DIGITAL FILTERS

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Ouvrage 3-540-66841-1 : DIGITAL FILTERS
It is the aim of this textbook to give insight into the characteristics and the design of digital filters. It briefly introduces to the theory of continuous-time systems and the design methods for analog filters. Discrete-time systems, the basic structures of digital filters, sampling theorem, and the design of IIR filters are widely discussed. Important parts of the book are devoted to the design of non-recursive filters and the effects of finite register length. The explanation of techniques like oversampling and noise shaping conclude the book. It is completed by an annex containing a selection of tables of filter parameters for Butterworth, Chebyshev, Cauer, and Bessel filters. Furthermore, several computer routines for filter design programs are given.
Table of Contents 1
Continuous-Time Systems
1
1.1
Relations in the Time Domain
1
1.2
Relations in the Frequency Domain
2
1.2.1
The Laplace Transform
3
1.2.2
The Fourier Transform
5
1.3
Transfer Functions
7
2
Analog Filters
13
2.1
The Ideal Low-Pass Filter
13
2.2
Butterworth Filters
15
2.3
Chebyshev Filters
19
2.4
Inverse Chebyshev Filters
24
2.5
Elliptic Filters
27
2.6
Bessel Filters
33
2.7
Filter Fransformations
41
2.7.1
Low-Pass High-Pass Transformation
42
2.7.2
Low-Pass Bandpass Transformation
43
2.7.3
Low-pass Bandstop Transformation
45
2.7.4
Filter Transformations in Practice
46
3
Discrete-Time Systems
51
3.1
Discrete-Time Convolution
52
3.2
Relations in the Frequency Domain
54
3.3
The z-Transform
58
3.3.1
Definition of the z-Transform
58
3.3.2
Properties of the z-Transform
59
3.3.3
Examples
61
3.4
Stability Criterion in the Time Domain
65
3.5
Further Properties of the Unit-Sample Response h(n)
66
4
Sampling Theorem
69
4.1
Introduction
69
4.2
Sampling
69
4.3
Band-Limited Signals
71
4.4
Practical Aspects of Signal Conversion
74
4.4.1
Analog-to-Digital Conversion
74
4.4.2
Digital-to-Analog Conversion
76
4.5
Mathematical Analysis of the D/A Conversion Process
77
5
Filter Structures
83
5.1
Graphical Representation of Discrete-Time Networks
83
5.2
FIR Filters
86
5.2.1
The Basic Structure of FIR Filters
86
5.2.2
Poles and Zeros of the FIR Filter
89
5.2.3
Linear-Phase Filters
91
5.3
IIR Filters
95
5.3.1
An Introductory Example
95
5.3.2
Direct-Form Filters
96
5.3.3
Poles and Zeros
98
5.4
Further Properties of Transfer Functions
101
5.4.1
Stability Considerations in the Frequency Domain
101
5.4.2
Minimum-Phase Filters
103
5.5
State-Space Structures
104
5.6
The Normal Form
109
5.7
Digital Ladder Filters
110
5.7.1
Lossless Networks
112
5.7.2
Realisability
113
5.7.3
Kirchhoff Ladder Filters
114
5.7.4
Wave Digital Filters
120
5.7.4.1
Wave Parameters
120
5.7.4.2
Decomposition of the Analog Reference Network into One-Ports and Multi-Ports
126
5.7.4.3
The Reflectance of Capacitor and Inductor
130
5.7.4.4
The Role of the Bilinear Transform
133
5.7.4.5
Observance of the Realisability Condition
135
5.7.4.6
Second-Order Wave Digital Filter Block
138
6
Design of IIR Filters
141
6.1
Introduction
141
6.2
Preservation Theorem of the Convolution
142
6.3
Approximation in the Time Domain
144
6.3.1
Impulse-Invariant Design
145
6.3.2
Step-Invariant Design
151
6.3.3
Ramp-Invariant Design
154
6.4
Approximation in the Frequency Domain
156
6.4.1
Difference Method
157
6.4.2
Bilinear Transform
160
6.4.3
Practical Filter Design Using the Bilinear Transform
165
6.4.4
IIR Filter with Bessel Characteristics
169
6.4.4.1
Design with Prescribed Delay t[subscript 0]
172
6.4.4.2
Design with Given 3-dB Cutoff Frequency
173
7
Design of FIR Filters
177
7.1
Introduction
177
7.2
Linear-Phase Filters
178
7.3
Frequency Sampling
181
7.4
Minimisation of the Mean-Square Error
188
7.4.1
Direct Solution
191
7.4.2
The Window Method
198
7.5
Chebyshev Approximation
203
7.5.1
The Remez Exchange Algorithm
203
7.5.2
Polynomial Interpolation
206
7.5.3
Introduction of a Weighting Function
210
7.5.4
Case 2, 3 and 4 Filters
213
7.5.5
Minimum-Phase FIR Filters
215
7.6
Maximally Flat (MAXFLAT) Design
219
8
Effects of Finite Register Length
227
8.1
Classification of the Effects
227
8.2
Number Representation
227
8.2.1
Fixed-Point Numbers
228
8.2.2
Floating-Point Numbers
231
8.3
Quantisation
233
8.3.1
Fixed-Point Numbers
233
8.3.2
Floating-Point Numbers
238
8.4
System Noise Behaviour
241
8.4.1
Signal-to-Noise Ratio
241
8.4.2
Noise Filtering by LSI Systems
243
8.4.3
Optimum Use of the Dynamic Range
245
8.5
Noise Optimisation with Fixed-Point Arithmetic
247
8.5.1
Noise Behaviour of First- and Second-Order Filter Blocks
247
8.5.2
Shaping of the Noise Spectrum
255
8.5.3
Noise in a Cascade of Second-Order Filter Blocks
260
8.6
Finite Coefficient Wordlength
264
8.6.1
Coefficient Sensitivity
265
8.6.2
Graphical Representation of the Pole Density
267
8.6.3
Increased Pole Density at Low and High Frequencies
272
8.6.4
Further Remarks Concerning Coefficient Sensitivity
277
8.6.5
A Concluding Example
278
8.7
Limit Cycles
281
8.7.1
Nonlinearities in Actual Implementations
282
8.7.2
Stability of the Linear Filter
284
8.7.3
Stability with Quantisation and Overflow Correction
289
8.7.4
A Constructive Approach for the Determination of Stability
290
8.7.4.1
The Linear System
290
8.7.4.2
Stability of the Nonlinear System
292
8.7.5
A Search Algorithm in the State Plane
304
8.7.6
Summary
311
9
Oversampling and Noise Shaping
313
9.1
D/A Conversion
313
9.1.1
Interpolation
314
9.1.2
Noise Filtering
316
9.1.3
Noise Shaping
317
9.2
A/D Conversion
320
9.2.1
Oversampling and Decimation
320
9.2.2
Noise Filtering
322
9.2.3
Oversampling [Delta Sigma] Converter
323
10
Appendix
327
10.1
Symbols and Abbreviations
327
10.2
References
329
10.3
Filter Design Tables
331
10.4
Filter Design Routines
346
Index
359

Auteur : SCHLICHTHARLE
Editeur : SPRINGER VERLAG
Nombre de pages : 361
Date de publication : 01 2002

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