What is meant by convolution

Convolution is a mathematical operation that combines two functions f(x)f(x)f(x) and g(x)g(x)g(x) to produce a third function, typically denoted as (f∗g)(x)(f * g)(x)(f∗g)(x), which expresses how one function modifies the shape of the other. Here’s a detailed explanation:

Definition:

In the context of continuous signals (1D), convolution is defined as: (f∗g)(x)=∫−∞∞f(τ)g(x−τ) dτ,(f * g)(x) = \int_{-\infty}^{\infty} f(\tau) g(x - \tau) \, d\tau,(f∗g)(x)=∫−∞∞?f(τ)g(x−τ)dτ, where:

• f(x)f(x)f(x) and g(x)g(x)g(x) are two functions (or signals).
• τ\tauτ is the integration variable that represents the time (or spatial) domain.
• xxx is the variable over which the resulting function (f∗g)(x)(f * g)(x)(f∗g)(x) is defined.

Interpretation:

Convolution can be understood as a measure of overlap between fff and a reversed and shifted version of ggg. Here’s how the operation unfolds:

1. Reversal and Shift: g(x−τ)g(x - \tau)g(x−τ) represents ggg reversed in time (or space) and shifted by xxx.

2. Multiplication: At each point xxx, f(τ)f(\tau)f(τ) and g(x−τ)g(x - \tau)g(x−τ) are multiplied together.

3. Integration: The product f(τ)g(x−τ)f(\tau) g(x - \tau)f(τ)g(x−τ) is integrated over all possible values of τ\tauτ, giving the value of (f∗g)(x)(f * g)(x)(f∗g)(x) at that point.

Properties:

• Commutativity: f∗g=g∗ff * g = g * ff∗g=g∗f
• Associativity: f∗(g∗h)=(f∗g)∗hf * (g * h) = (f * g) * hf∗(g∗h)=(f∗g)∗h
• Distributivity: f∗(g+h)=(f∗g)+(f∗h)f * (g + h) = (f * g) + (f * h)f∗(g+h)=(f∗g)+(f∗h)
• Identity Element: f∗δ=ff * \delta = ff∗δ=f, where δ(x)\delta(x)δ(x) is the Dirac delta function.

Applications:

Convolution finds extensive use in various fields:

• Signal Processing: Filtering, smoothing, and extracting features from signals.
• Image Processing: Blurring, edge detection, and image enhancement.
• Mathematics and Engineering: Solving differential equations and analyzing linear systems.
• Probability and Statistics: Computing probability distributions and estimating expectations.

Convolution vs. Cross-Correlation:

Convolution and cross-correlation are closely related operations, differing primarily in the treatment of the second function ggg. In convolution, ggg is typically reversed, whereas in cross-correlation, it is not reversed. The convolution operation is fundamental to understanding how systems and signals interact and transform in various domains of science and engineering.