convolutions

Convolution is a mathematical operation that combines two functions to produce a third function expressing how the shape of one is modified by the other. In analysis and signal processing, it describes how the profile of a signal is smoothed, sharpened, or otherwise transformed by a kernel or filter.

For continuous-time functions f and g, the convolution is defined by (f * g)(t) = ∫_{-∞}^{∞} f(τ) g(t − τ) dτ. For discrete-time sequences f[n] and g[n], the discrete convolution is (f * g)[n] = ∑_{k=-∞}^{∞} f[k] g[n − k], summed over all integers k.

Convolution is commutative and associative; it has the delta function δ as an identity: f * δ = f. It also distributes over addition.

The Fourier transform links convolution to multiplication: F{f * g} = F{f} · F{g}. In the discrete case, the same convolution–multiplication relationship holds with the discrete Fourier transform, enabling fast computation via the Fast Fourier Transform (FFT).

Practically, one must consider boundary conditions in finite signals, using padding or truncation. Convolution kernels have finite support in practice, and fast algorithms (like FFT-based methods) are used for large inputs.

Applications span signal processing (filters, smoothing, impulse responses), image processing (blurring, sharpening, edge detection), probability (distribution of sums), and machine learning (convolutional neural networks), where kernels slide over inputs to extract features; in many frameworks this operation is implemented as cross-correlation.