1. What is Convolution?

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. It's widely used in signal processing, image processing, and differential equations.

2. How Does the Calculator Work?

The calculator computes the convolution integral:

Where:

• \( f \) — First input function
• \( g \) — Second input function
• \( \tau \) — Dummy integration variable
• \( t \) — Time shift parameter

Explanation: The integral "slides" one function past the other, multiplying and accumulating the product at each position.

3. Importance of Convolution

Details: Convolution is fundamental in systems theory, where it describes the output of a linear time-invariant system to any input, given its impulse response.

4. Using the Calculator

Tips: Enter two functions in terms of the variable (typically 't'), and specify the integration limits. Use standard mathematical notation (e.g., e^(-t), sin(t), u(t) for unit step).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between convolution and multiplication?
A: Convolution combines functions through integration over their product at all possible shifts, while multiplication simply multiplies their values pointwise.

Q2: What are common applications of convolution?
A: Signal filtering, probability distributions of sums of random variables, solving differential equations, and image processing (blurring, edge detection).

Q3: What functions can be convolved?
A: Any functions where the integral converges. Common pairs include exponential, trigonometric, polynomial, and piecewise functions.

Q4: How does convolution relate to Fourier transforms?
A: Convolution in time domain equals multiplication in frequency domain, and vice versa.

Q5: What's the convolution of a function with a delta function?
A: It returns the original function (shifted if the delta is shifted) - this is the sifting property.