Section 2: Euler's Formula, Exponentials, Sinusoids, and Spectra

• Reading:
  • Chapter 3 (Proof of Euler's Theorem) of Mathematics of the DFT

  • Chapter 4 (Sinusoids and Exponentials)

  • Assignment 2

• Lecture Videos (Total Viewing Time $\approx$ 4 Hours):
  • Euler's Identity Proof [30:21]

  • DFT Overview, Euler's Identity on the Complex Plane, Additive Synthesis of a Square Wave by Fourier's Theorem [12:03]

  • Additive Synthesis of a Square Wave at Codepen

  • Sinusoidal Motion as Projection of Circular Motion [2:08]

  • Sinusoidal Motion as Projection of Circular Motion, Continued [0:53]

  • Mathematics of Sinusoids, Complex Sinusoids [10:30]

  • Spectra of Sinusoids [15:43]

  • Amplitude Modulation (AM) and its Spectral Effects, AM Demo, ``Ring Modulation,'' Perception of AM, ``Beats'' demo in Matlab (beatcon.m), Hearing Mechanics, Critical Bands [49:33]

  • Editable Online Javascript for Additive Synthesis Visualized

  • Additive Synthesis Visualized with Audio and More Controls

  • 3D Animation of Euler's Identity (GIF)

  • Circular Motion Projection and Superposition [1:43 (Kitty)]

  • Time Constant of Decay, Plotting Exponentials [1:00]

  • Euler's Identity in the Complex Plane [5:10]

  • Supplementary: Euler's Identity Viewed as ``Actions'' [6:13]

  • Generalized Complex Sinusoids [6:09]

  • Generalized Complex Sinusoids II [6:56]

  • Demo of Exponentially Decaying Sinusoids in Matlab, Spectrograms, DTMFs [9:01 (Kitty)]

  • Making a sinusoid in Pure Data (Pd) [6:23]

  • Drawing the word ``minimum'' in a spectrogram using various vocalizations [5:21]

  • Exponentials [13:03]

  • AM Review, Frequency Modulation (FM) [12:28]

  • FM Spectra, Bessel Functions [18:09]

  • Generalized Complex Sinusoids Review, The S-Plane [11:03]

  • Laplace Transforms [5:27]

  • Sampled Generalized Complex Sinusoids [5:00]

  • Sampled Generalized Complex Sinusoids II, Z Transforms [7:39]

  • Domain of Z Transforms, The Sampled Sinusoid jⁿ, Normalized Frequency [5:57]

  • Mapping s to z, Bilinear Transform Preview [6:44]

  • Overview and Demos Presented 10/9/2014: Sinusoidal AM Demos, Continuous/Discrete Fourier Transforms (Four Cases), Generalized Sinusoids, Laplace Transform, Z Transform [10:40]

  • You should now understand the twist in the complex-plane versus time display used in the WebGL Spectrum Analyzer

  • Supplementary flipped-class lecture on the meanings of points in the s and z planes as generators of generalized complex sinusoids, and derivation of the Laplace and Z Transforms via projection onto them:
    1. Part 1/2 [45:26]
    2. Part 2/2 [47:05]