convolution, a mathematical operation performed on two functions that yields a function that is a combination of the two original functions. Convolutions have been used in mathematics since the 18th century, but the term convolution was first used to describe the concept in 1934 by mathematician Aurel Wintner. Convolutions have applications in digital signal processing, image processing, natural language processing, and electrical engineering.

Calculation and properties

The convolution h of two functions f and g is defined as$h\left(t\right) = f * g = \underset{-\infty }{\overset{\infty }{\int }}f\left(x\right)g(t-x)dx = \underset{-\infty }{\overset{\infty }{\int }}g\left(x\right)f\left(t-x\right)dx$

The operation involves inverting, or mirroring, one of the functions and lagging it (delaying it in time) across the other function while running the integration. The inverse of the convolution operation is called deconvolution. The convolution operation follows the properties of multiplication:

• Commutative: fg = gf
• Associative: (fg)h = f(gh)
• Distributive: h(f+g) = hf + hg

Convolution with the Dirac delta function δ(t)—a unit impulse function that is 0 when t ≠ 0, is +∞ for t = 0, and has an integral equal to 1 over any interval containing 0—returns the original function.h(t) * δ(t) = h(t)Delaying the impulse function by a certain time T causes a delay in the resulting convolution function.h(t) * δ(t − T) = h(t − T)The convolution operation is directly related to the Fourier transform ℱ, which transforms a function of time to a function of frequency. The Fourier transform of the convolution of two functions f(t) and g(t) is equal to the product of their Fourier transforms.ℱ (f * g) = ℱ (f) ∙ ℱ (g)The converse also holds true. The inverse Fourier transform of the convolution of two functions is equal to the multiplication of the inverse Fourier transforms of the two individual functions.ℱ⁻¹(f * g) = ℱ⁻¹(f) ∙ ℱ⁻¹(g)

Applications

Convolution operations are frequently used in mathematics, such as in probability theory. If two independent random variables X and Y have probability density functions f and g, then the probability density function of X + Y is the convolution of their probability density functions—i.e., f * g. The convolution is also used in the calculation of moving averages.

Convolution operations are used in image processing for the blurring and unblurring of images. A blurred image is typically the convolution of the original image with a blurring function, and deconvolution of the blurred image using the blurring function yields back the original image. Because convolution is related to the Fourier transform, the Fourier transform can thus be used to reduce the number of computations needed to run a convolution. Convolutional neural networks, artificial neural networks that use a series of convolutions to filter inputs, have applications in speech and image processing.

Convolutional encoding is a method of channel encoding used in several forms of communication to add redundancy to transmitted signals and ensure that errors from transmission noise can be corrected at the receiver end. Trellis-coded modulation, which forms an essential part of modern voiceband modems, uses convolutional codes along with modulation.

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