Hybrid Active/Passive Wireless Network Aided by Intelligent Reflecting Surface: System Modeling and Performance Analysis

Research gap

• Large-scale IRS deployment remains insufficiently characterized for multi-cell networks with randomly located active BSs, passive IRSs, and inter-cell interference.Prior studies mainly optimize link-level systems or assume fixed BS and IRS locations, while the distribution of SINR and spatial throughput in randomly deployed hybrid networks had not been investigated.Source: §I · p. 2

Analytical framework

• The framework derives signal, interference, SINR, coverage, and spatial-throughput distributions under spatially correlated channels with IRS-induced channel hardening.Gamma approximation, interpolation for non-integer shape parameters, and normal approximation for large shape parameters support efficient SINR and throughput characterization.Source: §I · pp. 2–3

System model

• The paper models BS and IRS locations as independent HPPPs and associates each UE with its nearest IRS when it lies within a threshold distance.The model focuses on downlink communication and uses a practical UE-to-IRS association rule for dedicated reflect beamforming.Source: §I · p. 2

Analytical framework

• Mean reflected channel power scales as O(N^2) for associated-IRS reflect beamforming and O(N) for random scattering by non-associated IRSs.Here N denotes the number of reflecting elements per IRS.Source: §I · p. 2

Findings

• IRS deployment significantly increases signal power while causing only marginally increased interference, thereby improving hybrid-network throughput over active-BS-only networks.The improvement is especially pronounced when BS density, network loading, or the number of IRS elements is large.Source: §I · p. 3
• An optimal IRS/BS density ratio maximizes throughput under a total deployment-cost constraint, whereas no IRS or excessively large IRS density is generally suboptimal per unit cost.The optimal ratio increases with the BS/IRS cost ratio and network loading factor.Source: §I · p. 3

Channel model

• The cascaded channel is modeled from independent BS-IRS, IRS-reflected, and IRS-UE components, with IRS phase shifts controlling the reflected paths.For the desired link, customized channel estimation enables phase alignment of the reflected paths at the UE.Source: B. BS-IRS-UE Channel Power Statistics · pp. 4–5
• The analysis assumes Rayleigh fading on the BS-IRS and IRS-UE links and far-field propagation to characterize achievable performance under a worse-case propagation condition.The BS-UE direct channel is also modeled as Rayleigh faded, and the height assumptions avoid unbounded power gain at zero horizontal distance.Source: B. BS-IRS-UE Channel Power Statistics · p. 4

Reflect beamforming

• Reflect beamforming aligns the N reflected paths, enabling a large-N Gaussian approximation for the cascaded channel amplitude.The approximation is reported as accurate for the considered setup when N > 25.Source: B. BS-IRS-UE Channel Power Statistics · p. 5

Channel power scaling

• With reflect beamforming, the average channel power scales as O(N^2), whereas random scattering produces linear O(N) scaling.The two cases correspond to coherent combining for the associated IRS and non-beamformed scattering by other IRSs.Source: B. BS-IRS-UE Channel Power Statistics · p. 5

Random scattering

• Random scattering yields a circularly symmetric complex Gaussian approximation for the combined cascaded channel at practically large N.Its in-phase and quadrature components are approximated as independent normal variables.Source: B. BS-IRS-UE Channel Power Statistics · p. 5

SINR and rate

• The received SINR combines the total signal power, interference power, and normalized receiver noise.The corresponding achievable rate is measured in bps/Hz.Source: C. SINR, Coverage Probability, and… · p. 6

Coverage probability

• Coverage probability averages the non-outage event over channel fading and random BS/IRS locations for a specified target rate.The target rate is converted into a minimum required SINR threshold.Source: C. SINR, Coverage Probability, and… · p. 6

Throughput analysis

• Spatial throughput is obtained from the SINR distribution, which depends on the signal-power and interference-power distributions.The analysis separates signal and interference characterization before deriving SINR and throughput.Source: C. SINR, Coverage Probability, and… · p. 6

Statistical dependence

• Instantaneous signal and interference powers are independent through independent small-scale fading, but their fading-averaged statistics remain dependent through shared BS and IRS locations.The serving-link distance affects both mean signal and mean interference under distance-based association.Source: C. SINR, Coverage Probability, and… · p. 6

Distribution approximation

• The signal-power distribution is more complicated than in conventional networks, so the analysis approximates it conditionally with a moment-matched Gamma distribution.This approximation is introduced because a direct conventional SINR-distribution method cannot be applied.Source: §III · p. 6

Conditional signal statistics

• The conditional signal-power moments depend on BS-UE, BS-IRS, and IRS-UE average channel gains and their associated link distances.The BS-IRS distance is related to the BS-UE and IRS-UE distances through the cosine law.Source: §III · p. 6

Geometric approximation

• Approximating the BS-IRS distance by the BS-UE distance produces closed-form moment approximations when IRSs lie near the UE.The approximation is justified by small IRS-UE distances and is reported to match Monte Carlo results well.Source: §III · p. 6

Signal construction

• The associated IRS co-phases its reflected channel with the direct BS-UE channel, while other IRSs randomly scatter their reflected signals.The overall signal is decomposed into the coherently combined direct-plus-associated-IRS component and the remaining IRS contributions.Source: 1) Gamma Approximation with Moment… · p. 7

Gamma approximation

• The conditional signal-power distribution is approximated by a Gamma distribution matched to its first two moments.The shape and scale parameters are formed from the conditional mean and variance.Source: 1) Gamma Approximation with Moment… · p. 7

Moment calculation

• The first and second conditional moments incorporate direct, coherently reflected, and randomly scattered components before determining the signal-power variance.These moments provide the inputs for the moment-matched Gamma parameters.Source: 1) Gamma Approximation with Moment… · p. 7

Mean signal power

• The conditional mean signal power factors into the direct-link gain and an IRS-location-dependent term.The direct factor depends on the serving BS location, while the IRS factor depends on the associated IRS location.Source: 1) Gamma Approximation with Moment… · p. 7
• The dominant beamforming contribution scales as O(N^2), reflecting coherent combining by the associated IRS.This term is identified as the dominant component in the conditional mean signal power.Source: 1) Gamma Approximation with Moment… · p. 7

• As d0 decreases, the IRS-provided power gain increases substantially, especially when N is large.Source: 2) Impact of IRS 0-UE 0 Distance d0 · p. 7
• Channel hardening strengthens as d0 decreases because kbf scales in O(N), reducing signal-power variation around its mean.The resulting nearly deterministic channel improves reliability with lower outage and reduces the need for frequent channel estimation.Source: 2) Impact of IRS 0-UE 0 Distance d0 · p. 7
• For small d0, increasing N raises mean signal power and strengthens channel hardening, benefiting system performance.Source: 2) Impact of IRS 0-UE 0 Distance d0 · p. 7
• The signal-power distribution is approximated by Gamma laws for reflect beamforming, scattering only, and no nearby IRS, respectively.These approximations match the first and second moments of S conditioned on l0 and d0.Source: 1) Gamma Approximation with Moment… · p. 8
• The distribution is generally discontinuous at the association boundaries D1 and D2, with smaller gaps when the boundary is larger.Source: 1) Gamma Approximation with Moment… · p. 8

• The mean signal power conditioned on l0 and d0 factors into the direct-link gain gd(l0) and an IRS-location-dependent scattering gain κsc(d0).Source: 2) Impact of the Nearest IRS 0 versus… · p. 8
• The nearest IRS 0 typically has the dominant impact on mean signal power because its gain Ngr(d0) exceeds the contribution from farther IRSs at practical densities.Source: 2) Impact of the Nearest IRS 0 versus… · p. 8
• Larger IRS density λI produces smaller average d0 and therefore larger IRS-provided signal-power gain.Source: 2) Impact of the Nearest IRS 0 versus… · p. 8
• The analysis averages conditional mean signal power over l0 and weights the three d0 cases by their occurrence probabilities to obtain the unconditional mean.Source: C. Mean Signal Power · pp. 8–9

• The interference-power characterization begins by conditioning on channel fading and BS/IRS locations, then derives the mean interference power.Source: §IV · p. 9
• Under the adopted approximations, each interfering BS’s composite channel remains CSCG, so its interference power is exponential and total interference follows a generalized Erlang distribution.Source: A. Interference Power Distribution… · p. 9
• IRS contributions to interference decay with IRS-UE distance as O(dj^-α) and become negligible beyond a certain distance.Source: A. Interference Power Distribution… · p. 9
• The analysis replaces η with its conditional mean across the three d0 regimes because the nearest IRS typically dominates and its impact decays with distance.Simulation results verify that this approximation is practically reasonable.Source: B. Conditional Laplace Transform and Cdf · p. 9
• The conditional interference distribution is characterized through its Laplace transform and inverse transformation to obtain the conditional CDF.Source: B. Conditional Laplace Transform and Cdf · p. 10

• The HPPP model converts the infinite interfering-BS sum into a spatial integral over the two-dimensional plane.Source: C. Mean Interference Power · p. 10
• For d0 ≤ D2, mean interference is proportional to κsc(d0)=1+Ngr(d0)+NEI1(d0), the IRS-dependent scattering gain.Source: C. Mean Interference Power · p. 10
• Because Gsc=N generally cannot offset IRS-UE pathloss, direct-link interference dominates and reflected interference is non-negligible only when d0 is sufficiently small.Source: C. Mean Interference Power · p. 10
• The conditional non-outage probability is expressed using the interference Laplace transform and its derivatives, then integrated over l0 and d0 to obtain coverage and spatial throughput.Source: §V · p. 10
• For non-integer kS, the method linearly interpolates between results at the adjacent integer shape parameters.Source: A. Conditional SINR Distribution and… · p. 11
• When kS is large, the Gamma-distributed signal power is approximated by its mean, reducing the need for higher-order Laplace-transform derivatives.Source: A. Conditional SINR Distribution and… · p. 11

• The coverage calculation integrates conditional non-outage probability over the distributions of l0 and d0.Source: B. Coverage Probability and Spatial… · p. 11
• The integration distinguishes reflect beamforming, IRS scattering only, and the absence of a nearby IRS.The corresponding distance regions are evaluated using conditional non-outage probabilities and their occurrence probabilities.Source: B. Coverage Probability and Spatial… · p. 11
• Monte Carlo simulations validate the conditional power distributions, mean powers, coverage probability, and spatial throughput while varying BS/IRS densities, N, and network loading.Each simulation averages over 2000 randomly generated topologies.Source: §VI · p. 11
• Figure 2 plots the conditional signal-power CDF under fixed BS and IRS densities and l0, while varying d0 and N.The plotted setting uses λB = 10λ0, λI = 100λ0, and l0 = 50 m.Source: Figure 2 · p. 12

2) Impact of λI and N on Coverage Probability:

• The associated IRS significantly enhances signal power and produces channel hardening as N increases or the IRS–UE horizontal distance d0 decreases.The conditional interference distribution changes only slightly with IRSs, even for d0 = 1 m and N = 8000, and becomes nearly indistinguishable from the no-IRS case for d0 ≥10 m.Source: 1) Conditional Signal/Interference… · p. 12, Figure 3 · p. 12
• For fixed total IRS-element density Q, smaller N slightly improves low-SINR coverage when Q is small, whereas larger N improves coverage across thresholds when Q is large.At large Q, IRS density already covers most UEs, making the O(N 2) passive beamforming gain more effective.Source: 2) Impact of λI and N on Coverage… · p. 12, Figure 4 · p. 12

3) Mean Signal/Interference Power and Spatial Throughput:

• 2.1 dB (4.0 dB) gain in mean signal (interference) power occurs when BS density rises from 20λ0 to 40λ0 in the BS-only network, so interference grows faster.In the hybrid network, increasing IRS density instead raises mean signal power significantly while increasing mean interference only marginally.Source: 3) Mean Signal/Interference Power and… · p. 12
• Increasing IRS density always enhances spatial throughput at a given BS density, with faster gains under higher BS density or network loading factor p.The reported mechanisms are stronger signal with marginal interference growth, shorter average BS–IRS distance, and more UEs served per IRS.Source: 3) Mean Signal/Interference Power and… · pp. 12–13, Figure 6 · p. 13

B. Spatial Throughput Subject to Total BS/IRS Cost

• Under a fixed total deployment cost, an optimal IRS/BS density ratio ζ∗ maximizes spatial throughput and outperforms both BS-only deployment and excessively large ζ.The optimum is roughly proportional to the BS/IRS cost ratio KN; excessive ζ leaves too few BSs for effective IRS reflection and passive beamforming.Source: B. Spatial Throughput Subject to Total… · pp. 13–14, Figure 7 · p. 13
• With optimized densities, spatial throughput increases with total cost and can grow almost linearly, significantly outperforming the BS-only network.For p = 1 and KN = 5, ζ∗ is approximately 2.5 once BS density exceeds about 40λ0.Source: B. Spatial Throughput Subject to Total… · p. 13, Figure 8 · p. 14, Figure 9 · p. 14

APPENDIX B FIRST AND SECOND MOMENTS OF h1, h2, |h1|2 AND |h2|2

• For h2, zero mean follows because it is a sum of independent CSCG random variables, while the moments of |h1|2 are obtained from the associated-link channel model.The conditional first and second moments of h2 are zero in the cited derivation.Source: Appendix B · p. 14
• The appendix derives conditional moments of |h1|2, |h2|2, and the signal power using integrals over the projected angle and IRS distances.Exact expressions are available but may not be expressible in closed form.Source: Appendix B · pp. 14–15
• The closed-form signal-moment approximations replace interfering-link gains with g_d,0 and use the radial gain expressions g_r(d_j).This approximation is introduced for analytical simplicity.Source: Appendix B · p. 15
• For h2, the reflected component is modeled as a zero-mean CSCG variable, so |h2|2 follows an exponential distribution with mean P_r.Its first and second conditional moments are then used in the subsequent power analysis.Source: Appendix B · p. 15
• The conditional interference-power distribution is characterized through its Laplace transform, with derivatives and expectations used in the derivation.The derivation invokes the probability generating functional of the HPPP.Source: Appendix C · p. 15

APPENDIX D PROOF OF LEMMA 1

• The complementary CDF of the Gamma-distributed signal power is used to establish the result in Lemma 1.The derivation begins from S ∼ Γ[k_S, θ_S].Source: Appendix D · p. 15
• High-order derivatives of exp(V(s)) are evaluated with Faà di Bruno’s formula because the outer function is exponential.This reduces the composite-function derivatives to Bell-polynomial expressions.Source: Appendix E · pp. 15–16
• The i-th complete Bell polynomial B_i(x_1, ..., x_i) supplies fixed coefficients for the derivative expansion.The remaining task is computing derivatives of the inner function V(s).Source: Appendix E · p. 16
• Derivatives of V(s) are obtained from derivatives of U(x), whose higher-order forms use falling factorials and polynomial fractions L_i(x).The polynomial fractions can be generated with symbolic tools such as Mathematica.Source: Appendix E · p. 16