How to simulate BER, capacity and outage probability of NOMA (Rayleigh fading) using MATLAB?

Simulation of BER of Non orthogonal multiple access (NOMA) in Rayleigh fading channel - MATLAB code with explanation

Next post: What happens when SIC is imperfect?

In the previous post, we saw how to simulate a simple two user NOMA network over an AWGN channel in MATLAB. We also plotted the BER vs SNR graphs of both the users and observed the waterfall trend. The AWGN assumption simplified the channel model and allowed us to learn the basic skeletal operation of NOMA. In this post, we will use MATLAB to simulate the capacity, outage and BER performance of a two user NOMA network by following a more realistic model namely, Rayleigh fading model. The MATLAB code to plot BER, capacity and outage probability of NOMA are also available for download in this page. So, let's get started!

Wireless channel is prone to multipath propagation and fading. Several channel models are available to capture the effects of fading. Each model deals with a particular scenario. One such model is Rayleigh fading model. Rayleigh fading model can be used when there is no line of sight (LOS) path between the transmitter and the receiver. In other words, all multipath components have undergone small scale fading effects like reflection, scattering, diffraction, shadowing etc.

We are going to consider an extreme case of Rayleigh fading where each transmitted bit undergoes a different attenuation and phase shift due to multipath transmission. In other words, the channel changes for every bit. We will first look at the system model and signal model of NOMA and then the MATLAB code for simulating it. We will simulate the bit error rate (BER), capacity and outage of NOMA. The MATLAB code for NOMA can be downloaded here.

Capacity and Outage probability for NOMA - MATLAB code
BER of NOMA - MATLAB code

See this post: How to do power allocation in NOMA?

System model

We are considering downlink transmission from base station (BS) to the two users (Can we have more than 2 NOMA users?). Our network will look like this.

User 1 is the far/weak as he is far away from the transmitting BS. User 2 is the near/strong user. Let $d_1$ and $d_2$ denote their distances from the BS.

Signal model

The BS has two distinct messages $x_1$ to user 1 (far user), and $x_2$ to user 2 (near user). $\alpha_1$ and $\alpha_2$ are the power allocation factors for the far and the near user respectively ($\alpha_1$ + $\alpha_2$ = 1). In NOMA, to promote user fairness, more power is given to the far user and less power to the near user. That is, $\alpha_1$ > $\alpha_2$. Throughout this post, we will use $\alpha_1$=0.75 and $\alpha_2$=0.25. This is an arbitrary choice.  Let $h_1$ and $h_2$ denote the channel from the BS to the far and the near user respectively. 

See this post to find how performance of NOMA varies with different choices of $\alpha_1$ and $\alpha_2$. To see how to optimize $\alpha_1$ and $\alpha_2$, see this post: How to do power allocation in NOMA?

NOMA encoding and transmission

As we know already, the superposition coded NOMA signal transmitted by the BS is, $$x = \sqrt{P}(\sqrt{\alpha_1}x_1 + \sqrt{\alpha_2}x_2)$$ where, $P$ is the transmit power. The copy of $x$ received at the far user after propagating through channel $h_1$ is, $$y_1 = h_1x + w_1$$ Similarly, the copy of $x$ received at the near user after propagating through channel $h_2$ is, $$y_2 = h_2x + w_2$$

NOMA decoding at User 1 (far user)

Expanding the received signal at user 1, \begin{align} y_1 &= h_1x + w_1\\ &= h_1\sqrt{P}(\sqrt{\alpha_1}x_1 + \sqrt{\alpha_2}x_2) + w_1 \\ &= \underbrace{h_1\sqrt{P}\sqrt{\alpha_1}x_1}_\textrm{desired & dominating} + \underbrace{h_1\sqrt{P}\sqrt{\alpha_2}x_2}_\textrm{interference & low power} + \underbrace{w_1}_\textrm{noise} \end{align} Since $\alpha_1$ > $\alpha_2$, direct decoding of $y_1$ would yield $x_1$. The term containing the $x_2$ component will be treated as an interference. The signal to interference and noise ratio for the far user is, $$\gamma_1 = \dfrac{|h_1|^2P\alpha_1}{|h_1|^2P\alpha_2+\sigma^2}$$ and his achievable data rate is, $$R_1 = log_2(1 + \gamma_1) = log_2(1+\dfrac{|h_1|^2P\alpha_1}{|h_1|^2P\alpha_2+\sigma^2})$$

NOMA decoding at User 2 (near user)

Expanding the received signal at user 2, \begin{align} y_2 &= h_2x + w_2\\ &= h_2\sqrt{P}(\sqrt{\alpha_1}x_1 + \sqrt{\alpha_2}x_2) + w_2\\ &= \underbrace{h_2\sqrt{P}\sqrt{\alpha_1}x_1}_\textrm{interference & dominating} + \underbrace{h_2\sqrt{P}\sqrt{\alpha_2}x_2}_\textrm{desired & low power} + \underbrace{w_2}_\textrm{noise} \end{align} User 2 must first perform successive interference cancellation (SIC) before decoding his own signal. SIC is carried out as follows:

1. $y_2$ is directly decoded to obtain $x_1$ or rather, an estimate of $x_1$, that is $\tilde{x}_1$.
2. $y_2' = y_2 - \sqrt\alpha_1 \tilde{x}_1$ is computed.
3. $y_2'$ is decoded to obtain an estimate of $x_2$.

We are making a perfect SIC assumption here. What happens if SIC is imperfect?
The signal to interference and noise ratio at the user 2 for decoding the user 1 signal (before SIC) is, $$\gamma_{1,2} = \dfrac{|h_2|^2P\alpha_1}{|h_2|^2P\alpha_2+\sigma^2}$$ The corresponding achievable data rate is, $$R_{1,2} = log_2(1 + \gamma_{1,2}) = log_2(1+\dfrac{|h_2|^2P\alpha_1}{|h_2|^2P\alpha_2+\sigma^2})$$ After after cancellation of user 1's signal using SIC, the signal to noise ratio at the user 2 for decoding its own signal is, $$\gamma_2 = \dfrac{|h_2|^2P\alpha_2}{\sigma^2}$$ The corresponding achievable data rate is, $$R_2 = log_2(1 + \gamma_2) = log_2(1+\dfrac{|h_2|^2P\alpha_2}{\sigma^2})$$

Here, we have considered only two NOMA users. We can also multiplex more than two users in the same frequency carrier to accommodate more users in our network.

Simulation using MATLAB

Now that we have seen the mathematical details of NOMA we can go ahead and write a MATLAB code to simulate it. We will write two codes. One to study the BER performance and another one to study the capacity and outage performance of NOMA. So, let's get started.

Capacity and outage probability of NOMA - MATLAB simulation

You can download the MATLAB code to plot capacity and outage probability of NOMA here. Capacity and Outage probability for NOMA - MATLAB code

1. First, let's declare the values of some parameters. Let's say the distances are, $d_1 = 1000$ meters and $d_2 = 500$ meters. Let's set the power allocation factors as $\alpha_1 = 0.75$ and $\alpha_2 = 0.25$. You can set any values for these parameters but make sure that the far user is given the higher fraction of power.
2. We want a plot of capacity and outage probability as a function of transmit power. So, let's initialize a range of transmit power from 0 dBm to 40 dBm
3. Let's set our system bandwidth as $B = 1MHz$. Let's calculate the thermal noise power as $No = kTB$, where $k = 1.38\times 10^{-23} J/K$,$T=300K$.
4. Next, we have to generate the Rayleigh fading coefficients $h_1$ and $h_2$ to simulate the channel between the two users. This can be done using the following MATLAB command $h_i = \sqrt{d_i^{-\eta}}(randn(1,N)+1i*randn(1,N))/\sqrt{2}$. Here, $\eta$ is called the path loss exponent.Typically we set $\eta=4$
5. To plot capacity, we need to calculate $R_1 = log_2(1+\dfrac{|h_1|^2P\alpha_1}{|h_1|^2P\alpha_2+\sigma^2})$, $R_{1,2} =log_2(1+\dfrac{|h_2|^2P\alpha_1}{|h_2|^2P\alpha_2+\sigma^2})$, and $R_2 = log_2(1+\dfrac{|h_2|^2P\alpha_2}{\sigma^2})$.
   
6. Find the average value of each of the above quantities for every transmit power
7. Calculating the above quantities for different power levels (for eg., from 0 to 40 dBm), we get a plot as shown below:



8. Next, we will plot the outage probabilities. Set target rate for each user. For example, for user 1, we set the target rate as 1 bps/Hz and for user 2, we set the target rate as 2 bps/Hz.
9. Count the number of times the values calculated in step 5 fall below the target rates and take the average. In other words, set a counter for user 1, which is incremented every time $R_1 \lt 1 bps/Hz$ and another counter for user 2 which increments every time $R_{1,2} \lt 1 bps/Hz$ or $R_2 \lt 2 bps/Hz$
10. Plot the outage probabilities as a function of transmit power and the graph would look like this:

BER of NOMA - MATLAB simulation

You can download the MATLAB code to plot BER of NOMA here. BER of NOMA - MATLAB code

1. Follow steps 1 to 4 of capacity and outage simulation explained above
2. Let's next generate noise samples for both users. The formula is, $w_i = \sqrt{noise power}(randn(1,N) + 1i*randn(1,N))/\sqrt{2}$
3. Generate random binary data for both users
4. Modulate the data using any digital modulation scheme. In the code given here, BPSK is used for both users.
5. Calculate the superposition coded signal $x$
6. Calculate $y_1$ and $y_2$
7. Equalize $y_1$ and $y_2$ by diving by $h_1$ and $h_2$ respectively
8. From the equalized version of $y_1$ perform direct BPSK demodulation to get $\tilde{x_1}$.
9. Compare $\tilde{x_1}$ with user 1's original data and estimate BER using the biterr function
10. Directly decode the equalized version of $y_2$ to estimate $x_1$
11. Remodulate $x_1$ and subtract the remodulated $x_1$ component after multiplying it with $\sqrt{\alpha_1}$ from the equalized version of $y_2$. Decode this signal to get $\tilde{x_2}$.
12. Compare $\tilde{x_2}$ with user 2's original data and estimate BER using the biterr function.
13. Plot the BERs as a function of transmit power and the graph would look like this:

Note

Make sure you use the linear value of power in all calculation and not the dBm value. (Especially while calculating $R_1$, $R_{1,2}$, $R_{2}$.

Next post: How to do power allocation in NOMA?

References:

1. Theoretical BER: F. Kara, H. Kaya, BER Performances of Downlink and Uplink NOMA in the Presence of SIC Errors over Fading Channels. IET Communications. Vol. 12, no. 15, pp. 1834-1844.
2. Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, and K. Higuchi, Non-Orthogonal Multiple Access (NOMA) for Cellular Future Radio Access, IEEE 77th Vehicular Technology Conference (VTC Spring) (Dresden, Germany), 2013, pp. 1–5.
3. A. Benjebbour, Y. Saito, Y. Kishiyama, A. Li, T. Harada, and T. Nakamura, Concept and Practical Considerations of NonOrthogonal Multiple Access (NOMA) for Future Radio Access, International Symposium on Intelligent Signal Processing andbCommunication Systems (ICPACS 2013) (Okinawa, Japan), 2013, pp. 770– 774.

If you have any doubts or if you find any mistakes, you can comment below. I'll try my best to help you. Thank you!