prob_antennas.tex

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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 an EM plane wave has an E-field
\[
    \Ebf(x,y,z,t) = E_0 \ebf_y \cos(2\pi f t - kx).
\]
\begin{enumerate}[label=(\alph*)]
  \item What is direction of motion?
  \item If the average power flux density is $10^{-8}\, \si{mW/m^2}$, what is $E_0$?
   Assume the characteristic impedance is $\eta_0 = \SI{377}{\ohm}$.
  \item If the frequency is $f=$ \SI{1.5}{GHz}, what is $k$?
  What are the units of $k$?
\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:}  When a transmitter is at the origin,
its E-field in the far field can often be represented as,
\[
    \Ebf = E_\theta \ebf_\theta + E_\phi \ebf_\phi,
\]
where $\ebf_\theta$ and $\ebf_\phi$ are the basis vectors in elevation and azimuth direction.
Complete the following MATLAB function that takes a $1\times 3$ position vector
\mcode{pos} and $n \times 1$ values of $E_\theta$ and $E_\phi$ and returns the
an $n \times 3$ matrix \mcode{E} representing the E-field values in cartesian coordinates.
You may use any built in MATLAB functions.  Be careful whether the methods use degrees
or radians.
\begin{lstlisting}
    function E = convert(Etheta, Ephi, pos)
\end{lstlisting}


\item \emph{Rotation matrices}:  In wireless systems, we often need to consider antennas that
can be in arbitrary rotations.  One way of specifying the orientation of an object
is through its so-called \emph{Euler} angles $(\alpha,\beta,\gamma)$ or
\emph{yaw, pitch} and \emph{roll}.  Let $R(\alpha,\beta,\gamma)$ be the rotation matrix
for a given set of Euler angles.  You can find the formulae for $R(\alpha,\beta,\gamma)$
in any reference such as wikipedia.
\begin{enumerate}[label=(\alph*)]
%  \item   Write a simple MATLAB function
%  as follows that computes the rotation matrix given the Euler angles in degrees.
%\begin{lstlisting}
%    function rot = rotMatrix(yaw,pitch,roll)
%\end{lstlisting}

\item Given elevation and azimuth angles $(\theta,\phi)$ find $(\alpha,\beta,\gamma)$
with $\gamma=0$ that rotates the $x$-axis to point in $(\theta,\phi)$.
\item Is $R(\alpha,0,0)^{-1} = R(-\alpha,0,0)$?  Explain.
\item Is $R(\alpha,\beta,0)^{-1} = R(-\alpha,-\beta,0)$?  Explain.
\end{enumerate}

\item \emph{Angular areas:}  Find the angular area in steradians of
following sets of angles where $\phi$ is the azimuth angle and $\theta$ is the elevation angles
in degrees:
\begin{enumerate}[label=(\alph*)]
  \item $A_1 = \left\{ (\phi,\theta) ~ | ~ \phi \in [-\ang{30},\ang{30}],~ \theta \in [-\ang{90},\ang{90}]\right\}$
  \item $A_2 = \left\{ (\phi,\theta) ~|~ \phi \in [-\ang{30},\ang{30}],~ \theta \in [-\ang{45},\ang{45}]\right\}$
\end{enumerate}
\item \emph{Directivity:}  Suppose an antenna radiates power uniformly in
the angular beam $\phi \in [-\ang{30},\ang{30}]$, and $\theta \in [-\ang{45},\ang{45}]$,
and radiates no power at other angles.  What is the maximum directivity of the antenna in dBi?
You can use the results from the previous problem.


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

\item \emph{Radiation integration:}  Suppose the  radiation intensity is
\[
  U(\phi,\theta) = A\cos^2(\theta), \quad A = 10\,\si{mW/sr},
\]
where $(\phi,\theta)$ are the azimuth and elevation angles.
find the total radiated power in dBm and maximum directivity in dBi.
You can look up any integrals you need.


\item \emph{Numerically integrating patterns:}
Suppose we are given the radiation intensity $U(\theta,\phi)$ at discrete points,
$(\theta_i,\phi_j)$ where
$\theta_i$, $i=1,\ldots,M$ is uniformly spaced on $[-\pi/2,\pi/2]$
and $\phi_j$, $j=1,\ldots,N$ is uniformly spaced on $[-\pi,\pi]$.
Assume $(\theta, \phi)$ are elevation and azimuth angles.  Write a short MATLAB function
to compute the radiated power $P_{\rm rad}$ and directivity $D(\theta_i,\phi_j)$
from a matrix of values $U(\theta_i,\phi_j)$.



\item \emph{Friis' Law}:  A transmitter radiates \SI{15}{dBm} at a carrier $f_c =$ \SI{2.1}{GHz}
with a directional gain of $G_t = 9$\,\si{dBi}.
Suppose the receiver is $d =$ \SI{200}{m} from the transmitter and the path is free space.
What is the received power in dBm if:
\begin{enumerate}[label=(\alph*)]
\item The effective received aperture is \SI{1}{cm^2}.
\item The receiver gain is $G_r =$ \SI{5}{dBi}.
\end{enumerate}


\end{enumerate}

\end{document}