{\LARGE \bfseries COM-303 - Signal Processing for Communications \\ Final Exam} \\
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{\large Wednesday, July 3 2013, 08:15 to 11:15}
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\LARGE \bfseries
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Verify that this exam has YOUR last name on top \\
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DO NOT OPEN THE EXAM UNTIL INSTRUCTED TO DO SO
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}
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\begin{itemize}
\item {\bf Write your name} on the top left corner of {\bf ALL the sheets you turn in}. When you are done, simply leave your solution on your desk \textbf{with this page on top} and exit the classroom.
\item You can have two A4 sheets of \emph{handwritten} notes (front and back). Please \textbf{no photocopies, no books and no electronic devices}. Turn off your phone and store it in your bag.
\item There are 6 problems for a total of 100 points; the number of points is indicated for each problem.
\item Please \textbf{write your derivations clearly}.
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\begin{exercise}{(12 points)}
Suppose you know that a system $\mathcal{H}$ is time-invariant. Below you can see the system's outputs $y_1[n]$ and $y_2[n]$ when the inputs are $x_1[n]$ and $x_2[n]$ respectively (all signals are infinite-length and all samples not shown in the figure are equal to zero). Based on the plots:
\begin{enumerate}
\item Can you tell if the system $\mathcal{H}$ is linear?
\item Compute the system's transfer function $H(z)$
\item Plot the system's poles and zeros on the complex plane
\item Sketch the magnitude of the system's frequency response $|H(e^{j\omega})|$
\item Find the transfer function of a {\em stable} filter $G(z)$ so that $|H(z)G(z)| = 1$
\item Would the system be stable if we removed the lowest branch (i.e. if we set $\alpha = 0$)?
\end{enumerate}
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\begin{exercise}{(11 points)}
Consider a finite-support sequence $x[n]$ and an LTI filter $\mathcal{H}$.
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\item Prove that if $\mathcal{H}$ is FIR then $y[n] = \mathcal{H}\{x[n]\}$ is a finite-support sequence as well.
\item Show with a counterexample that the converse is not true, i.e. show that, for a filter $\mathcal{H}$, if the the input $x[n]$ is finite-support and the output $y[n]$ is also finite-support, this does not imply that $\mathcal{H}$ is FIR.
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%\begin{exercise}{( points)}
%Consider a processing chain, shown below, constituted of:
%\begin{itemize}
% \item a sampler with sampling period $T_1$
% \item a downsampler by 3
% \item an interpolator with interpolation period $T_2$
%Assume $x(t)$ is bandlimited to 4000Hz. Find values for $T_1$ and $T_2$ so that $y(t) = x(t)$ and explain your choice.
%\end{exercise}
\begin{exercise}{(25 points)}
Let $x[n]$ be the discrete-time version of an audio signal, originally bandlimited to 20KHz and sampled at 40KHz; assume that we can model $x[n]$ as an i.i.d. process with variance $\sigma^2_x$. The signal is converted to continuous time, sent over a noisy analog channel and resampled at the receiving end using the following scheme, where both the ideal interpolator and sampler work at a frequency $F_s = 40$KHz:
The channel introduces zero-mean, additive white Gaussian noise. At the receiving end, after the sampler, assume that the effect of the noise introduced by the channel can be modeled as a zero-mean white Gaussian stochastic signal $\eta[n]$ with power spectral density $P_\eta(e^{j\omega}) = \sigma_0^2$.
\begin{enumerate}
\item What is the signal to noise ratio of $\hat{x}[n]$, i.e. the ratio of the power of the ``good'' signal and the power of the noise? (Call this signal to noise ratio $\mbox{SNR}_1$)
\end{enumerate}
The SNR obtained with the transmission scheme above is too low for our purposes. Unfortunately the power constraint of the channel prevents us from simply amplifying the audio signal (in other words: the total power $\int_{-\pi}^{\pi}P_x(e^{j\omega})$ cannot be greater than $2\pi\sigma_x^2$). In order to improve the quality of the received signal, we modify the transmission scheme as shown below, by adding pre-processing and post-processing digital blocks at the transmitting and receiving ends, while $F_s$ is still equal to $40000$Hz.
\item Design the processing blocks A and B so that the signal to noise ratio of $\hat{x}[n]$ is at least three times better (i.e. $\mbox{SNR}_{2} \geq 3\mbox{SNR}_{1}$). You should use upsamplers, downsamplers and lowpass filters only. Please specify all the details. {(\em Hint: the idea is to transmit the signal more ``slowly'' and trade bandwidth for SNR...)}
\item Using your new scheme, how long does it take to transmit a 3-minute song?
Below are the pole-zero plots of four different transfer functions, where zeros are indicated by a dot and poles are indicated by a cross. For each plot, say if the corresponding filter could be a real-valued equiripple filter designed by the Parks-McClellan algorithm. If your answer is yes, specify the length of the associated impulse response; if your answer is no, briefly explain why.