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Exact Outage Probability and Ergodic Capacity Analysis of NOMA in Rayleigh Fading Channels

This paper derives exact closed-form expressions for the outage probability and ergodic capacity of the near user in two-user downlink NOMA over Rayleigh fading channels by accounting for the previously overlooked dependencies between noise and fading after successive interference cancellation, thereby revealing the significant inaccuracies of legacy Gaussian and residual-interference-factor models.

Original authors: Arafat Al-Dweik, Alok Kumar Shukla, Sami Muhaidat

Published 2026-04-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Arafat Al-Dweik, Alok Kumar Shukla, Sami Muhaidat

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Super-Listener" Problem

Imagine a radio station (the Base Station) broadcasting a message to two listeners: Alice (the "Near User" who lives right next to the tower) and Bob (the "Far User" who lives far away).

To save time and money, the station sends both messages at the same time, mixed together like a smoothie.

Bob's message is loud and heavy (high power) because he is far away and needs a strong signal to hear it.
Alice's message is quiet and light (low power) because she is close and doesn't need much volume.

The Challenge: Alice needs to hear her quiet message, but it's drowned out by Bob's loud message. To fix this, Alice uses a special trick called SIC (Successive Interference Cancellation).

The Trick:

Alice listens to the mix.
She focuses on the loud part (Bob's message) and tries to decode it first.
Once she understands Bob's message, she subtracts it from the mix, like removing the strawberry flavor from a smoothie to get to the banana.
Now, only her quiet message remains, and she can hear it clearly.

The Problem: When the Trick Fails

The paper argues that most engineers assume Alice's trick works perfectly every time. They assume that if she decodes Bob's message, she removes it 100%.

But in the real world, things go wrong. Sometimes, the signal is weak, or there is static (noise), and Alice guesses Bob's message wrong.

The Old Way: Engineers used a "fudge factor." They said, "Okay, if she gets it wrong, let's just assume a little bit of Bob's message is still there, maybe 10% or 20%." They treated this "leftover mess" as a random, independent variable they could just tweak.
The Reality: The paper shows that this "fudge factor" is a lie. When Alice makes a mistake, the leftover noise isn't random. It changes the very nature of the signal and the static around it. It's not just a little bit of Bob's voice; it's a distorted, chaotic version of it that depends entirely on how the mistake happened.

The Paper's Discovery: The "Dependent Duo"

The authors (Al-Dweik, Shukla, and Muhaidat) did the math to show something surprising: After Alice tries to cancel the interference, the "Noise" and the "Signal" become best friends who can't be separated.

Before the trick: The noise (static) and the signal (fading) are strangers. They don't care about each other.
After the trick: If Alice makes a mistake, the noise and the signal become dependent. The amount of static left over depends on how strong the signal was, and vice versa.

The Analogy:
Imagine you are trying to clean a muddy puddle (the signal) by scooping out the mud (the interference).

Old Model: We assumed that if you scoop out the mud, the water left behind is just "clean water" with a little bit of "mud residue" added randomly.
New Model: The authors show that if you scoop wrong, the water you are left with isn't just "water + mud." The act of scooping changes the water's temperature, color, and flow. The water and the mud are now chemically bonded in a specific way that you can't predict unless you know exactly how the scoop went wrong.

Why Does This Matter?

The paper proves that the old way of calculating performance (using a simple "imperfection factor") is dangerously inaccurate, especially when:

The signal is weak (Low-to-moderate SNR).
The power balance is uneven (One user gets way more power than the other).

The Consequences of the Old Math:

False Confidence: Engineers might think a system works great when it actually fails often.
Wasted Resources: They might allocate power inefficiently because their math says one thing, but reality says another.

The Solution: A New, Exact Map

The authors created a new mathematical map (formulas) that accounts for this "Dependent Duo."

They calculated the exact probability of failure (Outage Probability).
They calculated the exact amount of data that can be sent (Ergodic Capacity).

The Result:
Their new formulas don't need any "fudge factors." They are "parameter-free," meaning they work based on the actual physics of the situation, not on guesses.

Key Takeaways for the General Public

Perfection is a Myth: In wireless networks, assuming you can perfectly cancel out interference is like assuming you can perfectly un-mix a smoothie. It's impossible.
Mistakes Change the Game: When a system makes a mistake (cancels the wrong signal), it doesn't just leave a little bit of noise; it fundamentally changes the environment the receiver is operating in.
Stop Guessing: The old way of estimating how bad a mistake is (using a number between 0 and 1) is wrong. That number actually changes depending on the weather (signal strength) and the volume settings (power allocation).
Better Design: By using the new math from this paper, engineers can design 5G and 6G networks that are more reliable and efficient, ensuring that your video call doesn't drop just because the math used to design the tower was slightly off.

In short: This paper fixes the blueprint for how we build future wireless networks by admitting that when we try to cancel out noise, we often make a mess, and that mess is more complicated than we thought. They provided the exact recipe to handle that mess.

1. Problem Statement

Non-Orthogonal Multiple Access (NOMA) is a key technology for 5G and beyond, allowing multiple users to share the same time-frequency resources. In downlink NOMA, the "Near User" (NU) employs Successive Interference Cancellation (SIC) to decode and remove the signal of the "Far User" (FU) before decoding its own.

The Core Issue:
Existing analytical models for NOMA performance (Outage Probability - OP, and Ergodic Capacity - EC) rely on simplified assumptions regarding imperfect SIC:

Legacy Models: Assume that if SIC fails, a fixed fraction of the FU signal remains as "residual interference." This is modeled using a tunable imperfection factor ( $\eta$ or $\zeta$ ) bounded between [0, 1].
The Flaw: These models treat the noise and channel fading as independent of the SIC outcome. They assume the noise remains Gaussian and the fading remains Rayleigh regardless of whether SIC succeeded or failed.
Consequence: This independence assumption is mathematically incorrect. When SIC fails, the detection error creates a specific correlation between the noise and the fading, altering their statistical distributions. Legacy models often misestimate performance, particularly at low-to-moderate Signal-to-Noise Ratios (SNRs) and under unbalanced power allocation.

2. Methodology

The authors propose a rigorous analytical framework that explicitly accounts for the statistical dependence between noise and fading induced by the SIC process.

System Model: A two-user downlink NOMA system over Rayleigh fading channels using BPSK modulation. The Base Station transmits a superposition of signals.
Conditional Analysis: Instead of assuming a fixed residual interference factor, the authors condition the analysis on the SIC outcome (Success $S$ $S$ or Failure $F$ $F$ ).
- Successful SIC: The interference is removed, but the noise distribution is truncated because the detection threshold was met.
- Failed SIC: The interference is not removed (and effectively doubled due to sign flipping in BPSK), and the noise distribution is heavily skewed.
Derivation of Joint PDFs:
- The paper derives the joint Probability Density Functions (PDFs) of the post-SIC noise and fading ( $\beta$ ).
- It proves that after SIC, the noise and fading become dependent random variables.
- From the joint PDFs, the authors derive the marginal PDFs for the fading coefficient ( $\beta_S, \beta_F$ ) and the noise ( $W, Z$ ) for both success and failure cases.
Performance Metrics:
- Outage Probability (OP): Derived by integrating the marginal PDFs over the region where the instantaneous SINR falls below a target threshold.
- Ergodic Capacity (EC): Derived using the expectation of the log-SINR. The authors provide both an exact single-integral expression and a highly accurate closed-form approximation using Chiani's approximation for the Q-function.
Extension to QPSK: The methodology is extended to Quadrature Phase Shift Keying (QPSK), noting that for 2D modulations, the real and imaginary components of the noise also become dependent after SIC.

3. Key Contributions

Exact Statistical Characterization: The paper is the first (to the authors' knowledge) to characterize the exact joint and marginal PDFs of noise and fading after SIC in Rayleigh fading, proving they are dependent and non-Gaussian/non-Rayleigh in the failure case.
Parameter-Free Formulas: The derived exact expressions for OP and EC are parameter-free. Unlike legacy models, they do not require an arbitrary "imperfect SIC factor" ( $\zeta$ ) to be tuned or guessed.
Debunking the Legacy Factor: The authors prove that the residual interference factor $\zeta$ is not an independent variable. It depends on the SNR, power allocation ( $\alpha_1, \alpha_2$ ), and the target rate. Furthermore, the range $[0, 1]$ is insufficient; for certain conditions, the effective residual interference can mathematically exceed the bounds assumed in legacy models.
Accurate Approximations: A closed-form approximation for Ergodic Capacity is derived that matches the exact numerical integration with negligible error (average error < 1.11%), whereas legacy formulas show errors up to 10.87%.
Insight on QPSK: The analysis reveals that for 2D modulations (QPSK), the real and imaginary noise components become dependent post-SIC, a factor ignored in previous literature.

4. Results and Validation

Validation: Monte Carlo simulations with $10^7$ runs confirm that the derived analytical expressions match the simulation results perfectly across all SNR ranges.
Comparison with Legacy Models:
- Outage Probability: Legacy models significantly underestimate the outage probability at low-to-moderate SNRs, especially when power allocation is unbalanced. The legacy model fails to capture the "divergence region" where the system performance degrades faster than predicted.
- Ergodic Capacity: Using $\zeta = 0$ (perfect SIC assumption) in legacy models actually provides a closer approximation to the exact capacity than using non-zero $\zeta$ values, because the legacy model's structure inherently misrepresents the noise statistics when $\zeta > 0$ .
- SIC Factor Behavior: The study shows that the required $\zeta$ to match the exact model varies dynamically with system parameters. For example, at high SNRs, $\zeta$ approaches 0, but at low SNRs, it can exceed 1 (violating the standard [0,1] bound), proving the inadequacy of the fixed-factor approach.
Optimal Power Allocation: The exact analysis reveals a concave-up relationship between power allocation ( $\alpha_1$ ) and outage probability, identifying specific optimal points that legacy monotonic models miss.

5. Significance

This work fundamentally shifts the understanding of NOMA performance analysis under imperfect SIC:

Accuracy: It eliminates the need for heuristic tuning of the SIC imperfection factor, providing a mathematically exact tool for system design and optimization.
Reliability: It highlights that legacy models can lead to over-optimistic reliability estimates, which is critical for mission-critical 5G applications.
Design Guidance: The findings suggest that for capacity analysis, assuming perfect SIC ( $\zeta=0$ ) in legacy formulas is often more accurate than trying to model imperfect SIC with a fixed factor. For outage analysis, the exact derived formulas must be used, particularly in low-SNR regimes.
Future Impact: The framework provides a foundation for analyzing more complex scenarios (multi-antenna, multi-user, higher-order modulations) where the statistical dependence of noise and fading is even more critical.

In summary, the paper replaces heuristic, parameter-dependent models with a rigorous, exact statistical framework, demonstrating that the "residual interference" in NOMA is a complex, state-dependent phenomenon rather than a simple additive noise term.