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INTRODUCTION TO DIGITAL FILTERS
WITH AUDIO APPLICATIONS

JULIUS O. SMITH III
Center for Computer Research in Music and Acoustics (CCRMA)

Preface
- Outline
- Book Series Overview
- Acknowledgments
- Errata
The Simplest Lowpass Filter
- Introduction
  - What is a Filter?
  - Why learn about filters?
- The Simplest Lowpass Filter
  - Definition of the Simplest Low-Pass
- Finding the Frequency Response
- An Easier Way
- Summary
- Elementary Filter Theory Problems
Matlab Filter Analysis
- Matlab Filter Implementation
- Matlab Sine-Wave Analysis
- Complex Sine-Wave Analysis
- Practical Frequency-Response Analysis
- Elementary Matlab Problems
Analysis of a Digital Comb Filter
- Difference Equation
- Signal Flow Graph
- Software Implementation in Matlab
  - Sample-Level Implementation in Matlab
- Software Implementation in C++
- Software Implementation in Faust
- Impulse Response
- Transfer Function
- Frequency Response
- Amplitude Response
- Phase Response
- Pole-Zero Analysis
- Alternative Realizations
- Summary
Linear Time-Invariant Filters
- Definition of a Signal
- Definition of a Filter
- Examples of Digital Filters
- Linear Filters
- Time-Invariant Filters
- Showing Linearity and Time Invariance
- Dynamic Range Compression
  - Why Dynamic Range Compression is Nonlinear
- A Musical Time-Varying Filter Example
- Analysis of Nonlinear Filters
- Conclusions
- Linearity and Time-Invariance Problems
Time Domain Representations
- Difference Equation
- Signal Flow Graph
- Causal Recursive Filters
- Filter Order
- Direct-Form-I Implementation
- Impulse-Response Representation
- Filter Stability
- Impulse Response Example
- Implications of Linear-Time-Invariance
- Convolution Representation
  - Convolution Representation Summary
- FIR Digital Filters
- Transient and Steady State Response
- Summary and Conclusions
- Time Domain Representation Problems
Transfer Function Analysis
- The Z Transform
- Existence of the Z Transform
- Shift and Convolution Theorems
  - Shift Theorem
  - Convolution Theorem
- Z Transform of Convolution
- Z Transform of Difference Equations
- Factored Form
- Series and Parallel Transfer Functions
  - Series Case
  - Parallel Case
    - Series Combination is Commutative
- Partial Fraction Expansion
- Problems
Frequency Response Analysis
- Frequency Response
- Amplitude Response
- Phase Response
- Polar Form of the Frequency Response
  - Separating the Transfer Function Numerator and Denominator
- Frequency Response as a Ratio of DTFTs
  - Frequency Response in Matlab
  - Example LPF Frequency Response Using freqz
- Phase and Group Delay
- Frequency Response Analysis Problems
Pole-Zero Analysis
- Filter Order = Transfer Function Order
- Graphical Amplitude Response
- Graphical Phase Response
- Stability Revisited
- Bandwidth of One Pole
- Time Constant of One Pole
- Unstable Poles--Unit Circle Viewpoint
  - Geometric Series
  - One-Pole Transfer Functions
- Poles and Zeros of the Cepstrum
- Conversion to Minimum Phase
- Hilbert Transform Relations
- Pole-Zero Analysis Problems
Implementation Structures
- The Four Direct Forms
- Series and Parallel Filter Sections
- Pole-Zero Analysis Problems
Filters Preserving Phase
- Linear-Phase Filters
- Zero-Phase Filters
- Odd Impulse Reponses
- Symmetric Linear-Phase Filters
  - Simple Linear-Phase Filter Examples
  - Software for Linear-Phase Filter Design
- Antisymmetric Linear-Phase Filters
- Forward-Backward Filtering
- Phase Distortion at Passband Edges
Minimum-Phase Filters
- Definition of Minimum Phase Filters
- Minimum-Phase Polynomials
- Maximum Phase Filters
  - Example
- Minimum Phase Means Fastest Decay
- Minimum-Phase/Allpass Decomposition
- Linear Phase Audio Filters
- Creating Minimum Phase
Conclusion
Background Fundamentals
- Signal Representation and Notation
- Complex and Trigonometric Identities
- Sinusoids as Eigenfunctions of LTI Systems
Elementary Audio Digital Filters
- Elementary Filter Sections
- Allpass Filter Sections
  - The Biquad Allpass Section
  - Allpass Filter Design
- DC Blocker
  - DC Blocker Frequency Response
  - DC Blocker Software Implementations
- Low and High Shelf Filters
  - Exercise
- Peaking Equalizers
- Time-Varying Two-Pole Filters
- Elementary Filter Problems
Allpass Filters
- Allpass Examples
- Paraunitary Filters
- MIMO Allpass Filters
  - Paraunitary MIMO Filters
  - Paraunitary Filter Examples
- Allpass Problems
Laplace Transform Analysis
- Existence of the Laplace Transform
- Analytic Continuation
- Relation to the z Transform
- Laplace Transform Theorems
  - Linearity
  - Differentiation
- Laplace Analysis of Linear Systems
  - Moving Mass
  - Mass-Spring Oscillator Analysis
Analog Filters
- Example Analog Filter
- Capacitors
  - Mechanical Equivalent of a Capacitor is a Spring
- Inductors
  - Mechanical Equivalent of an Inductor is a Mass
- RC Filter Analysis
- RLC Filter Analysis
- Relating Pole Radius to Bandwidth
- Quality Factor (Q)
- Analog Allpass Filters
  - Lossless Analog Filters
Matrix Filter Representations
- Introduction
- General Causal Linear Filter Matrix
- General LTI Filter Matrix
- Cyclic Convolution Matrix
- Inverse Filters
- State Space Realization
  - State Space Filter Realization Example
- Time Domain Filter Estimation
  - Effect of Measurement Noise
  - Matlab System Identification Example
State Space Filters
- Markov Parameters
- Response from Initial Conditions
- Complete Response
- Transfer Function of a State Space Filter
  - Example State Space Filter Transfer Function
- Transposition of a State Space Filter
- Poles of a State Space Filter
- Difference Equations to State Space
- Similarity Transformations
- Modal Representation
- Repeated Poles
  - Jordan Canonical Form
- Digital Waveguide Oscillator Example
  - Finding the Eigenstructure of A
  - Choice of Output Signal and Initial Conditions
- References
- State Space Problems
Linear Time-Varying Filters
- Introduction
- Derivation
- Summary
Recursive Digital Filter Design
- Lowpass Filter Design
- Butterworth Lowpass Design
  - Butterworth Lowpass Poles and Zeros
  - Example: Second-Order Butterworth Lowpass
- Bilinear A/D Transformation
- Equation-Error Filter Design
Matlab Utilities
- Time Plots: myplot.m
- Frequency Plots: freqplot.m
- Saving Plots to Disk: saveplot.m
- Frequency Response Plots: plotfr.m
- Partial Fraction Expansion: residuez.m
  - Method
  - Example with Repeated Poles
- Partial Fraction Expansion: residued.m
  - Method
- Parallel SOS to Transfer Function: psos2tf
- Group Delay Computation: grpdelay.m
- Matlab listing: fold.m
- Matlab listing: clipdb.m
- Matlab listing: mps.m and test program
- Signal Plots: swanalplot.m
- Frequency Response Plot: swanalmainplot
Digital Filtering in Faust and PD
- A Simple Faust Program
- Generating Faust Block Diagrams
- Testing a Faust Filter Section
- A Look at the Generated C++ code
- Generating a Pure Data (PD) Plugin
- Generating a LADSPA Plugin via Faust
- Generating a VST Plugin via Faust
  - Bypassing Windows
- Generating a MIDI Synthesizer for PD
- MIDI Synthesizer Test Patch
Links to Online Resources
Bibliography
Index for this Document
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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024年09月03日 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
CCRMA