Fourier Theorems for the DFT

This chapter derives various Fourier theorems for the case of the DFT. Included are symmetry relations, the shift theorem, convolution theorem, correlation theorem, power theorem, and theorems pertaining to interpolation and downsampling. Applications related to certain theorems are outlined, including linear time-invariant filtering, sampling rate conversion, and statistical signal processing.

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Subsections

• The DFT and its Inverse Restated
  • Notation and Terminology
  • Modulo Indexing, Periodic Extension
  
  
• Signal Operators
  • Operator Notation
  • Flip Operator
  • Shift Operator
    • Examples
  • Convolution
    • Commutativity of Convolution
    • Convolution as a Filtering Operation
    • Convolution Example 1: Smoothing a Rectangular Pulse
    • Convolution Example 2: ADSR Envelope
    • Convolution Example 3: Matched Filtering
    • Graphical Convolution
    • Polynomial Multiplication
    • Multiplication of Decimal Numbers
  • Correlation
  • Stretch Operator
  • Zero Padding
  • Causal (Periodic) Signals
  • Causal Zero Padding
  • Zero Padding Applications
  • Ideal Spectral Interpolation
  • Interpolation Operator
  • Repeat Operator
  • Downsampling Operator
  • Alias Operator
  
  
• Even and Odd Functions
• Fourier Theorems
  • Linearity
  • Conjugation and Reversal
  • Symmetry
  • Shift Theorem
    • Linear Phase Terms
    • Linear Phase Signals
    • Zero Phase Signals
    • Application of the Shift Theorem to FFT Windows
  • Convolution Theorem
  • Dual of the Convolution Theorem
  • Correlation Theorem
  • Power Theorem
    • Normalized DFT Power Theorem
  • Rayleigh Energy Theorem (Parseval's Theorem)
  • Stretch Theorem (Repeat Theorem)
  • Downsampling Theorem (Aliasing Theorem)
    • Illustration of the Downsampling/Aliasing Theorem in Matlab
  • Zero Padding Theorem (Spectral Interpolation)
  • Interpolation Theorems
    • Relation to Stretch Theorem
    • Bandlimited Interpolation of Time-Limited Signals
  
  
• DFT Theorems Problems

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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.