NOMA Downlink: Two-User Power Allocation, Sum Rate and Fairness

For a channel snapshot, let $g_s=|h_s|^2\ge g_w=|h_w|^2$ and let $N$ be the equal noise power integrated over bandwidth $B$. The full-power received SNRs are $\rho_k=P g_k/N$. The selected weak-user fraction is $\alpha$ and the strong-user fraction is $1-\alpha$.

The two-user Shannon model follows Islam et al., Power-Domain NOMA in 5G Systems: Potentials and Challenges, arXiv manuscript pp. 7–9, and the two WiredWhite articles linked below:

For sensitivity to imperfect cancellation, the strong-user SINR is modeled as

Here $\varepsilon$ is a power residual fraction. This extension treats the residual as additional uncorrelated interference; it does not simulate SIC decoding errors.

User spectral efficiencies are $C_k=\log_2(1+\gamma_k)$ in bit/s/Hz. Rates are $R_k=B C_k$ with $B$ in Hz. Because the channels are ordered and the noise powers are equal, $\gamma_{w\to s}\ge\gamma_w$; the strong receiver supports the weak-layer coding rate. The displayed headroom is $C_{w\to s}-C_w$.

Equal-rate allocation solves $\gamma_s=\gamma_w$, which maximizes the smaller rate because one rate falls and the other rises with $\alpha$. With $t=1-\alpha$, define $q=\rho_s^{-1}+\rho_w^{-1}+2\varepsilon$ and $c=\rho_s^{-1}+\varepsilon$. The derived quadratic is $(1-\varepsilon)t^2+qt-c=0$, evaluated stably as

At perfect SIC this agrees with Taricco (2023), eqs. (12)–(16). The residual-SIC form is derived here.

Jain fairness is $J=(C_w+C_s)^2/[2(C_w^2+C_s^2)]$. The OMA reference gives each user half the time at full power and bandwidth: $C_{k,\mathrm{OMA}}=\tfrac12\log_2(1+\rho_k)$. Its average transmit power is the same as NOMA. The rate-tradeoff chart varies OMA time sharing. A short small-SINR series evaluates the logarithm without cancellation at very low SNR.

Chart sampling combines a uniform grid with logarithmically spaced samples near both allocation endpoints and a dense neighborhood of the equal-rate split. The exact equal-rate and selected splits are included. Complementary power fractions are retained separately to preserve precision near an endpoint.

• Two users share one downlink resource with a common effective channel for both superposed layers at each receiver. Gains are linear power ratios, include the chosen channel snapshot, and remain fixed during the calculation.
• Both receivers have equal integrated noise power. The strong user is ordered by channel-to-noise quality, not by received power of its own layer.
• Gaussian signaling, Shannon rate adaptation, perfect channel knowledge and synchronization. No implementation or coding gap is applied.
• Residual SIC is modeled as a fixed fraction of canceled received weak-layer power. The weak layer is decoded before the strong user's own message.
• All fractions from 0 to 1 are allowed to show single-user limits. A practical finite-constellation receiver may impose additional power-ratio and decoding constraints; a weak fraction above 0.5 alone does not prove reliable SIC.
• Bandwidth changes hold the entered total noise power fixed. Update that input if the noise spectral density is held fixed instead.

This is an instantaneous two-user rate and allocation model. It does not calculate a fading average, an outage probability, BER/BLER, finite-blocklength performance, practical MCS selection, antenna patterns, beamforming or multi-cell scheduling. Under residual SIC, the plotted rate tradeoff is the stated interference model, not a claim about the broadcast-channel capacity region.

Equal rates maximize Jain's index and the smaller modeled rate; they do not generally maximize sum rate. NOMA is not guaranteed to exceed the equal-time OMA reference at every allocation or SIC residual. Coding, pilot and control overhead reduce delivered payload throughput.

Reference audit: Abu Mahady (2025), PDF p. 31 (printed p. 11), reports 4.53 bit/s NOMA and 3.38 bit/s OMA at 0/20 dB, which disagree with its own equations. Those totals are not used as expected values. The 10/30 dB example is retained as a published approximate benchmark. Its reported NOMA sum 9.52 bit/s is compatible with truncating the correct 9.525521 bit/s. Its reported OMA sum 6.72 bit/s contains a small arithmetic discrepancy: the equations give 6.713329 bit/s, which rounds or truncates to 6.71. The absolute reference tolerance of 0.01 bit/s explicitly accommodates the OMA discrepancy of 0.006671 bit/s (about 0.10%) and the NOMA reporting precision. The correct formulas are kept. Taricco's Fig. 2 labels the strong SNR as 13 dB and reports a strong-user allocation of 0.1608; exact 13 dB gives about 0.16106. The check allows the figure's SNR rounding.

Source integrity: attachment s4 was attached with an incorrect title; its actual subject is holographic superconductors, and it is excluded from this calculator. The verified NOMA manuscript is attachment s6. Only the author can remove the unrelated attachment.