Analysis and Optimization of Outage Probability in Multi-Intelligent Reflecting Surface-Assisted Systems

• Multi-IRS-assisted systems remain insufficiently understood because prior work considered only one IRS, leaving joint phase adjustment and scaling with IRS count unresolved.Source: §I · p. 2
• The study considers a single-antenna source and destination assisted by multiple multi-element IRSs under Rician fading, with phase shifts adapting only to LoS components and slow NLoS fading.Source: §I · p. 2
• It obtains the outage-probability expression and optimizes all IRS phase shifts to minimize outage probability.Source: §I · p. 2
• Optimal outage probability decreases with the numbers of IRSs and elements per IRS when LoS components are stronger than NLoS components.Source: §I · p. 2
• The high-SNR asymptotically optimal outage probability decreases with the powers of the LoS components, providing design insights for practical multi-IRS systems.Source: §I · p. 2

• A single-antenna source serves a single-antenna destination with help from K multi-element IRSs, whose elements assist transmission through adjustable phase shifts.The k-th IRS contains N_k elements; the model illustration uses K = 3.Source: §II · p. 3, Figure 1 · p. 3
• The direct source-destination channel and source-to-IRS-to-destination links follow Rician fading with LoS and NLoS components, while inter-IRS channels are ignored.The inter-IRS channel is neglected when IRSs are aligned or sufficiently far apart.Source: §II · pp. 3–4
• The NLoS components vary slowly and only their distributions, together with fixed channel parameters, are known at the source; phase shifts do not track instantaneous NLoS realizations.Source: §II · p. 4
• The received signal is modeled using transmit power, an equivalent channel, and complex Gaussian noise, from which channel capacity is defined.Source: §II · p. 4
• Because capacity is random under unknown NLoS realizations, outage probability is defined as the probability that capacity falls below a target rate R.Source: §II · p. 5

Analysis

• The outage probability is derived analytically for the multi-IRS-assisted system under slow NLoS fading.The equivalent channel separates LoS and NLoS power contributions, with the NLoS term following a non-central chi-squared distribution.Source: §III · p. 5
• When the reflected-link path-loss factors increase by the same factor, the outage probability decreases.Source: §III · p. 6

Numerical Validation

• Analytical outage-probability curves closely match Monte Carlo results, verifying the derived expression.The comparison is made in Fig. 2 for varying phase shifts and transmit SNR.Source: §III · p. 6, Figure 2 · p. 6
• Outage probability decreases with transmit SNR and changes significantly with the phase shifts θ.This sensitivity motivates optimizing θ to minimize outage probability.Source: §III · p. 6

Optimization Formulation

• The optimization reduces the outage-probability problem to maximizing the LoS power of the equivalent channel under the phase-alignment structure.The resulting LoS term represents the maximum achievable LoS-component power.Source: §IV · pp. 6–7

Optimal Phase Shifts

• Theorem 2 gives optimal phase shifts that align reflected LoS components with the direct-link LoS component when the direct link has LoS.When the direct link lacks LoS, reflected LoS components should instead be mutually aligned.Source: §IV · pp. 6–7

Impact of IRS Deployment

• The optimized outage probability decreases with the number of IRSs and the number of elements per IRS when each relevant LoS component is stronger than its average NLoS counterpart.This condition is the basis of Lemma 1 and is illustrated in Fig. 3.Source: §IV · pp. 7–8, Figure 3 · p. 8
• The multi-IRS system has much lower optimal outage probability than the system without IRSs, especially for large K and N_k.Source: §IV · p. 8

High-SNR Characterization

• The asymptotically optimal outage probability is characterized as a high-SNR approximation to the optimized outage probability.The approximation is evaluated as a function of SNR under fixed system parameters.Source: §V · p. 8

Numerical Validation

• The asymptotic expression approaches the exact optimized outage probability as SNR increases.Fig. 4 compares the analytical and asymptotic curves.Source: Figure 4 · p. 9

Asymptotic Scaling

• The optimized and asymptotically optimized outage probabilities are inversely proportional to SNR in the high-SNR regime.Source: SNR ) · p. 9

Scope

• The paper notes that some derivation material is omitted due to page limitations.Source: SNR ) · p. 9

Asymptotic Properties

• The asymptotically optimal outage probability decreases with the power of each LoS component.This dependence is illustrated in Fig. 4(b).Source: SNR ) · p. 9

Comparison Without IRSs

• The asymptotic outage probability is much smaller than that of the system without IRSs, especially at large κ(r,d).Source: SNR ) · p. 9

• The analysis assumes a multi-IRS-assisted system under Rician fading with phase shifts adapting only to LoS components.Source: §VI · p. 10
• The study derives the outage probability as a function of all IRS phase shifts and obtains closed-form phase shifts that minimize it.Source: §VI · p. 10
• It characterizes the asymptotically optimal outage probability in the high-SNR regime.Source: §VI · p. 10
• When LoS components are stronger than NLoS components, optimal outage probability decreases as the numbers of IRSs and elements per IRS increase.Source: §VI · p. 10
• The paper characterizes the impact of each LoS component's power on optimal outage probability.Source: §VI · p. 10