Discrete convolution: (f*g)[n] = Σₖ f[k]g[n−k]. Output length = len(f)+len(g)−1. Equivalent to polynomial multiplication. In frequency domain: F{f*g} = F{f}·F{g}. Foundation of FIR filtering and linear systems.
How to Use
Enter sequences:
- f: First sequence
- g: Second sequence
- Output: (f*g)[n]
Properties
- Commutative: f*g = g*f
- Associative: f*(g*h) = (f*g)*h
- Distributive: f*(g+h) = f*g + f*h
- Identity: f*δ = f
FFT Convolution
Direct: O(n²). FFT method: O(n log n). Pad to power of 2, FFT both, multiply, inverse FFT. Used for large sequences.
Step-by-Step Instructions
- 1Enter sequence f.
- 2Enter sequence g.
- 3View (f*g)[n].
- 4See step calculations.
- 5Check output length.