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hw03.tex
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\documentclass[12pt,a4paper,fleqn]{article}
\usepackage{../styles/defsDSPcourse}
\title{COM-303 - Signal Processing for Communications}
\author{Homework \#3}
\date{}
\begin{document}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{DFT in matrix form}
Consider the DFT expressed as a as a matrix/vector multiplication and call $\mathbf{W}$ the $N\times N$ DFT matrix. Is $\mathbf{W}$ Hermitian-symmetric for all values of $N$?
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{Derivative in frequency}
Let $x[n] \longleftrightarrow X(e^{j\omega})$ be a DTFT transform-pair.
\begin{enumerate}
\item Assume $X(e^{j\omega})$ to be differentiable, compute the inverse DTFT of $j\frac{d}{d \omega} X(e^{j\omega})$.
\item Compute the inverse DTFT of $\frac{d}{d\omega} \left(\frac{X(e^{j\omega})}{\pi}\right)-2$. Which property of the DTFT allows you to simplify the calculation?
\end{enumerate}
\textit{Hint: use integration by parts.}
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{DTFT visual inspection}
The real and imaginary parts of $X(e^{j\omega})$ are shown in the figure below. By visual inspection of the plots, prove that
\begin{enumerate}
\item $x[n]$ is $0$-mean, i.e., $\sum_{n\in\mathbb{Z}} x[n]=0$;
\item $x[n]$ is real valued.
\end{enumerate}
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\begin{dspPlot}[width=6cm,xtype=freq,ylabel={$\Im\{X(e^{j\omega})\}$}]{-1,1}{-1.1,1.1}
\dspFunc{x 3 mul dup dup mul -1 mul 2.7 exch exp mul 2 mul}
\end{dspPlot}
&
\begin{dspPlot}[width=6cm,xtype=freq,ylabel={$\Re\{X(e^{j\omega})\}$}]{-1,1}{-1.1,1.1}
\dspFunc{x \dspPorkpie{0}{1} x x mul mul 2 mul}
\end{dspPlot}
\end{tabular}
\end{figure}
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{DTFT properties}
Derive the time-reversal and time-shift properties of the DTFT.
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{Placherel-Parseval equaltiy}
Let $x[n]$ and $y[n]$ be two complex valued sequences and $X(e^{j\omega})$ and $Y(e^{j\omega})$ their corresponding DTFTs.
\begin{enumerate}
\item Show that
\[
\left\langle x[n],y[n] \right\rangle = \frac{1}{2\pi} \left\langle X(e^{j\omega}),Y(e^{j\omega}) \right\rangle
\]
where we use the inner products for $\ell_2(\mathbb{Z})$ and $\mathcal{L}_2([-\pi,\pi])$ respectively.
\item What is the physical meaning of the above formula when $x[n]=y[n]$?
\end{enumerate}
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exercise}{DTFT,DFT, and numerical computations}
Consider the following infinite non-periodic discrete time signal, where $M\in\mathbb{N}$:
\[
x[n]=\begin{cases}
0 & n<0,\\
1 & 0\leq n< M,\\
0 & n\geq M.
\end{cases}
\]
The goal is to plot the magnitude of $X(e^{j\omega})$ using a numerical package. Using a numerical package implies that we will only obtain an approximation of the true value.
\begin{enumerate}
\item Compute $X(e^{j\omega})$ analytically
\item Using a numerical package plot $|X(e^{j\omega})|$ over $[-\pi \pi]$ for $M=20$ using 10,000 points.
\item Now generate a finite sequence $\hat{x}[n]$ of length $N=30$ such that $\hat{x}[n]=x[n]$ for $n=0,1, \ldots, N-1$. Compute its DFT using the numerical package's FFT algorithm and plot its magnitude. Compare it with the plot obtained previously.
\item Repeat now for different values of $N = 50, 100, 1000$. What can you conclude?
\end{enumerate}
\end{exercise}
\end{document}

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