Short-Term Fourier Analysis 

The discrete Fourier transform (DFT) is defined as:

where:

Where T is the sampling period and is the sampling frequency.

The inverse transform is defined by:

Note that is continuous - that is can take on any real value in the range 0 to .

, the DFT is periodic in with period - and therefore periodic in f with period .

  
Figure 13: A DFT illustrating the periodic nature

The amplitude spectrum is the magnitude of each component in the DFT, . The power spectrum is the square of the components in the amplitude spectrum:

 
• Properties of the DFT
  • Linearity
  • Time Shift
  • Frequency Shift
  • Convolution
  • Real valued input
• Windowing
• The short-term Fourier transform
• Zero padding
• Fast Fourier transforms
• Practical application of the short-term Fourier transform
• Overlap and add for linear filtering
  • Example: Spectral subtraction
• Cepstral analysis
• Homomorphic filtering
• Mel scaled analysis
• The Autocorrelation from the FFT