NOMA Downlink System Model
For orthogonal multiple access (e.g., OFDMA), each orthogonal resource block is allocated to a single user. If a user’s channel condition is poor, the entire subcarrier bandwidth may be assigned to that user, which is inefficient. Non-Orthogonal Multiple Access (NOMA) addresses this by allocating more transmit power to users with weaker channels and less power to users with stronger channels. This ensures efficient spectrum usage and fairness.
The transmit SNR is defined as:
$$ \rho_s = \frac{P_s}{\sigma^2}, $$
where $P_s$ is the signal power and $\sigma^2 = 1$ is the noise power (assuming unit-variance Gaussian noise).
Transmit Signal
In downlink NOMA, the base station transmits:
$$ x_s = \sqrt{a_1 \rho_s}\, x_1 + \sqrt{a_2 \rho_s}\, x_2 $$
- $x_1, x_2$: unit-power transmitted symbols
- $\rho_s$: total transmit SNR
- $a_1 + a_2 = 1$ (power allocation)
- Typically $a_2 > a_1$ (weak user receives more power)
Received Signal at User $i$
$$ y_i = h_i x_s + n_i $$
Substituting:
$$ y_i = h_i \left( \sqrt{a_1\rho_s} x_1 + \sqrt{a_2\rho_s} x_2 \right) + n_i $$
Define channel gain:
$$ \beta_i = |h_i|^2 $$
User 1 (Strong User) Decoding
The received signal is:
$$ y_1 = h_1\left( \sqrt{a_1\rho_s} x_1 + \sqrt{a_2\rho_s} x_2 \right) + n_1 $$
Step 1: Decode user 2's signal (SIC)
$$ \gamma_{u_1}^{x_2} = \frac{a_2 \rho_s \beta_1}{a_1 \rho_s \beta_1 + 1} $$
User 1 then subtracts user 2’s signal.
Step 2: Decode own signal
$$ \gamma_{u_1}^{x_1} = a_1 \rho_s \beta_1 $$
Achievable rate:
$$ R_1 = \log_2(1 + a_1\rho_s\beta_1) $$
User 2 (Weak User) Decoding
The received signal is:
$$ y_2 = h_2\left( \sqrt{a_1\rho_s} x_1 + \sqrt{a_2\rho_s} x_2 \right) + n_2 $$
User 2 cannot decode the strong user’s low-power signal and treats $x_1$ as noise.
SNR for decoding its own signal
$$ \gamma_{u_2}^{x_2} = \frac{a_2 \rho_s \beta_2}{a_1 \rho_s \beta_2 + 1} $$
Rate:
$$ R_2 = \log_2\left(1 + \gamma_{u_2}^{x_2} \right) $$
QoS Constraints
Let $\tilde{R}_1$ and $\tilde{R}_2$ be the target minimum rates.
$$ R_1 \ge \tilde{R}_1, \qquad R_2 \ge \tilde{R}_2 $$
NOMA Flow Summary
- Base Station:
$$ x_s = \sqrt{a_1\rho_s}x_1 + \sqrt{a_2\rho_s}x_2 $$ - User 1 (Strong):
- Decodes $x_2$ first (high power)
- Performs SIC (subtracts $x_2$)
- Decodes its own signal $x_1$
- User 2 (Weak):
- Treats $x_1$ as noise
- Decodes $x_2$ directly
- QoS conditions: $$ R_1 \ge \tilde{R}_1,\quad R_2 \ge \tilde{R}_2 $$