Discrete Convolution

A discrete Convolution, is a convolution applied to Discrete-Time functions, rather than Continuous-Time functions. We assume that our input signals have finite lengths and start at index zero, due to the Digital Signal Processing context.

$$\begin{array}{r}y[n]=\sum _{k=0}^{\text{length}(h)-1}h[k]\cdot x[n-k]\\ \\ \text{length}(y)=\text{length}(x)+\text{length}(h)-1\end{array}$$

Here $h[k]$ is the Impulse Response of the system; this is the discrete analog of Duhamel’s Principle, where $y_p(t)=h\ast f$.