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Lecture 1: Difference Equations

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

\begin{displaymath}y[k+n] + a_{n-1} y[k+n-1] + \cdots a_{1} y[k+1] + a_0 y[k] = b_m
f[k+m] + b_{m-1}f[k+m-1] + \cdots + b_1 f[k+1] + b_0 f[k] \end{displaymath}

(advance operator form) It could also be written as (by $k\rightarrow k-n$ )
\begin{displaymath}y[k] + a_{n-1} y[k-1] + \cdots a_{1} y[k-n+1] + a_0 y[k-n] = b_n
f[k] + b_{n-1}f[k-1] + \cdots + b_1 f[k-n+1] + b_0 f[k-n] \end{displaymath}
(delay operator form).

Numerically it is straightforward to find solutions of the system equation given some initial conditions:

\begin{displaymath}y[k] = -a_{n-1}y[k-1] - a_{n-2}y[k-2] - \cdots - a_0 y[k-n] + b_n
f[k] + b_{n-1}f[k-1] + \cdots + b_0 f[k-n] \end{displaymath}
Just propagate forward.

\begin{example}Fibonacci series: $y[1] = 1$, $y[2] = 1$, $y[n] = y[n-1] +
y[n-2]$.
\end{example}

\begin{example}$y[k] -0.5 y[k-1] = f[k] $. Suppose we know $y[-1] = 16$ and
$f[...
...h}\begin{displaymath}y[2] = 0.5(5) + 4 = 6.5 \end{displaymath}etc.
\end{example}

We will use $E$ to indicate the advance operator: $Ef[k] = f[k+1]$ , etc.


\begin{example}$y[k+2] + \frac{1}{4}y[k+1] + \frac{1}{16}y[k] = f[k+2]$. In
oper...
...{displaymath}(A lot like what we did for differential equations.)
\end{example}

We can write the general $n$ th order difference equation as

\begin{displaymath}(E^n + a_{n-1}E^{n-1} + \cdots + a_{1} E + a_0)y[k] = (b_n E^n +
b_{n-1}E^{n-1} + \cdots + b_1 E + b_0)f[k] \end{displaymath}
or
\begin{displaymath}Q[E]y[k] = P[E]f[k] \end{displaymath}

Now we do the same kinds of things we did before: the zero-input response, then the zero-state response.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 16). Lecture 1: Difference Equations. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/node3.html. This work is licensed under a Creative Commons License Creative Commons License