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\begin{document}

\title{Problems: Antennas and Free-Space Propagation\\
EL-GY 6023. Wireless Communications}
\author{Prof.\ Sundeep Rangan}
\date{}

\maketitle

In all the problems below, unless specified otherwise, $\phi$ is the
azimuth angle and $\theta$ is elevation angle.

\begin{enumerate}
\item \emph{EM wave}:  Suppose the E-field is,
\[
    \Ebf(x,y,z,t) = E_0 \ebf_y \cos(2\pi f t - kx).
\]
\begin{enumerate}[label=(\alph*)]
  \item What is direction of motion?
  \item If the average power flux is $10^{-8}$ mW/m$^2$, what is $E_0$?
   Assume the characteristic impedance is $\eta_0 = 377 \Omega$.
  \item If the frequency is $f=$ 1.5 GHz, what is $k$?
  What are the units of $k$?
\end{enumerate}

\item \emph{EM polarization}:  An EM plane wave has
a power flux of $10^{-8}$ mW/m$^2$ in a horizontal polarization (along the $x$-axis)
and $2(10)^{-8}$ mW/m$^2$ in a vertical polarization (along the $y$-axis).
Assume the characteristic impedance is $\eta_0 = 377 \Omega$.
\begin{enumerate}[label=(\alph*)]
  \item What is combined $E$-field value and its direction?
  \item What is the direction of motion?
  \item What is the total power flux?
\end{enumerate}


\item \emph{dBm to linear conversions:}
\begin{enumerate}[label=(\alph*)]
\item Convert the following to mW:   17 dBm, -73 dBm, -97 dBW.
\item Convert the following to dBm: 250 mW, $8(10)^{-8}$ W, $5(10)^{-6}$ mW
\end{enumerate}


\item \emph{Spherical-cartesian conversions:}
\begin{enumerate}[label=(\alph*)]
  \item Convert $(r,\phi,\theta)=(2,45^\circ,30^\circ)$ to $(x,y,z)$.
  \item Convert $(x,y,z) = (1,2,3)$ to $(r,\phi,\theta)$.
  \item Convert $(x,y,z) = (1,-2,3)$ to $(r,\phi,\theta)$.
\end{enumerate}

\item \emph{Rotation matrices}:  The \emph{rotation matrix}
$R(\theta,\phi)$ is the $3 \times 3$ matrix such that $R(\theta,\phi)\rbf$ rotates
the vector $\rbf$ by an angle pair $(\theta,\phi)$.  
You can read more about this on wikipedia or other sources.
\begin{enumerate}[label=(\alph*)]
  \item Find the entries of $R(\theta,\phi)$.
  \item Show $R(\theta,\phi)$ is \emph{orthogonal} meaning
  $R(\theta,\phi)^{-1}=R(\theta,\phi)\tran$.
  \item Is
\[
    R(\theta_1,\phi_1)R(\theta_2,\phi_2) = R(\theta_2,\phi_2)R(\theta_1,\phi_1)?
\]
That is, do the order of rotations matter?
Prove or find a counter-example.
\end{enumerate}


\item \emph{Angular areas:}  Find the angular area in steradians of
following sets of angles:
\begin{enumerate}[label=(\alph*)]
  \item $A_1 = \left\{ (\phi,\theta) ~ | ~ \phi \in [-30,30],~ \theta \in [-90,90]\right\}$
  \item $A_2 = \left\{ (\phi,\theta) ~|~ \phi \in [-30,30], ~ \theta \in [-45,45]\right\}$
\end{enumerate}

\item \emph{Directivity:}  Suppose an antenna radiates power uniformly in
the angular beam $\phi \in [-30,30], \theta \in [-45,45]$ and radiates
no power at other angles.  What is the directivity of the antenna?

\item \emph{Radiation intensity:}  A 170 cm x 40 cm object
(roughly the size of a human) is 800m from a base station.
If the antenna transmits 250~mW isotropically, how much power
reaches the human?  Use reasonable approximations that the human is far from the
transmitter.

\item \emph{Radiation integration:}  The radiation density for some antenna is,
\[
    U(\phi,\theta) = A\cos^2(\phi), \quad A = 100 \mbox{mW/sr}.
\]
\begin{enumerate}[label=(\alph*)]
\item What is the total radiated power in mW and dBm?
\item What is the directivity of the antenna in linear scale and in dBi?
\end{enumerate}

\item \emph{Numerically integrating patterns:}
Write a short MATLAB function,
\begin{lstlisting}
    function [totPow, dir] = powerDirectivity(az,el,E,eta)
\end{lstlisting}
that computes the total power and directivity as a function of the $E$-field
values.  The E field is specified as a complex matrix \mcode{E(i,j)}
at angles \mcode{az(i),el(j)}.  You can assume that the angles are uniformly
spaced over  the total angular space.   The total power output
 \mcode{totPow} should be a
scalar representing the total radiated power in dBm and the directivity output
should be \mcode{dir(i,j)} should be a matrix.

\item \emph{Friis' Law}:  A transmitter radiates 100 mW at a carrier $f_c = 2.1$ GHz
with a directional gain of $G_t = 10$ dBi.
Suppose the receiver is $d = 200$ m from the transmitter and the path is free space.
What is the received power if:
\begin{enumerate}[label=(\alph*)]
\item The effective received aperture is 1 cm$^2$.
\item The receiver gain is $G_r = 5$ dBi.
\end{enumerate}

\end{enumerate}

\end{document}