FILTER DESIGN FOR SIGNAL PROCESSING USING MATLAB AND MATHEMATICAL

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1 FILTER DESIGN FOR SIGNAL PROCESSING USING MATLAB AND MATHEMATICAL Miroslav D. Lutovac The University of Belgrade Belgrade, Yugoslavia Dejan V. Tosic The University of Belgrade Belgrade, Yugoslavia Brian L. Evans The University of Texas at Austin Austin, Texas PRENTICE HALL Upper Saddle River, New Jersey 07458

2 CONTENTS Foreword XXI Prpfare 1 SIGNALS XXV Signal Classification Continuous or Analog Signals Discrete-Time Signals Digital Signals Deterministic Signals Random (Nondeterministic) Signals The Sampling Theorem Basic Continuous-Time Signals Functions Sinusoidal Signals Real-Valued Exponential Signals Unit Step Signal Pulse Signals Unit Ramp Signal Unit Impulse Signal Causal Signals Combining Signals Basic Discrete-Time Signals Sequences Sinusoidal Sequences Real-Valued Exponential Sequences 18

3 vi Contents Unit Step Sequence Unit Ramp Sequence Unit Impulse Sequence Causal Sequences Continuous-Time Signals in MATLAB Sequences in MATLAB Continuous-Time Signals in Mathematica Sequences in Mathematica 31 Problems 35 MATLAB Exercises 38 Mathematica Exercises 39 2 SYSTEMS Basic Definitions Block Diagrams System Properties State and Relaxed Systems Causality and Realizable Systems Stability Time Invariance Linearity and Superposition Memoryless Systems Sensitivity Optimal and Adaptive Systems Invertibility Linear Time-Invariant Systems The Differential Operator Response of Continuous-Time LTI Systems Transient System Specifications The Shifting Operator for Difference Equations Discrete-Time LTI Systems Properties of LTI Systems 57 Problems 58 MATLAB Exercises 62 Mathematica Exercises 62

4 Contents vii 3 TRANSFORMS Phasor Transformation The Representation of a Sinusoid by a Phasor Properties of the Phasor Transform Application of the Phasor Transform Sinusoidal Steady-State Response Nonsinusoidal Steady-State Response Transfer Function and Frequency Response Application Example of Phasor Method Fourier Series and Harmonic Analysis The Fourier Series Complex Form of the Fourier Series Parseval's Identity Harmonics of Periodic Signals Gibbs Phenomenon Amplitude and Phase Spectrum of Periodic Signals Steady-State Response of a System to a Nonsinusoidal Periodic Stimulus Fourier Transform Definition of the Fourier Transform Properties of the Fourier Transform Convolution Parseval's Theorem and Energy Spectral Density Properties of the Fourier Transform of Real Signals Causal Signals and the Hilbert Transform Application of the Fourier Transform Fourier Transform of Sampled Signals and the Sampling Theorem Fourier Transform of Periodic Signals and Power Spectral Density Fourier Transform and the Phasor Method Laplace Transform Definition of the Laplace Transform Properties of the Laplace Transform The Inverse Laplace Transform of Rational Functions Transfer Function of Continuous-Time Systems Discrete Fourier Transform Definition of the Discrete Fourier Transform Properties of the Discrete Fourier Transform Computation of the Fourier Series Coefficients by DFT Computation of the Fourier Integral by DFT Frequency Response of Discrete-Time Systems The z Transform Definition of the z Transform Properties of the z Transform Transfer Function of Discrete-Time Systems Analysis of LTI Systems by Transform Method Continuous-Time LTI Systems Discrete-Time LTI Systems 126

5 viii Contents Analog LTI Circuits 131 Problems 134 MATLAB Exercises 137 Mathematica Exercises CLASSIC ANALOG FILTER DESIGN Introduction to Analog Filters Basic Filter Transfer Functions Second-Order Transfer Functions Decomposition of Transfer Functions 4.4 Pole-Zero Pairing 4.5 Optimum Cascading Sequence 4.6 Sensitivity Basic Definitions Sensitivity of Second-Order Transfer Function Sensitivity to Passive Components Gain-Sensitivity Product (GSP) Analog Filter Realizations 4.8 Op Amp Active RC Filters Low-Q-Factor Biquadratic Realizations Medium-Q-Factor Biquadratic Realizations High-Q-Factor Biquadratic Realizations Switched-Capacitor (SC) Filters Mode 1 SC Realization Mode 2 SC Realization Mode 3 SC Realization Mode 4 SC Realization Mode 5 SC Realization Modes 6 and 7 SC Realizations Low-Sensitive Lowpass Notch Realization Programmable Lowpass/Highpass SC Filters Passive RLC Filters Singly Terminated Ladder Realization Doubly Terminated Ladder Networks Quality Factor of Inductors and Capacitors Operational Transconductance Amplifier (OTA) Filters Biquadratic OTA-C Realizations Current-Conveyor (CC) Filters Current-Conveyor Filters Based on Passive RLC Filters Current-Conveyor Filters Derived from Op Amp Active RC Filters 233 Problems MATLAB Exercises Mathematica Exercises

6 Contents ix 5 ADVANCED ANALOG FILTER DESIGN CASE STUDIES Basic Definitions Specification of an Analog Filter Approximation Problem Design Space Basic Design Alternatives Design Dl Design D Design D3a Design D3b Design D4a Design D4b Remarks on Design Alternatives Visualization of Design Space SC Filter Advanced Design Example Introduction Mode 1 Operation Modification to Mode 1 Operation Cascaded Filter Realization Practical Implementation Concluding Remarks 273 Problems MATLAB Exercises Mathematica Exercises ADVANCED ANALOG FILTER DESIGN ALGORITHMS Introduction Notation Design Equations and Procedures Specification Special and Auxiliary Functions Transfer Function Order Zeros, Poles, and Q -Factors Discrimination Factor, Elliptic Rational Function, and Characteristic Function Normalized Lowpass Elliptic Transfer Function Selectivity Factor, Ripple Factor, and Edge Frequencies 289 Design Dl Design D2 Design D3a Design D3b Design D4a Design D4b Desien D

7 Contents 6.11 Time Response and Frequency Response 6.12 Highpass Filter 6.13 Bandpass Filter 6.14 Bandreject Filter 6.15 Concluding Remarks 7 MULTICRITERIA OPTIMIZATION OF ANALOG FILTER DESIGNS Introduction Notation Objective Function Deviation from a Desired Magnitude Response Deviation from an Ideal Phase Response Deviation from Desired Quality Factor Deviation from Desired Peak Overshoot in the Step Response The Complete Objective Function Constraints Example Filter Designs Allpole Filter with Near-Linear Phase Response and Minimal Peak Overshoot Filter with Near-Linear Phase Response and Minimal Peak Overshoot Filter Simultaneously Optimized for Four Criteria Validation and Verification of the Automated Framework Verification of Formulas and Code Generation Generating Working MATLAB Programs Validating the MATLAB Programs Conclusion CLASSIC DIGITAL FILTER DESIGN 1.1 Introduction to Digital Filters 1.2 Basic Filter Transfer Functions Second-Order Transfer Functions Decomposition of Transfer Functions 1.4 Pole-Zero Pairing >.5 Optimum Cascading Sequence i.6 Finite Wordlength Effects Coefficient Quantization Basic Definitions of Sensitivity Sensitivity of Second-Order Transfer Function Overflow Arithmetic Operations Limit Cycles and Overflow Oscillations Signal Quantization Error Roundoff Noise 359

8 Contents xi 8.7 Digital Filter Realizations Direct-Form I Realization Direct-Form II Realization Transposed Direct-Form II Realization Ansari-Liu Allpass Realizations Wave Digital Filter Realizations Comparison of Realizations 378 Problems 382 MATLAB Exercises 385 Mathematica Exercises ADVANCED DIGITAL FILTER DESIGN CASE STUDIES Basic Definitions Specification of a Digital Filter Approximation Step Design Space Basic Design Alternatives Design Dl Design D Design D3a Design D3b Design D4a Design D4b Remarks on Design Alternatives Visualization of Design Space Elliptic Half-Band IIR Filters Definition of Half-Band Filters Elliptic Minimal Q-Factor Transfer Functions /3-Band IIR Filters Comparison of FIR and IIR Half-Band Filters Parallel Realization of Elliptic Half-Band IIR Filters Hilbert Transformer Half-Band Filter and Hilbert Transformer Design of Hilbert Transformer Multiplierless Elliptic IIR Filters Introduction Amplitude Response Sensitivity for Allpass Realization of IIR Filters Sensitivity Functions Phase Tolerance Scheme EMQF Transfer Function Multiplier Coefficients in Allpass Sections js max Coefficient in Allpass Sections Multiplierless Elliptic IIR Filter Design The Choice of a The Choice of /S max 446

9 xii Contents The Choice of an and ft Applications Conclusion Linear Phase IIR Filters Filtering Finite Length Sequences Filtering Infinite Sequences 459 Problems 464 MATLAB Exercises 467 Mathematica Exercises ADVANCED DIGITAL FILTER DESIGN ALGORITHMS 10.1 Introduction 10.2 Notation 10.3 Design Equations and Procedures Specification Special and Auxiliary Functions Transfer Function Order Bilinear Transformation Zeros, Poles, and Q-Factors Discrimination Factor, Elliptic Rational Function, and Characteristic Function Lowpass Elliptic Transfer Function Selectivity Factor, Ripple Factor, and Edge Frequencies Design Dl 10.5 Design D Design D3a 10.7 Design D3b 10.8 Design D4a 10.9 Design D4b Design D Frequency Response Highpass Filter Bandpass Filter Bandreject Filter Concluding Remarks MULTICRITERIA OPTIMIZATION OF DIGITAL FILTER DESIGNS Introduction 11.2 Notation 11.3 Objective Function Deviation in Magnitude Response Deviation in the Phase Response Filter Quality Factor The Complete Objective Function 500

10 Contents xiii 11.4 Constraints Example Conclusion ELLIPTIC FUNCTIONS Legendre Elliptic Integral Jacobi Elliptic Functions Periods of Elliptic Functions Series Representation and Modular Constant Chebyshev Polynomials Elliptic Rational Function Nesting Property of Jacobi Elliptic Functions Nesting Property of Rational Elliptic Functions Poles of Normalized Transfer Function Exact Formulas for Transfer Function Poles Exact Formulas for Zeros of Elliptic Rational Functions Approximate Determination of Zeros The Degree Equation Discrimination Factor Elliptic Filters with Minimal Q-Factors 564 Problems 567 MATLAB Exercises 567 Mathematica Exercises ELLIPTIC RATIONAL FUNCTION Chebyshev Polynomials Second-Order Elliptic Rational Function Elliptic Rational Functions of Order 2' Third-Order Elliptic Rational Functions Elliptic Rational Functions of Order 2' 3^ Fifth-Order Elliptic Rational Functions Modified Elliptic Rational Functions Poles, Zeros, and Selectivity Factor Exact Formulas for Zeros Approximate Determination of Zeros Discrimination Factor Selectivity Factor Versus Discrimination Factor Transfer Function Poles First-Order r -Function Second-Order t -Function Third-Order f -Function Higher-Order t -Function Alternative Method for Computing Transfer Function Poles 624

11 xiv Contents Transfer Function with Minimal Q-Factors 625 Problems 630 MATLAB Exercises 630 Mathematica Exercises 631 A EXAMPLE MATHEMATICA NOTEBOOKS 633 A.I Analysis by Transform Method of Analog LTI Circuits 634 A.2 Analysis by Transform Method of Discrete-Time LTI Systems 638 A.3 Switched Capacitor Filter - Mode 3a Analysis and Design 644 A.4 OTA-C General Biquad Analysis and Design 651 A.5 Lowpass-Medium-Q-Factor Active RC Filter Analysis and Design 656 A.6 General Purpose High-Q-Factor RC Filter Analysis and Design 662 A.7 Advanced Analog Filter Design Case Studies 670 A.8 Classical Digital Filters Transpose Direct Form II IIR 2nd-Order Realization 685 A.9 Advanced Digital Filter Design Case Studies 698 A.10 Jacobi Elliptic Functions 713 A.ll Elliptic Rational Function 716 A.12 Transfer Function with Minimal Q-Factors 728 References 742 Index 749

DISCRETE-TIME SIGNAL PROCESSING

THIRD EDITION DISCRETE-TIME SIGNAL PROCESSING ALAN V. OPPENHEIM MASSACHUSETTS INSTITUTE OF TECHNOLOGY RONALD W. SCHÄFER HEWLETT-PACKARD LABORATORIES Upper Saddle River Boston Columbus San Francisco New