Fourier Transforms for Continuous/Discrete Time/Frequency

The Fourier transform can be defined for signals which are

• discrete or continuous in time, and
• finite or infinite in duration.

This results in four cases. Quite naturally, the frequency domain has the same four cases,

• discrete or continuous in frequency, and
• finite or infinite in bandwidth.

When time is discrete, the frequency axis is finite, and vice versa.

This book has been concerned almost exclusively with the discrete-time, discrete-frequency case (the DFT), and in that case, both the time and frequency axes are finite in length. In the following sections, we briefly summarize the other three cases. Table B.1 summarizes all four Fourier-transform cases corresponding to discrete or continuous time and/or frequency.

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Subsections

• Discrete Time Fourier Transform (DTFT)
• Fourier Transform (FT) and Inverse
  • Existence of the Fourier Transform
  • The Continuous-Time Impulse
  
  
• Fourier Series (FS)
  • Relation of the DFT to Fourier Series

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Order a Hardcopy of Mathematics of the DFT

Previous: FFT Software
Next: Discrete Time Fourier Transform (DTFT)

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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.