Last updated October 10, 2026 · Report an error
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.