Enter two discrete sequences — Sequence X and Sequence H — as space-separated numbers, and this Convolution Calculator computes the full convolution output y(n) using the standard discrete convolution formula. You get the complete result sequence displayed term by term, along with a bar chart visualizing the output values. Also try the Limit Definition Calculator.
Convolution Output y(n)
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Output Sequence Length
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Length of x(n)
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Length of h(n)
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Convolution is a mathematical operation that combines two sequences (or functions) to produce a third sequence. For discrete sequences x(n) and h(n), the convolution y(n) = Σ x(k)·h(n−k). It is widely used in signal processing, filtering, and systems analysis to describe the output of a linear time-invariant system. See also our Derivative Calculator.
For finite sequences where x(n) has M values and h(n) has H values, the output y(n) has length M + H − 1. For example, convolving a 4-element sequence with a 3-element sequence produces a 6-element output.
Simply type the sample values separated by spaces (e.g. '1 2 3 4'). Both positive and negative decimal numbers are supported. The calculator will automatically parse your input and compute the full convolution.
Linear (aperiodic) convolution, which this calculator performs, treats the sequences as finite and produces an output of length M+H−1. Circular convolution, used in DFT-based processing, wraps the sequences periodically and produces an output of the same length as the inputs. For most filtering applications, linear convolution is what you need. You might also find our use the Chain Rule Calculator useful.
Yes. In signal processing, convolving a signal x(n) with a filter's impulse response h(n) gives the filtered output y(n). Simply enter the input signal as x(n) and the filter coefficients as h(n) to compute the filter output.
No — convolution is commutative, meaning x(n) * h(n) = h(n) * x(n). The resulting output sequence y(n) will be identical regardless of which sequence you label as x(n) and which as h(n).
If either sequence is all zeros, the convolution output y(n) will also be all zeros. This reflects the mathematical property that any sequence convolved with the zero sequence yields zero.
The discrete unit impulse δ(n) is the sequence [1, 0, 0, …]. Any sequence convolved with the unit impulse returns itself unchanged, making δ(n) the identity element of convolution. You can test this by entering '1' as one of the sequences.