On the Performance Analysis of RIS-Empowered Communications Over Nakagami-m Fading

I. INTRODUCTION

• Optimal phase shifting requires perfect CSI at the source and faces practical phase-shifter resolution constraints, motivating analysis of random phase shifting.The paper notes that these constraints can make optimal phase shifting unsuitable in practical cases.Source: §I · p. 1
• The analysis covers random and optimal phase shifting over Nakagami-m fading, with exact outage and binary-modulation BER expressions and accurate ergodic-capacity approximations.It also derives simpler expressions that become tight for large RIS element counts and analyzes diversity order.Source: §I · p. 2

II. SYSTEM AND CHANNEL MODELS

• The modeled link has a source, an RIS with N reflecting elements, and a destination, with the received signal depending on channel coefficients and RIS element responses.The model also includes additive Gaussian noise.Source: §II · p. 2
• The source-to-RIS, RIS-to-destination, and direct-link channel envelopes are independent Nakagami-m random variables, with average powers specified using distance-dependent path loss.The model also assumes independent channel phases and envelopes; link distances follow the cosine law.Source: §II · p. 2
• Random phase shifting is used when the channel phases are unavailable, whereas optimal phase shifting matches RIS phases to the channel phases under perfect CSI.The optimal design also matches the direct-link phase.Source: §II · p. 2

III. END-TO-END SNR STATISTICS

• Under random phase shifting, the end-to-end SNR statistics require analyzing a sum of N complex random variables whose amplitudes follow a double Nakagami-m distribution.The analysis also uses the moments of those amplitudes.Source: §III · p. 2
• Assuming the random phase φ_n is uniform on [0, 2π] makes the random-phase SNR PDF derivable; the paper reports this approximation yields results nearly indistinguishable from the exact phase distribution.The phase PDF is difficult to derive in closed form because it combines a uniform random variable with two channel-phase random variables.Source: §III · p. 2
• Hankel transforms of the direct-link envelope and cascaded amplitudes are used to obtain the random-phase SNR CDF.The derivation applies the relevant transforms and then obtains the CDF after algebraic operations.Source: §III · p. 2

• For RPS, the large-N model makes |Y|^2 exponentially distributed, yielding an SNR mean proportional to N.The derivation assumes equal fading parameters across RIS elements and approximately equal distances.Source: D. SELIMIS et al.: PERFORMANCE… · p. 3
• For OPS, independence gives a product-form characteristic function, while moments of the resulting SNR can be expressed using component moments.Source: D. SELIMIS et al.: PERFORMANCE… · p. 3
• Under OPS, the large-N SNR grows quadratically with N, and this scaling extends to arbitrary fading when the cascaded-channel transform is available.The derivation uses a central-limit approximation leading to a non-central chi-square model.Source: D. SELIMIS et al.: PERFORMANCE… · p. 3

• Outage probability is the probability that instantaneous SNR falls below the outage threshold, and equals the SNR CDF evaluated there.Source: 1) Outage Probability (OP) · p. 3
• For both RPS and OPS, outage probability follows by evaluating the corresponding SNR CDF; the RPS integral can be computed reliably with standard numerical routines despite its oscillatory integrand.Source: 1) Outage Probability (OP) · p. 3
• For large N without a direct link, the paper recommends using the preceding large-N analysis to calculate outage probability efficiently.Source: 1) Outage Probability (OP) · p. 3

• Average BER for binary modulations is expressed using the SNR CDF, with parameter pairs specified for BPSK, BFSK, and BDPSK.Source: 2) Average Bit Error Rate (BER) · p. 3
• For RPS, substituting the SNR CDF yields a single-integral BER expression, while a closed-form approximation applies at large N without a direct link.With a direct link, BER can instead be evaluated using the CHF-based approach.Source: 2) Average Bit Error Rate (BER) · pp. 3–4
• For OPS, the paper gives a BER expression for coherent binary modulation and uses its CHF-based approach for large-N evaluation.Source: 2) Average Bit Error Rate (BER) · p. 4

• The stated diversity-order analysis assumes the direct link is absent.Source: 3) Diversity Order Analysis · p. 4
• For RPS, the diversity order is one.Source: 3) Diversity Order Analysis · p. 4
• For OPS, the diversity order is that of equal-gain combining under double Nakagami-m fading.Source: 3) Diversity Order Analysis · p. 4

4) Ergodic Capacity (EC):

• The Taylor-series approximation around the first moment of SNR provides an EC approximation for both RPS and OPS.The approximation is reported to be tight across a wide range of system parameters.Source: 4) Ergodic Capacity (EC) · p. 4
• For large N, RPS admits a closed-form EC approximation.The expression is obtained by evaluating the expectation for large N.Source: 4) Ergodic Capacity (EC) · p. 4
• At high SNR, the EC slope is one, while the power offset scales logarithmically with N.The high-SNR approximation is expressed using ln(ρ), ln(NΩgΩh), and Euler's constant.Source: 4) Ergodic Capacity (EC) · p. 4

V. NUMERICAL AND SIMULATION RESULTS

• Unless otherwise stated, simulations use a 2.45 GHz carrier, α = 2.5, and noise power of -85 dBm.The LOS example uses K = 10 and approximates Ricean fading with Nakagami-m fading.Source: §V · p. 4
• The uniform phase approximation is practically indistinguishable from the exact phase distribution; OP evaluations match simulations, and the EC approximation is tight across the SNR region.Large-N and high-SNR analyses are also reported to be accurate for all considered test cases.Source: §V · p. 4
• Without a direct link, OPS significantly outperforms RPS for BPSK BER, even with a small number of RIS elements.The RPS diversity order is independent of N and fading parameters; adding a direct link significantly improves performance, while RPS gains from increasing N are small.Source: §V · p. 4
• At rh ≈ 50 m, EC is minimized across the considered designs; quantized phase shifting performs better than RPS and, at N = 320, is about 0.6 b/s/Hz below OPS.The distance comparison considers the source-to-RIS link with no direct link.Source: §V · p. 4

D. SELIMIS et al.: PERFORMANCE ANALYSIS OF RIS-EMPOWERED COMMUNICATIONS OVER NAKAGAMI-m FADING 5

• Figure 1 plots OP and EC against transmit power P for several N values, with RPS in a LOS environment, K = 10, and no direct link.The caption identifies the system conditions and variables, but does not state a plotted outcome.Source: Figure 1 · p. 5
• Together, the captions cover OP, EC, and BER as P and N vary, with Figure 1 limited to RPS and Figure 2 comparing phase designs.Figure 1 specifies a LOS setting with K = 10; Figure 2 specifies BPSK and Nakagami-m fading.Source: Figure 1 · p. 5, Figure 2 · p. 5
• Figure 2 plots average BPSK BER against P for several N values, comparing RPS and OPS over Nakagami-m fading.The caption specifies mhn = 1.5 and mgn = 2.5 for all RIS elements.Source: Figure 2 · p. 5

VI. CONCLUSIONS

• The random phase scheme has diversity order one, independent of the number of RIS elements and the fading parameters.The conclusion contrasts this behavior with the optimal design.Source: §VI · p. 5
• The analysis presents exact analytical expressions and accurate closed-form approximations for important performance metrics.The systems considered use two phase-shifting designs over Nakagami-m fading.Source: §VI · p. 5
• Numerical and simulated results highlight tradeoffs between performance and complexity.Source: §VI · p. 5