# Decomposition of a Signal into Scaled Unit Impulses

Learn to build a discrete-time signal from separate unit impulses.

## We'll cover the following

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.

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