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\begin{document}

\title{Problems:  Non-LOS Propagation and Link Budget Analysis\\
EL-GY 6023. Wireless Communications}
\author{Prof.\ Sundeep Rangan}
\date{}

\maketitle

\begin{enumerate}

\item \emph{Noise:}
Suppose a receiver consists of an low noise amplifier with a gain of 30 dB
and noise figure of 2 dB, followed by a second stage of amplification of
another 20 dB with a noise figure of 10 dB.
\begin{enumerate}[label=(\alph*)]
\item What is the total noise figure and gain of the system?
\item Suppose a 10 dB attenuator is placed at the input of the LNA.
What is the resulting overall gain and noise figure?
\item What if the attenuator is placed at the output of the LNA?
\end{enumerate}

\item \emph{SINR:} 
Suppose when a transmitter, TX1, sends data to a receiver RX
without interference, the SNR is 10 dB.  Now suppose that TX2
starts transmitting resulting in interference.
Suppose TX2 transmits at the same power as TX1 and the path loss from
TX2 to RX is 5 dB greater than the path loss from TX1 to RX.
What is the resulting SINR from TX1 to RX when TX2 is transmitting?


\item \emph{Reflection loss:}  Let $\Gamma$ be the reflection coefficient from
an interface going from a characteristic impedance $\eta_1$ to $\eta_2$ and let 
$\Gamma'$ be the reflection coefficient from
 $\eta_2$ to $\eta_1$.  Show that if the angle of incidence is
zero $\theta_i=0$, $\Gamma = -\Gamma'$.  


\item \emph{SNR requirements:} 
A signal is received at power of $P = $-100 dBm and the noise power density
(including the noise figure) is $N_0 = $ -170 dBm/Hz.
If a transmission of $b=100$ bits requires an $E_b/N_0$
of 6~dB, what is the time to transmit the packet?

\item \emph{Simulating a statistical model:}  
Write short MATLAB code to do the following.  You do not need to run the code,
just write the code.
\begin{enumerate}[label=(\alph*)]
\item Drop \mcode{nrx=1000} RX locations randomly in a circle of radius \mcode{radius = 100}m.
\item Assuming the transmitter is at the origin at a height \mcode{htx=2}m higher
than the RX, compute the distances \mcode{dist} to the RXs.
\item Assuming a path loss model,
\[
    PL = 32.4 + 14.3\log_{10}(d) + 20\log_{10}(f_c)  + \xi,
    \quad \xi \sim \mathcal{N}(0,\sigma^2),
\]
generate random path losses to the RXs.  Assume $\sigma = 4$ and 
$f_c = 2.3$ GHz.
\item Compute the SNR, $E_s/N_0$, to the RX.  State all the parameters that you would need.
\end{enumerate}


\item \emph{Simulating a statistical model:}
Consider an indoor propagation model where the path loss is,
\[
    PL = \mathrm{FSPL}(d) + L_0 N, \quad N \sim \mathrm{Poisson}(\alpha d),
\]
where $d$ is the distance between the TX and RX,
$\mathrm{FSPL}(d)$ is the free-space path loss, $N$ is 
the number of walls between a transmitter and receiver and $N$ is modeled as 
a Poisson random variable where $\alpha$ is average number of walls
per meter of distance.  
Write a MATLAB function 
\begin{lstlisting}
    function pl = indoorPL(d, ... )
\end{lstlisting}
to generate random path loss values as function of a vector of distances.  
State all the other parameters this function needs.  You may assume you have
access to a function \mcode{fspl(lambda,d)} for the free-space path loss.
  
\item \emph{Outage probability:} Suppose that a link has the following properties:
\begin{itemize}
\item TX power, $P_{tx} =$ 20 dBm
\item Bandwidth, $W =$ 20 MHz
\item Noise power density (including noise figure) $N_0=$ -170 dBm/Hz.
\end{itemize}
Answer the following:
\begin{enumerate}[label=(\alph*)]
\item What is the maximum path loss, $PL_{max}$, that the link can
have to meet an SNR target of 10 dB?
\item Suppose that the path loss is lognormally distributed with
\[
    PL = PL_0 + \xi, \quad \xi \sim {\mathcal N}(0,\sigma^2),
\]
where $PL_0 = $ 110 dB and $\sigma$ = 8~dB.
What is the outage probability $P_{out}=\Pr(PL \geq PL_{max})$
using the value $PL_{max}$ from part (a)?
Your answer should have
a $Q$-function.  You can evaluate it with MATLAB's function \mcode{qfunc}.

\item A common model for indoor path loss is given by
\[
    PL = PL_0 + \xi + DN, \quad \xi \sim {\mathcal N}(0,\sigma^2),
\]
where $N$ is the number of walls that the signal must pass through
and $D$ is the loss per wall.  Suppose that we model the number of
walls as a random variable with distribution:
\[
    P(N=n) = \begin{cases}
        0.5 & \mbox{if } n=0 \\
        0.3 & \mbox{if } n=1 \\
        0.2 & \mbox{if } n=2 \\
        0 & \mbox{else, }
        \end{cases}
\]
and the loss per wall is $D=$ 7~dB.
What is the outage probability $P_{out}$?
\end{enumerate}

\end{enumerate}

\end{document}