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  2. Convolution - Wikipedia

    en.wikipedia.org/wiki/Convolution

    The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two polynomials, then the coefficients of the ordinary product of the two polynomials are the convolution of the original two

  3. Convolution theorem - Wikipedia

    en.wikipedia.org/wiki/Convolution_theorem

    In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).

  4. Cauchy product - Wikipedia

    en.wikipedia.org/wiki/Cauchy_product

    The Cauchy product may apply to infinite series [1] [2] or power series. [3] [4] When people apply it to finite sequences [5] or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution).

  5. Circular convolution - Wikipedia

    en.wikipedia.org/wiki/Circular_convolution

    Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discrete-time Fourier transform (DTFT). In particular, the DTFT of the product of two discrete sequences ...

  6. Generating function - Wikipedia

    en.wikipedia.org/wiki/Generating_function

    A sequence of convolution polynomials defined in the notation above has the following properties: The sequence n! · f n (x) is of binomial type; Special values of the sequence include f n (1) = [z n] F(z) and f n (0) = δ n,0, and; For arbitrary (fixed) ,,, these polynomials satisfy convolution formulas of the form

  7. Discrete Fourier transform - Wikipedia

    en.wikipedia.org/wiki/Discrete_Fourier_transform

    The convolution theorem for the discrete-time Fourier transform (DTFT) indicates that a convolution of two sequences can be obtained as the inverse transform of the product of the individual transforms.

  8. Young's convolution inequality - Wikipedia

    en.wikipedia.org/wiki/Young's_convolution_inequality

    In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, [1] named after William Henry Young. Statement

  9. Cross-correlation - Wikipedia

    en.wikipedia.org/wiki/Cross-correlation

    Visual comparison of convolution, cross-correlation and autocorrelation.For the operations involving function f, and assuming the height of f is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point.