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Convolution Calculator

Compute the discrete linear convolution of two finite sequences. Convolution is fundamental to signal processing, probability, and polynomial multiplication.

Reviewed by Chase FloiedUpdated

This free online convolution calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

First input sequence

Second input sequence

Number of elements in A

Number of elements in B

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Convolution Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Convolution Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Convolution Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Convolution Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Convolution Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Compute the discrete linear convolution of two finite sequences. Convolution is fundamental to signal processing, probability, and polynomial multiplication. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Convolution Calculator

Convolution is a mathematical operation that combines two sequences (or functions) to produce a third sequence expressing how one is modified by the other. In discrete mathematics, convolution of two finite sequences of lengths M and N produces an output of length M + N − 1. Convolution is foundational in digital signal processing (filtering), probability theory (distribution of sums of random variables), image processing (blurring, edge detection), and polynomial multiplication. The convolution theorem states that convolution in the time domain equals pointwise multiplication in the frequency domain, enabling fast computation via the Fast Fourier Transform (FFT). This calculator performs direct (brute-force) linear convolution of two finite sequences, showing each output element as the sum of products of overlapping elements.

The Math Behind It

Given sequences f of length M and g of length N, their linear convolution (f * g) has length M + N − 1 and is defined by (f * g)[n] = Σ_{k=max(0,n-N+1)}^{min(n,M-1)} f[k] × g[n − k]. Direct computation requires O(MN) multiplications. The FFT-based approach computes convolution in O((M+N) log(M+N)) time by: (1) zero-padding both sequences to length M+N−1, (2) computing their FFTs, (3) multiplying pointwise, (4) inverse-FFT. Convolution is commutative (f*g = g*f), associative ((f*g)*h = f*(g*h)), and distributive over addition. The identity element is the unit impulse δ[n]. Circular (cyclic) convolution arises naturally with the DFT and differs from linear convolution by aliasing. Polynomial multiplication is equivalent to convolving the coefficient arrays.

Formula Reference

Discrete Convolution

(f * g)[n] = Σₖ f[k] × g[n − k]

Variables: f, g = input sequences; output length = len(f) + len(g) − 1

Worked Examples

Example 1: Convolve [1, 2, 3] with [1, 1]

Compute the linear convolution of A = [1, 2, 3] and B = [1, 1].

Step 1:Output length = 3 + 2 − 1 = 4
Step 2:(A*B)[0] = 1×1 = 1
Step 3:(A*B)[1] = 1×1 + 2×1 = 3; (A*B)[2] = 2×1 + 3×1 = 5; (A*B)[3] = 3×1 = 3

Result = [1, 3, 5, 3]

Common Mistakes & Tips

  • !Confusing linear convolution with circular (cyclic) convolution — they differ in boundary handling.
  • !Forgetting that the output length is M + N − 1, not max(M, N).
  • !Reversing the wrong sequence — convolution flips g, not f (though result is the same due to commutativity).

Related Concepts

Sequence Calculator

Evaluate and explore general sequences.

Sum of Series

Summation is a component of each convolution output element.

Frequently Asked Questions

What is the difference between convolution and correlation?

Convolution flips one sequence before sliding and multiplying; correlation does not flip. For real symmetric sequences, they produce the same result.

Why is convolution used in signal processing?

The output of a linear time-invariant (LTI) system is the convolution of the input signal with the system's impulse response, making convolution the fundamental tool for filter analysis.