# The Spectrum of the Unit Impulse

Learn to derive the DFT of a unit impulse and observe the fundamental result.

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Understanding the DFT starts with the spectrum of a unit impulse. This case gives us an interesting result, as we will now see.

## The signal

A unit impulse is a signal that is $0$ everywhere, except at time index $0$ where its value is $1$. In mathematical terms, we say:

$\begin{equation*} \delta[n] = \left\{ \begin{array}{l} 1, \quad n = 0 \\ 0, \quad n \neq 0 \\ \end{array} \right. \end{equation*}$

This is drawn in the figure below:

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