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unit03_fading/prob/prob_fading.tex
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238 | 238 | | |
239 | 239 | | \end{enumerate} |
240 | 240 | | |
241 | | - | \item A received signal has two paths: |
| 241 | + | \item \emph{Two path channel:} A received signal has two paths: |
242 | 242 | | \begin{itemize} |
243 | 243 | | \item Path 1: power -100 dBm and Doppler shift 100 Hz, |
244 | 244 | | \item Path 2: power -103 dBm and Doppler shift -50 Hz. |
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250 | 250 | | \item What is the fraction of time the power is greater than -101 dBm? |
251 | 251 | | \end{enumerate} |
252 | 252 | | |
253 | | - | \item (*) Suppose that a narrowband complex channel $h(t)$ be modeled as |
| 253 | + | \item \emph{Two path channel:} Suppose that a narrowband complex channel $h(t)$ be modeled as |
254 | 254 | | a wide-sense stationary random process. |
255 | 255 | | \begin{enumerate}[label=(\alph*)] |
256 | 256 | | |
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262 | 262 | | Write $\rho(\tau)$ in terms of the autocorrelation function |
263 | 263 | | $R(\tau) = \Exp h(t)h^*(t-\tau)$. |
264 | 264 | | |
265 | | - | \item If $h(t)$ follows a Jakes' spectrum with maximum Doppler $f_{max} = $ 200 Hz, |
| 265 | + | \item If $h(t)$ follows a Jakes' spectrum with a uniform angular distribution, |
| 266 | + | the autocorrelation is given by , |
| 267 | + | \[ |
| 268 | + | R(\tau) = R(0)J_0(2\pi f_{max}\tau), |
| 269 | + | \] |
| 270 | + | where $f_{max}$ is the maximum Doppler shift and $J_0(\cdot)$ is the Bessel |
| 271 | + | of the first kind. Plot $R(\tau)/R(0)$ vs.\ $\tau f_{max}$ for $\tau f_{max} \in [0,5]$. |
| 272 | + | |
| 273 | + | \item Using the autocorrelation function in the previous part, |
| 274 | + | if $f_{max} = $ \SI{200}{Hz}, |
266 | 275 | | what is the time it takes the channel to change by 10\%? |
267 | | - | You will need MATLAB to evaluate the Bessel function. |
268 | 276 | | \end{enumerate} |
269 | 277 | | |
270 | | - | \item (*) Consider a multipath fading channel of the form |
| 278 | + | \item \emph{Auto-correlation.} Consider a multipath fading channel of the form |
271 | 279 | | \[ |
272 | 280 | | y(t) = \frac{1}{\sqrt{L}}\sum_{\ell = 1}^L g_\ell |
273 | 281 | | e^{2\pi if_\ell} x(t-\tau_\ell), \quad f_\ell = f_{max}\cos(\theta_\ell), |
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301 | 309 | | R(\delta f) = \Exp\left[ H(t,f)H^*(t,f-\Delta f) \right]. |
302 | 310 | | \] |
303 | 311 | | |
304 | | - | \item Suppose we define the 3~dB coherence bandwidth as the frequency $W$ |
| 312 | + | \item Suppose we define the \SI{3}{dB} coherence bandwidth as the frequency $W$ |
305 | 313 | | where the frequency response changes by less than 3~dB. That is, |
306 | 314 | | \[ |
307 | 315 | | \Exp|H(t,f)-H(t,f+W)|^2 \leq \beta \Exp|H(t,f)|^2, \quad \beta = 0.5. |
308 | 316 | | \] |
309 | 317 | | Under the above exponential delay model, |
310 | | - | if the delay spread is $\lambda =$ 100 ns, what is the 3~dB bandwidth? |
311 | | - | \end{enumerate} |
312 | | - | |
313 | | - | \item |
314 | | - | \begin{enumerate}[label=(\alph*)] |
315 | | - | \item (*) Consider the time-varying frequency response \eqref{eq:Htf}. |
316 | | - | Suppose that the AoAs are uniformly distributed in $\theta_\ell \in [-\Delta/2,\Delta/2]$. |
317 | | - | Write an expression for the autocorrelation over time, |
318 | | - | \[ |
319 | | - | R(\tau) := \Exp H(t,f)H^*(t-\tau,f). |
320 | | - | \] |
321 | | - | Your expression should have an integral. Do not evaluate this integral. |
322 | | - | |
323 | | - | \item In 3D channel models, the paths arrive with an \emph{elevation} (vertical) |
324 | | - | angle $\omega_\ell$ to the $z$-axis |
325 | | - | and \emph{azimuth} (horizontal) angle $\varphi_\ell$ to the $x$-axis on the |
326 | | - | $xy$-plane. |
327 | | - | Suppose that the receiver moves along the positive $x$-axis. What is the Doppler |
328 | | - | shift of a path arriving at angles $(\omega_\ell,\varphi_\ell)$. |
329 | | - | |
| 318 | + | if the delay spread is $\lambda =$ \SI{100}{ns}, what is the \SI{3}{dB} bandwidth? |
330 | 319 | | \end{enumerate} |
| 320 | + | % |
| 321 | + | %\item |
| 322 | + | %\begin{enumerate}[label=(\alph*)] |
| 323 | + | %\item (*) Consider the time-varying frequency response \eqref{eq:Htf}. |
| 324 | + | %Suppose that the AoAs are uniformly distributed in $\theta_\ell \in [-\Delta/2,\Delta/2]$. |
| 325 | + | %Write an expression for the autocorrelation over time, |
| 326 | + | %\[ |
| 327 | + | % R(\tau) := \Exp H(t,f)H^*(t-\tau,f). |
| 328 | + | %\] |
| 329 | + | %Your expression should have an integral. Do not evaluate this integral. |
| 330 | + | % |
| 331 | + | %\item In 3D channel models, the paths arrive with an \emph{elevation} (vertical) |
| 332 | + | %angle $\omega_\ell$ to the $z$-axis |
| 333 | + | %and \emph{azimuth} (horizontal) angle $\varphi_\ell$ to the $x$-axis on the |
| 334 | + | %$xy$-plane. |
| 335 | + | %Suppose that the receiver moves along the positive $x$-axis. What is the Doppler |
| 336 | + | %shift of a path arriving at angles $(\omega_\ell,\varphi_\ell)$. |
| 337 | + | % |
| 338 | + | %\end{enumerate} |
331 | 339 | | |
332 | 340 | | \end{enumerate} |
333 | 341 | | |
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