Section 4: Linear and Minimum Phase Filters, Recursive Digital Filter Design, Butterworth Filters, More on Allpass Filters, More on State Space

• Reading:
  • Chapter 11 (Filters Preserving Phase)

  • Chapter 12 (Minimum-Phase Filters)

  • Appendix F (Matrix Filter Representations)

  • Appendix G (State Space Filters)

  • Supplementary: The Digital Waveguide Oscillator

  • State-Space Formulation of Multi-Input, Multi-Output (MIMO) Linear Filters

  • Digitizing the State-Variable Filter

  • Appendix I (Recursive Digital Filter Design)

  • Appendix K (Digital Filtering in FAUST and Pd)
    Supplementary: Audio Signal Processing in FAUST

  • Supplementary: State-Space Canonical Forms

  • Supplementary: State Variable Filter used in the ARP2500 analog synthesizer

  • Assignment 4

• Lecture Videos
  • Butterworth Filter Properties: Maximum Flatness at Infinity, Low Ringing, Mild Phase Response, Poles on a Circle; Spectral Factorization, Series Biquad Realization, Elliptic Function Filters, Chebyshev Optimality, Remez Exchange (Parks-McLellan), firpm in matlab, cvx for Convex Optimization [35:19]

  • Butterworth Power Response, Analytic Continuation, Butterworth Poles, Matlab butter() Function [7:02]

  • Example Butterworth Filter of Order 2, Digitization via Bilinear Transform, Frequency Prewarping [18:49]

  • Digital Filter Design and Implementation in Matlab: Noise Removal via Lowpass Filtering, Create Sinusoid and Noise, Matlab's butter(), filter(), fdatool (Filter Design and Analysis tool), Simulating a Telephone Channel Bandwidth [10:30]

  • Matlab: freqz(), freqs(); Continuous Butterworth Filter Analysis; Converting to Second-Order Sections in Matlab using tf2sos(); Viewing Butterworth Poles in Matlab using zplane(); Excess Delay at Filter Cutoff; grpdelay(); Elliptic Filters using ellip(); Ripple; impz(); Zero Phase versus Minimum Phase (Pre-Ring versus Post-Ring); Minimum-Delay Property of Minimum-Phase Filters; Partial Fraction Expansion in Matlab using residue(), residuez(), or residued() [31:10]

  • Supplementary Note on Repeated Poles [4:40]

  • Supplementary: MicroModeler Digital Filter Design Tools [5:23]

  • Minimum Phase Filters and Signals; Allpass Filters: Poles and Zeros, Graphical Amplitude and Phase Response, Biquad Realization, Phasing; Allpass-Minimum-Phase Decomposition [28:44]

  • Allpass Filters in z and s Planes; Instability as Noncausality; Laurent Series; Bilateral DTFT; Cepstrum; Converting Arbitrary Spectra to Minimum-Phase Form [39:13]

  • The State Space Formulation of Linear Systems [2016 flipped-class review]
    • Adding Feedback around the Integrator Chain, Derivation of the State Space Formulation [10:46]
    • State Space Formulation, Continued [1:33]
    • State Space Overview [11:50]
    • Force-Driven Mass Revisited [4:49]
    • General Discussion of State Space [8:48]
    • Defining State Variables [4:51]
    • State Variable Choice Summary [5:14]
    • General State Space Model and Digitization via Backward Euler [5:16]
    • State Space Converts Nth-Order to Vector First-Order [2:08]

  • Supplementary Review (because we already did this in Section 2): State Variable Lowpass, Bandpass, and Highpass
    • Normalized Biquad Lowpass Filter, Continuous Time [19:29]
    • State Variable Realization of Normalized Biquad Lowpass [31:22]
    • State Variable Filter Lowpass/Bandpass/Highpass [3:00]
    • State Variable Filter LP/BP/HP Frequency Scaling and Digitization [11:35]