Convolution

Convolution is the correlation function of f(τ) with the reversed function g(t-τ).

The convolution operator is the asterisk symbol * .

Continuous convolution

The convolution of f(t) and g(t) is equal to the integral of f(τ) times f(t-τ):

$f(t)*g(t)=\int_{-\infty }^{\infty }f(\tau )g(t-\tau )d\tau$

Discrete convolution

Convolution of 2 discrete functions is defined as:

$f(n)*g(n)=\sum_{k=-\infty }^{\infty }f(k)\: g(n-k)$

2D discrete convolution

2 dimensional discrete convolution is usually used for image processing.

$f(n,m)*g(n,m)=\sum_{j=-\infty }^{\infty }\sum_{k=-\infty }^{\infty }f(j,k)\: g(n-j,m-k)$

Filter implementation with convolution

We can filter the discrete input signal x(n) by convolution with the impulse response h(n) to get the output signal y(n).

y(n) = x(n) * h(n)

Convolution theorem

The Fourier transform of a multiplication of 2 functions is equal to the convolution of the Fourier transforms of each function:

ℱ{f ⋅ g} = ℱ{f} * ℱ{g}

The Fourier transform of a convolution of 2 functions is equal to the multiplication of the Fourier transforms of each function:

ℱ{f * g} = ℱ{f}⋅ ℱ{g}

Convolution theorem for continuous Fourier transform

ℱ{f (t) ⋅ g(t)} = ℱ{f(t)} * ℱ{g(t)}= F(ω) * G(ω)

ℱ{f (t) * g(t)} = ℱ{f(t)} ⋅ ℱ{g(t)}= F(ω) ⋅ G(ω)

Convolution theorem for discrete Fourier transform

ℱ{f (n) ⋅ g(n)} = ℱ{f(n)} * ℱ{g(n)}= F(k) * G(k)

ℱ{f (n) * g(n)} = ℱ{f(n)} ⋅ ℱ{g(n)}= F(k) ⋅ G(k)

Convolution theorem for Laplace transform

ℒ{f (t) * g(t)} = ℒ{f(t)} ⋅ ℒ{g(t)}= F(s) ⋅ G(s)

Convolution is a fundamental mathematical operation with applications spanning various disciplines, from signal processing to image analysis. In its essence, convolution combines two functions to produce a third, describing how one function modifies the shape of the other.

Convolution finds extensive use in various fields, including engineering, physics, and computer science. In signal processing, for instance, convolution helps analyze how signals respond to system inputs. In image processing, convolution plays a pivotal role in tasks like blurring and edge detection.

Mastering convolution is crucial for understanding these applications and enhancing problem-solving skills. Our guide aims to demystify convolution in mathematics, offering clear explanations and practical insights to empower you in your mathematical journey. Explore the versatility of convolution and its real-world applications with our comprehensive guide.