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{\LARGE\bf EE 341: Linear Systems Analysis}
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{\large{\bf Assignment 4 : Using the FFT}}\\
\noindent Due Date: Wednesday, March 3, 1999
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When using a digital computer, spectral analysis means using a Fast Fourier
Transform (FFT).
This necessitates we spend some time
becoming familiar with using the FFT to study the spectral contents of
a sequence.
\begin{enumerate}
\item{\bf MATLAB function FFT}
In this problem you will learn how to use the MATLAB command FFT. First,
use the HELP feature in MATLAB to learn the syntax of the FFT function.
The FFT function determines the Discrete Fourier Transform (DFT) of a
sequence.
In general the FFT of a sequence will be a complex function so you will
need to look at the magnitude and phase separately. The MATLAB commands ABS
and ANGLE are useful for obtaining the magnitude and phase of
a complex valued sequence. If you are not familiar with these commands,
type HELP. You can plot a discrete signal using the STEM command.
Plot magnitude and phase of
the FFT of the following sequences. The FFT outputs
over the range $0 \leq \omega \leq 2\pi$ so label the axis
accordingly.
\begin{enumerate}
\item $x[n] = 5 \sin(2\pi fn)$ where $f = 0.65$ and $0 \leq n \leq 127$.
\item $x[n] = 5(1 + \cos(2\pi fn))$ where $f = 0.65$ and $0 \leq n \leq 127$.
\end{enumerate}
\underline{Describe what the magnitude plots show.} Don't
hand in these plots. You are
probably more familiar with seeing the spectrum
plotted over the range $-\pi \leq \omega \leq \pi$. The FFTSHIFT
function can be used for this purpose. \underline{Hand in plots of
the following sequences and their respective}
\underline{magnitude and phase
plots.} Make sure the magnitude and phase plots are over the range
$-\pi \leq \omega \leq \pi$.
\begin{enumerate}
\item $x[n] = sinc(f[n-64])$ where $f = 0.4$ and $0 \leq n \leq 127$.
Note: sinc is the function {\tt sinc.m} that
generates the sinc function -- you will receive it by email.
\item $x[n] = sinc(f[n-64])cos(2\pi fn)$ where $f = 0.4$ and $0 \leq n \leq 127$.
\end{enumerate}
\underline{What type of signals are these (low pass, high pass, etc.)?}
\underline{What important
electrical engineering property is taking place}
\underline{ with the
second signal above?} (Hint: compare
with the spectrum of the
first signal).
\end{enumerate}
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