. 75
3.3 The discrete Fourier transform (DFT)	p. 78
3.3.1 Expressing the Fourier transform of a discrete sequence	p. 78
3.3.2 Relations between the Laplace and Fourier z-transforms	p. 80
3.3.3 The inverse Fourier transform	p. 81
3.3.4 The discrete Fourier transform	p. 82
3.4 The fast Fourier transform (FFT)	p. 86
3.5 The fast Fourier transform for a time/frequency/energy representation of a non-stationary signal	p. 90
3.6 Frequential characterization of a continuous-time system	p. 91
3.6.1 First and second order filters	p. 91
3.6.1.1 1st order system	p. 91
3.6.1.2 2nd order system	p. 93
3.7 Frequential characterization of discrete-time system	p. 95
3.7.1 Amplitude and phase frequential diagrams	p. 95
3.7.2 Application	p. 96
Chapter 4 Continuous-Time and Analog Filters	p. 99
4.1 Introduction	p. 99
4.2 Different types of filters and filter specifications	p. 99
4.3 Butterworth filters and the maximally flat approximation	p. 104
4.3.1 Maximally flat functions (MFM)	p. 104
4.3.2 A specific example of MFM functions: Butterworth polynomial filters	p. 106
4.3.2.1 Amplitude-squared expression	p. 106
4.3.2.2 Localization of poles	p. 107
4.3.2.3 Determining the cut-off frequency at -3 dB and filter orders	p. 110
4.3.2.4 Application	p. 111
4.3.2.5 Realization of a Butterworth filter	p. 112
4.4 Equiripple filters and the Chebyshev approximation	p. 113
4.4.1 Characteristics of the Chebyshev approximation	p. 113
4.4.2 Type I Chebyshev filters	p. 114
4.4.2.1 The Chebyshev polynomial	p. 114
4.4.2.2 Type I Chebyshev filters	p. 115
4.4.2.3 Pole determination	p. 116
4.4.2.4 Determining the cut-off frequency at -3 dB and the filter order	p. 118
4.4.2.5 Application	p. 121
4.4.2.6 Realization of a Chebyshev filter	p. 121
4.4.2.7 Asymptotic behavior	p. 122
4.4.3 Type II Chebyshev filter	p. 123
4.4.3.1 Determining the filter order and the cut-off frequency	p. 123
4.4.3.2 Application	p. 124
4.5 Elliptic filters: the Cauer approximation	p. 125
4.6 Summary of four types of low-pass filter: Butterworth, Chebyshev type I, Chebyshev type II and Cauer	p. 125
4.7 Linear phase filters (maximally flat delay or MFD): Bessel and Thomson filters	p. 126
4.7.1 Reminders on continuous linear phase filters	p. 126
4.7.2 Properties of Bessel-Thomson filters	p. 128
4.7.3 Bessel and Bessel-Thomson filters	p. 130
4.8 Papoulis filters (optimum (O[subscript n]))	p. 132
4.8.1 General characteristics	p. 132
4.8.2 Determining the poles of the transfer function	p. 135
4.9 Bibliography	p. 135
Chapter 5 Finite Impulse Response Filters	p. 137
5.1 Introduction to finite impulse response filters	p. 137
5.1.1 Difference equations and FIR filters	p. 137
5.1.2 Linear phase FIR filters	p. 142
5.1.2.1 Representation	p. 142
5.1.2.2 Different forms of FIR linear phase filters	p. 147
5.1.2.3 Position of zeros in FIR filters	p. 150
5.1.3 Summary of the properties of FIR filters	p. 152
5.2 Synthesizing FIR filters using frequential specifications	p. 152
5.2.1 Windows	p. 152
5.2.2 Synthesizing FIR filters using the windowing method	p. 159
5.2.2.1 Low-pass filters	p. 159
5.2.2.2 High-pass filters	p. 164
5.3 Optimal approach of equal ripple in the stop-band and passband	p. 165
5.4 Bibliography	p. 172
Chapter 6 Infinite Impulse Response Filters	p. 173
6.1 Introduction to infinite impulse response filters	p. 173
6.1.1 Examples of IIR filters	p. 174
6.1.2 Zero-loss and all-pass filters	p. 178
6.1.3 Minimum-phase filters	p. 180
6.1.3.1 Problem	p. 180
6.1.3.2 Stabilizing inverse filters	p. 181
6.2 Synthesizing IIR filters	p. 183
6.2.1 Impulse invariance method for analog to digital filter conversion	p. 183
6.2.2 The invariance method of the indicial response	p. 185
6.2.3 Bilinear transformations	p. 185
6.2.4 Frequency transformations for filter synthesis using low-pass filters	p. 188
6.3 Bibliography	p. 189
Chapter 7 Structures of FIR and IIR Filters	p. 191
7.1 Introduction	p. 191
7.2 Structure of FIR filters	p. 192
7.3 Structure of IIR filters	p. 192
7.3.1 Direct structures	p. 192
7.3.2 The cascade structure	p. 209
7.3.3 Parallel structures	p. 211
7.4 Realizing finite precision filters	p. 211
7.4.1 Introduction	p. 211
7.4.2 Examples of FIR filters	p. 212
7.4.3 IIR filters	p. 213
7.4.3.1 Introduction	p. 213
7.4.3.2 The influence of quantification on filter stability	p. 221
7.4.3.3 Introduction to scale factors	p. 224
7.4.3.4 Decomposing the transfer function into first- and second-order cells	p. 226
7.5 Bibliography	p. 231
Chapter 8 Two-Dimensional Linear Filtering	p. 233
8.1 Introduction	p. 233
8.2 Continuous models	p. 233
8.2.1 Representation of 2-D signals	p. 233
8.2.2 Analog filtering	p. 235
8.3 Discrete models	p. 236
8.3.1 2-D sampling	p. 236
8.3.2 The aliasing phenomenon and Shannon's theorem	p. 240
8.3.2.1 Reconstruction by linear filtering (Shannon's theorem)	p. 240
8.3.2.2 Aliasing effect	p. 240
8.4 Filtering in the spatial domain	p. 242
8.4.1 2-D discrete convolution	p. 242
8.4.2 Separable filters	p. 244
8.4.3 Separable recursive filtering	p. 246
8.4.4 Processing of side effects	p. 249
8.4.4.1 Prolonging the image by pixels of null intensity	p. 250
8.4.4.2 Prolonging by duplicating the border pixels	p. 251
8.4.4.3 Other approaches	p. 252
8.5 Filtering in the frequency domain	p. 253
8.5.1 2-D discrete Fourier transform (DFT)	p. 253
8.5.2 The circular convolution effect	p. 255
8.6 Bibliography	p. 259
Chapter 9 Two-Dimensional Finite Impulse Response Filter Design	p. 261
9.1 Introduction	p. 261
9.2 Introduction to 2-D FIR filters	p. 262
9.3 Synthesizing with the two-dimensional windowing method	p. 263
9.3.1 Principles of method	p. 263
9.3.2 Theoretical 2-D frequency shape	p. 264
9.3.2.1 Rectangular frequency shape	p. 264
9.3.2.2 Circular shape	p. 266
9.3.3 Digital 2-D filter design by windowing	p. 271
9.3.4 Applying filters based on rectangular and circular shapes	p. 271
9.3.5 2-D Gaussian filters	p. 274
9.3.6 1-D and 2-D representations in a continuous space	p. 274
9.3.6.1 2-D specifications	p. 276
9.3.7 Approximation for FIR filters	p. 277
9.3.7.1 Truncation of the Gaussian profile	p. 277
9.3.7.2 Rectangular windows and convolution	p. 279
9.3.8 An example based on exploiting a modulated Gaussian filter	p. 280
9.4 Appendix: spatial window functions and their implementation	p. 286
9.5 Bibliography	p. 291
Chapter 10 Filter Stability	p. 293
10.1 Introduction	p. 293
10.2 The Schur-Cohn criterion	p. 298
10.3 Appendix: resultant of two polynomials	p. 314
10.4 Bibliography	p. 319
Chapter 11 The Two-Dimensional Domain	p. 321
11.1 Recursive filters	p. 321
11.1.1 Transfer functions	p. 321
11.1.2 The 2-D z-transform	p. 322
11.1.3 Stability, causality and semi-causality	p. 324
11.2 Stability criteria	p. 328
11.2.1 Causal filters	p. 329
11.2.2 Semi-causal filters	p. 332
11.3 Algorithms used in stability tests	p. 334
11.3.1 The jury Table	p. 334
11.3.2 Algorithms based on calculating the Bezout resultant	p. 339
11.3.2.1 First algorithm	p. 340
11.3.2.2 Second algorithm	p. 343
11.3.3 Algorithms and rounding-off errors	p. 347
11.4 Linear predictive coding	p. 351
11.5 Appendix A: demonstration of the Schur-Cohn criterion	p. 355
11.6 Appendix B: optimum 2-D stability criteria	p. 358
11.7 Bibliography	p. 362
List of Authors	p. 365
Index	p. 367

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