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\begin{center}
{\large{\bf The Z-plane }}\\
\end{center}
The purpose of this assignment is to learn about the system function of
a discrete time system and its
relationship to the Z-plane, and filtering of discrete signals.
You will need the m-file {\tt frevalz.m} and can get it from the
web page.
The function you'll use, {\tt frevalz.m},
is similar to {\tt frevals.m} that you used on a prior lab.
Like {\tt frevals},
{\tt frevalz} takes in a system function, then computes and plots the
frequency and impulse responses. {\tt frevalz} has the additional feature of
plotting the pole/zero diagram.
The syntax of the two functions are quite different so
type help frevalz (in MATLAB) to learn more.
\begin{enumerate}
\item{\bf First Order System}
\begin{enumerate}
\item Use {\tt frevalz} to examine a first-order system that consists
of
a single pole on the real axis:
\[
H(z) = \frac{1}{1-az^{-1}}
\]
\underline{How does the sign of the pole affect the frequency
response of the system?}
\underline{What happens to the impulse response when the
pole has absolute value greater than 1? }
\item Compare the low pass system you can generate with the simple
system above to the following improved system:
\[
H(z) = \frac{1+z^{-1}}{1-az^{-1}}
\]
Pick $ a = 0.75$.
\underline{How is this system an improvement over the system described in
part (a)?}
\underline{Hand in a hard copy of the summary plot from the IMPROVED system.}
\end{enumerate}
\item{\bf Second Order System}
\begin{enumerate}
\item Use {\tt frevalz} to design a second-order low pass
system.
\underline{State where you placed your poles and zeros.}
\underline{Hand in a hardcopy of the summary plot of your system.}
\item Use {\tt frevalz} to design a second-order bandpass
system. Try to design a system that passes frequencies around
0.25 and rejects all others.
\underline{Explain where you chose to place your poles
and zeros and why.}
\underline{Hand in a hardcopy of the summary plot of
your system.}
\end{enumerate}
\end{enumerate}
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