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.

Time shift

Next, we multiply it with an impulse shifted by $1$, or $\delta[n-1]$ ...