In mathematics, a convolution measures the overlap between two functions when one is flipped and shifted. For continuous functions, such as $f, g: \mathbb{R}^d o \mathbb{R}$, it is defined by the integral: $(f * g)(\mathbf{x}) = \int f(\mathbf{z}) g(\mathbf{x}-\mathbf{z}) d\mathbf{z}$ For discrete sequences over $\mathbb{Z}$, the integral becomes a sum. The one-dimensional discrete convolution of functions $f$ and $g$ is: $(f * g)(i) = \sum_a f(a) g(i-a)$ In signal processing, this operation connects an input signal $x[n]$ and an impulse response $h[n]$ to produce an output $(x * h)[n] = \sum_{k=-\infty}^{+\infty} x[k] h[n-k]$, with bounded summation for finite-length signals.