Convolution Calculator

If you've ever stared at a signals and systems problem wondering how two sequences combine to produce a new one, you're not alone. Convolution is one of those mathematical operations that feels abstract until you actually need to compute it by hand — and then it becomes painfully tedious. Whether you're a student preparing for an exam, an engineer working on digital filters, or someone exploring audio processing, you'll often need to convolve two discrete signals quickly and accurately. That's exactly where this free online convolution calculator comes in: you enter two sequences, and it instantly returns the convolved result along with a full step-by-step breakdown so you can check your work and truly understand what's happening.

How to Use the Convolution Calculator

Using this tool is straightforward. Here's how to get your convolved signal in just a few steps:

1. Enter Signal A — In the first textarea, type the numbers of your first discrete sequence separated by commas. For example: 1, 2, 3.
2. Enter Signal B — In the second textarea, type the numbers of your second discrete sequence separated by commas. For example: 0, 1, 0.5.
3. Click "Calculate Convolution" — The tool immediately processes both signals using the discrete convolution formula and displays the resulting convolved signal below.
4. Review the result — You'll see the convolved signal as an array (or numbered list if you prefer) along with its total length in samples. The result updates automatically as you type, so you can experiment with different values.
5. Explore the step-by-step breakdown — Click "Show Step-by-Step Breakdown" to see a table that walks you through each index of the convolution, showing exactly which products are summed to produce each output value.
6. Tweak advanced options — Use the "Advanced Options" panel to change decimal places, rounding mode (standard, ceiling, or floor), and output format (array or numbered list).
7. Clear and start over — Hit the "Clear" button to reset all fields and results.

Formula

The discrete convolution of two finite sequences is defined mathematically as:

(A ∗ B)[n] = Σ_k A[k] · B[n − k]

Where A and B are the two input sequences, n ranges from 0 to (len(A) + len(B) − 2), and the sum runs over all k where both A[k] and B[n−k] are defined. The length of the resulting convolved signal is always len(A) + len(B) − 1.

Let's walk through a practical example. Suppose Signal A = [1, 2, 3] and Signal B = [0, 1, 0.5]. The output length will be 3 + 3 − 1 = 5. For index n=0, only k=0 works: 1·0 = 0. For n=1, we have k=0 and k=1: 1·1 + 2·0 = 1. For n=2, k=0,1,2: 1·0.5 + 2·1 + 3·0 = 2.5. For n=3, k=1,2: 2·0.5 + 3·1 = 4. For n=4, k=2: 3·0.5 = 1.5. So the final convolution is [0, 1, 2.5, 4, 1.5]. The tool does all this instantly, with any rounding you choose.

What is a Convolution Calculator?

A convolution calculator is a digital tool that computes the discrete convolution of two finite sequences — a fundamental operation in signal processing, systems theory, and applied mathematics. At its core, convolution combines two signals to produce a third signal that represents how the shape of one is modified by the other. This is essential for understanding how linear time-invariant (LTI) systems respond to inputs, how digital filters work, and how echoes or reverberation are modeled in audio.

Students in electrical engineering, computer science, and physics use convolution constantly — for instance, when analyzing the output of a filter given an input signal, or when performing cross-correlation for pattern matching. Engineers working with digital audio, image processing, or control systems also rely on convolution daily. The beauty of this calculator is that it removes the drudgery of manual computation while preserving the educational value: you see every intermediate step, so you can verify your understanding and catch mistakes. Instead of spending ten minutes on a single convolution by hand, you get the answer in milliseconds, plus a detailed breakdown that shows exactly how each output sample is built from the input values.

Frequently Asked Questions

What if my two signals have very different lengths?

No problem at all. The convolution formula handles sequences of any length — the output length will always be len(A) + len(B) − 1, regardless of whether one signal is much longer than the other. The calculator automatically loops over all valid index pairs, so you can enter a 3-sample signal and a 10-sample signal, and it will compute the 12-sample result correctly.

Can I use this calculator for circular convolution or only linear?

This calculator performs standard linear (aperiodic) convolution, which is the most common type in introductory signals and systems courses and in many practical applications. If you need circular convolution (for example, when working with the Discrete Fourier Transform and cyclic prefixes), you would need to zero-pad the sequences to the same length first — this tool doesn't wrap around the ends. For most homework and real-world filtering tasks, linear convolution is exactly what you need.

Why does the result have more samples than either of my input signals?

That's a fundamental property of convolution. When you convolve two sequences of lengths L and M, the resulting sequence has length L + M − 1. This happens because each output sample is a sum over shifted products — the signals slide across each other, and the overlap region grows and then shrinks. It's not an error; it's how convolution works. The step-by-step breakdown table makes this visually clear by showing which pairs of indices contribute to each output value.