Convolution operation¶
\[\begin{split}
s(t) = \int x(a) w(t-a)da \\
s(t) = \sum_{a=-\infty}^ tx(a) w(t-a) \\
s(t) = (x * w)(t)
\end{split}\]
\(w\) is a valid probability density
in the discrete case in general we do not need the infinite sum since in most cases we have only finite number of elements.
Example \(x(a)\) measures a position, this measurement in noisy, and we take multiple measurements to reduce the noise. \(w(a)\) is a weighted average operation that puts more weight on recent measurements.