Simplified Convolution

Let’s revisit the definition of the convolution expression between the two signals $x[n]$ and $h[n]$.

$$y[n] = \sum_m x[m]h[n-m]$$

We now know how it can be more easily implemented in discrete-time systems.

Discrete-time implementation

A simplified method implements this procedure right out of the definition itself.

$$y[n] = \cdots + x[-1]h[n+1] + x[0]h[n-0]+x[1]h[n-1]+x[2]h[n-2]+\cdots$$