Section 1: Linearity and Time Invariance; Time-Domain Representations

• Demos:
  • Interactive display and hearing of classic digital filters by java@falstad.com

• Reading:
  • Chapters 1 and 2 of Introduction to Digital Filters

  • Chapter 4 (Linearity and Time Invariance) and
    Chapter 5 (Time Domain Filter Representations) of Introduction to Digital Filters

  • First section of Chapter 9 (Implementation Structures) on the Four Direct Forms

  • Matrix Filter Representations

  • Optionally peruse the Music 421 overheads pertaining to acyclic convolution

  • Supplementary: Audio Signal Processing in FAUST

  • Assignment 1

• Lecture Videos (Total Viewing Time $\approx$ 2 Hours):

  IMPORTANT NOTICE: The videos are hosted on YouTube and they use annotations for corrections and supplementary information. These annotations are not supported on mobile devices. It is therefore unfortunately important to view these videos in a Web browser on a desktop/laptop computer.

  • Linear Time-Invariant (LTI) Filters, Convolution, Linearity, Nonlinear Example, Time-Varying Example, Ideal Lowpass Filter, Pass Band, Cutoff Frequency, Guard Band, Stop Band, Transition Band, Simplest Lowpass Filter, Impulse Response, DTFT, Frequency Response, Amplitude Response, Phase Response, Linear Phase, Sinewave Analysis [38:36]

  • Derivation of Convolution from Linearity and Time-Invariance (LTI) (Superposition) [2015] [29:08]

  • General Linear [Causal] [Time-Invariant] Filters -- Matrix Representations [19:03]

  • Recursive Filters, Simplest Lowpass, Phase Delay, Group Delay [28:01]

  • Supplementary: FAUST in the Classroom [41:00]

  • Supplementary: FAUST Intro [26:00]

  • Supplementary: FAUST Implementation of the Simplest Lowpass Filter [18:22]

  • Simplest RECURSIVE LPF, Pole Gain, PFE, Time-Constant of a Pole, Stability Pole, Bandwidth, Laplace Transform, s-plane poles and zeros, s-plane pole corresponds to exponential [38:37]

  • What does the Laplace Transform really tell us? A visual explanation (plus applications) [20:24]
    Nice 3D plot of two poles at t=512 []

  • Direct Form Digital Filters, Transposing a Flow Graph, Transposed Direct Forms 1 and 2, Direct Form 1 Biquad, Direct Form 2 Biquad, Transposed Direct Form 2 Biquad, Interpolated Delay-Line Read, Interpolated Delay-Line Write = Transpose of Read [14:35]

  • Simplest Mechanical LPF: Ideal Mass on Frictionless Surface, Newton's law of motion f=ma, Analog Transfer Function for Driving-Force Input, Velocity Output, Admittance (Mobility) of a Mass [5:31]

  • Simplest Mechanical LPF: Ideal Mass on Frictionless Surface, Differentiation Theorem for Laplace Transforms, Transfer Function of the Force-Driven Mass: Frequency Response, Poles and Zeros,, Amplitude Response, 6dB per octave roll off, Bode Plot, Harald Bode, Phase Response [27:29]