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Lecture 10: Discrete-Time

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Schedule :: Perspective :: Transfer Functions :: Laplace Transform :: Poles and Eigenvalues :: Time Domain :: Linear Transformations :: Special Transformation :: Controllability :: Discrete-Time

Most of what can be said for continuous time can also be said for discrete time:

\begin{displaymath}\xbf[k+1] = A \xbf[k] + B \fbf[k]
\end{displaymath}

\begin{displaymath}\ybf[k] = C \xbf[k] + D \fbf[k].
\end{displaymath}

Solution:
\begin{displaymath}\xbf[k] = A^k \xbf[0] + \sum_{j=0}^{k-1} A^{k-1-j} B \fbf[j].
\end{displaymath}

(Show how this works by recursion), starting from
\begin{displaymath}\xbf[1] = A \xbf[0] + B \fbf[0].
\end{displaymath}

Z-transform:

\begin{displaymath}zX(z) - z\xbf[0] = AX(z) + BF(z).
\end{displaymath}

\begin{displaymath}X(z) = (zI-A)^{-1} [z\xbf[0] + B F(z)]
\end{displaymath}

\begin{displaymath}H(z) = C(zI-A)^{-1}B + D
\end{displaymath}
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 08). Lecture 10: 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/10_8node8.html. This work is licensed under a Creative Commons License Creative Commons License