This is a list of linear_transformations of functions related to the Fourier_transform. Such transformations map a function to a set of coefficients of basis_functions, where the basis_functions are sinusoidal and are therefore strongly localized in the frequency_spectrum. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component. Applied to functions of continuous arguments, Fourier-related transforms include:
  • Fourier_transform (FT), with special cases:
  • Cosine_transform and sine_transform (for functions of even/odd symmetry)
  • Fourier_series (for periodic functions)
  • Hartley_transform
  • Short-time_Fourier_transform (or short-term Fourier transform) (STFT) For usage on computers, discrete arguments (e.g. functions of a series of discrete samples) are more appropriate, and are handled by the transforms (analogous to the continuous cases above):
  • The Z-transform is a more general transform, of which the DFT is a special case.
  • Discrete_Fourier_transform (DFT), with special cases:
  • Discrete_cosine_transform (DCT)
  • Discrete_sine_transform (DST)
  • Modified_discrete_cosine_transform (MDCT)
  • Discrete_Hartley_transform (DHT)
  • Also the discretized STFT (see above). The usage of all of these transforms is greatly facilitated by the existence of efficient algorithms based on a fast_Fourier_transform (FFT). The Nyquist-Shannon_sampling_theorem is critical for understanding the output of such discrete transforms. See also related information in:
  • Wavelet_transform
  • Fourier_transform_spectroscopy
  • Harmonic_analysis
  • List_of_transforms
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