Decomposition of a Signal into Scaled Unit Impulses

Let’s explore how any length-$N$ discrete-time signal can be decomposed into $N$ separate scaled and shifted unit impulses.

Role of the unit impulse

A signal $x[n]$ is drawn in the figure below. Let’s multiply it with a unit impulse $\delta[n]$ at time $0$ and observe the output. Clearly, the value of $x[n]$ at time $0$ (i.e., $x[0]$) is the result. This is because all the other values in the impulse were zero, and the multiplication of the remaining values with zero is zero.