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Lecture 1: Useful Signal Models for Discrete-Time

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Schedule :: Intro :: Signal Models for Discrete-Time :: Signal Operations :: Difference Equations :: Zero-Input :: Zero-State :: Natural & Forced :: System Stability

Some Useful Signal Models for Discrete-Time

Discrete-time unit impulse $\delta[k]$ :
\begin{displaymath}\delta[k] =
\begin{cases}
1 & k = 0 \\
0 & k \neq 0
\end{cases}\end{displaymath}

Discrete-time unit step $u[k]$ :

\begin{displaymath}u[k] =
\begin{cases}
1 & k \geq 0 \\
0 & k < 0
\end{cases}\end{displaymath}

Discrete-time exponential
$\gamma^k$ .

$\gamma^k$ grows when $\vert\gamma\vert > 1$ , and decays when $\vert\gamma\vert < 1$ . Plot on unit circle.

What about when $\vert\gamma\vert = 1$ ? We can write this as $e^{j \Omega k}$ for some $\Omega$ . This is a rotating phasor.

Discrete-time sinusoid $\cos(\Omega k + \theta)$ .

Note: Not all discrete-time sinusoids are periodic (unlike in the continuous-time case):

\begin{displaymath}f[k] = f[k+N_0]
\end{displaymath}

This is only possible if $\Omega N_0$ is an integral multiple of $2\pi$ :

\begin{displaymath}\cos(\Omega k) = \cos(\Omega(k + N_0))
\end{displaymath}

iff

\begin{displaymath}\Omega N_0 = 2 \pi m
\end{displaymath}
or

\begin{displaymath}\frac{\Omega}{2\pi} = \frac{m}{N_0}
\end{displaymath}

Must therefore have $\Omega/2\pi$ be a rational number.

Note: There is non-uniqueness of discrete-time sinusoids. For example:

\begin{displaymath}\cos(9.6 \pi k + \theta) = \cos(8\pi k + 1.6 \pi k + \theta) ...
...\theta) = \cos(-1.4 \pi k + \theta) = \cos(0.4 \pi k
- \theta)
\end{displaymath}

All these frequencies 'look' the same!
Exponentially damped sinusoid
$\gamma^k \cos(\Omega k + \theta)$
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 16). Lecture 1: Useful Signal Models for Discrete-Time. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/1_1node1.html. This work is licensed under a Creative Commons License Creative Commons License