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Analysis of Uplink IRS-Assisted NOMA under Nakagami-m Fading via Moments Matching
Bashar Tahir, Stefan Schwarz, Markus Rupp
TL;DR
Prior IRS-NOMA analyses often used restricted direct/reflection-link arrangements or Rayleigh fading, despite the relevance of LOS propagation. This paper models direct and reflected links under Nakagami-m fading and uses Gamma moment matching for tractable outage analysis; in its example, the NLOS user benefits from IRS deployment, while the LOS-dominated user can see degraded performance under specified conditions.
Problem
Prior IRS-NOMA studies often assumed restricted direct and reflected links or Rayleigh fading, although IRS deployments and user pairing can involve strong LOS links.
Source: §I · p. 1Method
The paper models both users' direct and reflected links with Nakagami-m fading and approximates their received powers as Gamma random variables via moments matching for tractable outage analysis under IC.
Source: §I · p. 1, §III · p. 2Results
The NLOS-dominated user always benefits from IRS deployment, whereas at low outage thresholds the LOS-dominated user might fare worse when the IRS boosts the other user, especially at low SNR.
Source: §VI · p. 5Takeaways & Limitations
In the analyzed scenario, IRS assistance is not uniformly beneficial: its configuration matters for the LOS-dominated user's outage, while the NLOS-dominated user benefits regardless of configuration.
Source: §V · p. 4, §VI · p. 5
Abstract
from arXiv · showhide
This letter investigates the uplink outage performance of intelligent reflecting surface (IRS)-assisted non-orthogonal multiple access (NOMA). We consider the general case where all users have both direct and reflection links, and all links undergo Nakagami-m fading. We approximate the received powers of the NOMA users as Gamma random variables via moments matching. This allows for tractable expressions of the outage under interference cancellation (IC), while being flexible in modeling various propagation environments. Our analysis shows that under certain conditions, the presence of an IRS might degrade the performance of users that have dominant line-of-sight (LOS) to the base station (BS), while users dominated by non-line-of-sight (NLOS) will always benefit from it.
I. INTRODUCTION
Prior IRS-NOMA studies reported promising outage and sum-rate gains, but often relied on restricted link configurations or Rayleigh fading. This work considers a two-user uplink with direct and reflected links for both users and Nakagami-m fading.
I. INTRODUCTION
- Earlier IRS-NOMA studies reported outage and sum-rate gains, but commonly assumed one user used only the IRS link and the other only the direct link, or used Rayleigh fading.The introduction notes that strong LOS links can occur in IRS deployments and user pairing.Source: §I · p. 1
- The paper studies a general two-UE IRS-NOMA uplink in which both users have direct and reflected links, with Nakagami-m fading on every link.Source: §I · p. 1
- Adjustable Nakagami-m fading parameters represent LOS and NLOS conditions, while moments-matched Gamma received powers support tractable outage analysis under interference cancellation.The analysis considers configuring the IRS to boost either UE and reports that IRS presence can degrade outage performance for a UE with dominant BS LOS under certain conditions.Source: §I · p. 1
II. SYSTEM MODEL
The system model combines direct and IRS-reflected links from two single-antenna users at a single-antenna base station. Nakagami-m fading models the links, and IRS phase shifts are selected to boost either user's received power.
II. SYSTEM MODEL
- The uplink has two single-antenna UEs and an N-element IRS, with both direct and reflected signals contributing to reception at a single-antenna BS.The model identifies fading coefficients and pathlosses for the UE–BS, BS–IRS, and UE–IRS links.Source: §II · p. 1
- All links are modeled with Nakagami-m fading, whose parameters describe the corresponding link distributions.The passage notes that Nakagami-m has Gamma-distributed power, making some results exact under certain conditions.Source: §II · p. 1
- The IRS phase shifts are set to maximize the received power of whichever UE is selected for boosting.The maximizing choice is justified using the triangular inequality on the received amplitude.Source: §II · pp. 1–2
- The paper introduces moments matching for approximating a nonnegative random variable using a Gamma random variable with matching first and second moments.The Gamma distribution's scaling property is also stated.Source: §II · p. 2
III. STATISTICS OF THE RECEIVED POWER
The section frames outage probability as the target quantity for each UE, both with the other UE's interference and after interference cancellation. Because the effective channel-power distributions are difficult to characterize exactly, it uses Gamma approximations obtained by moments matching.
III. STATISTICS OF THE RECEIVED POWER
- The section targets each UE's outage probability when the other UE's signal interferes, and also defines an SNR outage when interference is removed by IC.Source: §III · p. 2
- Exact effective channel-power distributions are difficult to characterize, so the received powers are approximated as Gamma random variables via moments matching for tractable outage evaluation.The Gamma distribution encompasses many power distributions as special cases.Source: §III · p. 2
- For analysis, the IRS is assumed to boost UE1, giving it coherent combining while UE2 has random combining under the assumption that the users' channels are uncorrelated.The alternative IRS configuration is obtained by switching the user indices.Source: §III · p. 2
A. Statistics of the Coherently Combined UE
For the coherently combined UE, the IRS-reflected signal is approximated using moments of a Gamma distribution. The mean received power grows quadratically with the number of IRS elements, while the composite pathloss sets a condition for realizing that gain.
A. Statistics of the Coherently Combined UE
- The positive, in-phase reflected components are approximated by a Gamma random variable using the central limit theorem and moment matching.The approximation uses the first two moments of the sum under unit-power i.i.d. conditions.Source: A. Statistics of the Coherently… · p. 2
- The Gamma approximation improves as N grows and fits the density well even with only four IRS elements.Figure 1 shows the density in log-scale for N = 4.Source: A. Statistics of the Coherently… · p. 2, Figure 1 · p. 2
- Coherent combining yields a quadratic increase in UE1 mean received power with N, but N must be large enough for the IRS to overcome composite pathloss.The reflected term contains the composite pathloss of the BS–IRS and IRS–UE1 channels.Source: A. Statistics of the Coherently… · p. 3
- After scaling by P1, UE1's received power is modeled as a Gamma random variable with parameters matched to its moments.The first two channel-power moments are given under independence.Source: A. Statistics of the Coherently… · p. 3
B. Statistics of the Randomly Combined UE
For the randomly combined UE, the IRS components add out of phase, so their sum is approximated by a complex Gaussian and the received power by a moment-matched Gamma random variable. The mean received power increases linearly with N, and the magnitude fit is less accurate in the tails than the coherent-case Gamma approximation.
B. Statistics of the Randomly Combined UE
- Assuming uncorrelated user channels makes UE2's combining random, and its channel-power moments are derived under independence.The derivation uses the fact that Nakagami fading becomes Rayleigh for m = 1.Source: B. Statistics of the Randomly Combined… · p. 3
- Because the reflected coefficients combine out of phase, their sum is approximated by a complex Gaussian using the central limit theorem.The resulting magnitude is compared with a Rayleigh fit.Source: B. Statistics of the Randomly Combined… · p. 3
- For N = 4, the Rayleigh fit is good overall but less accurate in the tails than the coherent-case Gamma approximation.Figure 2 compares the magnitude of the sum with the Rayleigh fit.Source: B. Statistics of the Randomly Combined… · p. 3, Figure 2 · p. 3
- Random combining gives UE2 a linear increase in mean received power with N, unlike the quadratic increase under coherent combining.The UE2 received power is modeled as a Gamma random variable with parameters matched to its moments.Source: B. Statistics of the Randomly Combined… · p. 4
IV. INTERFERENCE CANCELLATION OUTAGE ANALYSIS
The outage analysis approximates received powers and interference-plus-noise with Gamma random variables to obtain tractable outage expressions. For interference cancellation, it considers parallel detection and approximates success through two detection paths, with a bound from the noise-only case.
IV. INTERFERENCE CANCELLATION OUTAGE ANALYSIS
- Without IC, the analysis gives an outage expression for Gamma-modeled UE powers, using the CDF of a beta-prime ratio and a moment-matched Gamma approximation for interference plus noise.The noise term prevents the denominator from being Gamma directly.Source: §IV · p. 4
- The parallel detection scheme allows either or both UEs to be detected first, then detects any remaining UE after IC without requiring an ordered cancellation sequence.This avoids the ordered statistics required under successive IC.Source: §IV · p. 4
- Under IC, success is approximated through either initial correct detection or correct detection of the other UE followed by interference-free detection; the result is capped by the noise-only case.The detection events are not fully independent, motivating the approximation.Source: §IV · p. 4
V. ANALYSIS OF AN EXAMPLE SCENARIO
The example compares two IRS configurations with a NOMA-only baseline at low and high SNR. The NLOS-dominated UE benefits across configurations, while the LOS-dominated UE can be harmed at low outage thresholds but performs better at sufficiently high thresholds.
V. ANALYSIS OF AN EXAMPLE SCENARIO
- The example uses a 32-element IRS, with UE1 received 10 dB above UE2 on the direct links; UE1 and the IRS have good BS LOS, while UE2 is near-Rayleigh.Both users have moderate LOS to the IRS.Source: §V · p. 4
- UE2 always benefits from IRS assistance, while UE1 can suffer at low outage thresholds—especially at low SNR—yet performs better at sufficiently high thresholds.The comparison covers both IRS configurations and the NOMA-only baseline at 20 dBm and 35 dBm.Source: §V · pp. 4–5, Figure 3 · p. 5, Figure 4 · p. 5
- Random IRS combining for UE1 adds components around its dominant LOS, widening the fading tail and flattening the outage slope.The analysis attributes this behavior to increased spread in the moment-matched Gamma distribution when a reflection path is added.Source: §V · p. 5
- Splitting the IRS surface between the users is proposed as a way to guarantee each user's performance and potentially reduce LOS-related degradation.The authors suggest this as an alternative to configuring the entire IRS to boost one UE.Source: §V · p. 5
VI. CONCLUSIONS
The letter models a general IRS-assisted NOMA uplink with direct and reflected links under Nakagami-m fading. Its example analysis finds consistent IRS benefits for the NLOS-dominated user, but possible low-threshold degradation for a LOS-dominated user.
VI. CONCLUSIONS
- Second-order moment matching approximates received user powers as Gamma random variables, enabling tractable outage expressions under interference cancellation.The approach accommodates direct and reflection links for all users under Nakagami-m fading.Source: §VI · p. 5
- The IRS always improves performance for the NLOS-dominated UE, regardless of IRS configuration.Source: §VI · p. 5
- At low outage thresholds, the IRS might degrade the LOS-dominated UE when configured to boost the other UE, especially at low SNR.The analysis is verified by simulations.Source: §VI · p. 5