Assume that we are using a QAM signaling scheme to communicate over a certain channel. If we want to decrease the error rate, which of the following steps can we take? Select all that applies.
Consider the QAM receiver shown below in which the discrete-time input signal $\hat{s}[n]$ is demodulated using two auxiliary signals $m_1[n]$ and $m_2[n]$.
\begin{center}
\centering
\includegraphics[width=0.7\textwidth]{receiver}
\end{center}
Determine which of the following choices for $m_1[n]$ and $m_2[n]$ allow for a correct demodulation of the signal. Assume that $\omega_c$ is much larger than $2\omega_0$, the effective bandwidth of the input signal.
Consider a simplified ADSL transmission scheme with 8 sub-channels of equal width, $\mbox{CH}_0$ to $\mbox{CH}_7$; assume that the power constraint is the same for all sub-channels. Each sub-channel $\mbox{CH}_i$ is centered at $\omega_i = \frac{i\pi}{N}$. Only on the sub-channels $\mbox{CH}_2$ to $\mbox{CH}_7$ are used for transmission.
QAM signaling is used on each of the allowed sub-channels, and the maximum achievable SNR's for each sub-channel are shown graphically here:
\begin{center}
\centering
\includegraphics[width=0.69\textwidth]{lineprobe}
\end{center}
\begin{enumerate}
\item Indicate the sub-channels with the lowest and highest throughput.
\item Consider the following SNR curves for QAM signaling:
\begin{center}
\centering
\includegraphics[width=0.69\textwidth]{snr}
\end{center}
Based on the sub-channels SNR's shown in the first figure, determine the maximum throughput for channels $\mbox{CH}_3$, $\mbox{CH}_4$ and $\mbox{CH}_7$ when the maximum accepted probability of error for any sub-channel is $P_{\it err}=10^{-6}$ and the sampling frequency of the system is $F_s=2~$MHz.
Consider the problem of designing a data communication system over an analog channel with a given bandwidth constraint:
\begin{enumerate}
\item Assume the usable bandwidth extends from $F_{\it min}=250~$MHz to $F_{\it max}=500~$MHz. To meet the bandwidth constraint, the signal is upsampled by a factor $K=4$ and interpolated at $F_s=1~$GHz before D/A conversion. Determine the Baud rate (in symbols/s) and the throughput (in bits/s), assuming the alphabet $\cal{A}$ has 32 symbols and all symbols are equiprobable.
\item Consider now a usable bandwidth extending from $F_{\it min}=400~$MHz to $F_{\it max}=600~$MHz. Choose among the possibilities below the combination of sampling frequency $F_s$ and upsampling factor $K$ that allows meeting the given bandwidth constraint in the analog domain:
Consider an analog channel whose power constraint results in a maximum achievable SNR of $30~$dB. If the channel's bandwidth is $3~$kHz, what is the maximum throughput $R$ (in bits/s) that can be achieved by QAM signaling using a square constellation if we can accept a probability of error of $10^{-6}$?
Consider a 32-PAM signaling scheme using the symbols shown in this diagram:
\begin{center}
\includegraphics[width=0.69\textwidth]{pam}
\end{center}
At the receiver, after demodulation, the symbols are affected by additive noise whose amplitude is uniformly distributed over the interval $[-100,\,100]$.
Find the minimum value of $G$ for which the error probability does not exceed $10^{-2}$.