Digital Filter Design and Realization

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RIVER PUBLISHERS SERIES IN SIGNAL, IMAGE AND SPEECH PROCESSING
Title Page - Digital Filter Design and Realization
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Contents
Preface
List of Figures
List of Tables
List of Abbreviations
Chapter 1 - Introduction
	1.1 Preview
	1.2 Terminology for Signal Analysis and Typical Signals
		1.2.1 Terminology for Signal Analysis
		1.2.2 Examples of Typical Signals
	1.3 Digital Signal Processing
		1.3.1 General Framework for Digital Signal Processing
		1.3.2 Advantages of Digital Signal Processing
		1.3.3 Disadvantages of Digital Signal Processing
	1.4 Analysis of Analog Signals
		1.4.1 The Fourier Series Expansion of Periodic Signals
		1.4.2 The Fourier Transform
		1.4.3 The Laplace Transform
	1.5 Analysis of Discrete-Time Signals
		1.5.1 Sampling an Analog Signal
		1.5.2 The Discrete-Time Fourier Transform
		1.5.3 The Discrete Fourier Transform (DFT)
		1.5.4 The z-Transform
	1.6 Sampling of Continuous-Time Sinusoidal Signals
	1.7 Aliasing
	1.8 Sampling Theorem
	1.9 Recovery of an Analog Signal
	1.10 Summary
	References
Chapter 2 - Discrete-Time Systems and z-Transformation
	2.1 Preview
	2.2 Discrete-Time Signals
	2.3 z-Transform of Basic Sequences
		2.3.1 Fundamental Transforms
		2.3.2 Properties of z-Transform
	2.4 Inversion of z-Transforms
		2.4.1 Partial Fraction Expansion
		2.4.2 Power Series Expansion
		2.4.3 Contour Integration
	2.5 Parseval’s Theorem
	2.6 Discrete-Time Systems
	2.7 Difference Equations
	2.8 State-Space Descriptions
		2.8.1 Realization 1
		2.8.2 Realization 2
	2.9 Frequency Transfer Functions
		2.9.1 Linear Time-Invariant Causal Systems
		2.9.2 Rational Transfer Functions
		2.9.3 All-Pass Digital Filters
		2.9.4 Notch Digital Filters
		2.9.5 Doubly Complementary Digital Filters
	2.10 Summary
	References
Chapter 3 - Stability and Coefficient Sensitivity
	3.1 Preview
	3.2 Stability
		3.2.1 Definition
		3.2.2 Stability in Terms of Poles
		3.2.3 Schur-Cohn Criterion
		3.2.4 Schur-Cohn-Fujiwara Criterion
		3.2.5 Jury-Marden Criterion
		3.2.6 Stability Triangle of Second-Order Polynomials
		3.2.7 Lyapunov Criterion
	3.3 Coefficient Sensitivity
	3.4 Summary
	References
Chapter 4 - State-Space Models
	4.1 Preview
	4.2 Controllability and Observability
	4.3 Transfer Function
		4.3.1 Impulse Response
		4.3.2 Faddeev’s Formula
		4.3.3 Cayley-Hamilton’s Theorem
	4.4 Equivalent Systems
		4.4.1 Equivalent Transformation
		4.4.2 Canonical Forms
		4.4.3 Balanced, Input-Normal, and Output-Normal State-Space Models
	4.5 Kalman’s Canonical Structure Theorem
	4.6 Hankel Matrix and Realization
		4.6.1 Minimal Realization
		4.6.2 Minimal Partial Realization
		4.6.3 Balanced Realization
	4.7 Discrete-Time Lossless Bounded-Real Lemma
	4.8 Summary
	References
Chapter 5 - FIR Digital Filter Design
	5.1 Preview
	5.2 Filter Classification
	5.3 Linear-phase Filters
		5.3.1 Frequency Transfer Function
		5.3.2 Symmetric Impulse Responses
		5.3.3 Antisymmetric Impulse Responses
	5.4 Design Using Window Function
		5.4.1 Fourier Series Expansion
		5.4.2 Window Functions
		5.4.3 Frequency Transformation
	5.5 Least-Squares Design
		5.5.1 Quadratic-Measure Minimization
		5.5.2 Eigenfilter Method
	5.6 Analytical Approach
		5.6.1 General FIR Filter Design
		5.6.2 Linear-Phase FIR Filter Design
	5.7 Chebyshev Approximation
		5.7.1 The Parks-McClellan Algorithm
		5.7.2 Alternation Theorem
	5.8 Cascaded Lattice Realization of FIR Digital Filters
	5.9 Numerical Experiments
		5.9.1 Least-Squares Design
			5.9.1.1 Quadratic measure minimization
			5.9.1.2 Eigenfilter method
		5.9.2 Analytical Approach
			5.9.2.1 General FIR filter design
			5.9.2.2 Linear-Phase FIR filter design
		5.9.3 Chebyshev Approximation
		5.9.4 Comparison of Algorithms’ Performances
	5.10 Summary
	References
Chapter 6 - Design Methods Using Analog Filter Theory
	6.1 Preview
	6.2 Design Methods Using Analog Filter Theory
		6.2.1 Lowpass Analog-Filter Approximations
			6.2.1.1 Butterworth approximation
			6.2.1.2 Chebyshev approximation
			6.2.1.3 Inverse-Chebyshev approximation
			6.2.1.4 Elliptic approximation
		6.2.2 Other Analog-Filter Approximations by Transformations
			6.2.2.1 Lowpass-to-lowpass transformation
			6.2.2.2 Lowpass-to-highpass transformation
			6.2.2.3 Lowpass-to-bandpass transformation
			6.2.2.4 Lowpass-to-bandstop transformation
		6.2.3 Design Methods Based on Analog Filter Theory
			6.2.3.1 Invariant impulse-response method
			6.2.3.2 Bilinear-transformation method
	6.3 Summary
	References
Chapter 7 - Design Methods in the Frequency Domain
	7.1 Preview
	7.2 Design Methods in the Frequency Domain
		7.2.1 Minimum Mean Squared Error Design
		7.2.2 An Equiripple Design by Linear Programming
		7.2.3 Weighted Least-Squares Design with Stability Constraints
		7.2.4 Minimax Design with Stability Constraints
	7.3 Design of All-Pass Digital Filters
		7.3.1 Design of All-Pass Filters Based on Frequency Response Error
		7.3.2 Design of All-Pass Filters Based on Phase Characteristic Error
		7.3.3 A Numerical Example
	7.4 Summary
	References
Chapter 8 - Design Methods in the Time Domain
	8.1 Preview
	8.2 Design Based on Extended Pade’s Approximation
		8.2.1 A Direct Procedure
		8.2.2 A Modified Procedure
	8.3 Design Using Second-Order Information
		8.3.1 A Filter Design Method
		8.3.2 Stability
		8.3.3 An Efficient Algorithm for Solving (8.35)
	8.4 Least-Squares Design
	8.5 Design Using State-Space Models
		8.5.1 Balanced Model Reduction
		8.5.2 Stability and Minimality
	8.6 Numerical Experiments
		8.6.1 Design Based on Extended Pade’s Approximation
		8.6.2 Design Using Second-Order Information
		8.6.3 Least-Squares Design
		8.6.4 Design Using State-Space Model (Balanced Model Reduction)
		8.6.5 Comparison of Algorithms’ Performances
	8.7 Summary
	References
Chapter 9 - Design of Interpolated and FRM FIR Digital Filters
	9.1 Preview
	9.2 Basics of IFIR and FRM Filters and CCP
		9.2.1 Interpolated FIR Filters
		9.2.2 Frequency-Response-Masking Filters
		9.2.3 Convex-Concave Procedure (CCP)
	9.3 Minimax Design of IFIR Filters
		9.3.1 Problem Formulation
		9.3.2 Convexification of (9.10) Using CCP
		9.3.3 Remarks on Convexification in (9.13)–(9.14)
	9.4 Minimax Design of FRM Filters
		9.4.1 The Design Problem
		9.4.2 A CCP Approach to Solving (9.23)
	9.5 FRM Filters with Reduced Complexity
		9.5.1 Design Phase 1
		9.5.2 Design Phase 2
	9.6 Design Examples
		9.6.1 Design and Evaluation Settings
		9.6.2 Design of IFIR Filters
		9.6.3 Design of FRM Filters
		9.6.4 Comparisons with Conventional FIR Filters
	9.7 Summary
	References
Chapter 10 - Design of a Class of Composite Digital Filters
	10.1 Preview
	10.2 Composite Filters and Problem Formulation
		10.2.1 Composite Filters
		10.2.2 Problem Formulation
	10.3 Design Method
		10.3.1 Design Strategy
		10.3.2 Solving (10.7) with y Fixed to y = yk
		10.3.3 Updating y with x Fixed to x = xk
		10.3.4 Summary of the Algorithm
	10.4 Design Example and Comparisons
	10.5 Summary
	References
Chapter 11 - FiniteWord Length Effects
	11.1 Preview
	11.2 Fixed-Point Arithmetic
	11.3 Floating-Point Arithmetic
	11.4 Limit Cycles—Overflow Oscillations
	11.5 Scaling Fixed-Point Digital Filters to Prevent Overflow
	11.6 Roundoff Noise
	11.7 Coefficient Sensitivity
	11.8 State-Space Descriptions with FiniteWord Length
	11.9 Limit Cycle-Free Realization
	11.10 Summary
	References
Chapter 12 - l2-Sensitivity Analysis and Minimization
	12.1 Preview
	12.2 l2-Sensitivity Analysis
	12.3 Realization with Minimal l2-Sensitivity
	12.4 l2-Sensitivity Minimization Subject to l2-Scaling Constraints Using Quasi-Newton Algorithm
		12.4.1 l2-Scaling and Problem Formulation
		12.4.2 Minimization of (12.18) Subject to l2-Scaling Constraints — Using Quasi-Newton Algorithm
		12.4.3 Gradient of J(x)
	12.5 l2-Sensitivity Minimization Subject to l2-Scaling Constraints Using Lagrange Function
		12.5.1 Minimization of (12.19) Subject to l2-Scaling Constraints — Using Lagrange Function
		12.5.2 Derivation of Nonsingular T from P to Satisfy l2-Scaling Constraints
	12.6 Numerical Experiments
		12.6.1 Filter Description and Initial l2-Sensitivity
		12.6.2 l2-Sensitivity Minimization
		12.6.3 l2-Sensitivity Minimization Subject to l2-Scaling Constraints Using Quasi-Newton Algorithm
		12.6.4 l2-Sensitivity Minimization Subject to l2-Scaling Constraints Using Lagrange Function
	12.7 Summary
	References
Chapter 13 - Pole and Zero Sensitivity Analysis and Minimization
	13.1 Preview
	13.2 Pole and Zero Sensitivity Analysis
	13.3 Realization with Minimal Pole and Zero Sensitivity
		13.3.1 Weighted Pole and Zero Sensitivity Minimization WithoutImposing l2-Scaling Constraints
		13.3.2 Zero Sensitivity Minimization Subject to Minimal Pole Sensitivity
	13.4 Pole Zero Sensitivity Minimization Subject to l2-Scaling Constraints Using Lagrange Function
		13.4.1 l2-Scaling Constraints and Problem Formulation
		13.4.2 Minimization of (13.37) Subject to l2-Scaling Constraints — Using Lagrange Function
		13.4.3 Derivation of Nonsingular T from P to Satisfyl2-Scaling Constraints
	13.5 Pole and Zero Sensitivity Minimization Subject to l2-Scaling Constraints Using Quasi-Newton Algorithm
		13.5.1 l2-Scaling and Problem Formulation
		13.5.2 Minimization of (13.68) Subject to l2-Scaling Constraints — Using Quasi-Newton Algorithm
		13.5.3 Gradient of J(x)
	13.6 Numerical Experiments
		13.6.1 Filter Description and Initial Pole and Zero Sensitivity
		13.6.2 Weighted Pole and Zero Sensitivity Minimization Without Imposing l2-Scaling Constraints
		13.6.3 Weighted Pole and Zero Sensitivity Minimization Subject to l2-Scaling Constraints Using Lagrange Function
		13.6.4 Weighted Pole and Zero Sensitivity Minimization Subject to l2-Scaling Constraints Using Quasi-Newton Algorithm
	13.7 Summary
	References
Chapter 14 - Error Spectrum Shaping
	14.1 Preview
	14.2 IIR Digital Filters with High-Order Error Feedback
		14.2.1 Nth-Order Optimal Error Feedback
		14.2.2 Computation of Autocorrelation Coefficients
		14.2.3 Error Feedback with Symmetric or Antisymmetric Coefficients
	14.3 State-Space Filter with High-Order Error Feedback
		14.3.1 Nth-Order Optimal Error Feedback
		14.3.2 Computation of Qi for i = 0, 1, · · · ,N − 1
		14.3.3 Error Feedback with Symmetric orAntisymmetric Matrices
	14.4 Numerical Experiments
		14.4.1 Example 1 : An IIR Digital Filter
		14.4.2 Example 2 : A State-Space Digital Filter
	14.5 Summary
	References
Chapter 15 - Roundoff Noise Analysis and Minimization
	15.1 Preview
	15.2 Filters Quantized after Multiplications
		15.2.1 Roundoff Noise Analysis and Problem Formulation
		15.2.2 Roundoff Noise Minimization Subject to l2-Scaling Constraints
	15.3 Filters Quantized before Multiplications
		15.3.1 State-Space Model with High-Order Error Feedback
		15.3.2 Formula for Noise Gain
		15.3.3 Problem Formulation
		15.3.4 Joint Optimization of Error Feedback and Realization
			15.3.4.1 The Use of Quasi-Newton Algorithm
			15.3.4.2 Gradient of J(x)
		15.3.5 Analytical Method for Separate Optimization
	15.4 Numerical Experiments
		15.4.1 Filter Description and Initial Roundoff Noise
		15.4.2 The Use of Analytical Method in Section 15.2.2
		15.4.3 The Use of Iterative Method in Section 15.3.4
	15.5 Summary
	References
Chapter 16 - Generalized Transposed Direct-Form II Realization
	16.1 Preview
	16.2 Structural Transformation
	16.3 Equivalent State-Space Realization
		16.3.1 State-Space Realization I
		16.3.2 State-Space Realization II
		16.3.3 Choice of {Δi} Satisfying l2-Scaling Constraints
	16.4 Analysis of Roundoff Noise
		16.4.1 Roundoff Noise of ρ-Operator Transposed Direct-Form II Structure
		16.4.2 Roundoff Noise of Equivalent State-Space Realization
	16.5 Analysis of l2-Sensitivity
		16.5.1 l2-Sensitivity of ρ-Operator Transposed Direct-Form II Structure
		16.5.2 l2-Sensitivity of Equivalent State-Space Realization
	16.6 Filter Synthesis
		16.6.1 Computation of Roundoff Noise and l2-Sensitivity
		16.6.2 Choice of Parameters {γi| i = 1, 2, · · · , n}
		16.6.3 Search of Optimal Vector γ = [γ1, γ2, · · · , γn]T
	16.7 Numerical Experiments
	16.8 Summary
	References
Chapter 17 - Block-State Realization of IIR Digital Filters
	17.1 Preview
	17.2 Block-State Realization
	17.3 Roundoff Noise Analysis and Minimization
		17.3.1 Roundoff Noise Analysis
		17.3.2 Roundoff Noise Minimization Subject to l2-Scaling Constraints
	17.4 l2-Sensitivity Analysis and Minimization
		17.4.1 l2-Sensitivity Analysis
		17.4.2 l2-Sensitivity Minimization Subject to l2-Scaling Constraints
			17.4.2.1 Method 1: using a Lagrange function
			17.4.2.2 Method 2: using a Quasi-Newton algorithm
			17.4.3 l2-Sensitivity Minimization Without Imposing l2-Scaling Constraints
			17.4.4 Numerical Experiments
	17.5 Summary
	References
Index
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