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
    • Sine-Wave Analysis
    • Mathematical Sine-Wave Analysis
    • Amplitude Response
    • Phase Response
  • An Easier Way
    • Complex Sinusoids
    • Complex Amplitude
    • Phasor Notation
    • Complex Sinusoids as Circular Motion
    • 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
    • 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
    • First-Order Parallel Sections
    • Parallel, Real, Second-Order Sections
    • Parallel Second-Order Signal Flow Graph
    • Series, Real, Second-Order Sections
  • Summary
  
  
• Linear Time-Invariant Filters
  • Definition of a Signal
  • Definition of a Filter
  • Examples of Digital Filters
  • Linear Filters
    • Scaling:
    • Superposition:
    • Real Linear Filtering of Complex Signals
  • 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
    • FIR impulse response
    • Convolution Representation of FIR Filters
    • The ``Finite'' in FIR
    • Causal FIR Filters
    • FIR Transfer Function
    • FIR Order
    • FIR Software Implementations
  • Transient and Steady State Response
    • FIR Example
    • IIR Example
    • Transient and Steady-State Signals
    • Decay Response, Initial Conditions Response
    • Complete 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
    • Example
    • Complex Example
    • PFE to Real, Second-Order Sections
    • Inverting the Z Transform
    • FIR Part of a PFE
      • Example: The General Biquad PFE
    • Alternate PFE Methods
    • Repeated Poles
      • Dealing with Repeated Poles Analytically
      • Example
      • Impulse Response of Repeated Poles
      • So What's Up with Repeated Poles?
    • Alternate Stability Criterion
    • Summary of the Partial Fraction Expansion
    • 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
    • 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
    • Phase Delay
    • Phase Unwrapping
    • Group Delay
      • Derivation of Group Delay as Modulation Delay
    • Group Delay Examples in Matlab
    • Vocoder Analysis
    • 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
    • Computing Reflection Coefficients
    • Step-Down Procedure
    • Testing Filter Stability in Matlab
  • 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
    • Direct-Form I
      • Two's Complement Wrap-Around
    • Direct Form II
      • More about Potential Internal Overflow of DF-II
    • Transposed Direct-Forms
    • Numerical Robustness of TDF-II
  • Series and Parallel Filter Sections
    • Series Second-Order Sections
      • Matlab Example
    • Parallel First and/or Second-Order Sections
      • First-Order Complex Resonators
      • Real Second-Order Sections
      • Implementation of Repeated Poles
    • Formant Filtering Example
    • Butterworth Lowpass Filter Example
    • Summary of Series/Parallel Filter Sections
  • Pole-Zero Analysis Problems
  
  
• Filters Preserving Phase
  • Linear-Phase Filters
  • Zero-Phase Filters
    • -Phase Filters
    • Phase in the Stopband
    • Example Zero-Phase Filter Design
    • Elementary Zero-Phase Filter Examples
  • 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
    • Units
    • Sinusoids
    • Spectrum
  • Complex and Trigonometric Identities
    • Complex Numbers
    • The Exponential Function
    • Trigonometric Identities
      • Trigonometric Identities, Continued
    • Half-Angle Tangent Identities
  • Sinusoids as Eigenfunctions of LTI Systems
    • Proof Using Trigonometry
    • Proof Using Complex Variables
    • Phasor Analysis
  
  
• Elementary Audio Digital Filters
  • Elementary Filter Sections
    • One-Zero
    • One-Pole
    • Two-Pole
      • Resonator Bandwidth in Terms of Pole Radius
    • Two-Zero
    • Complex Resonator
      • Two-Pole Partial Fraction Expansion
    • The BiQuad Section
    • Biquad Software Implementations
  • 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
    • Normalizing Two-Pole Filter Gain at Resonance
    • Constant Resonance Gain
    • Peak Gain Versus Resonance Gain
    • Constant Peak-Gain Resonator
    • Four-Pole Tunable Lowpass/Bandpass Filters
  • Elementary Filter Problems
  
  
• Allpass Filters
  • Allpass Examples
  • Paraunitary Filters
  • MIMO Allpass Filters
    • Paraunitary MIMO Filters
      • MIMO Paraconjugate
      • MIMO Paraunitary Condition
      • Properties of Paraunitary Systems
      • Properties of Paraunitary Filter Banks
    • 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
    • Driving Point Impedance
    • Transfer Function
    • Impulse Response
    • The Continuous-Time Impulse
    • Poles and Zeros
  • RLC Filter Analysis
    • Driving Point Impedance
    • Transfer Function
    • Poles and Zeros
    • Impulse Response
  • Relating Pole Radius to Bandwidth
  • Quality Factor (Q)
    • Q of a Complex Resonator
    • Q of a Real Second-Order Resonator
      • Critical Damping and Related Terms
    • Damping Ratio
    • Decay Time is Q Periods
    • Q as Energy Stored over Energy Dissipated
  • 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
    • Converting to State-Space Form by Hand
    • Signal Flow Graph to State Space Filter
    • Controllability and Observability
    • A Short-Cut to Controller Canonical Form
    • Matlab Direct-Form to State-Space Conversion
    • State Space Simulation in Matlab
    • Other Relevant Matlab Functions
    • Matlab State-Space Filter Conversion Example
  • Similarity Transformations
  • Modal Representation
    • Diagonalizing a State-Space Model
    • Finding the Eigenvalues of A in Practice
    • Example of State-Space Diagonalization
    • Properties of the 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
    • Bilinear Transformation
    • Frequency Warping
    • Analog Prototype Filter
    • Examples
  • Equation-Error Filter Design
    • Equation Error Formulation
    • Error Weighting and Frequency Warping
    • Stability of Equation Error Designs
    • An FFT-Based Equation-Error Method
    • Prony's Method
    • The Padé-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
    • 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
    • Matlab listing: mps.m
    • Matlab listing: tmps.m
    • 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
    • Generating the PD Plugin
    • Generating a PD Plugin-Wrapper Abstraction
    • A PD Test Patch for the Plugin Wrapper
  • 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
• About this document ...