Impulse Signals

• Non-zero signals concentrated around a specific moment in time

• Has an area of 1

• Not a function, but can be approximated with the

  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 $\delta_{\Delta} = 1$

  $$\int_{-\Delta}^{\Delta}\frac{1}{2\Delta} \ \mathrm{d}t = 1$$

  • 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 $\Delta \rightarrow 0$

Sampling Property

• A function multiplied by an impulse scales the function by its value at the time of the impulse
  • $f(t)\delta(t-t_{0}) = f(t_{0})\delta(t-t_{0})$
• Basically, you’re getting the value of $f(t)$ specifically only at $t_0$, and 0 everywhere else

Example of Impulse Sampling

Simplify this as much as possible: $x_{2}(t) = e^{-(t-5)}u(t-5)\delta(t-10)$