Dirac Delta Function

A limiting case of a rectangular pulse function, $\delta_{\Delta}(t)$, as the width $\Delta \rightarrow 0$.

Rectangular Pulse Function

• For $-\Delta \le t \le \Delta$, the rectangular pulse has a value of $\frac{1}{2\Delta}$
• Outside this range, the function is 0

Normalization Property

• Regardless of the choice of $\Delta$, the area of $\delta_{\Delta} = 1$

• This ensures that the height of the pulse increases proportionally as the width becomes narrower

Dirac Delta as a Limit

• $\delta(t)$ is defined as the limit of this rectangular pulse as $\Delta \rightarrow 0$