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                   INTRODUCTION TO DIGITAL FILTERS
                       WITH AUDIO APPLICATIONS

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

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  * Preface
      + Outline
      + Book Series Overview
      + Acknowledgments
      + Errata

  * The Simplest Lowpass Filter
      + Introduction
          o What is a Filter?
          o Why learn about filters?
      + The Simplest Lowpass Filter
          o Definition of the Simplest Low-Pass
      + Finding the Frequency Response
          o Sine-Wave Analysis
          o Mathematical Sine-Wave Analysis
          o Amplitude Response
          o Phase Response
      + An Easier Way
          o Complex Sinusoids
          o Complex Amplitude
          o Phasor Notation
          o Complex Sinusoids as Circular Motion
          o Rederiving the Frequency Response
      + 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
          o 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
          o First-Order Parallel Sections
          o Parallel, Real, Second-Order Sections
          o Parallel Second-Order Signal Flow Graph
          o Series, Real, Second-Order Sections
      + Summary

  * Linear Time-Invariant Filters
      + Definition of a Signal
      + Definition of a Filter
      + Examples of Digital Filters
      + Linear Filters
          o Scaling:
          o Superposition:
          o Real Linear Filtering of Complex Signals
      + Time-Invariant Filters
      + Showing Linearity and Time Invariance
      + Dynamic Range Compression
          o 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
          o Convolution Representation Summary
      + FIR Digital Filters
          o FIR impulse response
          o Convolution Representation of FIR Filters
          o The ``Finite'' in FIR
          o Causal FIR Filters
          o FIR Transfer Function
          o FIR Order
          o FIR Software Implementations
      + Transient and Steady State Response
          o FIR Example
          o IIR Example
          o Transient and Steady-State Signals
          o Decay Response, Initial Conditions Response
          o Complete Response
      + Summary and Conclusions
      + Time Domain Representation Problems

  * Transfer Function Analysis
      + The Z Transform
      + Existence of the Z Transform
      + Shift and Convolution Theorems
          o Shift Theorem
          o Convolution Theorem
      + Z Transform of Convolution
      + Z Transform of Difference Equations
      + Factored Form
      + Series and Parallel Transfer Functions
          o Series Case
          o Parallel Case
              # Series Combination is Commutative
      + Partial Fraction Expansion
          o Example
          o Complex Example
          o PFE to Real, Second-Order Sections
          o Inverting the Z Transform
          o FIR Part of a PFE
              # Example: The General Biquad PFE
          o Alternate PFE Methods
          o Repeated Poles
              # Dealing with Repeated Poles Analytically
              # Example
              # Impulse Response of Repeated Poles
              # So What's Up with Repeated Poles?
          o Alternate Stability Criterion
          o Summary of the Partial Fraction Expansion
          o Software for Partial Fraction Expansion
              # Example 2
              # Polynomial Multiplication in Matlab
              # Polynomial Division in Matlab
      + Problems

  * Frequency Response Analysis
      + Frequency Response
      + Amplitude Response
      + Phase Response
      + Polar Form of the Frequency Response
          o Separating the Transfer Function Numerator and
            Denominator
      + Frequency Response as a Ratio of DTFTs
          o Frequency Response in Matlab
          o Example LPF Frequency Response Using freqz
      + Phase and Group Delay
          o Phase Delay
          o Phase Unwrapping
          o Group Delay
              # Derivation of Group Delay as Modulation Delay
          o Group Delay Examples in Matlab
          o Vocoder Analysis
          o Numerical Computation of Group Delay
      + Frequency Response Analysis Problems

  * Pole-Zero Analysis
      + Filter Order = Transfer Function Order
      + Graphical Amplitude Response
      + Graphical Phase Response
      + Stability Revisited
          o Computing Reflection Coefficients
          o Step-Down Procedure
          o Testing Filter Stability in Matlab
      + Bandwidth of One Pole
      + Time Constant of One Pole
      + Unstable Poles--Unit Circle Viewpoint
          o Geometric Series
          o 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
          o Direct-Form I
              # Two's Complement Wrap-Around
          o Direct Form II
              # More about Potential Internal Overflow of DF-II
          o Transposed Direct-Forms
          o Numerical Robustness of TDF-II
      + Series and Parallel Filter Sections
          o Series Second-Order Sections
              # Matlab Example
          o Parallel First and/or Second-Order Sections
              # First-Order Complex Resonators
              # Real Second-Order Sections
              # Implementation of Repeated Poles
          o Formant Filtering Example
          o Butterworth Lowpass Filter Example
          o Summary of Series/Parallel Filter Sections
      + Pole-Zero Analysis Problems

  * Filters Preserving Phase
      + Linear-Phase Filters
      + Zero-Phase Filters
          o $ \pi $ -Phase Filters
          o Phase $ \pi $ in the Stopband
          o Example Zero-Phase Filter Design
          o Elementary Zero-Phase Filter Examples
      + Odd Impulse Reponses
      + Symmetric Linear-Phase Filters
          o Simple Linear-Phase Filter Examples
          o 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
          o Example
      + Minimum Phase Means Fastest Decay
      + Minimum-Phase/Allpass Decomposition
      + Linear Phase Audio Filters
      + Creating Minimum Phase

  * Conclusion
  * Background Fundamentals
      + Signal Representation and Notation
          o Units
          o Sinusoids
          o Spectrum
      + Complex and Trigonometric Identities
          o Complex Numbers
          o The Exponential Function
          o Trigonometric Identities
              # Trigonometric Identities, Continued
          o Half-Angle Tangent Identities
      + Sinusoids as Eigenfunctions of LTI Systems
          o Proof Using Trigonometry
          o Proof Using Complex Variables
          o Phasor Analysis

  * Elementary Audio Digital Filters
      + Elementary Filter Sections
          o One-Zero
          o One-Pole
          o Two-Pole
              # Resonator Bandwidth in Terms of Pole Radius
          o Two-Zero
          o Complex Resonator
              # Two-Pole Partial Fraction Expansion
          o The BiQuad Section
          o Biquad Software Implementations
      + Allpass Filter Sections
          o The Biquad Allpass Section
          o Allpass Filter Design
      + DC Blocker
          o DC Blocker Frequency Response
          o DC Blocker Software Implementations
      + Low and High Shelf Filters
          o Exercise
      + Peaking Equalizers
      + Time-Varying Two-Pole Filters
          o Normalizing Two-Pole Filter Gain at Resonance
          o Constant Resonance Gain
          o Peak Gain Versus Resonance Gain
          o Constant Peak-Gain Resonator
          o Four-Pole Tunable Lowpass/Bandpass Filters
      + Elementary Filter Problems

  * Allpass Filters
      + Allpass Examples
      + Paraunitary Filters
      + MIMO Allpass Filters
          o Paraunitary MIMO Filters
              # MIMO Paraconjugate
              # MIMO Paraunitary Condition
              # Properties of Paraunitary Systems
              # Properties of Paraunitary Filter Banks
          o Paraunitary Filter Examples
      + Allpass Problems

  * Laplace Transform Analysis
      + Existence of the Laplace Transform
      + Analytic Continuation
      + Relation to the z Transform
      + Laplace Transform Theorems
          o Linearity
          o Differentiation
      + Laplace Analysis of Linear Systems
          o Moving Mass
          o Mass-Spring Oscillator Analysis

  * Analog Filters
      + Example Analog Filter
      + Capacitors
          o Mechanical Equivalent of a Capacitor is a Spring
      + Inductors
          o Mechanical Equivalent of an Inductor is a Mass
      + RC Filter Analysis
          o Driving Point Impedance
          o Transfer Function
          o Impulse Response
          o The Continuous-Time Impulse
          o Poles and Zeros
      + RLC Filter Analysis
          o Driving Point Impedance
          o Transfer Function
          o Poles and Zeros
          o Impulse Response
      + Relating Pole Radius to Bandwidth
      + Quality Factor (Q)
          o Q of a Complex Resonator
          o Q of a Real Second-Order Resonator
              # Critical Damping and Related Terms
          o Damping Ratio
          o Decay Time is Q Periods
          o Q as Energy Stored over Energy Dissipated
      + Analog Allpass Filters
          o Lossless Analog Filters

  * Matrix Filter Representations
      + Introduction
      + General Causal Linear Filter Matrix
      + General LTI Filter Matrix
      + Cyclic Convolution Matrix
      + Inverse Filters
      + State Space Realization
          o State Space Filter Realization Example
      + Time Domain Filter Estimation
          o Effect of Measurement Noise
          o Matlab System Identification Example

  * State Space Filters
      + Markov Parameters
      + Response from Initial Conditions
      + Complete Response
      + Transfer Function of a State Space Filter
          o Example State Space Filter Transfer Function
      + Transposition of a State Space Filter
      + Poles of a State Space Filter
      + Difference Equations to State Space
          o Converting to State-Space Form by Hand
          o Signal Flow Graph to State Space Filter
          o Controllability and Observability
          o A Short-Cut to Controller Canonical Form
          o Matlab Direct-Form to State-Space Conversion
          o State Space Simulation in Matlab
          o Other Relevant Matlab Functions
          o Matlab State-Space Filter Conversion Example
      + Similarity Transformations
      + Modal Representation
          o Diagonalizing a State-Space Model
          o Finding the Eigenvalues of A in Practice
          o Example of State-Space Diagonalization
          o Properties of the Modal Representation
      + Repeated Poles
          o Jordan Canonical Form
      + Digital Waveguide Oscillator Example
          o Finding the Eigenstructure of A
          o 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
          o Butterworth Lowpass Poles and Zeros
          o Example: Second-Order Butterworth Lowpass
      + Bilinear A/D Transformation
          o Bilinear Transformation
          o Frequency Warping
          o Analog Prototype Filter
          o Examples
      + Equation-Error Filter Design
          o Equation Error Formulation
          o Error Weighting and Frequency Warping
          o Stability of Equation Error Designs
          o An FFT-Based Equation-Error Method
          o Prony's Method
          o The Pade-Prony Method

  * 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
          o Method
          o Example with Repeated Poles
      + Partial Fraction Expansion: residued.m
          o 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
          o Matlab listing: mps.m
          o Matlab listing: tmps.m
          o Matlab diary: tmps.d
      + 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
          o Generating the PD Plugin
          o Generating a PD Plugin-Wrapper Abstraction
          o A PD Test Patch for the Plugin Wrapper
      + Generating a LADSPA Plugin via Faust
      + Generating a VST Plugin via Faust
          o Bypassing Windows
      + Generating a MIDI Synthesizer for PD
      + MIDI Synthesizer Test Patch

  * Links to Online Resources
  * Bibliography
  * Index for this Document
  * About this document ...

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``Introduction to Digital Filters with Audio Applications'', by 
Julius O. Smith III, (September 2007 Edition)
Copyright (c) 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   
Stanford University
CCRMA