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On the Impact of Phase Shifting Designs on IRS-NOMA

Zhiguo Ding, Robert Schober, H. Vincent Poor

arXiv:2001.10909v1cs.IT

TL;DR

IRS-NOMA phase-shifting design presents a reliability-versus-complexity question: coherent shifting performs well but requires phase/CSI acquisition, while random shifting reduces overhead. The paper develops analytical approximations and bounds, then compares both designs with relaying and IRS-OMA through simulations.

  • Problem

    The paper asks how coherent and random phase shifting affect the performance and complexity of IRS-assisted NOMA.

  • Method

    The paper develops outage analyses using approximations and an upper bound, and evaluates them against simulations and benchmark schemes.

  • Results

    The two phase-shifting designs achieve different tradeoffs between system performance and complexity, and the analytical approximations and upper bound are evaluated for accuracy.

  • Takeaways & Limitations

    Random phase shifting reduces CSI overhead and avoids complicated phase control, while phase selection can significantly improve IRS transmission performance.

Abstract

from arXiv · show

In this letter, the impact of two phase shifting designs, namely random phase shifting and coherent phase shifting, on the performance of intelligent reflecting surface (IRS) assisted non-orthogonal multiple access (NOMA) is studied. Analytical results are developed to show that the two designs achieve different tradeoffs between reliability and complexity. Simulation results are provided to compare IRS-NOMA to conventional relaying and IRS assisted orthogonal multiple access, and also to verify the accuracy of the obtained analytical results.

I. INTRODUCTION

The paper studies IRS-NOMA with coherent and random phase shifting in a cooperative two-user setting, comparing their performance, complexity, and outage behavior with relaying and IRS-OMA.

  • Phase-Shifting Designs: Coherent phase shifting matches each reflecting element’s phase to its incoming and outgoing fading channels, but requires phase information at the source.Finite phase-shifter resolution and CSI acquisition overhead motivate considering random phase shifting.
  • Phase-Shifting Designs: Random phase shifting avoids perfect phase adjustment and reduces the CSI acquisition overhead required at the source.
  • System Model: The cooperative scenario contains one source and two users, with no direct source-to-U1 link because of severe blockage.
  • IRS-NOMA: The source broadcasts a superimposed message so IRS-NOMA serves both users simultaneously.The signals use power allocation coefficients with c1 ≥ c2 and c1^2 + c2^2 = 1.
  • System Model: IRS-NOMA outage analysis depends on the diagonal phase-shifting matrix, whose design creates different performance-complexity tradeoffs.The IRS has N reflecting elements, while the fading vectors and path-loss parameters determine the received signals.
  • IRS-NOMA: For scenarios where the reflecting path is much weaker than the direct link, U2’s outage probability is similar to conventional NOMA without IRS.With d2 = dr, dr2 = 10 m, and α = 4, the reflecting path loss is 10^4 times larger than the direct-link path loss.
  • Benchmark Schemes: The benchmarks include cooperative OMA without IRS and IRS-assisted cooperative OMA, both organized into two transmission phases.The conventional-relaying comparison uses equal source and relay powers, P1 = Pr = P2 = Ps.

3) IRS-OMA:

The IRS-OMA benchmark serves users separately with IRS assistance and uses the same transmit-power reference as IRS-NOMA for fair comparison.

  • IRS-OMA: IRS-OMA assigns P0 to the transmit power of s1 and assumes P0 = Ps for a fair comparison.
  • Performance Analysis: U2’s outage performance in IRS-NOMA is much better than in the benchmarking schemes.The analysis therefore focuses mainly on U1’s outage probability because U2’s analysis resembles the no-IRS case.
  • Comparison Basis: The outage probabilities of IRS-NOMA are evaluated according to the phase-shifting matrix design, with two designs offering different performance-complexity tradeoffs.

A. Coherent Phase Shifting

For coherent phase shifting, the effective channel gain is formed by phase-aligning the IRS-reflected components; its outage analysis uses approximations because the exact density is difficult to obtain.

  • Effective Channel Gain: The effective channel gain ξN sums the IRS-reflected components after applying the reflecting elements’ phase shifts.gi,n and g0,n are the nth fading-vector elements, while θn is the nth reflecting element’s phase shift.
  • Coherent Phase Shifting: Coherent phase shifting matches the IRS phase shifts with the phases of the IRS fading gains.The source is assumed able to acquire the phase of g0,ngi,n before selecting the shifts.
  • Outage Analysis: The outage calculation requires the probability density function of ξN, which is difficult to obtain exactly.The difficulty arises because the density of the component magnitude contains a modified Bessel function.
  • CLT-based Approximation: Because the component magnitudes are i.i.d., the central limit theorem motivates approximating ξN as a Gaussian random variable.The CLT approximation requires the mean and variance of |g0,ngi,n|.
  • CLT-based Approximation: The resulting Gaussian approximation is used to approximate the outage probability of IRS-NOMA.The analysis assumes c1^2 > εc2^2 in the stated operating condition.

2) An upper bound:

The section develops an upper bound on IRS-NOMA outage probability and uses it to characterize achievable diversity. At high SNR, the bound yields a diversity order of 4N, while full diversity N is also achievable.

  • 2) An upper bound:: An upper bound on the IRS-NOMA outage probability is developed under the even-N assumption.The bound is stated using the incomplete Gamma and Gamma functions.
  • 2) An upper bound:: At high SNR, the outage-bound approximation is obtained as ϵ1 → 0 and exponential equality.This approximation leads directly to the stated diversity-order result.
  • 2) An upper bound:: 4N is an achievable diversity order for IRS-NOMA under the derived high-SNR approximation.The result is expressed through the asymptotic outage behavior of the upper bound.
  • 2) An upper bound:: The full diversity gain N is also achievable, established using the inequality ξN ≥ |g0,ngi,n| for each reflecting element.The resulting bound is useful for diversity analysis but is looser than Lemma 1’s bound, especially for large N.

B. Random Phase Shifting

Random phase shifting avoids perfect phase adjustment and lowers CSI overhead, but it does not efficiently exploit IRS spatial degrees of freedom. Its effective channel admits a Gaussian approximation as N grows, and the resulting IRS transmission has diversity order one.

  • B. Random Phase Shifting: Random phase shifting reduces the need for perfect phase adjustment and lowers CSI acquisition overhead at the source.Each reflecting element uses a randomly chosen phase shift.
  • B. Random Phase Shifting: For random phase shifting, ξN is a sum of complex-valued random variables whose real and imaginary parts are correlated, so the CLT is not directly applicable.For small N, ξN is not complex Gaussian distributed.
  • B. Random Phase Shifting: As N →∞, ξN can be approximated as a zero-mean complex Gaussian random variable with variance N.This approximation supports the subsequent outage-probability analysis.
  • B. Random Phase Shifting: IRS transmission with random phase shifting realizes a diversity order of one.The outage probabilities are approximated using the large-N Gaussian model.
  • B. Random Phase Shifting: Random phase shifting cannot effectively use the spatial degrees of freedom, despite its low implementation complexity.Phase shift selection is proposed as a higher-performance complexity tradeoff using Q random phase-shift sets and pilot signals.
  • B. Random Phase Shifting: Simple phase shift selection can significantly improve IRS transmission performance, although dependent effective channel gains complicate analytical treatment.The gains from different random-phase sets are not independent.

IV. NUMERICAL STUDIES

Simulations compare IRS-NOMA under coherent and random phase shifting with conventional relaying and IRS-OMA, while testing analytical approximations. Phase selection substantially improves random-phase IRS transmission, including a 10 dBm power reduction at outage probability 10^-3 for Q = 4.

  • Simulation setup: The simulations evaluate three transmission schemes and assess the accuracy of the developed analytical approximations and upper bound.The setup uses fixed channel, distance, rate, power, and noise parameters, including NOMA target rate R1 = 1.8 BPCU and noise power −70 dBm.
  • Coherent phase shifting: Conventional relaying can outperform both IRS transmission schemes at low SNR because IRS transmission suffers severe path loss.Increasing transmission power or the number of reflecting elements eventually makes IRS schemes outperform conventional relaying.
  • Approximation accuracy: The CLT-based approximation is accurate in the low-SNR regime, whereas the developed upper bound is more accurate in the high-SNR regime.The simulations also verify the approximation based on Lemma 2.
  • Scheme comparison: IRS-NOMA consistently outperforms IRS-OMA across the reported comparisons.This relationship is observed in the coherent-phase and random-phase studies.
  • Random phase shifting: Random phase shifting leaves IRS transmission below conventional relaying, although increasing N reduces the performance gap.The reported reason is that random phase shifting cannot efficiently utilize the spatial degrees of freedom offered by the IRS.

APPENDIX A PROOF FOR LEMMA 1

The proof bounds the amplitude distribution using an upper bound on a Bessel-function-containing density, then derives an upper bound for the sum of independent terms. The result yields an upper-bounded outage expression for IRS-OMA.

  • Density bound: The proof identifies the Bessel function in f_|g0,ngi,n|(yi) as the main source of analytical difficulty.An upper bound on the Bessel function is introduced to simplify the density.
  • Density bound: Replacing the Bessel function with its upper bound produces an upper bound g(y) for the density of each amplitude term.The resulting expression is then used to bound the distribution of the sum.
  • Sum distribution: The proof obtains the Laplace transform of g(y) and uses the i.i.d. property of yi to bound the density of their sum.The sum-density bound is stated after transforming and combining the individual terms.
  • Sum distribution: Assuming N is even and defining N̄ = N/2, the proof derives a further upper bound on the sum density.This even-N assumption is part of the bounding construction.
  • Outage probability: The bounded sum density is inserted into the IRS-OMA outage analysis to obtain an upper-bounded outage probability.The appendix also states the direct IRS-OMA outage-probability expression before presenting its upper bound.

APPENDIX B PROOF FOR PROPOSITION 1

The proof analyzes the single-element effective channel by decomposing complex Gaussian fading into real components and applying unitary transformations. It then derives the CDF and PDF of the real part of the effective gain.

  • Complex-variable decomposition: The complex Gaussian fading variables are decomposed into real and imaginary components that are independent identically distributed Gaussian variables.The proof writes g0,n and gi,n using real variables an, bn, cn, and dn.
  • Complex-variable decomposition: For one reflecting element, ξ1 is expressed through rotated combinations of products of the real Gaussian components and the random phase θn.The real and imaginary parts use complementary sine and cosine combinations.
  • Distribution analysis: The real and imaginary parts of ξ1 are identically distributed, so the proof focuses on Re{ξ1}.A unitary transformation preserves the Gaussian statistical properties of the transformed variables.
  • Distribution analysis: The transformed linear combinations remain Gaussian, allowing the CDF of Re{ξ1} to be simplified and differentiated into the proposition’s PDF.The proof uses Gaussian-sum reasoning and a change involving an exponentially distributed quantity.

APPENDIX C PROOF FOR LEMMA 2

The proof approximates the effective channel gain for random phase shifting through characteristic functions rather than directly invoking the CLT. It establishes Gaussian behavior for the real part and joint Gaussian structure for the complex gain.

  • Characteristic-function approximation: The proof represents Re{ξN} as a sum of independent real parts of randomly phase-rotated channel products.Independence holds across reflecting elements with distinct indices.
  • Characteristic-function approximation: The characteristic function of Re{ξN} is obtained by multiplying the single-element characteristic functions.The approximation follows from the limiting form of the exponential function.
  • Gaussian approximation: As N grows, the characteristic function approaches e^(-Nt^2/4), the characteristic function of a Gaussian random variable.This establishes the Gaussian approximation for Re{ξN} without directly applying the CLT.
  • Joint distribution: The proof constructs arbitrary linear combinations of Re{ξN} and Im{ξN} to establish that they are jointly Gaussian distributed.The argument uses the Gaussianity of the linear combinations formed from transformed component variables.
  • Joint distribution: Because the real and imaginary parts are identically distributed and uncorrelated, the proof establishes their independence.The independence conclusion follows after proving joint Gaussianity and zero correlation.
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