Decomposition of a Signal into Scaled Unit Impulses

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

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

Role of the unit impulse

A signal x[n]x[n] is drawn in the figure below. Let’s multiply it with a unit impulse δ[n]\delta[n] at time 00 and observe the output. Clearly, the value of x[n]x[n] at time 00 (i.e., x[0]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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