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INTRODUCTION TO DIGITAL FILTERS
WITH AUDIO APPLICATIONS

JULIUS O. SMITH III

Center for Computer Research in Music and Acoustics (CCRMA)
Department of Music, Stanford University, Stanford, California 94305 USA

August 2006 Edition

Preface
- Outline
- Book Series Overview
- Acknowledgments
The Simplest Lowpass Filter
- Introduction
  - What is a Filter?
  - Why should a musician 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++
- 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
- FIR Digital Filters
- Transient and Steady State Response
  - Steady State Signals
  - Decay Response
- Matrix Filter Representations
- Time Domain Filter Estimation
  - Effect of Measurement Noise
  - Matlab System Identification Example
- 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 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
- Complex Sinewave Analysis Revisited
- Real Sinewave Analysis
- From Sinewaves to Arbitrary Signals
- Phase of the Ideal Lowpass Filter
- Phase Response Applications
- Frequency Response Analysis Problems
Pole-Zero Analysis
- Graphical Amplitude Response
- Graphical Phase Response
- Stability Revisited
- Poles and Zeros of the Cepstrum
- Pole-Zero Analysis Problems
Implementation Structures
- The Four Direct Forms
- Series and Parallel Filter Sections
- Pole-Zero Analysis Problems
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 Shelving Filters
  - Exercise
- Peaking Equalizers
- Time-Varying Two-Pole Filters
- Elementary Filter Problems
Filters Preserving Phase
- Zero-Phase Filters
- Example Zero-Phase Filters
- Odd Impulse Reponses
- Symmetric Linear-Phase Filters
- Example Linear-Phase Filters
- Antisymmetric Linear-Phase Filters
- Forward-Backwards Filtering
- Phase Distortion at Passband Edges
Minimum Phase Digital Filters
- Definition of Minimum Phase
- Minimum-Phase/Allpass Decomposition
- Linear Phase Audio Filters
- Minimum Phase Filters and Signals
Conclusion
Background Fundamentals
- Signal Representation and Notation
- Complex and Trigonometric Identities
- Sinusoids as Eigenfunctions of LTI Systems
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)
  - Decay Time is Q Periods
  - Q as Energy Stored over Energy Dissipated
- Analog Allpass Filters
  - Lossless Analog Filters
Allpass Filters
- Allpass Examples
- Paraunitary Filters
- MIMO Allpass Filters
  - Paraunitary MIMO Filters
  - Paraunitary Filter Examples
- Allpass Problems
State Space Models
- Markov Parameters
- Response from Initial Conditions
- Complete Response
- Transfer Function of a State Space Model
  - Example State Space Model Transfer Function
- Transposition of a State Space Model
- Poles of a State Space Model
- 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.m
- 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.m
FIR Filter Implementation in C++
- Header file FIR.h
- Class definition file FIR.cpp
Links to Online Resources
Bibliography
Index
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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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