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  • unit03_fading/prob/prob_fading.pdf
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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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