Convolution Calculator

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Discrete convolution combines two finite sequences into a third sequence by systematically summing overlapping products — one of the most fundamental operations in digital signal processing, where it models how a filter transforms an input signal over time (an audio equalizer, image blur/sharpen filter, or an FIR digital filter is literally computing a convolution). It also appears in probability theory (the distribution of a sum of two independent random variables is the convolution of their individual distributions) and in pure algebra, where convolving the coefficient lists of two polynomials is exactly equivalent to multiplying those polynomials together.

How it works

Enter both sequences as comma-separated numbers, f and g. The calculator computes y[n] = Σ f[k]·g[n-k], summing over every k where both f[k] and g[n-k] are defined, for every valid output index n — showing each individual term in the sum so you can see exactly which pairs of input values contributed to each output value.

Enter sequence f (comma-separated).
Enter sequence g (comma-separated).
Click Calculate to see your results.

Examples

f = [1, 2, 3], g = [0, 1]

y = [0, 1, 2, 3] — since g just shifts and scales f by 1, the result is the same as shifting f by one position and adding a leading zero.

f = [1, 1], g = [1, 1]

y[0]=1×1=1. y[1]=1×1+1×1=2. y[2]=1×1=1. Result: y = [1, 2, 1] — matching the coefficients of (x+1)² = x²+2x+1, since convolving two sequences is equivalent to multiplying their corresponding polynomials.

Common mistakes to avoid

Expecting the output to be the same length as the inputs — convolution always produces a longer sequence (length len(f)+len(g)-1), unless one input has length 1.
Reversing the wrong sequence mentally — the formula flips g (via the n-k index), not f, though the commutative property means you'll still get the right final answer either way.
Confusing convolution with simple element-wise multiplication of the two sequences, which is a completely different (and much simpler) operation.

Frequently asked questions

Its length is len(f) + len(g) - 1 — for example, two length-3 sequences convolve into a length-5 output.

If you treat each sequence as the coefficients of a polynomial (in order from lowest to highest power), convolving the two sequences produces exactly the coefficients of the product polynomial — this is a handy way to multiply polynomials by hand-checkable arithmetic.

No — convolution is commutative, so convolving f with g always gives the same result as convolving g with f.

Every blur, sharpen, or edge-detection filter applied to an image is a 2D convolution between the image's pixel grid and a small "kernel" matrix — the same core operation as this 1D calculator, just extended to two dimensions.

Convolving with a length-1 sequence just scales every term of the other sequence by that single value and leaves the length unchanged, since y[n]+len-1 with len=1 reduces to the original sequence's length.

The entire output becomes all zeros too, since every term in the convolution sum ends up multiplied by at least one zero from that sequence.

Free Convolution Calculator