Uplink NOMA - Outage Probability in MATLAB

 Till now, we were dealing with downlink NOMA (where the base station (BS) transmits NOMA signal to the end users). In this post, let's see how to perform NOMA in the uplink. Uplink communication is when the users transmit to the BS.

Download the MATLAB code here

 

Network model

Let's consider the following network consisting of two users, who want to transmit their data to the BS.

Fig. 1: Uplink NOMA network model

In this example, we are taking $U_f$ as the far user and $U_n$ as the near user. Let $d_f$ and $d_n$ denote their distances from the BS. $h_f$ and $h_n$ represent their corresponding Rayleigh fading coefficients. ($d_f \gt d_n$ and $|h_f|^2 \lt |h_n|^2$)

How to do power domain multiplexing in uplink NOMA?

The power domain multiplexing part is done in a slightly different way for uplink NOMA. As we know, in downlink NOMA, the BS used superposition coding to perform power domain multiplexing. But in uplink, the transmit power of the users is limited only by their battery capacity. That is, both users can transmit with their maximum possible power. The differentiation in power domain is brought about at the receiver side (BS) as a result of the differences in the channel gains of the users. Let's dig in to see how this works.

Signal model

Let $x_f$ and $x_n$ denote the messages to be transmitted by $U_f$ and $U_n$ respectively. Let's assume that both the users transmit their signals with the same power. The signal transmitted by $U_f$ is, $\sqrt{P}x_f$. Similarly, the signal transmitted by $U_n$ is $\sqrt{P}x_n$. 

By now, the following two questions will rise in our mind.

If both users transmit with the same power, then how can the BS differentiate their signals in the power domain?

How SIC is done in uplink NOMA?

To answer these questions, let's write the equation for the received signal at the BS. $$y = \underbrace{\sqrt{P}x_fh_f}_\textrm{far user term} + \underbrace{\sqrt{P}x_nh_n}_\textrm{near user term} + w$$

As per our initial assumptions, $U_n$ is closer to the BS, which means that he has a strong channel gain compared to $U_f$. i.e., $|h_n|^2 \gt |h_f|^2$. So, in the received signal, the power of the $U_n$'s term will be dominating. $$y = \sqrt{P}x_fh_f + \underbrace{\sqrt{P}x_nh_n}_\textrm{dominating} + w$$

This means, the BS can treat the $U_f$'s term as interference and directly decode the $x_n$ term. Next, successive interference cancellation can be done to retrieve the $x_f$.

Now we see the power domain differentiation coming into play. This is where uplink NOMA differs from its downlink counterpart. If the users are sufficiently far apart from each other, they would have distinct channel gains. As a result, the BS can successfully differentiate their signals, (even without power control a.k.a superposition coding). In other words, the power control is brought about by the inherent differences in the channel gains and not by intentional superposition coding as in downlink NOMA.

It is important to note that, for uplink NOMA to be successful, the channel gains of the users must be distinct enough. If both the users have similar channel gains, then there is no way for the BS to distinguish their signals in the power domain. In that case, power control must be done. That is, the users must transmit with different powers.

Another important observation from the above discussion is that, the SIC order is reversed in uplink NOMA. The near user's signal is decoded first. Whereas in downlink NOMA, the far user's signal is decoded first.

Achievable Rates of uplink NOMA

The near user's signal is decoded first, by treating the far user's signal as interference. Therefore, the achievable rate at the BS to decode the near user data is, $$R_n = log_2\left(1 + \dfrac{P|h_n|^2}{P|h_f|^2+\sigma^2}\right)$$

After successive interference cancellation (SIC), the far user's achievable rate is given by, $$R_f = log_2\left(1 + \dfrac{P|h_f|^2}{\sigma^2}\right)$$

MATLAB Simulation

Download the MATLAB code here

Next, let's simulate our uplink NOMA network in MATLAB and study its outage performance. In this simulation, no power control is implemented. The inherent differences in the channel gains are exploited to aid the power domain multiplexing. The simulation is done for two different near user - far user distance pairs (800, 500)m and (800, 200)m. When we plot the outage, we get the following graph.

Fig. 2: Outage of uplink NOMA

Before inferring the graph, we can make the following observation: the channel gains of the users are more distinct for the (800,200)m pair than for the (800,500)m pair. 

We can see from the above graph that, both the users experience lesser outage for the distance pair (800,200)m. For the (800,500)m pair, the outage probability is high for both the users. This confirms the reasoning that we made before. That is, NOMA gives superior performance when the channel conditions between the users become more and more distinct.

Further improvement in performance can be achieved by power control, which we will see in the next post. Thank you!

References

Y. Liu, M. Derakhshani and S. Lambotharan, "Outage analysis and power allocation in uplink non-orthogonal multiple access systems," IEEE Communication Letters, vol. 22, no. 2, pp. 336-339, 2018.

H. Tabassum, M. S. Ali, E. Hossain, M. J. Hossain and D. I. Kim, "Uplink Vs. Downlink NOMA in Cellular Networks: Challenges and Research Directions," 2017 IEEE 85th Vehicular Technology Conference (VTC Spring), 2017, pp. 1-7, doi: 10.1109/VTCSpring.2017.8108691.