Convolution Calculator

Convolution combines two sequences to describe how one pattern modifies, filters, or overlaps another. For finite discrete sequences, every output sample is the sum of products formed by sliding one sequence across the other. If the first sequence has length m and the second has length n, their full linear convolution has length m + n - 1. This calculator performs that complete operation and displays samples from the earliest overlap through the latest overlap. The operation appears throughout digital signal processing. A sampled input signal convolved with a system's impulse response produces the system output when the system is linear and time invariant. Audio equalizers, smoothing filters, echo models, image kernels, and communication channels all use this idea. In probability, convolution also gives the distribution of a sum of independent random variables, while polynomial multiplication uses the same coefficient-by-coefficient pattern. To calculate an output sample, pair values whose indices add to that sample's index, multiply each pair, and add the products. Near the ends, only a few values overlap. In the middle, more values contribute. The calculator accepts integers, decimals, and negative numbers, so it can model many practical finite signals without requiring special notation. It computes linear convolution rather than circular convolution; circular convolution wraps indices around a fixed period and can produce a different result. Convolution is commutative, so exchanging the two input sequences does not change the output. It is also associative and distributive, properties that make cascaded filters and decomposed systems easier to analyze. A one-sample impulse with value one acts as an identity: convolving any sequence with it returns the original sequence. Scaling either input scales every output by the same factor. Use the displayed result as an exact sample list for classroom exercises, quick filter checks, or verification of hand calculations. Very long sequences are often processed with fast Fourier transform methods, but direct summation is clear and reliable for the short sequences typically entered here. Always keep the sample order consistent, and include explicit zeros when they represent meaningful delays.