MUS421 Lecture 2
Review of the Discrete Fourier Transform (DFT)

Julius O. Smith III (jos at ccrma)
Center for Computer Research in Music and Acoustics (CCRMA)
Department of Music, Stanford University

Detailed Contents (and Navigation)

• Domains of Definition
• Geometric Interpretation of the Fourier Transform
• Signal and Transform Notation
• The Discrete Fourier Transform
  • Inverse DFT
  • The DFT, Cont'd
  
  
• Properties of the DFT
  • Linearity
  • Symmetries for Real Signals
  • Time Reversal
  • Shift Theorem
  • Convolution
  
  
• Linear Convolution of Short Signals
  • FFT Convolution
  • Example FFT Convolution
  • FFT Convolution vs. Direct Convolution
  • Correlation
  • Autocorrelation
  • Power Theorem
  • Parseval's Theorem
  • Vector Cosine
  • Stretch
  • Repeat or Scale
  • Zero-Padding $\leftrightarrow$ Interpolation
  • Downsampling $\leftrightarrow$ Aliasing
  
  
• Ideal Spectral Interpolation
  • Meaning of Spectral Interpolation
  • Interpolating a DFT
  • Practical Zero Padding
  • Reasons for Zero Padding
    (Spectral Interpolation)
  
  
• Zero Padding Examples
  • Critically Sampled Hamming Window Transform
  • 2X Oversampled Hamming Window Transform
  • Oversampled Spectral Peaks
  
  
• Zero-Centered Zero-Padding
  • Zero-Centered Spectra
  • fftshift
  
  
• About this document ...

For more details, see

• Mathematics of the DFT (Music 320 text):
  http://ccrma.stanford.edu/~jos/mdft/
• Chapter 2 and Appendix B of Spectral Audio Signal Processing (our text):
  http://ccrma.stanford.edu/~jos/sasp/