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convolution (computer science) | Visual explanation of convolution. 1.Express each function in terms of a dummy variable τ. 2.Transpose one of the functions: g(τ)→g( − τ). 3.Add a time-offset, t, which allows g(t − τ) to slide along the τ-axis. 4.Start t at -∞ and slide it all the way to +∞. Wherever the two functions intersect, find the integral of their product. In other words, compute a sliding, weighted-average of function f(τ), where the weighting function is g( − τ). The resulting waveform (not shown here) is the convolution of functions f and g. If f(t) is a unit impulse, the result of this process is simply g(t), which is therefore called the impulse response. | | dummy variable | unit impulse | impulse response | mathematics | functional analysis | operation | functions | cross-correlation | statistics | computer vision | image | signal processing | electrical engineering | differential equations | groups | Euclidean space | circular convolution | periodic functions | circle | integers | numerical analysis | numerical linear algebra | finite impulse response | integral transform | commutativity | Circular convolution | circular, cyclic, or periodic convolution | commutativity | polynomials | sequences | Cauchy product | circular, cyclic, or periodic convolution | cyclic group | integers modulo N | multiplication | Digital signal processing | fast Fourier transform | circular convolution theorem | circular convolution | Schönhage-Strassen algorithm | rings | compactly supported | continuous functions | locally integrable | Lebesgue integrable functions | L1(Rd) | Tonelli's theorem | rapidly decreasing functions | Schwartz functions | Distribution (mathematics) | distribution | Convolution algebra | linear space | commutative algebra | identity | closed | Commutativity | Associativity | Distributivity | Multiplicative identity | approximations to the identity | derivative | partial derivative | difference operator | convolution theorem | Fourier transform | normalization | Fourier_transform#Some_Fourier_transform_properties | Laplace transform | two-sided Laplace transform | Z-transform | Mellin transform | Titchmarsh convolution theorem | linear operator | time-invariant systems | LTI system theory | impulse response | Convolution power#Convolution inverse | inverse element | abelian group | group operation | group | measure | locally compact | Hausdorff | topological group | Haar measure | integrable | circle group | Hilbert space | compact | normal | spectral theory | characters | multiplication operator | harmonic analysis | representation theory | Peter-Weyl theorem | Lie groups | clarify | Borel subsets | absolutely continuous | Lebesgue measure | so that each has a density function | probability measures | probability distribution | independent | random variables | electrical engineering | input | impulse response | linear time-invariant system | statistics | moving average | probability theory | probability distribution | independent | random variables | optics | shape | bokeh | digital image processing | algorithms | edge detection | acoustics | reverberation | digital signal processing | impulse response | invariant | linear system | Dirac delta | LTI system theory | digital signal processing | fluorescence spectroscopy | physics | linear system | superposition principle | Navier–Stokes equations | Clay Mathematics Millennium Problem | LTI_system_theory#Impulse_response_and_convolution | Toeplitz matrix | Circulant matrix | Cross-correlation | Deconvolution | Dirichlet convolution | Titchmarsh convolution theorem | Convolution power | Analog signal processing | List of convolutions of probability distributions | Hörmander, L. | MR | ISBN 3-540-12104-8 | Knuth, Donald | ISBN 0-201-89684-2 | Encyclopaedia of Mathematics | ISBN 978-1556080104 | Stein, Elias | ISBN 0-691-08078-X | ISBN 0849382734 | Titchmarsh, E | ISBN 978-0828403245 | ISBN 0-521-82646-2 | | Wiktionary | MathWorld | Categories | Functional analysis | Image processing | Binary operations | Fourier analysis | Wikipedia articles needing clarification |
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