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E-raamat: Digital Filter Design and Realization

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(Hiroshima University, Japan), (University of Victoria, Canada)
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Digital Filter Design and RealizationSuurem pilt
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Analysis, design, and realization of digital filters have experienced major developments since the 1970s and have now become an integral part of the theory and practice in the field of contemporary digital signal processing.

Digital Filter Design and Realization presents an up-to-date and comprehensive account of the analysis, design, and realization of digital filters. It is intended to be used as a text for graduate students as well as a reference book for practitioners in the field. Prerequisites for this book include basic knowledge of calculus, linear algebra, signal analysis, and linear system theory.

Technical topics discussed in the book include:
- Discrete-Time Systems and z-Transformation
- Stability and Coefficient Sensitivity
- State-Space Models
- FIR Digital Filter Design
- Frequency-Domain Digital Filter Design
- Time-Domain Digital Filter Design
- Interpolated and Frequency-Response-Masking FIR Digital Filter Design
- Composite Digital Filter Design
- Finite Word Length Effects
- Coefficient Sensitivity Analysis and Minimization
- Error Spectrum Shaping
- Roundoff Noise Analysis and Minimization
- Generalized Transposed Direct-Form II
- Block-State Realization
Preface xvii
List of Figures xix
List of Tables xxv
List of Abbreviations xxvii
1 Introduction 1(22)
1.1 Preview
1(1)
1.2 Terminology for Signal Analysis and Typical Signals
1(3)
1.2.1 Terminology for Signal Analysis
1(2)
1.2.2 Examples of Typical Signals
3(1)
1.3 Digital Signal Processing
4(2)
1.3.1 General Framework for Digital Signal Processing
4(1)
1.3.2 Advantages of Digital Signal Processing
5(1)
1.3.3 Disadvantages of Digital Signal Processing
5(1)
1.4 Analysis of Analog Signals
6(4)
1.4.1 The Fourier Series Expansion of Periodic Signals
6(1)
1.4.2 The Fourier Transform
7(1)
1.4.3 The Laplace Transform
8(2)
1.5 Analysis of Discrete-Time Signals
10(4)
1.5.1 Sampling an Analog Signal
10(1)
1.5.2 The Discrete-Time Fourier Transform
11(2)
1.5.3 The Discrete Fourier Transform (DFT)
13(1)
1.5.4 The z-Transform
13(1)
1.6 Sampling of Continuous-Time Sinusoidal Signals
14(2)
1.7 Aliasing
16(1)
1.8 Sampling Theorem
17(3)
1.9 Recovery of an Analog Signal
20(1)
1.10 Summary
21(1)
References
22(1)
2 Discrete-Time Systems and z-Transformation 23(34)
2.1 Preview
23(1)
2.2 Discrete-Time Signals
23(2)
2.3 z-Transform of Basic Sequences
25(4)
2.3.1 Fundamental Transforms
25(2)
2.3.2 Properties of z-Transform
27(2)
2.4 Inversion of z-Transforms
29(4)
2.4.1 Partial Fraction Expansion
30(1)
2.4.2 Power Series Expansion
31(1)
2.4.3 Contour Integration
32(1)
2.5 Parseval's Theorem
33(1)
2.6 Discrete-Time Systems
34(3)
2.7 Difference Equations
37(3)
2.8 State-Space Descriptions
40(2)
2.8.1 Realization 1
40(1)
2.8.2 Realization 2
41(1)
2.9 Frequency Transfer Functions
42(12)
2.9.1 Linear Time-Invariant Causal Systems
42(1)
2.9.2 Rational Transfer Functions
43(2)
2.9.3 All-Pass Digital Filters
45(3)
2.9.4 Notch Digital Filters
48(5)
2.9.5 Doubly Complementary Digital Filters
53(1)
2.10 Summary
54(1)
References
55(2)
3 Stability and Coefficient Sensitivity 57(10)
3.1 Preview
57(1)
3.2 Stability
57(7)
3.2.1 Definition
57(1)
3.2.2 Stability in Terms of Poles
58(2)
3.2.3 Schur-Cohn Criterion
60(1)
3.2.4 Schur-Cohn-Fujiwara Criterion
60(1)
3.2.5 Jury-Marden Criterion
61(1)
3.2.6 Stability Triangle of Second-Order Polynomials
62(1)
3.2.7 Lyapunov Criterion
62(2)
3.3 Coefficient Sensitivity
64(1)
3.4 Summary
65(1)
References
66(1)
4 State-Space Models 67(30)
4.1 Preview
67(1)
4.2 Controllability and Observability
67(3)
4.3 Transfer Function
70(3)
4.3.1 Impulse Response
70(1)
4.3.2 Faddeev's Formula
71(2)
4.3.3 Cayley-Hamilton's Theorem
73(1)
4.4 Equivalent Systems
73(8)
4.4.1 Equivalent Transformation
73(1)
4.4.2 Canonical Forms
74(5)
4.4.3 Balanced, Input-Normal, and Output-Normal State-Space Models
79(2)
4.5 Kalman's Canonical Structure Theorem
81(4)
4.6 Hankel Matrix and Realization
85(6)
4.6.1 Minimal Realization
85(2)
4.6.2 Minimal Partial Realization
87(2)
4.6.3 Balanced Realization
89(2)
4.7 Discrete-Time Lossless Bounded-Real Lemma
91(4)
4.8 Summary
95(1)
References
96(1)
5 FIR Digital Filter Design 97(38)
5.1 Preview
97(1)
5.2 Filter Classification
98(2)
5.3 Linear-phase Filters
100(8)
5.3.1 Frequency Transfer Function
100(1)
5.3.2 Symmetric Impulse Responses
101(3)
5.3.3 Antisymmetric Impulse Responses
104(4)
5.4 Design Using Window Function
108(6)
5.4.1 Fourier Series Expansion
108(2)
5.4.2 Window Functions
110(1)
5.4.3 Frequency Transformation
111(3)
5.5 Least-Squares Design
114(3)
5.5.1 Quadratic-Measure Minimization
114(2)
5.5.2 Eigenfilter Method
116(1)
5.6 Analytical Approach
117(3)
5.6.1 General FIR Filter Design
117(1)
5.6.2 Linear-Phase FIR Filter Design
118(2)
5.7 Chebyshev Approximation
120(4)
5.7.1 The Parks-McClellan Algorithm
120(1)
5.7.2 Alternation Theorem
121(3)
5.8 Cascaded Lattice Realization of FIR Digital Filters
124(4)
5.9 Numerical Experiments
128(5)
5.9.1 Least-Squares Design
128(1)
5.9.1.1 Quadratic measure minimization
128(1)
5.9.1.2 Eigenfilter method
128(1)
5.9.2 Analytical Approach
129(2)
5.9.2.1 General FIR filter design
129(1)
5.9.2.2 Linear-Phase FIR filter design
130(1)
5.9.3 Chebyshev Approximation
131(1)
5.9.4 Comparison of Algorithms' Performances
132(1)
5.10 Summary
133(1)
References
134(1)
6 Design Methods Using Analog Filter Theory 135(16)
6.1 Preview
135(1)
6.2 Design Methods Using Analog Filter Theory
135(14)
6.2.1 Lowpass Analog-Filter Approximations
136(4)
6.2.1.1 Butterworth approximation
136(1)
6.2.1.2 Chebyshev approximation
136(1)
6.2.1.3 Inverse-Chebyshev approximation
137(1)
6.2.1.4 Elliptic approximation
138(2)
6.2.2 Other Analog-Filter Approximations by Transformations
140(1)
6.2.2.1 Lowpass-to-lowpass transformation
140(1)
6.2.2.2 Lowpass-to-highpass transformation
140(1)
6.2.2.3 Lowpass-to-bandpass transformation
140(1)
6.2.2.4 Lowpass-to-bandstop transformation
141(1)
6.2.3 Design Methods Based on Analog Filter Theory
141(10)
6.2.3.1 Invariant impulse-response method
141(2)
6.2.3.2 Bilinear-transformation method
143(6)
6.3 Summary
149(1)
References
150(1)
7 Design Methods in the Frequency Domain 151(22)
7.1 Preview
151(1)
7.2 Design Methods in the Frequency Domain
151(13)
7.2.1 Minimum Mean Squared Error Design
151(4)
7.2.2 An Equiripple Design by Linear Programming
155(2)
7.2.3 Weighted Least-Squares Design with Stability Constraints
157(4)
7.2.4 Minimax Design with Stability Constraints
161(3)
7.3 Design of All-Pass Digital Filters
164(7)
7.3.1 Design of All-Pass Filters Based on Frequency Response Error
164(3)
7.3.2 Design of All-Pass Filters Based on Phase Characteristic Error
167(3)
7.3.3 A Numerical Example
170(1)
7.4 Summary
171(1)
References
172(1)
8 Design Methods in the Time Domain 173(40)
8.1 Preview
173(2)
8.2 Design Based on Extended Pade's Approximation
175(3)
8.2.1 A Direct Procedure
176(1)
8.2.2 A Modified Procedure
177(1)
8.3 Design Using Second-Order Information
178(12)
8.3.1 A Filter Design Method
178(4)
8.3.2 Stability
182(3)
8.3.3 An Efficient Algorithm for Solving (8.35)
185(5)
8.4 Least-Squares Design
190(6)
8.5 Design Using State-Space Models
196(8)
8.5.1 Balanced Model Reduction
196(3)
8.5.2 Stability and Minimality
199(5)
8.6 Numerical Experiments
204(6)
8.6.1 Design Based on Extended Pade's Approximation
204(1)
8.6.2 Design Using Second-Order Information
205(3)
8.6.3 Least-Squares Design
208(1)
8.6.4 Design Using State-Space Model (Balanced Model Reduction)
209(1)
8.6.5 Comparison of Algorithms' Performances
209(1)
8.7 Summary
210(1)
References
211(2)
9 Design of Interpolated and FRM FIR Digital Filters 213(26)
9.1 Preview
213(1)
9.2 Basics of IFIR and FRM Filters and CCP
213(5)
9.2.1 Interpolated FIR Filters
213(1)
9.2.2 Frequency-Response-Masking Filters
214(3)
9.2.3 Convex-Concave Procedure (CCP)
217(1)
9.3 Minimax Design of IFIR Filters
218(4)
9.3.1 Problem Formulation
218(1)
9.3.2 Convexification of (9.10) Using CCP
219(2)
9.3.3 Remarks on Convexification in (9.13)-(9.14)
221(1)
9.4 Minimax Design of FRM Filters
222(3)
9.4.1 The Design Problem
222(1)
9.4.2 A CCP Approach to Solving (9.23)
223(2)
9.5 FRM Filters with Reduced Complexity
225(2)
9.5.1 Design Phase 1
225(1)
9.5.2 Design Phase 2
226(1)
9.6 Design Examples
227(7)
9.6.1 Design and Evaluation Settings
227(1)
9.6.2 Design of IFIR Filters
227(2)
9.6.3 Design of FRM Filters
229(5)
9.6.4 Comparisons with Conventional FIR Filters
234(1)
9.7 Summary
234(2)
References
236(3)
10 Design of a Class of Composite Digital Filters 239(14)
10.1 Preview
239(1)
10.2 Composite Filters and Problem Formulation
240(3)
10.2.1 Composite Filters
240(1)
10.2.2 Problem Formulation
241(2)
10.3 Design Method
243(5)
10.3.1 Design Strategy
243(1)
10.3.2 Solving (10.7) with y Fixed to y = Yk
243(1)
10.3.3 Updating y with x Fixed to x = xk
244(3)
10.3.4 Summary of the Algorithm
247(1)
10.4 Design Example and Comparisons
248(2)
10.5 Summary
250(1)
References
250(3)
11 Finite Word Length Effects 253(20)
11.1 Preview
253(1)
11.2 Fixed-Point Arithmetic
254(3)
11.3 Floating-Point Arithmetic
257(1)
11.4 Limit Cycles-Overflow Oscillations
257(3)
11.5 Scaling Fixed-Point Digital Filters to Prevent Overflow
260(2)
11.6 Roundoff Noise
262(1)
11.7 Coefficient Sensitivity
263(1)
11.8 State-Space Descriptions with Finite Word Length
264(2)
11.9 Limit Cycle-Free Realization
266(4)
11.10 Summary
270(1)
References
270(3)
12 l2-Sensitivity Analysis and Minimization 273(26)
12.1 Preview
273(1)
12.2 l2-Sensitivity Analysis
274(3)
12.3 Realization with Minimal l2-Sensitivity
277(3)
12.4 l2-Sensitivity Minimization Subject to l2-Scaling Constraints Using Quasi-Newton Algorithm
280(5)
12.4.1 l2-Scaling and Problem Formulation
280(1)
12.4.2 Minimization of (12.18) Subject to l2-Scaling Constraints - Using Quasi-Newton Algorithm
281(2)
12.4.3 Gradient of J (x)
283(2)
12.5 l2-Sensitivity Minimization Subject to l2-Scaling Constraints Using Lagrange Function
285(3)
12.5.1 Minimization of (12.19) Subject to l2-Scaling Constraints - Using Lagrange Function
285(2)
12.5.2 Derivation of Nonsingular T from P to Satisfy l2-Scaling Constraints
287(1)
12.6 Numerical Experiments
288(6)
12.6.1 Filter Description and Initial l2-Sensitivity
288(2)
12.6.2 l2-Sensitivity Minimization
290(1)
12.6.3 l2-Sensitivity Minimization Subject to l2-Scaling Constraints Using Quasi-Newton Algorithm
291(2)
12.6.4 l2-Sensitivity Minimization Subject to l2-Scaling Constraints Using Lagrange Function
293(1)
12.7 Summary
294(2)
References
296(3)
13 Pole and Zero Sensitivity Analysis and Minimization 299(28)
13.1 Preview
299(1)
13.2 Pole and Zero Sensitivity Analysis
300(6)
13.3 Realization with Minimal Pole and Zero Sensitivity
306(4)
13.3.1 Weighted Pole and Zero Sensitivity Minimization Without Imposing l2-Scaling Constraints
306(3)
13.3.2 Zero Sensitivity Minimization Subject to Minimal Pole Sensitivity
309(1)
13.4 Pole Zero Sensitivity Minimization Subject to l2-Scaling Constraints Using Lagrange Function
310(2)
13.4.1 l2-Scaling Constraints and Problem Formulation
310(1)
13.4.2 Minimization of (13.37) Subject to l2-Scaling Constraints - Using Lagrange Function
310(2)
13.4.3 Derivation of Nonsingular T from P to Satisfy l2-Scaling Constraints
312(1)
13.5 Pole and Zero Sensitivity Minimization Subject to l2-Scaling Constraints Using Quasi-Newton Algorithm
312(3)
13.5.1 l2-Scaling and Problem Formulation
312(1)
13.5.2 Minimization of (13.68) Subject to l2-Scaling Constraints - Using Quasi-Newton Algorithm
313(1)
13.5.3 Gradient of J(x)
314(1)
13.6 Numerical Experiments
315(8)
13.6.1 Filter Description and Initial Pole and Zero Sensitivity
315(1)
13.6.2 Weighted Pole and Zero Sensitivity Minimization Without Imposing l2-Scaling Constraints
316(2)
13.6.3 Weighted Pole and Zero Sensitivity Minimization Subject to l2-Scaling Constraints Using Lagrange Function
318(3)
13.6.4 Weighted Pole and Zero Sensitivity Minimization Subject to l2-Scaling Constraints Using Quasi-Newton Algorithm
321(2)
13.7 Summary
323(1)
References
324(3)
14 Error Spectrum Shaping 327(30)
14.1 Preview
327(1)
14.2 UR Digital Filters with High-Order Error Feedback
328(10)
14.2.1 Nth-Order Optimal Error Feedback
328(2)
14.2.2 Computation of Autocorrelation Coefficients
330(2)
14.2.3 Error Feedback with Symmetric or Antisymmetric Coefficients
332(6)
14.3 State-Space Filter with High-Order Error Feedback
338(11)
14.3.1 Nth-Order Optimal Error Feedback
338(3)
14.3.2 Computation of Q, for i = 0, 1, ··· , N - 1
341(1)
14.3.3 Error Feedback with Symmetric or Antisymmetric Matrices
342(7)
14.4 Numerical Experiments
349(6)
14.4.1 Example 1: An IIR Digital Filter
349(1)
14.4.2 Example 2: A State-Space Digital Filter
350(5)
14.5 Summary
355(1)
References
355(2)
15 Roundoff Noise Analysis and Minimization 357(26)
15.1 Preview
357(1)
15.2 Filters Quantized after Multiplications
358(6)
15.2.1 Roundoff Noise Analysis and Problem Formulation
358(4)
15.2.2 Roundoff Noise Minimization Subject to l2-Scaling Constraints
362(2)
15.3 Filters Quantized before Multiplications
364(9)
15.3.1 State-Space Model with High-Order Error Feedback
364(2)
15.3.2 Formula for Noise Gain
366(2)
15.3.3 Problem Formulation
368(1)
15.3.4 Joint Optimization of Error Feedback and Realization
368(4)
15.3.4.1 The Use of Quasi-Newton Algorithm
368(2)
15.3.4.2 Gradient of J(x)
370(2)
15.3.5 Analytical Method for Separate Optimization
372(1)
15.4 Numerical Experiments
373(6)
15.4.1 Filter Description and Initial Roundoff Noise
373(1)
15.4.2 The Use of Analytical Method in Section 15.2.2
374(1)
15.4.3 The Use of Iterative Method in Section 15.3.4
375(4)
15.5 Summary
379(1)
References
380(3)
16 Generalized Transposed Direct-Form II Realization 383(28)
16.1 Preview
383(1)
16.2 Structural Transformation
384(4)
16.3 Equivalent State-Space Realization
388(5)
16.3.1 State-Space Realization I
388(2)
16.3.2 State-Space Realization II
390(2)
16.3.3 Choice of {Di} Satisfying l2-Scaling Constraints
392(1)
16.4 Analysis of Roundoff Noise
393(4)
16.4.1 Roundoff Noise of p-Operator Transposed Direct-Form II Structure
393(3)
16.4.2 Roundoff Noise of Equivalent State-Space Realization
396(1)
16.5 Analysis of l2-Sensitivity
397(7)
16.5.1 l2-Sensitivity of p-Operator Transposed Direct-Form II Structure
397(3)
16.5.2 l2-Sensitivity of Equivalent State-Space Realization
400(4)
16.6 Filter Synthesis
404(2)
16.6.1 Computation of Roundoff Noise and l2-Sensitivity
404(1)
16.6.2 Choice of Parameters {gammi|i = 1, 2, ··· , n}
405(1)
16.6.3 Search of Optimal Vector gamma = [ gamma1, gamma2, ···, gamman]T
405(1)
16.7 Numerical Experiments
406(3)
16.8 Summary
409(1)
References
410(1)
17 Block-State Realization of IIR Digital Filters 411(34)
17.1 Preview
411(1)
17.2 Block-State Realization
412(7)
17.3 Roundoff Noise Analysis and Minimization
419(4)
17.3.1 Roundoff Noise Analysis
419(3)
17.3.2 Roundoff Noise Minimization Subject to l2-Scaling Constraints
422(1)
17.4 l2-Sensitivity Analysis and Minimization
423(18)
17.4.1 l2-Sensitivity Analysis
423(6)
17.4.2 l2-Sensitivity Minimization Subject to l2-Scaling Constraints
429(5)
17.4.2.1 Method 1: using a Lagrange function
429(3)
17.4.2.2 Method 2: using a Quasi-Newton algorithm
432(2)
17.4.3 l2-Sensitivity Minimization Without Imposing l2-Scaling Constraints
434(1)
17.4.4 Numerical Experiments
435(6)
17.5 Summary
441(1)
References
442(3)
Index 445(8)
About the Authors 453
Takao Hinamoto, Wu-Sheng Lu
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
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