11 11 \usepackage{hyperref} 12 12 \usepackage{tikz} 13 13 \usepackage{enumitem} 14 + \usepackage{siunitx} 14 15 \usetikzlibrary{shapes,arrows} 15 16 \usepackage{mdframed} 16 17 \usepackage{mcode} 177 178 178 179 \begin{document} 179 180 180 - \title{Problems : Antennas and Free-Space Propagation\\ 181 + \title{Problem Solutions : Antennas and Free-Space Propagation\\ 181 182 EL-GY 6023. Wireless Communications} 182 183 \author{Prof.\ Sundeep Rangan} 183 184 \date{} 188 189 azimuth angle and $\theta$ is elevation angle. 189 190 190 191 \begin{enumerate} 191 - \item \emph{EM wave}: Suppose the E-field is , 192 + \item \emph{EM wave}: Suppose an EM plane wave has an E-field 192 193 \[ 193 194 \Ebf(x,y,z,t) = E_0 \ebf_y \cos(2\pi f t - kx). 194 195 \] 195 196 \begin{enumerate}[label=(\alph*)] 196 197 \item What is direction of motion? 197 - \item If the average power flux is $10^{-8}$ mW/m$ ^2$, what is $E_0$? 198 - Assume the characteristic impedance is $\eta_0 = 377 \Omega $. 199 - \item If the frequency is $f=$ 1.5 GHz, what is $k$? 198 + \item If the average power flux density is $10^{-8}\ , \ si { mW/m^2} $, what is $E_0$? 199 + Assume the characteristic impedance is $\eta_0 = \ SI { 377} { \ohm } $. 200 + \item If the frequency is $f=$ \ SI { 1.5} { GHz} , what is $k$? 200 201 What are the units of $k$? 201 202 \end{enumerate} 202 203 203 - \item \emph{EM polarization}: An EM plane wave has 204 - a power flux of $10^{-8}$ mW/m$^2$ in a horizontal polarization (along the $x$-axis) 205 - and $2(10)^{-8}$ mW/m$^2$ in a vertical polarization (along the $y$-axis). 206 - Assume the characteristic impedance is $\eta_0 = 377 \Omega$. 207 - \begin{enumerate}[label=(\alph*)] 208 - \item What is combined $E$-field value and its direction? 209 - \item What is the direction of motion? 210 - \item What is the total power flux? 211 - \end{enumerate} 212 - 213 - 214 204 \item \emph{dBm to linear conversions:} 215 205 \begin{enumerate}[label=(\alph*)] 216 206 \item Convert the following to mW: 17 dBm, -73 dBm, -97 dBW. 218 208 \end{enumerate} 219 209 220 210 221 - \item \emph{Spherical-cartesian conversions:} 222 - \begin{enumerate}[label=(\alph*)] 223 - \item Convert $(r,\phi,\theta)=(2,45^\circ,30^\circ)$ to $(x,y,z)$. 224 - \item Convert $(x,y,z) = (1,2,3)$ to $(r,\phi,\theta)$. 225 - \item Convert $(x,y,z) = (1,-2,3)$ to $(r,\phi,\theta)$. 226 - \end{enumerate} 227 211 228 - \item \emph{Rotation matrices}: The \emph{rotation matrix} 229 - $R(\theta,\phi)$ is the $3 \times 3$ matrix such that $R(\theta,\phi)\rbf$ rotates 230 - the vector $\rbf$ by an angle pair $(\theta,\phi)$. 231 - You can read more about this on wikipedia or other sources. 232 - \begin{enumerate}[label=(\alph*)] 233 - \item Find the entries of $R(\theta,\phi)$. 234 - \item Show $R(\theta,\phi)$ is \emph{orthogonal} meaning 235 - $R(\theta,\phi)^{-1}=R(\theta,\phi)\tran$. 236 - \item Is 212 + \item \emph{Spherical-cartesian conversions:} When a transmitter is at the origin, 213 + its E-field in the far field can often be represented as, 237 214 \[ 238 - R(\theta_1,\phi_1)R(\theta_2,\phi_2) = R(\theta_2,\phi_2)R(\theta_1,\phi_1)? 215 + \Ebf = E_\theta \ebf_\theta + E_\phi \ebf_\phi, 239 216 \] 240 - That is, do the order of rotations matter? 241 - Prove or find a counter-example. 242 - \end{enumerate} 217 + where $\ebf_\theta$ and $\ebf_\phi$ are the basis vectors in elevation and azimuth direction. 218 + Complete the following MATLAB function that takes a $1\times 3$ position vector 219 + \mcode{pos} and $n \times 1$ values of $E_\theta$ and $E_\phi$ and returns the 220 + an $n \times 3$ matrix \mcode{E} representing the E-field values in cartesian coordinates. 221 + You may use any built in MATLAB functions. Be careful whether the methods use degrees 222 + or radians. 223 + \begin{lstlisting} 224 + function E = convert(Etheta, Ephi, pos) 225 + \end{lstlisting} 243 226 244 227 245 - \item \emph{Angular areas:} Find the angular area in steradians of 246 - following sets of angles: 228 + \item \emph{Rotation matrices}: In wireless systems, we often need to consider antennas that 229 + can be in arbitrary rotations. One way of specifying the orientation of an object 230 + is through its so-called \emph{Euler} angles $(\alpha,\beta,\gamma)$ or 231 + \emph{yaw, pitch} and \emph{roll}. Let $R(\alpha,\beta,\gamma)$ be the rotation matrix 232 + for a given set of Euler angles. You can find the formulae for $R(\alpha,\beta,\gamma)$ 233 + in any reference such as wikipedia. 247 234 \begin{enumerate}[label=(\alph*)] 248 - \item $A_1 = \left\{ (\phi,\theta) ~ | ~ \phi \in [-30,30],~ \theta \in [-90,90]\right\}$ 249 - \item $A_2 = \left\{ (\phi,\theta) ~|~ \phi \in [-30,30], ~ \theta \in [-45,45]\right\}$ 235 + % \item Write a simple MATLAB function 236 + % as follows that computes the rotation matrix given the Euler angles in degrees. 237 + %\begin{lstlisting} 238 + % function rot = rotMatrix(yaw,pitch,roll) 239 + %\end{lstlisting} 240 + 241 + \item Given elevation and azimuth angles $(\theta,\phi)$ find $(\alpha,\beta,\gamma)$ 242 + with $\gamma=0$ that rotates the $x$-axis to point in $(\theta,\phi)$. 243 + \item Is $R(\alpha,0,0)^{-1} = R(-\alpha,0,0)$? Explain. 244 + \item Is $R(\alpha,\beta,0)^{-1} = R(-\alpha,-\beta,0)$? Explain. 250 245 \end{enumerate} 251 246 247 + \item \emph{Angular areas:} Find the angular area in steradians of 248 + following sets of angles where $\phi$ is the azimuth angle and $\theta$ is the elevation angles 249 + in degrees: 250 + \begin{enumerate}[label=(\alph*)] 251 + \item $A_1 = \left\{ (\phi,\theta) ~ | ~ \phi \in [-\ang{30},\ang{30}],~ \theta \in [-\ang{90},\ang{90}]\right\}$ 252 + \item $A_2 = \left\{ (\phi,\theta) ~|~ \phi \in [-\ang{30},\ang{30}],~ \theta \in [-\ang{45},\ang{45}]\right\}$ 253 + \end{enumerate} 252 254 \item \emph{Directivity:} Suppose an antenna radiates power uniformly in 253 - the angular beam $\phi \in [-30,30], \theta \in [-45,45]$ and radiates 254 - no power at other angles. What is the directivity of the antenna? 255 + the angular beam $\phi \in [-\ ang { 30} ,\ ang { 30} ]$ , and $ \theta \in [-\ ang { 45} ,\ ang { 45} ]$, 256 + and radiates no power at other angles. What is the maximum directivity of the antenna in dBi ? 257 + You can use the results from the previous problem. 258 + 255 259 256 - \item \emph{Radiation intensity:} A 170 cm x 40 cm object 257 - (roughly the size of a human) is 800m from a base station. 258 - If the antenna transmits 250~ mW isotropically, how much power 260 + \item \emph{Radiation intensity:} A \ SI { 170} { cm} $ \ times $ \ SI { 40} { cm} object 261 + (roughly the size of a human) is \ SI { 800 } { m } from a base station. 262 + If the base station antenna transmits \ SI { 250} { mW} isotropically, how much power 259 263 reaches the human? Use reasonable approximations that the human is far from the 260 264 transmitter. 261 265 262 - \item \emph{Radiation integration:} The radiation density for some antenna is, 266 + \item \emph{Radiation integration:} Suppose the radiation intensity is 263 267 \[ 264 - U(\phi,\theta) = A\cos^2(\phi ), \quad A = 100 \mbox {mW/sr}. 268 + U(\phi,\theta) = A\cos^2(\theta ), \quad A = 10 \ , \si {mW/sr}, 265 269 \] 266 - \begin{enumerate}[label=(\alph*)] 267 - \ item What is the total radiated power in mW and dBm ? 268 - \item What is the directivity of the antenna in linear scale and in dBi? 269 - \end{enumerate} 270 + where $(\phi,\theta)$ are the azimuth and elevation angles. 271 + find the total radiated power in dBm and maximum directivity in dBi . 272 + You can look up any integrals you need. 273 + 270 274 271 275 \item \emph{Numerically integrating patterns:} 272 - Write a short MATLAB function, 273 - \begin{lstlisting} 274 - function [totPow, dir] = powerDirectivity(az,el,E,eta) 275 - \end{lstlisting} 276 - that computes the total power and directivity as a function of the $E$-field 277 - values. The E field is specified as a complex matrix \mcode{E(i,j)} 278 - at angles \mcode{az(i),el(j)}. You can assume that the angles are uniformly 279 - spaced over the total angular space. The total power output 280 - \mcode{totPow} should be a 281 - scalar representing the total radiated power in dBm and the directivity output 282 - should be \mcode{dir(i,j)} should be a matrix. 276 + Suppose we are given the radiation intensity $U(\theta,\phi)$ at discrete points, 277 + $(\theta_i,\phi_j)$ where 278 + $\theta_i$, $i=1,\ldots,M$ is uniformly spaced on $[-\pi/2,\pi/2]$ 279 + and $\phi_j$, $j=1,\ldots,N$ is uniformly spaced on $[-\pi,\pi]$. 280 + Assume $(\theta, \phi)$ are elevation and azimuth angles. Write a short MATLAB function 281 + to compute the radiated power $P_{\rm rad}$ and directivity $D(\theta_i,\phi_j)$ 282 + from a matrix of values $U(\theta_i,\phi_j)$. 283 283 284 - \item \emph{Friis' Law}: A transmitter radiates 100 mW at a carrier $f_c = 2.1$ GHz 285 - with a directional gain of $G_t = 10$ dBi. 286 - Suppose the receiver is $d = 200$ m from the transmitter and the path is free space. 287 - What is the received power if: 284 + 285 + 286 + \item \emph{Friis' Law}: A transmitter radiates \SI{15}{dBm} at a carrier $f_c =$ \SI{2.1}{GHz} 287 + with a directional gain of $G_t = 9$\,\si{dBi}. 288 + Suppose the receiver is $d =$ \SI{200}{m} from the transmitter and the path is free space. 289 + What is the received power in dBm if: 288 290 \begin{enumerate}[label=(\alph*)] 289 - \item The effective received aperture is 1 cm$ ^2$ . 290 - \item The receiver gain is $G_r = 5$ dBi. 291 + \item The effective received aperture is \ SI { 1} { cm^2} . 292 + \item The receiver gain is $G_r =$ \ SI { 5} { dBi} . 291 293 \end{enumerate} 294 + 292 295 293 296 \end{enumerate} 294 297 295 298 \end{document} 296 299 297 300 298 - 299 -
Sundeep Rangan
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3 years ago
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