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Lecture 5: Some Useful Functions

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Schedule :: Fourier Series Formulas :: Nonperiodic Functions :: Useful Functions :: Simple Functions :: Fourier Transforms :: Properties :: System Analysis :: Signal Distortion :: Filters :: Parseval's Formulas

The unit gate or unit pulse is defined by
\begin{displaymath}\rect(x) = \left\{\begin{array}{ll} 0 & \vert x\vert > \frac{...
...rac{1}{2} \\
1 & \vert x\vert < \frac{1}{2}\end{array}\right.
\end{displaymath}

Some interesting variations: Plot $\rect(x/2)$ , $\rect((x-1)/2)$ , $\rect((x-a)/T)$ . Note that the total width is $T$ and the function is centered around $a$ . The unit triangle function, also known as the hat function (or the chapeau function), is defined by
\begin{displaymath}\Delta(x) = \left\{\begin{array}{ll} 0 & \vert x\vert > \frac...
...- 2\vert x\vert & \vert x\vert < \frac{1}{2}\end{array}\right.
\end{displaymath}

Plot. Also look at $\Delta(x/T)$ . Notice that the total width of the function is $T$ . The sinc function we have already met in the context of the F.S. of a square wave. To reiterate,
\begin{displaymath}\sinc(x) = \frac{\sin(x)}{x}
\end{displaymath}

Observe the following:
  1. $\sinc(x)$ is even.
  2. $\sinc(x)$ is zero for $x = \pm \pi, \pm 2\pi, \pm 3\pi,\ldots$ .
  3. $\sinc(0) = 1$
  4. $\sinc(x)$ decays as $1/x$ .
The signum function is defined by
\begin{displaymath}\sgn(t) = \left\{ \begin{array}{ll} 1 & t > 0 \\
-1 & t < 0\end{array}\right.
\end{displaymath}
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 26). Lecture 5: Some Useful Functions. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/5_3node3.html. This work is licensed under a Creative Commons License Creative Commons License