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

$$ \begin{align} \cos(25\pi t)(\delta(t)+\delta(t-0.01)) &= \cos(25\pi t)\delta(t) + \cos(25\pi t)\delta(t-0.01) \\ &= \cos(0)\delta(t) + \cos(0.25\pi t)\delta(t-0.01) \\ &= \delta(t) + 0.707\delta(t-0.01) \end{align} $$

Example of Impulse Sampling

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

$$ \begin{align} x_{2}(t) &= e^{-(10-5)}u(10-5)\delta(t-10) \\ &= e^{-5} \cdot 1 \cdot \delta(t-10) \\ &= e^{-5}\delta(t-10) \end{align} $$

Step Signals

• Signals that are nonzero for $t \ge 0$

$$ u(t) = \begin{cases} 0 & t \lt 0 \\ 1 & t \ge 0 \end{cases} $$

• Unit step signals have a height of 1
• Used to create one sided signals (usually right-sided)

$$ \begin{align} x(t) &= 3.72e^{-12.8t} \\ x_{\mathrm{rs}}(t) &= 3.72e^{-12.8t} \cdot u(t) \end{align} $$

• Discontinuous at $t = 0$ due to the piecewise nature of the function

Pulse Signals

• Pulses are only non-zero between $t_1$ and $t_2$
• Unit pulses have a height of 1
• Technically a difference of step signals

$$ p(t) = u(t-t_{1})-u(t-t_{2})=\begin{cases} 1 & t_{1} \le t \le t_{2} \\ 0 & t \lt t_{1} \ \mathrm{or} \ \gt t_{2} \end{cases} $$

• Multiplying a function by a pulse gives us a finite duration signal

$$ x(t) = 5\cos(3\pi t) \cdot (u(t)-u(t-2)) $$

• This gives us a cosine wave with $f = 3 \pi$ rad/s and $A = 5$ for $t \in [0, 2]$