# Rate Adaptation for Multiuser Networks with Limited Feedback

## Rate Adaptation for a Single-User MISO network with Limited Feedback

Rate adaptation for a single-user MISO networks can be implemented as follows:

1. 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 $$\mathcal{Q}_{b}$$.

2. 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.

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{aligned} \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{aligned} 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.

## 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:

1. 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.

2. To design efficient quantizers for the optimal transmit rates and power allocations.

# 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 (Ding 2014).

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{aligned} y_n = h_n \sqrt{P}\sum_{m = 1}^N \sqrt{\alpha_m} x_m + v_n. \nonumber\end{aligned} 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,

### References

1. Z. Ding, Z. Yang, P. Fan, H. V. Poor. On the Performance of Non-Orthogonal Multiple Access in 5G Systems with Randomly Deployed Users. IEEE Signal Processing Letters 21, 1501-1505 (2014).