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\head{CSG250}{Wireless Networks}{Fall 2007}{11 September 2007}{1}

\begin{center}
\Large{\bf Problem Set 1 (due Friday, September 21)}
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{\bf 0. (due now!)} Follow the instructions on the webpage below to
join the class mailing list.

{\tt https://lists.ccs.neu.edu/bin/listinfo/csg250}

{\bf 1. (10 points) 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.

\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.
\end{itemize}

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

{\bf 2. (4 points) Bandwidth, signal element, and SNR}

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?

{\bf 3. (6 points) Effect of transmission frequency and distance on
attenuation}

Under the free-space path-loss model, find the transmit power required
to obtain a received power of 1 dBm for a wireless system with
isotropic antennas (gain = 1) and a carrier frequency $f = $ 5 GHz,
assuming a distance $d = 10$ m.  Repeat for $d = 100$ m, keeping $f =
$ 5 GHz.  Repeat for $f = 50 $ GHz, keeping $d = 10$ m.

{\bf 4. (5 points) Received power using half-wave dipoles}

Problem 5.9.

{\bf 5. (7 points) Modulation schemes and $E_b/N_0$}

Problem 6.2.

{\bf 6. (5 points) Telephone channel bandwidth}

Problem 6.6.

{\bf 7. (5 points) Sampling}

Problem 6.10.

{\bf 8. (8 points) Delta modulation}

Problem 6.13.

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