INTRODUCTION TO DIGITAL FILTERS
WITH AUDIO APPLICATIONS

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• 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 Shelving 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)
    • 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.m
  • 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.m
  
  
• 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 MIDI Synthesizer for PD
  • MIDI Synthesizer Test Patch
  
  
• Links to Online Resources
• Bibliography
• Index
• About this document ...

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

dear sir,

is there any way to have soft copy of this book?

Sunny

An E-book version is planned, but not yet available. I'll try to remember to post here when it is.