Section 4: Fourier Theorems, Convolution and Correlation

• Reading:
  • Chapter 7 (Fourier Theorems for the DFT) of Mathematics of the DFT

  • Appendix B (Fourier Transforms for Continuous/Discrete Time/Frequency)

  • Filters and Convolution

  • Assignment 4

• Lecture Videos (Total Viewing Time $\approx$ 4 Hours):
  • Signal and Spectra Notation for the DFT Theorems [5:14]

  • Review of DFT as a change of coordinates, Periodic Extension versus Time Limited Signal Windows, DFT interpolation between spectral samples, ``spectral splatter'' in DFT of non-DFT-sinusoids [20:44]

  • Windowed Signal Segment Gives One Time Sample in the Time-Frequency Distribution [1:28]

  • DFT Linearity, Flip Operator (index reversal), Flip Theorem, Real Signals have Hermitian (conjugate symmetric) Spectra, Fourier Duality [23:31]

  • DFT Symmetry Theorems, Even and Odd Functions, DCT & DST, ``Zero-Phase'' Spectra [34:18]

  • Shift Operator, Shift Theorem, Linear Phase Terms, Convolution Thm Preview [11:16]

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

  • Circular Convolution, Commutativity of Convolution, Graphical Convolution, Convolution Reverb, Impulse Response, Convolution Representation of Linear Time-Invariant (LTI) Filters, Convolution Theorem Stated [36:59]

  • Aliasing Demo [4:13]

  • Continuous Graphical Convolution Demo [11:15]

  • Convolution Theorem Proof, FFT Convolution, Filter Frequency Response [21:41]

  • Dual of Convolution Theorem, Application to Time-Domain Windowing [6:00]

  • Review/FAQ Presented 11/11/2014: Why Vector Spaces, Signal Energy as Length Squared, Power, RMS, Shift Theorem, Convolution Theorem [26:34]

  • Correlation, Lagged Product, Correlation Thm, Autocorrelation $\Leftrightarrow$ Power Spectrum [6:42]

  • Power Thm, Parseval's Thm [7:50]

  • Normalized DFT (NDFT) [2015] [7:04]

  • Review of Linearity, Flip, Symmetry, Shift, Convolution, Correlation, and Power Theorems [19:51]

  • [Optional] Intuitive Explanation of the Sampling Theorem [14:57]

  • Scaling Theorem (continuous time), Stretch Operator, Stretch Thm, Filter Guard Bands, Discrete-Time Stretch Thm, Downsampling, Aliasing, Downsampling Theorem [31:36]

  • FFT Bandlimited Interpolation [9:39]