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\author[ ]{Xiaoyi Liu}
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\section{Rate Adaptation for Multiuser Networks with Limited Feedback}
\subsection{Rate Adaptation for a Single-User MISO network with Limited Feedback}
Rate adaptation for a single-user MISO networks can be implemented as follows:
\begin{enumerate}
\item Offline-designed quantizers for the transmitter's rate and transmit beamforming vectors are known at the transmitter's and receiver's sides, denoted by $\mathcal{Q}_{R}$ and $\mathcal{Q}_{b}$.
\item The receiver obtains the channel vector ${\pmb h}$, picks the appropriate rate and transmit beamforming vector determined by $\mathcal{Q}_R$ and $\mathcal{Q}_b$, and sends their indices to the transmitter.
\end{enumerate}
Let $\mathcal{C}_{R} = \left\{R_0 = 0, R_1, R_2, \ldots, R_{2^{B_1}-1}\right\}$ be the codebook for the fixed-length quantizer $\mathcal{Q}_{R}$. The selected rate for the transmitter will be $R_{s}$ such that $R_{s} < \left|\left|\pmb h\right|\right|^2 \leq R_{s+1}$, where $0 \leq s\leq 2^{B_1}-1$. Let $\mathcal{Q}_{b}$ be the variable-length quantizer in our previous work. Then, for ${\pmb h}$ and the selected rate $R_s$, we can find an appropriate beamforming vector such that $\left|{\pmb h}^{+}{\pmb w}\right|^2 \geq R_s$, and the average feedback rate is finite.
The remaining problem is to design the codebook $\mathcal{C}_b$. The average achieved rate of the MISO network is
\begin{align}
\sum_{i=0}^{2^{B_1}-2}\text{Prob}\left\{R_i < \left|\left|\pmb h\right|\right|^2 \leq R_{i+1}\right\} \times R_i + \text{Prob}\left\{ \left|\left|\pmb h\right|\right|^2 > R_{2^{B_1}-1}\right\} \times R_{2^{B_1}-1}.\nonumber
\end{align}
The optimal values for $R_1, \ldots, R_{2^{B_1}-1}$ can be found by maximizing the average achieved rate above.
Rate adaptation schemes for other single-user networks (such as MIMO networks and amplify-and-forward relay networks) can be designed in a similar way.
\subsection{Challenges for Rate Adaptation for Multiuser Networks with Limited Feedback}
Take the decode-and-forward relay network (one source ${\sf S}$, $N$ decode-and-forward relays ${\sf R}_1, \ldots, {\sf R}_N$ and one destination $\sf D$) for example. The challenges here include:
\begin{enumerate}
\item To find optimal transmit rates and power allocations for the source and the relays to achieve the maximum achieved data rate at $\sf D$ for each channel state without outage.
\item To design efficient quantizers for the optimal transmit rates and power allocations.
\end{enumerate}
\section{Non-Orthogonal Multiple-Access (NOMA) Systems with Limited Feedback}
In the future 5G communication systems, a promising downlink multiple access scheme is the non-orthogonal multiple access (NOMA) which achieves high spectral efficiencies by combining superposition coding at the transmitter with successive
interference cancellation (SIC) at the receivers \cite{NOMA}.
We consider the system model with one base station ${\sf B}$ and $N$ downlink users ${\sf U}_1, \ldots, {\sf U}_N$; all terminals are equipped with a single antenna. In the traditional orthogonal multiple access methodology, $\sf B$ serves only one users at any time slot. In NOMA, $\sf B$ simultaneously serve all users by using the entire bandwidth to transmit data via a superposition
coding technique at the transmitter side and SIC techniques at the users. More specifically, the transmit signal of $\sf B$ is $\sqrt{P}\sum_{n = 1}^N \sqrt{\alpha_n} x_n$, where $P$ is the transmit power, $\alpha_n$ is the power allocation coefficient and $x_n$ is the message for ${\sf U}_n$. The received signal at ${\sf U}_n$ is
\begin{align}
y_n = h_n \sqrt{P}\sum_{m = 1}^N \sqrt{\alpha_m} x_m + v_n.
\nonumber
\end{align}
When the channels are ordered as $\left|h_1\right|^2 \leq \left|h_2\right|^2 \leq \cdots \left|h_N\right|^2$, SIC will be performed at the users. Therefore, ${\sf U}_n$ will detect ${\sf U}_i$'s message when $i < n$, and then remove the detected message from its received signal $y_n$, in a successive way. The messages for $i > n$ are treated as noises. As a result,
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