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{\LARGE\bf EE 341: Linear Systems Analysis}
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{\large{\bf Assignment 3 : Digital Filters}}\\
\noindent Due Date: Wednesday, February 17, 1999
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In this assignment you will again be using the {\tt frevalz} to look
at the behavior of system functions in the z-plane. This time the
system functions you will examine will be various digital filters
that you use MATLAB to design. This will give you some insight into how digital
filters are designed and what are the properties of different digital
filter design algorithms. You will not be expected to understand the
details of
how these algorithms work; you only need to evaluate their behavior.
\begin{enumerate}
\item{\bf FIR (Finite Impulse Response) Digital Filters}
MATLAB has numerous built-in functions for generating discrete time filters.
In this problem we are going to use a few to look at how higher order
systems behave in the z-plane.
\begin{enumerate}
\item Use the MATLAB function FIR1 to create a
low pass FIR filter of order 8 with cutoff frequency
of $.4 \pi$. The function
FIR1 works with normalized frequencies between the
range of $-1 \leq \omega \leq 1$, which corresponds to the
unnormalized range $-\pi \leq \omega \leq \pi$.
Use {\tt frevalz} to study the system.
Save your filter
in a vector since you'll use it again in part 3.
\item Use the MATLAB function FIR1 to create a
high pass FIR filter of order 8 with cutoff frequency
of $.4 \pi$.\underline{ Comment on the placement of}
\underline{ the zeros in this
filter.}
\end{enumerate}
\item{\bf IIR (Infinite Impulse Response) Digital Filters}
\begin{enumerate}
\item Use the MATLAB function BUTTER
to create a low pass butterworth filter with
cut-off frequency $.4 \pi$ and filter order of 8.
\underline{ How does the performance of this filter compare to that of
the FIR filter?}
In this context, performance refers to how close
a filter matches an ideal low pass filter:
\[
H_{ideal}(e^{j\omega}) = \left\{
\begin{array}{ll}
1 & |\omega|\leq .4 \pi \\
0 & \mbox{otherwise}.
\end{array} \right.
\]
\underline{Comment on any differences in the phase of the two filters.}
You do not need to hand in any plots.
Save your filter in another vector since you'll use it again in
part 3.
\item Repeat part (a) of this problem for a high pass filter
with cut-off frequency of $.4 \pi$.
\underline{Compare the performance
of this IIR filter to that of the FIR.} That is, how close
is the performance of the two filters to that of an ideal
high pass filter:
\[
H_{ideal}(e^{j\omega}) = \left\{
\begin{array}{ll}
1 & |\omega| \geq .4 \pi\\
0 & \mbox{otherwise}.
\end{array} \right.
\]
Again, you do not need to hand in any plots.
\end{enumerate}
\item{\bf Filter Implementation}
Use the MATLAB function FILTER to implement the two low pass
filters you produced in part (a) of problems 1 and 2. Let the
input to the filters be a pulse of length ten:
\[
x[n] = u[n] - u[n-10]
\]
but let $x[n]$ be of length 60 (so append 50
zeros on the end).
\underline{ Comment on the differences in the output of the two filters.}
\underline{ Hand in a plot of the output of both filters.}
\end{enumerate}
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