Fourier Theorems for the DTFT

This section states and proves selected Fourier theorems for the DTFT. A more complete list of Fourier theorems for the Discrete Fourier Transform case (DFT) may be found in [243].^3.2

As introduced in §2.1 above, the Discrete-Time Fourier Transform (DTFT) is defined as

We say that is the spectrum of .

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Subsections

• Linearity of the DTFT
• Time Reversal
• Symmetry of the DTFT for Real Signals
  • Real Even (or Odd) Signals
  
  
• Shift Theorem
• Convolution Theorem
• Correlation Theorem
• Autocorrelation
• Power Theorem
• Stretch Operator
• Repeat (Scaling) Operator
• Stretch/Repeat (Scaling) Theorem
• Downsampling and Aliasing
  • Proof of Aliasing Theorem
  
  
• Differentiation Theorem Dual

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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.