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On the Performance of NOMA-Based Cooperative Relaying Systems over Rician Fading Channels
Ruicheng Jiao, Linglong Dai, Jiayi Zhang, Richard MacKenzie, Mo Hao
TL;DR
The paper addresses performance evaluation for NOMA-based cooperative relaying over Rician fading channels, where average-rate calculation is difficult. It derives exact achievable-rate expressions and a Gauss-Chebyshev approximation, with analytical results matching Monte Carlo simulations and exceeding traditional CRS achievable rate.
Problem
NOMA-based cooperative relaying over Rician fading channels requires performance analysis despite the integration difficulty caused by Rician-distribution Bessel functions.
Method
The paper derives exact average achievable-rate expressions using Taylor expansion and incomplete Gamma functions, then approximates them with Gauss-Chebyshev Integration.
Results
The analytical results match Monte Carlo simulations, and the NOMA-based CRS achieves a higher achievable sum rate than the traditional CRS.
Takeaways & Limitations
NOMA-based cooperative relaying provides a tractable analytical framework and higher achievable rate than traditional cooperative relaying within the studied Rician-fading setting.
Abstract
from arXiv · showhide
Non-orthogonal multiple access (NOMA) is a promising technique for the fifth generation (5G) wireless communications. As users with good channel conditions can serve as relays to enhance the system performance by using successive interference cancellation (SIC), the integration of NOMA and cooperative relaying has recently attracted increasing interests. In this paper, a NOMA-based cooperative relaying system is studied, and an analytical framework is developed to evaluate its performance. Specifically, the performance of NOMA over Rician fading channels is studied, and the exact expression of the average achievable rate is derived. Moreover, we also propose an approximation method to calculate the achievable rate by using the Gauss-Chebyshev Integration. Numerical results confirm that our derived analytical results match well with the Monte Carlo simulations.
I. INTRODUCTION
The paper motivates combining NOMA with cooperative relaying for 5G and investigates its performance over Rician fading channels. It derives analytical achievable-rate results and validates them against Monte Carlo simulations.
- Motivation: NOMA supports simultaneous transmission over shared time/frequency resources using different power levels and SIC.This can accommodate more users, with increased receiver complexity from SIC.
- Motivation: Strong NOMA users can decode other users’ messages and serve as cooperative relays.The paper connects this capability with cooperative relaying’s potential to enhance cellular-network performance.
- Contribution: The paper investigates a NOMA-based cooperative relaying scheme over Rician fading channels, where Bessel-function densities complicate average-rate integration.It derives exact expressions using Taylor expansion and the incomplete Gamma function, then introduces an approximation method.
- Contribution: Gauss-Chebyshev Integration is proposed to simplify achievable-rate calculation, and analytical results match Monte Carlo simulations.The paper presents this approximation as a response to the computational difficulty of evaluating incomplete Gamma functions.
II. SYSTEM MODEL
The system contains a source, a half-duplex decode-and-forward relay, and a destination with Rician-faded links. Unlike traditional relaying, the NOMA design transmits two symbols across two slots, using SIC and relay forwarding.
- System configuration: The cooperative relaying system comprises a source, a half-duplex decode-and-forward relay, and a destination connected by three available links.The links are S-to-D, S-to-R, and R-to-D, with independent Rician fading coefficients.
- Traditional CRS: The traditional system transmits one symbol from source to relay and destination, then forwards it from relay to destination in the second slot.The destination therefore receives one signal over two time slots.
- NOMA-based CRS: The NOMA-based system transmits a superposition of two data symbols to the relay and destination in the first slot.It can deliver two different signals over two time slots, unlike the traditional configuration.
- Signal model: The symbols have normalized unit power, total transmit power is P_t, and power coefficients satisfy a_1 + a_2 = 1 with a_1 > a_2.The passage also links the allocation ordering to the channel-power relationship.
- Detection and forwarding: The destination decodes s_1 while treating s_2 as noise, whereas the relay uses SIC to acquire s_2.In the second slot, the relay forwards decoded s_2 to the destination with full power P_t, assuming perfect relay decoding.
III. ACHIEVABLE RATE ANALYSIS AND APPROXIMATION
The paper derives exact average achievable-rate expressions for the NOMA-based cooperative relaying system over Rician fading and proposes Gauss-Chebyshev Integration for easier numerical evaluation.
- Rate analysis: Exact and approximated achievable rates are calculated for the NOMA-based cooperative relaying system over Rician fading.The approximation addresses the difficulty of numerically evaluating the exact expressions.
- Approximation: Gauss-Chebyshev Integration is used to simplify numerical calculation of the incomplete Gamma function in the achievable-rate expressions.The method adapts finite-interval integration to the infinite intervals appearing in the exact formulas.
A. Achievable Rate Analysis
The paper derives exact average achievable rates for both NOMA signals over Rician fading by combining link statistics, CDFs, and special-function analysis. The derivation accounts for the minimum-rate constraint across the relevant relay and destination links.
- Rate formulation: The achievable rate of each signal is limited by the lower rate of its two required decoding links.Both the relay and destination must successfully decode the signals, so the rate is determined by the minimum of the corresponding link rates.
- Rate formulation: The analysis models z1=min{λSR, λSD} and z2=min{a2λSR, λRD} before deriving their cumulative distribution functions.These variables represent the bottleneck channel gains for the two signals.
- Rician fading analysis: Rician-channel CDF expressions use an expansion of the incomplete Gamma function and parameters determined by average powers and Rician factors.The paper identifies the link-specific parameters and notes that the incomplete Gamma expansion is used in the CDF derivation.
- Rician fading analysis: The infinite summation in the derived CDF converges because its final value does not change as the summation indices increase.This establishes convergence of the series used in the analytical expressions.
- Exact-rate derivation: Exact achievable-rate expressions for s1 and s2 are obtained through integration involving incomplete Gamma functions and repeated integration by parts.The derivation uses a lemma and repeated integration by parts when β is a root of Jm(x).
- Exact-rate derivation: Although exact expressions are derived, their incomplete Gamma functions make direct numerical evaluation difficult.This computational difficulty motivates the approximation method in the following section.
B. Achievable Rate Approximation
The paper approximates the exact achievable-rate expressions by transforming the infinite integration interval and applying Gauss-Chebyshev Integration. The resulting formulas are intended to make numerical calculation convenient while retaining accuracy for validation.
- Approximation method: Gauss-Chebyshev Integration is introduced to simplify numerical evaluation of incomplete Gamma functions over infinite intervals.Because the standard integration rule applies on [-1, 1], the paper first transforms the original interval.
- Approximation method: The transformed integral is substituted into the exact rate expression to obtain an approximation for the achievable rate of s1.The approximation order is denoted by n.
- Approximation method: The same approximation procedure is applied to the achievable rate of s2.The paper states that the expression for s2 can be approximated similarly.
- Approximation method: The approximated rates in (27) and (29) can be calculated numerically and are subsequently validated by simulation results.The stated purpose is convenient numerical computation followed by accuracy validation.
IV. NUMERICAL RESULTS AND SIMULATIONS
The simulations evaluate achievable rates for the NOMA-based CRS over Rician fading and compare it with traditional cooperative relaying. Analytical and simulation results agree, while NOMA-based CRS provides higher sum rates under the reported settings.
- Simulation setup: 10^5 Rician realizations and Gauss-Chebyshev approximation order 100 are used to validate the analytical results.The numerical study compares analytical results with Monte Carlo simulations.
- NOMA-based CRS over Rician fading: As a2 increases, s2's achievable rate rises while s1's rate decreases.Increasing a2 allocates more power to s2.
- NOMA-based CRS over Rician fading: The Gauss-Chebyshev analytical results match the simulation results for the NOMA-based CRS over Rician fading.Figure 2 reports the achievable rates of s1, s2, and their sum rate.
- NOMA-based CRS over Rician fading: The sum rate first increases and then slowly decreases as a2 increases, implying an optimal power allocation coefficient.The reported behavior reflects the trade-off between the two signals' achievable rates.
- Comparison with traditional CRS: The NOMA-based CRS achieves a higher achievable sum rate than traditional CRS in the reported transmit-SNR comparison.The comparison uses a2 = 0.4, ΩSD = 9, ΩRD = 36, and ΩSR = 144; NOMA transmits two signals in two slots while traditional CRS transmits one.
V. CONCLUSIONS
The paper derives exact achievable-rate expressions for NOMA-based cooperative relaying and proposes a Gauss-Chebyshev approximation. Simulations verify agreement with Monte Carlo results and report higher achievable rates than traditional CRS.
- Exact analytical expressions are derived for the achievable rates of a NOMA-based cooperative relaying system.
- Gauss-Chebyshev Integration provides an efficient approximation whose sum series converges quickly.
- Monte Carlo simulations verify the analytical results, and NOMA-based CRS achieves a higher achievable rate than traditional CRS.