# Convolution Integral Calculator (Numeric)

Numeric (f*g)(t) = ∫ f(τ)g(t−τ)dτ by composite Simpson, in causal or direct-window mode, at up to 8 output points.

> Canonical page: https://elysiatools.com/en/tools/convolution-integral-calculator

- **Category:** Math & Numbers

- **Keywords:** convolution, convolution integral, signal processing, lti systems, impulse response, simpson rule, numerical integration, laplace product, gaussian convolution, calculus

## Overview

The Convolution Integral Calculator numerically evaluates the continuous convolution integral (f*g)(t) = ∫ f(τ)g(t−τ)dτ using the composite Simpson rule. It supports both causal signals and direct-window integration across up to 8 user-defined time points.

## Inputs

- **f(τ) expression (variable x)** (text): First signal, written in the variable x. Supports + - * / ^, parentheses, and functions like exp, sin, cos, sqrt, ln.
- **g(τ) expression (variable x)** (text): Second signal, written in the variable x (it is evaluated at t − τ inside the integral).
- **Mode** (select)
- **Window start τmin** (number): Lower integration bound.
- **Window end τmax** (number): Upper integration bound.
- **Output t values (comma-separated)** (text): 1–8 evaluation points t at which (f*g)(t) is reported.
- **Simpson intervals** (number): Number of Simpson subintervals per integral (even, 100–20000).
- **Decimal Places** (number)

## When to use

- Verifying analytical convolution integrals and Laplace transform multiplication in signals and systems coursework.
- Evaluating continuous-time LTI system responses to non-standard or complex input signals lacking simple closed-form solutions.
- Computing the overlap and smoothing effects of bilateral continuous functions like Gaussian kernels.

## How it works

- Enter the mathematical expressions for f(x) and g(x) using the variable x, standard operators, and supported functions such as exp, sin, cos, sqrt, and ln.
- Select causal mode for signals active only when t ≥ 0, or direct window mode to integrate over the full \[τmin, τmax\] interval.
- Specify the integration window bounds (τmin, τmax), output evaluation points t (comma-separated, up to 8), Simpson intervals, and decimal precision.
- The tool performs composite Simpson quadrature to compute and report the numerical convolution value for each specified t.

## Use cases

- Evaluating system impulse responses for causal exponential decay signals in linear circuits.
- Calculating overlapping Gaussian pulse interactions in optical and signal processing models.
- Cross-checking manual convolution steps against high-precision numerical quadrature.

## Frequently asked questions

### What variable name must be used in function expressions?

Always use x as the variable in both f(x) and g(x) expressions.

### What is the difference between causal and direct window mode?

Causal mode restricts the integration bounds to [max(τmin, 0), min(τmax, t)] assuming signals are zero for negative values, while direct window mode integrates across the entire [τmin, τmax] range.

### How many output time points can be calculated at once?

You can evaluate between 1 and 8 comma-separated time values in a single calculation.

### How does the number of Simpson intervals impact the result?

Higher interval counts increase numerical integration accuracy for steep or rapidly oscillating signals, with a default setting of 2000.

### Which mathematical functions are supported?

Supported operations include +, -, *, /, ^, parentheses, and standard functions like exp, sin, cos, sqrt, and ln.

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