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\documentclass[12pt,a4paper,fleqn]{article}
\usepackage{../styles/defsDSPcourse}
\title{COM-303 - Signal Processing for Communications}
\author{Homework \#8}
\date{}
\begin{document}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{Autocorrelation function of a random process}
Consider the following real-valued random process
\[
x[n] = A\cos(\omega_{0}n) + w[n]
\]
where $A$ is a Gaussian random variable with zero mean and variance $\sigma_{A}^{2}$ and where $w[n]$ is a zero-mean white noise process, independent of $A$, with variance $\sigma_{w}^{2}$.
\begin{enumerate}
\item What is the autocorrelation of $x[n]$?
\item Can we define the power spectral density of the process?
\item Repeat (a) and (b) in the case when the cosine starts with a random phase offset, uniformly distributed over $[-\pi, \pi]$ and independent of all the other random variables.
\end{enumerate}
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{More white noise}
A real-valued random process is defined by
\[
x[n] = s[n] + w_{0}[n]
\]
where $w_{0}[n]$ is a unit-variance, zero-mean white noise process and where $s[n]$ is defined as
\[
s[n] = as[n-1] + w_{1}[n]
\]
with $a\in\mathbb{R}$ and $w_{1}[n]$ a unit variance, zero-mean white noise process independent of $w_{0}[n]$.
\begin{enumerate}
\item Determine the autocorrelation $r_{x}[k]$.
\item Determine the power spectral density function $P_{x}(e^{j\omega})$.
\end{enumerate}
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{Filtering a Sequence of Independent Random Variables in Python}
Let $x[n]$ be a real-valued white Gaussian random process, with zero mean and variance $\sigma_x^2=3$. We filter the process with the FIR filter $h[n]$ where
\[
h[1]=1/2, \quad h[2]=1/4, \quad h[3]=1/4, \quad
h[n]=0~\forall~n\neq 1,2,3
\]
Moreover, at the output of the filter, we add white Gaussian noise $z[n]$ with unit variance. The system is shown in the
following diagram:
\begin{center}
\begin{dspBlocks}{1}{0.6}
& & $z[n]$ \\
$x[n]$~~ & \BDfilter{$H(z)$} & \BDadd & $~~y[n]$
\psset{linewidth=1.5pt}
\ncline{->}{2,1}{2,2}\ncline{->}{2,2}{2,3}\ncline{->}{2,3}{2,4}
\ncline{->}{1,3}{2,3}
\end{dspBlocks}
\end{center}
\begin{enumerate}
\item Write a routine in Python to generate $N$ samples of the input process, $N$ samples of the additive Gaussian noise and compute the output of the system.
\item write a routine to estimate the power spectral density of the output
\item compare the numerical estimation of the PSD with its theoretical value
\end{enumerate}
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{Analytic Signals \& Modulation.}
In this exercise we will explore a modulation-demodulation scheme
commonly used in data transmission systems. Consider two real
sequences $x[n]$ and $y[n]$, which represent \emph{two} data
streams we want to transmit. Assume that their spectrum is of
lowpass type, i.e.\ $X(e^{j\omega}) = Y(e^{j\omega}) = 0$ for
$|\omega| > \omega_c$. Consider further the following derived
signal:
\[
c[n] = x[n] + jy[n]
\]
and the modulated signal:
\[
r[n] = c[n]e^{j\omega_0 n}, \quad \omega_c < \omega_0 < \pi - \omega_c
\]
\begin{enumerate}
\item Set $\omega_c = \pi/6$, $\omega_0 = \pi/2$ and sketch
$|R(e^{j\omega})|$ for whatever shapes you choose for
$X(e^{j\omega}), Y(e^{j\omega})$. Verify from your plot that
$r[n]$ is an analytic signal.
\end{enumerate}
The signal $r[n]$ is called a \emph{complex passband signal}. Of
course it cannot be transmitted as such, since it is complex. The
transmitted signal is, instead,
\[
s[n] = \Re\{r[n]\}.
\]
This modulated signal is an example of Quadrature Amplitude
Modulation (QAM).
\begin{enumerate}
\setcounter{enumi}{1} \item Write out the expression for $s[n]$ in
terms of $x[n], y[n]$. Now you can see the reason behind the term
QAM, since we are modulating with two carriers in quadrature
(i.e.\ out of phase by 90 degrees).
\end{enumerate}
Now we want to recover $x[n]$ and $y[n]$ from $s[n]$. To do so,
follow these steps:
\begin{enumerate}
\setcounter{enumi}{2} \item Show that $s[n] + j(h[n] \ast s[n]) =
r[n]$, where $h[n]$ is the Hilbert filter. In other words, we have
recovered the analytic signal $r[n]$ from its real part only.
\item Once you have $r[n]$, show how to extract $x[n]$ and $y[n]$.
\end{enumerate}
\end{exercise}
\end{document}

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