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\documentclass[a4paper,13pt]{article}
\usepackage{defsDSPcourse}
\title{COM-303 - Signal Processing for Communications}
\author{Homework \#2}
\date{Assigned on Feb 29, 2016.}
\begin{document}
\maketitle
\begin{exercise}{Bases}
Let $\{\mathbf{x}^{(k)}\}_{k=0,\ldots,N-1}$ be a basis for a
subspace $S$. Prove that any vector $\mathbf{z}\in S$ is
\emph{uniquely} represented
in this basis.\\\\
\emph{Hint:} prove by contradiction.
\end{exercise}
\begin{exercise}{Fourier Basis}
Consider the {\it Fourier basis} $\{ \wb^{(k)}\}_{k=0,
\ldots,N-1}$, defined as:
\[
\wb_n^{(k)}=e^{-j\frac{2\pi}{N}nk}.
\]
\begin{enumerate}
\item Prove that it is an {\it orthogonal basis} in $\setC^N$.
\item Normalize the vectors in order to get an {\it orthonormal
basis}.
\end{enumerate}
\end{exercise}
\begin{exercise}{DFT in matrix form}
\begin{enumerate}
\item
Express the DFT and inverse DFT (IDFT) formulas (analysis and synthesis) as
a matrix - vector multiplication.
\item
Is the DFT matrix hermitian?
\end{enumerate}
\end{exercise}
\begin{exercise}{DFT of elementary functions}
Derive the formula for the DFT of the length-$N$ signal
\[
x[n] = \cos((2\pi/N)Ln + \phi).
\]
\end{exercise}
\begin{exercise}{DFT Example}
Consider a length-64 signal $x[n]$ which is the sum of the three
sinusoidal signals plotted in Figure~\ref{dftFig}. Compute the DFT
coefficients $X[k], k = 0, 1, \ldots, 63$.
\begin{figure}[h!]
\centering \psfig{figure=DFTsigs.eps, width=10cm,height=4cm}
\caption{Three sinusoidal signals.}\label{dftFig}
\end{figure}
\end{exercise}
\begin{exercise}{Structure of DFT formulas}
The DFT and IDFT formulas are similar, but not
identical. Consider a length-$N$ signal $x[n], N = 0, \ldots,
N-1$; what is the length-$N$ signal $y[n]$ obtained as
\[
y[n] = \mbox{DFT}\{\mbox{DFT}\{x[n]\}\}
\]
(i.e. by applying the DFT algorithm twice in a row)?
\end{exercise}
\begin{exercise}{Plancherel-Parseval Equality}
Let $x[n]$ and $y[n]$ be two complex valued sequences and
$X[k]$ and $Y[k]$ their corresponding DFTs.
\begin{enumerate}
\item Show that
\[
\sum_{n=0}^{N-1} x[n]y^*[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k]Y^*[k],
\]
where $*$ denotes the Hermitian transpose.
\item What is the physical meaning of the above formula when
$x[n]=y[n]$ ?
\end{enumerate}
\end{exercise}
%\begin{exercise}{DFT \& Matlab}
%\begin{enumerate}
%\item
%Consider the signal $x(n)=\cos{(2\pi f_0 n)}$. Draw the DFT of the signal in $N=128$ points without running Matlab, for $f_0=21/128$.
%
%\item In Matlab, use the \textit{fft} function to compute and draw the DFT for $f_0=21/128$ and $f_0=21/127$.
%Explain the differences that we can see in these two signal spectra.
%
%\item
%Repeat the drawing in Matlab, this time using the \textit{dftmtx} function and check that
%the results are the same. What is the preferred option between the two?
%\end{enumerate}
%\end{exercise}
\end{document}

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