Convolution Integral Calculator

Convolution is a mathematical operation that combines two functions to produce a third function that expresses how the shape of one is modified by the other. This calculator computes the convolution integral of two functions, which is widely used in signal processing, probability theory, and physics.

What is a Convolution Integral?

The convolution integral is a mathematical operation that combines two functions to produce a third function that expresses how the shape of one is modified by the other. It is defined as:

(f * g)(t) = ∫[from -∞ to ∞] f(τ) g(t - τ) dτ

Where:

f(t) and g(t) are the two functions to be convolved
τ is the integration variable
The result is a new function that represents the convolution of f and g

Convolution is commutative, meaning f * g = g * f, and associative, meaning (f * g) * h = f * (g * h).

In signal processing, convolution is used to model the effect of a system on an input signal. In probability theory, it's used to combine independent random variables.

How to Use This Calculator

Enter the first function f(t) in the first input field
Enter the second function g(t) in the second input field
Specify the time range for the convolution
Click "Calculate" to compute the convolution integral
View the result and visualization

This calculator supports common mathematical functions including polynomials, exponentials, trigonometric functions, and constants.

Convolution Integral Formula

The convolution of two functions f(t) and g(t) is given by:

(f * g)(t) = ∫[from -∞ to ∞] f(τ) g(t - τ) dτ

For discrete functions, this becomes a sum:

(f * g)[n] = Σ[from k=-∞ to ∞] f[k] g[n - k]

The calculator uses numerical integration to approximate this integral for given functions.

Applications of Convolution

Convolution integrals are used in various fields:

Signal Processing: Modeling system responses to input signals
Probability Theory: Combining independent random variables
Image Processing: Applying filters to images
Control Systems: Analyzing system responses
Quantum Mechanics: Describing particle interactions

In signal processing, convolution is used to determine the output of a linear time-invariant system given an input and the system's impulse response.

FAQ

What is the difference between convolution and correlation?

Convolution involves flipping one of the functions before integrating, while correlation does not. Mathematically, correlation is convolution with one of the functions reversed in time.

When is convolution commutative?

Convolution is always commutative, meaning f * g = g * f for all functions f and g.

What are the units of a convolution?

The units of the convolution are the product of the units of the two functions being convolved.

Can convolution be used with complex functions?

Yes, convolution can be extended to complex-valued functions by treating the real and imaginary parts separately.