Derivation of the Discrete Fourier Transform (DFT)

This chapter derives the Discrete Fourier Transform (DFT) as a projection of a length signal onto the set of sampled complex sinusoids generated by the th roots of unity.

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Subsections

• Geometric Series
• Orthogonality of Sinusoids
  • Nth Roots of Unity
  • DFT Sinusoids
  
  
• Orthogonality of the DFT Sinusoids
• Norm of the DFT Sinusoids
• An Orthonormal Sinusoidal Set
• The Discrete Fourier Transform (DFT)
• Frequencies in the ``Cracks''
• Spectral Bin Numbers
• Fourier Series Special Case
• Normalized DFT
• The Length 2 DFT
• Matrix Formulation of the DFT
• DFT Problems

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Order a Hardcopy of Mathematics of the DFT

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Next: Geometric Series

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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.