Low-Complexity Random Rotation-based Schemes for Intelligent Reflecting Surfaces

I. INTRODUCTION

• IRSs use reconfigurable passive elements to control incident-signal phases, enabling programmable reflections that can extend communication range and improve spectral and energy efficiency.Source: §I · p. 1
• Existing IRS approaches mainly optimize phase shifts with CSI, increasing complexity and power consumption while channel estimation can be impractical because of limited IRS resources.Source: §I · p. 2
• Four low-complexity schemes are developed through coding-based and selection-based approaches using random phase rotations with no or limited CSI requirements.Source: §I · p. 2
• The analytical framework characterizes outage probability, energy efficiency, diversity order, and coding gain, and compares the proposed schemes with coherent beamforming under perfect or imperfect CSI.Source: §I · p. 2, §II · p. 3
• The coding-based schemes generate time-varying channels through random rotations, while selection-based schemes activate IRS partitions according to received signal power at the destination.Source: §I · p. 2

II. SYSTEM MODEL

• The system model assumes a single-antenna source and destination, Rayleigh fading, no instantaneous source CSI, and random IRS phase shifts over a coherence period of T channel uses.Source: §II · pp. 2–3

III. CODING-BASED IRS SCHEMES

• Both coding-based schemes are independent of a particular code design because performance is evaluated through outage probability rather than error probability.Source: §III · p. 3, A. Random Rotations Coding-based Scheme · p. 3

A. Random Rotations Coding-based Scheme

• Random phase shifts create artificial fast fading across T transmissions, allowing receiver combining such as MRC to convert spatial diversity into time diversity without source CSI.Source: A. Random Rotations Coding-based Scheme · p. 3
• The RRC outage analysis uses an independence approximation and a CLT-based exponential approximation for channel gains, with the latter yielding simpler expressions for diversity and coding-gain analysis.Source: A. Random Rotations Coding-based Scheme · p. 4
• The two RRC approximations can describe small-M behavior well, while the more accurate and simpler formulations converge asymptotically as M increases.Source: A. Random Rotations Coding-based Scheme · p. 4
• RRC achieves an expected rate independent of T, and its energy efficiency accounts for amplifier efficiency and static source, destination, and IRS power consumption.Source: A. Random Rotations Coding-based Scheme · p. 4

B. One-bit Feedback Scheme

• One-bit feedback retains phase updates only when they increase the received SNR, then fixes the best phases for the remaining channel uses.The scheme uses a distributed ascent algorithm requiring only one-bit feedback during a training period τ ≤ T.Source: B. One-bit Feedback Scheme · pp. 4–5
• The algorithm updates each element with a uniformly random phase perturbation bounded by the maximum step size ∆.At each training instant, the controller accepts or rejects the perturbed phases according to the destination's feedback.Source: B. One-bit Feedback Scheme · p. 5
• The outage probability is obtained analytically from the channel-gain distribution after the random phase-update process.Source: B. One-bit Feedback Scheme · p. 5
• For sufficiently large τ and T, the algorithm converges to the beamforming gain regardless of ∆, but with fixed τ the maximum gain depends strongly on ∆.This dependence is illustrated by expected-SNR curves versus training duration in Fig. 2.Source: B. One-bit Feedback Scheme · p. 5, Figure 2 · p. 5
• The scheme's energy-efficiency analysis assumes negligible one-bit feedback power and approximates the effective feedback at time t as b = κt.The constant κ ∈ (0, 1] captures the fraction of feedback bits that effectively improve performance.Source: B. One-bit Feedback Scheme · p. 5

C. Diversity Analysis

• RRC achieves diversity order min(T, M) because it virtually behaves as a T × M Rayleigh product channel.The independent random phase rotations create the equivalent diversity mechanism across channel uses.Source: C. Diversity Analysis · p. 6
• OBF achieves diversity order min(τ, M), since it employs RRC during its τ-channel-use training period.Source: C. Diversity Analysis · p. 6

IV. SELECTION-BASED IRS SCHEMES

• Selection-based schemes partition the IRS into N non-overlapping sub-surfaces and use training feedback to identify a suitable active partition.Each sub-surface is activated sequentially during training, after which the destination feeds back its selected index.Source: §IV · p. 6

A. Transmit Diversity Scheme

• TD selects the sub-surface with the highest destination SNR and requires ⌈log2(N)⌉ feedback bits.Source: A. Transmit Diversity Scheme · p. 6
• The TD outage probability follows from the distribution of the highest ordered SNR among the N sub-surfaces.The random-selection outage distribution is used with M = m for each sub-surface.Source: A. Transmit Diversity Scheme · p. 6
• The TD expected rate is derived analytically from the selected sub-surface's received-SNR distribution.Source: A. Transmit Diversity Scheme · p. 6
• TD becomes more energy efficient as M increases because only m elements are active at each time slot, reducing IRS power consumption.The scheme uses P_IRS = mP_E rather than activating all M elements.Source: A. Transmit Diversity Scheme · p. 6

B. Adaptive Transmit Diversity Scheme

• ATD tests sub-surfaces sequentially and stops when one reaches SNR threshold ψ, using one-bit feedback to accept or reject each candidate.If none of the first N − 1 sub-surfaces meets ψ, the N-th is selected regardless of its SNR.Source: B. Adaptive Transmit Diversity Scheme · p. 6
• ATD performance improves with N when ρ ≤ ψ, whereas for ρ > ψ the second outage term increases with N.The case ψ = ρ corresponds to the TD scheme.Source: B. Adaptive Transmit Diversity Scheme · p. 7
• The ATD scheme can stop after K ≤ N − 1 sub-surfaces, with the analyzed N-case providing an upper performance bound.The analysis generalizes by setting N = K + 1.Source: B. Adaptive Transmit Diversity Scheme · p. 7

C. Diversity Analysis

• TD achieves full spatial diversity order d_TD = N.Source: C. Diversity Analysis · p. 7
• ATD has diversity order N when ρ ≤ ψ, but diversity order one when ρ > ψ.Equality holds at ψ = ρ for the boundary case.Source: C. Diversity Analysis · p. 7

D. Limiting Distribution

• As N and M grow, the limiting distribution of the largest TD order statistic is Gumbel.The normalization constants are a_N = σ_m^2 and b_N = σ_m^2 log(N).Source: D. Limiting Distribution · p. 7

• CT channel estimation uses an MMSE protocol with M sequential sub-phases, activating one IRS element per sub-phase and transmitting orthogonal pilot symbols.Source: §V · pp. 7–8
• RRC avoids training and its associated rate loss and power consumption because it requires no CSI, whereas OBF requires τ-bit feedback signaling.Source: §V · p. 8
• TD uses N<M training sub-phases and RSS-based selection, reducing training power relative to CT while avoiding estimation errors.Source: §V · p. 8
• ATD resembles TD, but its pre-log factor depends on which sub-surface reaches the required threshold ψ.Source: §V · p. 8
• Random phase rotations keep IRS complexity low regardless of phase resolution, while CT, RRC, and OBF require continuous phase shifts and selection schemes do not.Source: §V · p. 8
• Coding- and selection-based schemes can be jointly implemented, such as combining RRC with TD on a selected sub-surface.Source: §V · p. 8

VI. NUMERICAL RESULTS

• 96% and 99% outage-probability reductions occur for RRC with T = 2 and T = 3 versus the conventional case with M = 10, while achieving full diversity order T for T < M.The simulations also closely match the theoretical results, with approximations tightening as M increases.Source: §VI · p. 9
• OBF outperforms ACCPM-based and perfect MISO beamforming for both training periods, while increasing τ and T brings it closer to the CT benchmark.For fixed τ, outage probability converges to a lower bound as T increases, so T need not greatly exceed τ in low-complexity operation.Source: §VI · p. 9
• TD selection improves performance as N increases, especially at high SNR, where it achieves full spatial diversity order.With equal active elements, CT outperforms TD using 10 active elements, whereas TD provides significant gains with only 4 elements.Source: §VI · p. 9
• ATD outage probability increases with N when ψ = 0.9 < ρ but decreases when ψ = 1.1 > ρ, because selection can respectively admit or exclude outage cases.TD outperforms ATD but may require more feedback bits; the limiting distribution still describes TD behavior well.Source: §VI · pp. 9–10
• Energy efficiency initially increases with M and then decreases because rate grows logarithmically while power consumption grows linearly with M.OBF has the highest energy efficiency among the compared schemes, while TD becomes more efficient than RRC and CT at larger M by activating fewer elements.Source: §VI · p. 10

VII. CONCLUSION

• The conclusion reports significant gains over the conventional case while retaining low complexity and high energy efficiency without instantaneous CSI.The four schemes provide analytical outage-probability and energy-efficiency expressions together with diversity-order and coding-gain analysis.Source: §VII · p. 10

APPENDIX A PROOF OF PROPOSITION 1

• For T = 1, random rotations remove the phase effect on channel gain, enabling outage evaluation from the distribution of H1.The derivation defines θ ≜ 2ρ − 1 and uses the CDF of H1.Source: Appendix A · p. 11

APPENDIX B PROOF OF THEOREM 1

• As M increases, the correlation between gains Ht decreases, approaching zero as M →∞.The correlation coefficient ζ between distinct channel-use gains is derived from their moments.Source: Appendix B · p. 11
• The outage expression is obtained by integrating the gain PDFs over the rate constraint and applying a variable transformation.The derivation uses the PDF of Ht, evaluates the integration limits sequentially, and transforms wt to 1 + Pzt/σ2_n.Source: Appendix B · p. 11

APPENDIX C PROOF OF LEMMA 1

• For large M, the in-phase and quadrature sums converge to independent Gaussian variables, so the gain Ht becomes exponentially distributed.The central-limit argument gives Xt and Yt variance σ2M/2 and an exponential Ht with parameter 1/(σ2M).Source: Appendix C · p. 11

APPENDIX D DIVERSITY OF RRC SCHEME

• For the single-channel-use case, the derived outage asymptotics show diversity order M.The conclusion follows from the small-argument approximation of the Bessel-function expression.Source: Appendix D · pp. 11–12
• The analysis also considers the T > M regime and the limiting high-SNR behavior as T tends to infinity.The appendix introduces a phase vector and a high-SNR approximation for this regime.Source: Appendix D · pp. 11–12
• The RRC scheme achieves spatial diversity order T when T < M, with coding gain GRRC determined by the high-SNR expansion.The result follows because the lowest-order term in the expansion dominates the outage probability.Source: Appendix D · p. 12

APPENDIX E PROOF OF PROPOSITION 4

• The expected rate of the highest received SNR is derived from the PDF of the largest order statistic among the sub-surfaces.The result is obtained by substituting the order-statistic PDF into the expected-rate integral.Source: Appendix E · p. 12

APPENDIX F PROOF OF PROPOSITION 5

• The ATD outage expression separates the cases ρ ≤ ψ and ρ > ψ according to whether selected sub-surfaces meet the outage threshold.Each case enumerates mutually exclusive selection and outage events and applies Proposition 1 to their probabilities.Source: Appendix F · p. 12

APPENDIX G DIVERSITY OF SELECTION-BASED SCHEMES

• The selection-based TD scheme achieves diversity order N, with coding gain equal to the expression given in (35).The result follows from the small-x expansion exp(−x) ≈ 1 − x.Source: Appendix G · p. 12
• Theorem 3 agrees with Eq. (69) for ΠCT(ρ), validating the corresponding analytical approach.The agreement is reported for θ = 1 and σ2_n = 0 dB.Source: Figure 9 · p. 13

APPENDIX H PROOF OF THEOREM 3

• For coherent transmission, phase alignment makes the channel gain a sum of M independent products of Rayleigh variables.The characteristic function of the sum is obtained by raising the single-product characteristic function to the Mth power.Source: Appendix H · p. 13
• The derivation uses the PDF of a product of two Rayleigh variables and standard logarithmic and inverse-hyperbolic identities.These identities support evaluation of the characteristic function in closed form.Source: Appendix H · p. 13
• Channel-estimation accuracy is modeled through σ2_e = 1/(1 + τP_o/M), which determines the estimated cascaded-channel distribution and resulting SNR.The outage probability is then evaluated from the induced PDF and characteristic function.Source: Appendix H · p. 13
• The final expansion has its first 2M − 1 terms equal to zero, a result validated by Fig. 9.The outage expression is obtained using the Gil-Pelaez inversion theorem and a Taylor expansion.Source: Appendix H · p. 13