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\head{CSG250}{Wireless Networks}{Fall 2008}{17 September 2008}{1}

\begin{center}
\Large{\bf Problem Set 1 (due Wednesday, September 24)}
\end{center}

{\bf 1. Applying low-pass and bandpass filters to a
digital signal}

A square periodic signal is represented as the following sum of
sinusoids:

\[
s(t) = \frac{2}{\pi} \sum_{k = 0}^{\infty} \frac{(-1)^k}{2k+1}
cos((2k+1)\pi t).
\]

(Note that this is just a rewriting of the formula we discussed in
class.)
\begin{itemize}
\item[{\bf (a)}]
Suppose that the signal is applied to an ideal low-pass filter with
bandwidth 15 Hz.  Plot the output from the low-pass filter and compare
to the original signal.  Repeat for 5 Hz; for 3 Hz.  What happens as
the bandwidth increases. [5]

\item[{\bf (b)}] 
Suppose that the signal is applied to a bandpass filter that passes
the frequencies from 5 to 9 Hz.  Plot the output from the filter and
compare to the original signal. [5]
\end{itemize}

For your plots, use an appropriate plotting tool.  One such tool is
gnuplot, available in Unix.

{\bf 2. Fourier Transforms}

\begin{itemize}
\item[{\bf (a)}]
Show that convolution in the time domain is multiplication in the frequency domain.
In other words show that if 
\[ y(t) = \int_{-\infty}^\infty x(\tau)n(t-\tau)d\tau\] then
\[Y(f) = X(f)N(f)\] [3]

\item[{\bf (b)}] 
Consider the signal $x(t) = e^{-at}, a > 0$. Plot the phase and magnitude of the Fourier 
transform as a function of $f$. [4]

\item[{\bf (c)}] 
Explain why the Fourier basis of complex exponentials is a good basis for representing 
(wireless) signals. In particular, why are complex exponentials superior to sinusoidals? [3]

\end{itemize}

{\bf 3. Sampling}

\begin{itemize}
\item[{\bf (a)}]
State and prove the Nyquist sampling theorem [5]

\item[{\bf (b)}] 
Consider the signal $x(t) = \cos(\frac{f_s}{2}t).$ If the sampling frequency is $f_s$ then 
what does the sampled signal look like? What does the reconstructed signal look like? 
Plot both. Why are we unable to reconstruct the original signal? [5]

\end{itemize}

{\bf 4. Shannon's theorem}

\begin{itemize}
\item[{\bf (a)}]
A digital signaling system is required to operate at 38.4 Kbps.  If a
signal element encodes a 8-bit word, what is the minimum required
bandwidth of the channel.  What signal-to-noise ratio is required to
achieve the desired capacity on the bandwidth that you have computed? [4]

\item[{\bf (b)}] 
Derive the ``spectral efficiency'' form of the Shannon theorem from the capacity version. Plot
the spectral efficiency (bits per Hertz) in terms of the bit-energy-to-noise-density. Indicate
the attainable region. Calculate the largest value (in dB) of bit-energy-to-noise-density below which
there can be no error-free communication. [6]
\end{itemize}

{\bf 5. Antenna}

\begin{itemize}
\item[{\bf (a)}]
Show that doubling the transmission frequency is equivalent to doubling the distance between the
sending and receiving antennae. Calculate, in dB, attenuation of power in either case. [4]

\item[{\bf (b)}] 
Suppose we used the thumbrule that the optical line of sight distance from an antenna $h$ meters high
to the horizon, in kilometers, is $4\sqrt{h}$. What is the radius of the earth in kilometers 
that would justify this approximation? Assume the earth to be a perfect sphere. [3]

\item[{\bf (c)}] 
Assume that two antennas are half-wave dipoles, each with a directive gain of 10 dB. 
If the transmitted power is 0.5W and the two antennae are separated by a distance of 5km, 
what is the received power? Assume perfect alignment of the antennae and a frequency of 300MHz. [3] 

\end{itemize}
 
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