prob_demod.tex

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\begin{document}

\title{Problems:  Demodulation}
\author{Prof.\ Sundeep Rangan}
\date{}

\maketitle

\begin{enumerate}

\item \emph{SNR.} 
Suppose that the receive power is -80 dBm, the symbol rate is 1 MHz and
the noise power is -170 dBm/Hz.
\begin{itemize}
\item[(a)] What is the symbol SNR?  If the system uses
16-QAM what is the data rate and the SNR per bit ($E_b/N_0$)?
\item[(b)] What if a second noise source is added at a level of -100 dBm?
What is the resulting symbol SNR and SNR per bit?
\end{itemize}

\item \emph{SNR.}
Consider a wireless communication system where the
transmit power is 10 dBm, the noise power is -170 dBm/Hz and the data rate is 10 Mbps.
What is the maximum path loss (transmit - receive power in dB) for $E_b/N_0$= 10 dB.

\item \emph{Detection theory.} 
Suppose that we attempt to detect a signal from the received power $y$.
Let $x=1$ if a signal is present and $x=1$ if it is present.  So, we want to detect $x$ from $y$.
Assume that $y$ is exponential with power $P_i$ if $x=i$, $i=0,1$.
\begin{enumerate}[(a)]
\item Write the log likelihood ratio,
\[
    L(y) = \ln \frac{p(y|x=1)}{p(y|x=0)}.
\]

\item Consider the likelihood ratio test detector,
\[
    \hat{x} = \begin{cases}
        1 &  \mbox{if } L(y) \geq \gamma \\
        0 &  \mbox{if } L(y) < \gamma.
    \end{cases}
\]
Find the probability of missed detection and false alarm in terms of $\gamma$,
\[
    P_{MD} = P(\hat{x}=0|x=1), \quad
    P_{FA} = P(\hat{x}=1|x=0).
\]

\item Find $\gamma$ so that the $P_{FA}=(10)^{-3}$.   Plot the missed detection rate $P_{MD}$ at this $\gamma$
as a function of the SNR $P_1/P_0$.  Your plot should have $P_1/P_0$ in dB and $P_{MD}$ in log scale.

\end{enumerate}

\item \emph{Bit error rate}.  
\begin{enumerate}[(a)]
\item Assuming Gray coding, derive the formula
for the BER for 16-QAM in terms of the symbol SNR $\gamma_s = E_s/N_0$ and 
bit SNR $\gamma_b = E_b/N_0$.

\item Find an approximate expression for the BER for large $\gamma_b$.
\end{enumerate}

\item \emph{Vector-valued channel.}
Suppose a transmitted symbol is $x \in \{-1,1\}$ and
the received vector in complex signal space is $\rbf=(r_1,r_2)^T$
with
\[
    r_1 = h + w_1, \quad r_2 = hx + w_2,
\]
where $h$ represents some unknown channel gain
and $\wbf = (w_1,w_2)$ is i.i.d.\
Gaussian noise $w_i \sim {\mathcal CN}(0,N_0)$.
You can think of the first symbol as being transmitted with a reference
and the second symbol as carrying the data.
\begin{enumerate}[(a)]
\item  What is the likelihood $p_{\rbf|h,x}(\rbf|h,x)$?
\item  Compute the ML estimate for $x$ based on
the most likely channel 
\[
    \xhat = \argmax_{x \in \{-1,1\} }
    \max_{h \in \C} p_{\rbf|h,x}(\rbf|h,x).
\]
Describe the decision regions for $\xhat$.
\item What is the 
error rate as a function of the SNR $\gamma = |h|^2/N_0$.
\end{enumerate}

\end{enumerate}

\end{document}