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prob_sig_space.tex
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prob_sig_space.tex
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\documentclass[11pt]{article}
\usepackage{fullpage}
\usepackage{amsmath, amssymb, bm, cite, epsfig, psfrag}
\usepackage{graphicx}
\usepackage{float}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{listings}
\usepackage{cite}
\usepackage{hyperref}
\usepackage{tikz}
\usepackage{enumerate}
\usepackage[outercaption]{sidecap}
\usetikzlibrary{shapes,arrows}
%\usetikzlibrary{dsp,chains}
%\restylefloat{figure}
%\theoremstyle{plain} \newtheorem{theorem}{Theorem}
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\begin{document}
\title{Problems: Signal Space}
\author{Prof.\ Sundeep Rangan}
\date{}
\maketitle
\begin{enumerate}
\item \emph{Vector spaces and bases in $\F^N$.} For each set $V$ below,
identify if it is a vector space or not. If it is a vector space, find a basis.
If not, state the property that fails to occur.
\begin{enumerate}[(a)]
\item $V = $ the set of $(x_1,x_2,x_3)$ such that $2x_1 + x_2 = 0$.
\item $V = $ the set of $\xbf \in \R^3$ with $\|\xbf\|\leq 1$.
\end{enumerate}
\item \emph{Vector spaces of functions.} For each set $V$ below,
state if $V$ is a subspace or not. Explain.
\begin{enumerate}[(a)]
\item Let $T> 0$ be some sampling period.
$V = $ the set of $f(t)$ such that $f(nT)=0$ for all $n$.
\item Let $f_{max} > 0$. $V$ is the set of $s(t)$ that are bandlimited so that
$S(f)=0$ for $|f| > f_{max}$.
\item $V=$ set of functions on $[0,\infty)$ of the form, $f(t) = Ae^{-(t-\tau)}$ for
some $A$ and $\tau$.
\item $V=$ set of functions on $[0,\infty)$ of the form, $f(t) = Ae^{-Bt}$ for some $A$ and $B$.
\end{enumerate}
\item \emph{Signal set and signal space}. Let $N$ and $K$ be constants and
consider the signal set ${\mathcal S}$ consisting
of signals $s[n]$ such that $s[n]=1$ in \emph{exactly} $K$ times $n\in [0,1,\ldots,N-1]$.
For all other $n$, $s[n]=0$.
\begin{enumerate}[(a)]
\item Find $M$, the number of signals in ${\mathcal S}$.
\item Find the number of degrees of freedom.
\item Find the rate of signal set.
\end{enumerate}
This type of signal set can encode information by the position of the non-zero elements.
\item \emph{Signal set and signal space.} Consider the following four functions:
\[
s(t) = e^{-At + B}, t \geq 0,
\]
where $A = 1$ or $2$ and $B=0$ or 1.
\begin{enumerate}[(a)]
\item Find a basis for a signal space containing the signal set.
Use a basis with a minimum number of signals.
\item Find the coordinates of each signals in the basis.
\end{enumerate}
\item \emph{Bandlimited channels.} Suppose that a communication system is allocated
a channel 2.29 to 2.31 GHz and has 10\% overhead.
\begin{enumerate}[(a)]
\item What are the (complex) degrees of freedom per second?
\item What is the spectral efficiency required for 40 Mbps?
\item What is the rate if the system uses 16-QAM on every degree of freedom?
\item If the signal is received at -100 dBm, what is the average energy per degree of freedom.
\end{enumerate}
\item \emph{Orthonormal bases} Suppose that a signal space has a basis $s_1(t)$, $s_2(t)$ with
\[
\|s_1\|^2 = \|s_2\|^2 = 1, \bkt{s_1,s_2} = \rho.
\]
\begin{enumerate}[(a)]
\item Using Gram-Schmidt, find an orthonormal basis $u_1$, $u_2$ for the signal space.
\item Write $s_1$ and $s_2$ in terms of $u_1$ and $u_2$.
\item Suppose a signal is transmitted as,
\[
s(t) = a_1 s_1(t) + a_2 s_2(t).
\]
Find the coordinates of $s(t)$ in the $u_1(t),u_2(t)$ basis.
That is, find $b_1,b_2$ in terms of $(a_1,a_2)$ such that $s(t) = b_1u_1(t) + b_2 u_2(t)$
\item Find $b_1,b_2$ for the four constellation points $a_1=a_2 = \pm 1$ and $\rho = 0.2$.
\end{enumerate}
\end{enumerate}
\end{document}