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\documentclass[12pt,a4paper,fleqn]{article}
\usepackage{../styles/defsDSPcourse}
\title{COM-303 - Signal Processing for Communications}
\author{Homework \#5}
\date{}
\begin{document}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{LTI Systems}
Consider the transformation $\mathcal{H}\{x[n]\} = nx[n]$. Does $\mathcal{H}$ define an LTI system?
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{Convolution}
Let $x[n]$ be a discrete-time sequence defined as
\[
x[n] = \left\{
\begin{array}{ll}
M-n & 0\leq n\leq M, \\
M+n & -M\leq n\leq 0, \\
0 & \mbox{otherwise}.
\end{array}
\right.
\]
for some odd integer M.
\begin{enumerate}
\item Show that $x[n]$ can be express as the convolution of a sequences $t[n]$ with itself. Check your result numerically (using Python or any other numerical package) for $M=11$. %
\item Using the results found in (a), compute the DTFT of $x[n]$.
\end{enumerate}
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{System Properties}
For the systems whose input-output relationship is described below, say whether the system is linear, time-invariant, BIBO stable, causal; when applicable, find the impulse response.
\begin{enumerate}
\item $y[n]=x[-n]$
\item $y[n]=e^{-j\omega n} x[n]$
\item $y[n]=\sum_{k=n-n_0}^{n+n_0}x[k]$
\item $y[n]=ny[n-1]+x[n]$, so that if $x[n]=0$ for $n<n_0$, then $y[n]=0$ for $n<n_0$. \\
(\emph{Hint}: Since the system is causal and satisfies initial-rest conditions, we can recursively find the response to any input as, for instance, $\delta [n]$.)
\end{enumerate}
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{Ideal Filters}
Derive the impulse response of a bandpass filter with center frequency $\omega_0$ and passband $\omega_b$:
\[
H_{bp}(e^{j \omega}) = \left\{
\begin{array}{ll}
1 & \omega_0 - \omega_b/2 \leq \omega \leq \omega_0 + \omega_b/2, \\
1 & -\omega_0 - \omega_b/2 \leq \omega \leq -\omega_0 + \omega_b/2, \\
0 & \mbox{elsewhere}.
\end{array}
\right.
\]
(\emph{Hint}: consider the following ingredients: a cosine at frequency $\omega_0$, a lowpass filter with bandwidth $\omega_b$ and
the Modulation Theorem.)
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{Signals and Systems}
Consider the system $\mathcal{H}$ implementing the input-output relation $\mathcal{H}\lbrace x[n] \rbrace = x^2[n]$.
\begin{enumerate}
\item Prove by example that the system is nonlinear.
\item Prove that the system is time-invariant.
\end{enumerate}
Now consider the following cascade
\begin{center}
\begin{dspBlocks}{1.5}{0.4}
$x[n]$~ & \BDfilter{$\mathcal{H}$} & \BDfilter{$\mathcal{G}$} & $~y[n]$ \\
\psset{linewidth=1.5pt}
\ncline{->}{1,1}{1,2}\ncline{->}{1,2}{1,3}\ncline{->}{1,3}{1,4}\ncline{->}{1,4}{1,5}
\end{dspBlocks}
\end{center}
where $\mathcal{G}$ is the ideal highpass filter:
\[
G(e^{j\omega}) = \begin{cases}
0 & \text{for }|\omega| < \pi/2,\\
2 & \text{otherwise}
\end{cases}
\]
(as per usual, $G(e^{j\omega})$ is $2\pi$-periodic (i.e. prolonged by periodicity outside of $[-\pi, \pi]$)). The output of the cascade is therefore $y[n] = \mathcal{G} \lbrace \mathcal{H} \lbrace x[n] \rbrace \rbrace$.
\begin{enumerate}
\setcounter{enumi}{2}
\item Compute $y[n]$ when $x[n] = \cos(\omega_0 n)$ for $\omega_0 = 3\pi/8$. How would you describe the transformation operated by the cascade on the input?
\item Compute $y[n]$ as before, with now $\omega_0 = 7\pi/8$.
\end{enumerate}
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{Linear Phase FIR Filters}
The frequency response of an $M$-tap FIR filter with real-valued impulse response $h[n]$ is
\[
H(e^{j\omega})=\sum_{n=0}^{M-1}h[n]e^{-jn\omega}
\]
where we have assumed that $h[n] = 0$ for $n < 0$ and $n \geq M$.
Show that if $M$ is odd and the impulse response is symmetric, then the filter has a linear phase response.
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{The Gibbs Phenomenon}
In this exercise you will verify by yourself the existence of the Gibbs phenomenon using Python (or any other numerical package). The idea is to plot a zoomed-in version of the
frequency response of a truncated ideal lowpass filter with cutoff frequency $\pi/2$:
\[
\hat H(e^{j\omega}) = \sum_{n=-N}^{N} (1/2)\mbox{sinc}(n/2) \; e^{-j\omega n}
\]
where we are interested in plotting the transform over a small interval around the cutoff frequency.
\begin{enumerate}
\item Plot $\hat H(e^{j\omega})$ over 2000 points in the interval $1.4 \leq \omega \leq 1.7$ for $N = 20$.
\item Repeat the above point for $N = 100$ and $N=200$ and verify that the peak of the magnitude is still approximately $9\%$ of the value of the discontinuity.
\end{enumerate}
\end{exercise}
\end{document}

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