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  2. Fourier transform - Wikipedia

    en.wikipedia.org/wiki/Fourier_transform

    Joseph Fourier introduced sine and cosine transforms (which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can be formally defined as an improper Riemann integral, making it an integral ...

  3. Hadamard transform - Wikipedia

    en.wikipedia.org/wiki/Hadamard_transform

    The Hadamard transform H m is a 2 m × 2 m matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2 m real numbers x n into 2 m real numbers X k.The Hadamard transform can be defined in two ways: recursively, or by using the binary (base-2) representation of the indices n and k.

  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. Discrete Fourier transform - Wikipedia

    en.wikipedia.org/wiki/Discrete_Fourier_transform

    As seen above, the discrete Fourier transform has the fundamental property of carrying convolution into componentwise product. A natural question is whether it is the only one with this ability. It has been shown [9] [10] that any linear transform that turns convolution into pointwise product is the DFT up to a permutation of coefficients ...

  6. Fast Fourier transform - Wikipedia

    en.wikipedia.org/wiki/Fast_Fourier_transform

    A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa.

  7. Transformation matrix - Wikipedia

    en.wikipedia.org/wiki/Transformation_matrix

    The resulting equation is known as eigenvalue equation. [5] ... Linear Transformation Calculator; Transformation Applet - Generate matrices from 2D transformations ...

  8. Discrete-time Fourier transform - Wikipedia

    en.wikipedia.org/.../Discrete-time_Fourier_transform

    The lower right corner depicts samples of the DTFT that are computed by a discrete Fourier transform (DFT). The utility of the DTFT is rooted in the Poisson summation formula, which tells us that the periodic function represented by the Fourier series is a periodic summation of the continuous Fourier transform: [b]

  9. Fourier analysis - Wikipedia

    en.wikipedia.org/wiki/Fourier_analysis

    Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis. To use Fourier analysis, data must be equally spaced. Different approaches have been developed for analyzing unequally spaced data, notably the least-squares spectral analysis (LSSA) methods that use a ...