Convolution
Compute the discrete convolution of two sequences using the formula (a ∗ b)[n] = Sigma a[k] b[n-k]. Free online convolution calculator for signal processing, probability, and sequence analysis with instant results.
About This Calculator
The Convolution Calculator computes the discrete convolution of two finite sequences. Convolution is a fundamental mathematical operation where one sequence is slid over another, and at each position, the sum of element-wise products is computed. The resulting sequence (a ∗ b) has length m + n - 1 when convolving sequences of lengths m and n.
The convolution formula c[n] = Sigma a[k] x b[n-k] combines the two input sequences by multiplying overlapping terms and summing them. This operation is central to signal processing (system response to any input), probability theory (sum of random variables), computer vision (convolutional neural networks), differential equations, and audio processing.
Enter two comma-separated sequences. For example, [1, 2, 3] convolved with [4, 5, 6] yields [4, 13, 28, 27, 18]. This calculator handles sequences of any length up to 15 terms each, supports decimals, and displays the full convolution result with all terms.
Frequently Asked Questions
What is convolution in mathematics?
Convolution is a mathematical operation on two sequences that produces a third sequence. For sequences a and b, the convolution (a ∗ b) at index n is the sum of products a[k] x b[n-k] for all k. It combines the two sequences by sliding one over the other and computing weighted sums.
How do you calculate discrete convolution?
For sequences a[0..m-1] and b[0..n-1], the convolution c has length m+n-1. Each term c[i] = Sigma a[k] x b[i-k] for k from max(0, i-n+1) to min(i, m-1). For example, [1,2,3] ∗ [4,5,6] = [4, 13, 28, 27, 18].
What is convolution used for in signal processing?
In signal processing, convolution describes how a linear time-invariant system responds to any input. Given the system's impulse response, convolving it with any input signal produces the system's output. Applications include audio filtering, image blurring/sharpening, and echo/reverb effects.
Is convolution commutative?
Yes, convolution is commutative: a ∗ b = b ∗ a. It is also associative: (a ∗ b) ∗ c = a ∗ (b ∗ c), and distributive: a ∗ (b + c) = a ∗ b + a ∗ c. These properties make convolution a linear operation, fundamental in signal processing and systems theory.
What is the relationship between convolution and probability?
In probability theory, the probability density function of the sum of two independent random variables is the convolution of their individual density functions. This makes convolution essential for understanding sums of random variables, central limit theorem applications, and statistical modeling.
How is convolution used in computer vision?
In computer vision and deep learning, convolutional neural networks (CNNs) use 2D convolution operations to detect features in images. A filter (kernel) slides across the image, computing dot products to detect edges, textures, and patterns at multiple scales.