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  2. Discrete Fourier transform - Wikipedia

    en.wikipedia.org/wiki/Discrete_Fourier_transform

    In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration ...

  3. Discrete-time Fourier transform - Wikipedia

    en.wikipedia.org/wiki/Discrete-time_Fourier...

    The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see § Sampling the DTFT), which is by far the most common method of modern Fourier analysis. Both transforms are invertible.

  4. DFT matrix - Wikipedia

    en.wikipedia.org/wiki/DFT_matrix

    In this case, if we make a very large matrix with complex exponentials in the rows (i.e., cosine real parts and sine imaginary parts), and increase the resolution without bound, we approach the kernel of the Fredholm integral equation of the 2nd kind, namely the Fourier operator that defines the continuous Fourier transform. A rectangular ...

  5. Fourier transform on finite groups - Wikipedia

    en.wikipedia.org/wiki/Fourier_transform_on...

    There is a direct relationship between the Fourier transform on finite groups and the representation theory of finite groups.The set of complex-valued functions on a finite group, , together with the operations of pointwise addition and convolution, form a ring that is naturally identified with the group ring of over the complex numbers, [].

  6. Fourier analysis - Wikipedia

    en.wikipedia.org/wiki/Fourier_analysis

    That is, it takes a function from the time domain into the frequency domain; it is a decomposition of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.

  7. Fourier transform - Wikipedia

    en.wikipedia.org/wiki/Fourier_transform

    The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the ...

  8. Discrete Fourier series - Wikipedia

    en.wikipedia.org/wiki/Discrete_Fourier_series

    In digital signal processing, a discrete Fourier series (DFS) is a Fourier series whose sinusoidal components are functions of discrete time instead of continuous time. A specific example is the inverse discrete Fourier transform (inverse DFT).

  9. Cooley–Tukey FFT algorithm - Wikipedia

    en.wikipedia.org/wiki/Cooley–Tukey_FFT_algorithm

    The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size = in terms of N 1 smaller DFTs of sizes N 2, recursively, to reduce the computation time to O(N log N) for highly composite N (smooth numbers).