Two-Timescale Design for Reconfigurable Intelligent Surface-Aided Massive MIMO Systems with Imperfect CSI

• Instantaneous-CSI RIS schemes incur pilot overhead proportional to the number of reflecting elements and require frequent beamforming computation and feedback.These costs are especially problematic because RISs generally contain many elements.Source: §I · p. 3
• In the two-timescale scheme, the BS uses instantaneous aggregated CSI whose dimension is independent of RIS size, so pilots need only exceed the number of users.Source: §I · p. 3
• The RIS is optimized using slowly varying statistical CSI and updated when large-scale channel information changes, reducing computational complexity, feedback overhead, and energy consumption.Source: §I · pp. 3–4
• Open questions include achievable-rate and power-scaling laws under imperfect CSI, realistic modeling of LoS and NLoS components, and deployment guidance.Source: §I · pp. 4–5
• The introduction also identifies spatial correlation among RIS elements and electromagnetic interference as unexamined factors that can impair channel estimation and end-to-end SINR.The stated concern is especially relevant for large RISs and weak useful signals.Source: §I · p. 5
• The paper analyzes uplink two-timescale transmission with imperfect aggregated CSI and adopts Rician fading to assess LoS and NLoS components.Source: §I · p. 5

I. INTRODUCTION

• For spatially independent fading, the analysis evaluates system parameters, asymptotic rates, and power-scaling laws, with a single-user specialization providing further engineering insights.The paper first uses spatial-independent Rician fading for tractable rate expressions and a theoretical framework for parameter and power-scaling analysis.Source: §I · pp. 5–6
• With RIS spatial correlation and EMI, the authors derive LMMSE channel estimates and a closed-form achievable-rate bound; correlation enhances RIS customization, while severe EMI may change power-scaling laws.The analysis explicitly examines how spatial correlation and EMI affect achievable rates and power scaling.Source: §I · pp. 5–6
• For spatially independent Rician fading, the authors estimate the aggregated channel with LMMSE and derive closed-form MRC use-and-then-forget achievable-rate bounds for arbitrary antenna and RIS-element counts.They also analyze estimator MSE and normalized MSE.Source: §I · p. 5
• For both channel models, an accelerated gradient-ascent algorithm maximizes the minimum user rate using a log-sum-exp-smoothed objective and angle gradients, with no projection loss under periodic angles and unit-modulus constraints.The single-user case also admits closed-form solutions.Source: §I · p. 6
• The single-user special case of the minimum-rate design problem has closed-form solutions.This result is stated for the accelerated gradient-ascent design discussion.Source: §I · p. 6
• Numerical results show that large RISs may benefit from spatial correlation, severe EMI may erase RIS gains over conventional massive MIMO, and RIS deployment near cell-edge users is especially beneficial.Similar trends hold when spatial correlation and EMI are moderate.Source: §I · p. 6

II. SYSTEM MODEL

• The uplink system has K single-antenna users, an M-antenna BS, and an N-element RIS; direct and RIS-reflected paths form the aggregated channel Q=G+D.Source: §II · p. 8
• The BS uses low-complexity MRC to detect transmitted symbols based on the estimated channel.Source: §II · pp. 8–9

A. Channel Model

• The direct user-BS channel is modeled as Rayleigh fading, while user-RIS and RIS-BS channels use Rician fading to represent their likely LoS paths.Source: A. Channel Model · p. 9

III. CHANNEL ESTIMATION

• Because the cascaded channel is double-Gaussian rather than Gaussian, exact MMSE estimation is difficult; the paper uses tractable LMMSE based on first- and second-order statistics.Source: §III · p. 11
• Estimating only aggregated Q, which includes direct and reflected channels and has dimension M×K, requires pilot sequences of length tau >= K and reduces overhead and computational complexity versus estimating MN individual channels.Source: §III · p. 13
• With deterministic, known LoS channels, the MSE matrix matches conventional massive MIMO because those channels introduce no additional estimation error.This is the stated limiting case when the LoS components are present.Source: §III · p. 13
• Increasing the number of RIS elements can play a role similar to longer pilots in reducing NMSE, and under p = Eu/N the NMSE converges to a limit below one.The paper also notes that increasing N can improve system rate while reducing NMSE, even at low pilot power.Source: §III · pp. 14–15
• When pilot length τ→∞, ˆqk→qk and ek→0; as N→∞, NMSE converges to zero, whereas LMMSE-estimator MSE converges to a non-zero constant.Source: §III · p. 14
• For a Rayleigh RIS-BS channel, MSE and NMSE decrease with pilot power; as N increases, MSE increases while NMSE decreases.The RIS adds paths without a corresponding increase in pilot length, raising MSE, while stronger channel gains improve normalized error relative to massive MIMO without an RIS.Source: §III · p. 15

IV. ANALYSIS OF THE ACHIEVABLE RATE

• Section IV derives achievable-rate lower bounds from Theorem 1 channel estimates, then uses the analytical expressions to optimize RIS phase shifts based on statistical CSI.Source: §IV · p. 16

A. Derivation of the Rate

• The derivation uses the UatF bound and accounts for pilot overhead while expressing the user rate through desired signal, signal leakage, and multi-user interference.Source: A. Derivation of the Rate · p. 16
• The auxiliary variables e_k1, e_k2, and e_k3 capture performance degradation from imperfect CSI, remain in [0, 1], approach 1 as τ_p or N grows, and approach 0 as τ_p tends to zero.Source: A. Derivation of the Rate · p. 16
• Theorem 2 gives a closed-form ergodic-rate lower bound that avoids inverse-matrix calculations and numerical integration, with low computational complexity even for large M and N.The expression relies only on statistical CSI and can serve as an objective for RIS phase-shift optimization.Source: A. Derivation of the Rate · pp. 17–18
• The analysis assumes perfectly known statistical CSI; mobility-related location or angular errors may degrade receiver and RIS beamforming design, an impact left for future work.Source: A. Derivation of the Rate · p. 18
• Compared with [49, Theorem 1], the impact of imperfect CSI is completely characterized by e_k1, e_k2, and e_k3.Source: A. Derivation of the Rate · p. 18
• The multi-user interference term I_ki scales as O(M^2), implying stronger multi-user interference in RIS-aided than conventional massive MIMO systems.Source: A. Derivation of the Rate · p. 18
• For N = 1, |f_k(Φ)| = 1 for any Φ; for N > 1, phase optimization makes any value from 0 to N achievable.Source: A. Derivation of the Rate · p. 19

Asymptotic behavior and fairness

• Aligning RIS phases to maximize one user’s signal makes that user’s rate unbounded but drives the other users’ rates to zero as N grows, motivating fairness constraints.As M grows, the signal and interference terms scale as O(M^2), and the rate converges to a finite limit.Source: B. Multi-user Case · p. 19

Power scaling laws

• With a Rician BS–RIS channel, transmit power can scale as 1/M while maintaining a non-zero rate for all users, whether user–RIS channels are Rician or Rayleigh.This scaling does not hold if the RIS-aided channels are absent or the BS–RIS channel is Rayleigh.Source: B. Multi-user Case · pp. 20–21

Rayleigh BS–RIS channel

• For a Rayleigh BS–RIS channel, the ergodic rate is independent of the RIS phase-shift matrix; if all user–RIS links are fully NLoS, RIS phase shifts need not be designed.When M and N both grow, rates increase without bound for all users, so fairness is implicitly guaranteed in this special case.Source: B. Multi-user Case · pp. 21–22

Power scaling laws

• With a Rayleigh BS–RIS channel, power can also scale as 1/sqrt(M) while maintaining a non-zero rate.This is the power-scaling result associated with Corollary 6; the subsequent discussion contrasts it with the 1/N scaling law.Source: B. Multi-user Case · p. 23
• With a Rayleigh BS–RIS channel, transmit power can scale as 1/N while maintaining a non-zero rate, independently of the user–RIS Rician factors.The desired-signal term scales as O(M), interference as O(1), and the rate grows logarithmically with the number of BS antennas.Source: B. Multi-user Case · pp. 23–24

B. Multi-user Case

• When δ = 0, rich scattering between the RIS and BS provides spatial multiplexing gains and helps mitigate multi-user interference; users at equal RIS distances achieve equal rates.Equal rates in this special case guarantee fairness.Source: B. Multi-user Case · p. 24
• With δ > 0 and Rayleigh user–RIS channels, Corollary 8 analyzes power scaling p = Eu/N as N → ∞.These conditions distinguish this case from the δ = 0 result.Source: B. Multi-user Case · p. 24
• For Rayleigh user–RIS channels, both desired signal and interference scale as O(M), so the rate remains bounded as M grows large.This behavior differs from the RIS–BS rich-scattering case described in Corollary 7.Source: B. Multi-user Case · pp. 24–25
• A small δ is beneficial for power-scaling laws because it corresponds to a high-rank RIS–BS channel that provides spatial diversity for multi-user communications.When δ = 0, the rate reduces to the expression in (58), confirming that the scaling law is unrelated to εk.Source: B. Multi-user Case · p. 25
• Placing the RIS close to users can yield a small δ and a high-rank RIS–BS channel, making this a good deployment choice.The analysis links the decrease in Rician factor with communication distance to the potential for small δ when the RIS is near users.Source: B. Multi-user Case · p. 25
• Near the BS, δ could be large and the RIS–BS channel could become rank-deficient; artificial scatterers or placing the RIS very close to the BS are proposed to improve its rank.Source: B. Multi-user Case · p. 25

C. Single-user Case

• With |f_k(Φ)| = N, desired-signal, leakage, and noise powers scale as O(M^2N^4), O(M^2N^3), and O(MN^2); rate is bounded as M → ∞ but can grow without bound as N → ∞.The phase condition is optimal for the single-user, large-N case.Source: C. Single-user Case · p. 25
• Corollary 9 gives rate lower bounds when p = E_u/(MN^2) as M,N → ∞ and when p = E_u/N^2 as N → ∞.Both scaling laws assume |f_k(Φ)| = N.Source: C. Single-user Case · p. 26
• For nonzero δ, the SNRs in (63) and (64) do not depend on τ, making τ = K = 1 optimal; with δ = 0, SNR increases with τ, so τ = 1 is not guaranteed optimal.Corollary 10 gives a non-zero-rate lower bound for p = E_u/N when δ = 0 and N → ∞.Source: C. Single-user Case · pp. 26–27
• The rates in (63) and (64) increase with Rician factors δ and ε_k; if either factor is zero, the 1/N^2 power scaling no longer holds and power must scale as 1/N for non-zero rate.The rates are maximized as both factors tend to infinity.Source: C. Single-user Case · p. 26

V. EXTENSION TO CORRELATED CHANNELS WITH EMI

• The extension examines how RIS spatial correlation and EMI affect the system and whether findings from Section IV hold when both are present.BS spatial correlation is ignored, while RIS spatial correlation is retained.Source: §V · p. 27

A. Channel Model with Spatial Correlation

• The channel model assumes user–RIS channels contain only LoS components, a tractable approximation for relatively large ε_k; analysis for arbitrary ε_k is deferred.The setup focuses on placing the RIS near users, where LoS components dominate those channels.Source: A. Channel Model with Spatial… · p. 27
• The correlated aggregated channel for user k is q_c,k = H_c,2Φh_k + d_k, while EMI reaches the BS after RIS reflection through H_c,2Φv.The model includes spatial correlation at the RIS in the EMI and RIS–BS NLoS channels.Source: A. Channel Model with Spatial… · p. 28
• Under isotropic scattering, the EMI and RIS NLoS spatial-correlation matrices are both R, whose entries depend on RIS element spacing.The model treats R as Hermitian.Source: A. Channel Model with Spatial… · p. 28

B. Channel Estimation

• The LMMSE estimator obtains each user’s aggregated channel from the pilot-correlated observation vector.The observation includes the aggregated channel together with RIS-BS channel, EMI, and noise contributions.Source: B. Channel Estimation · pp. 28–29
• Spatial correlation can improve RIS-aided channel-estimation accuracy, whereas without spatial correlation the MSE no longer depends on RIS phase shifts.The phase-shift dependence arises because the correlation matrices are not identity matrices.Source: B. Channel Estimation · p. 29

C. Achievable Rate

• Theorem 4 gives the UatF achievable-rate bound for each user under spatial correlation and EMI, accounting for desired signal, leakage, noise, and EMI terms.The bound is derived using MRC detection based on the estimated channel.Source: C. Achievable Rate · pp. 30–32
• Spatial correlation lets the RIS shape a rate-related channel term that is fixed at N when correlation is negligible.With correlation, the term can be adjusted by the RIS, enhancing its ability to shape the channel.Source: C. Achievable Rate · p. 32
• With EMI, the power-scaling law proportional to 1/M is not guaranteed: in the stated counterexample, SINR tends to zero as M grows.The counterexample identifies desired-signal and EMI terms that scale as O(M^2).Source: C. Achievable Rate · p. 33
• With only NLoS components and no spatial correlation, the power-scaling law proportional to 1/N is also not guaranteed to hold in the presence of EMI.The cited special case considers transmit power proportional to 1/N as N grows.Source: C. Achievable Rate · p. 33
• As transmit power scales down with increasing M or N, useful signal power at the RIS weakens while EMI power remains unaffected and can dominate.The passage notes that EMI may exceed thermal noise without necessarily exceeding multi-user interference when M or N is not very large.Source: C. Achievable Rate · pp. 33–34

VI. DESIGN OF THE RIS PHASE SHIFTS

• Statistical-CSI-based phase-shift optimization enables less frequent RIS updates, reducing channel-acquisition overhead and computational cost, especially in sub-6 GHz.Source: §VI · p. 34

A. Single-user Case

• In the single-user case, the optimal design selects |fk(Φ)| = N or 0 based on SNR conditions; as N → ∞, |fk(Φ)| = N is optimal.When the SNR is nonmonotonic over the feasible domain, the optimum is at an endpoint, determined by comparing SNRk(0) and SNRk(N^2).Source: A. Single-user Case · pp. 35–36

B. Multi-user Case

• For multiple users, the RIS phases are optimized to maximize the minimum user rate using gradient ascent over real-valued phase variables θ.The unit-modulus constraint holds automatically because c = e^jθ, avoiding the projection used by complex-variable projected gradient ascent [46].Source: B. Multi-user Case · pp. 36–37
• The nondifferentiable minimum function is approximated in the objective in (92), with μ controlling accuracy; the approximation error is smaller than ln K, and a method increases gradient-method convergence speed.Using real variables makes the algorithm more robust to the choice of this tuning parameter.Source: B. Multi-user Case · pp. 37, 41
• Steps 6–7 of the multi-user algorithm for optimizing fc(θ) correspond to Nesterov’s acceleration method.Source: B. Multi-user Case · p. 41

1) Quality of the LMMSE Channel Estimation:

• As N grows in general Rician channels, estimation MSE increases while NMSE decreases; in purely LoS RIS-assisted channels, both are independent of N.More communication paths without a corresponding increase in pilot length raise estimation error, while stronger channel gains reduce normalized error.Source: Figure 2 · p. 42, 1) Quality of the LMMSE Channel… · p. 42
• Increasing pilot length from 8 to 30 lowers NMSE, but the same NMSE can be achieved with 8 pilot symbols by using a larger N; NMSE also tends to a limit below 1 when power scales as 100/N.As N approaches infinity, MSE tends to an upper bound and NMSE tends to zero.Source: 1) Quality of the LMMSE Channel… · pp. 42–43

2) Single-user Case:

• The instantaneous-CSI RIS design outperforms the two-timescale design under the same assumed rate-loss factor, especially for large N.The instantaneous design exploits both LoS and NLoS components, whereas the statistical-CSI design averages out fast-fading NLoS information.Source: Figure 3 · p. 43, 2) Single-user Case · p. 43
• With actual channel-estimation overhead, the two-timescale scheme outperforms the instantaneous-CSI design, whose pilot length grows with N and leaves fewer symbols for data.The resulting estimation overhead causes a severe rate decrease for the instantaneous-CSI scheme at large N.Source: 2) Single-user Case · pp. 43–44
• In Rician fading, rate converges to a limit when power scales as 1/N^2; in NLoS-only channels, that scaling drives rate to zero, while 1/N scaling maintains non-zero rate.The limiting rate is maximized in LoS-only RIS-assisted channels.Source: 2) Single-user Case · p. 44

3) Multi-user Case:

• For multi-user performance, placing the RIS near users is favorable when accounting for spatial diversity; near-BS deployments may require scatterers or a spherical-wave model to increase channel rank.Increasing ε_k by raising the RIS strengthens user-RIS LoS components, while high rank in the RIS-BS channel remains important.Source: 3) Multi-user Case · p. 45
• In the multi-user case, RIS deployment improves rate, with greater gains as N increases; achievable rate decreases with δ and increases with each ε_k.The RIS-BS LoS component becomes rank one as δ grows, limiting simultaneous multi-user transmission.Source: 3) Multi-user Case · pp. 44–45
• The design uses simple MRC at the BS, while the LMMSE channel estimator requires the same overhead as conventional massive MIMO.Source: 3) Multi-user Case · p. 45
• A conventional massive-MIMO BS with 200 antennas achieves the same rate as RIS-aided massive MIMO with 100 BS antennas and 64 RIS elements, or 64 BS antennas and 400 RIS elements.Source: 3) Multi-user Case · p. 45
• With a purely NLoS RIS-BS channel (δ = 0), the rate remains non-zero under power scaling as M grows, and the asymptotic limit improves with more RIS elements.For N growth, the transmit power is scaled proportionally to 1/N; the rate remains non-zero as N → ∞ under the stated NLoS conditions.Source: 3) Multi-user Case · pp. 45–46
• For δ = 0, increasing BS antennas from M = 64 to M = 400 significantly improves the asymptotic limit as N → ∞.The high-rank RIS-BS channel reduces user spatial correlation and mitigates multi-user interference.Source: 3) Multi-user Case · p. 46
• For εk = 0, ∀k, increasing BS antennas from M = 64 to M = 400 only marginally increases the asymptotic limit as N → ∞.The results indicate that spatial diversity between the RIS and BS can provide a good rate in RIS-aided massive MIMO systems.Source: 3) Multi-user Case · p. 46

B. Spatial-correlated Channels in the Presence of EMI

• With RIS inter-element spacing λ/2 and light EMI (ρ = 30 dB), spatial correlation can be safely ignored; at λ/8, it cannot be ignored and has a non-negligible impact on rate.Source: B. Spatial-correlated Channels in the… · p. 47
• EMI has negligible rate impact when ρ < 60 dB, but becomes dominant as ρ increases and can make RIS-aided systems perform worse than conventional systems.The power-scaling law is confirmed for mild EMI but does not hold under strong EMI.Source: Figure 10 · p. 47, B. Spatial-correlated Channels in the… · pp. 47–48
• For mild EMI, the transmit-power scaling law is confirmed; the rate behavior with RIS spacing agrees with the correlation results.The illustrated power is scaled as p = 10/N.Source: Figure 11 · p. 48, B. Spatial-correlated Channels in the… · p. 48
• The accelerated gradient method converges faster and within 100 iterations, and outperforms the two benchmark methods especially for large N under spatial correlation.The benchmarks are a genetic algorithm and gradient ascent using complex-valued variables.Source: Figure 12 · p. 48, B. Spatial-correlated Channels in the… · pp. 48–49

C. Randomly Distributed Users in a Large Area

• For 500 random user locations in a 100 m × 100 m area, RIS deployment still provides some performance gains, though smaller than with optimized RIS deployment near cell-edge users.The achievable rate is low when the RIS phase-shift matrix is not optimized.Source: C. Randomly Distributed Users in a… · pp. 49–50

VIII. CONCLUSION

• The study combines LMMSE estimation of aggregated user-BS channels, MRC detection, a UatF achievable-rate bound, and gradient-ascent optimization of RIS phase shifts for spatially independent and correlated channels.The correlated-channel analysis includes EMI and accounts for channel-estimation errors.Source: §VIII · p. 50
• With spatially independent channels, non-zero rate is maintained by scaling power as 1/M for RIS-BS Rician channels, or as 1/√M or 1/N for Rayleigh channels under the corresponding large-M or large-N limits.Source: §VIII · p. 50
• The conclusion identifies spatial correlation as beneficial for shaping wireless channels and recommends placing the RIS near cell-edge users; sufficiently strong EMI can invalidate scaling laws derived without EMI.The placement recommendation is tied to compensating for product path loss in the far-field region.Source: §VIII · p. 50

APPENDIX A SOME USEFUL RESULTS

• For zero-mean i.i.d. entries of variance vx, E[XWX^H] is diagonal with every diagonal entry equal to vx Tr{W}.The same result yields corresponding random-vector identities by setting m = 1 or n = 1.Source: Appendix A · p. 51
• The proof also derives the stated complex-Gaussian moment results by using independent, zero-mean entries of ˜H2.For unequal indices the relevant expectation vanishes by independence and zero mean; the equal-index case uses an earlier identity.Source: Appendix A · p. 51
• Appendix A also establishes Gaussian matrix-moment identities and applies them to derive results involving the channel matrices and beamforming terms.The supplied passages identify Lemma 9 and subsequent applications, but their displayed formulas are incomplete.Source: Appendix A · pp. 52–53

APPENDIX B

• Appendix B derives the mean of qk and its covariance with observation yk from the channel definition, independence, and Lemma 7.The derivation assumes the listed variables are mutually independent and have zero-mean entries, then introduces auxiliary variables to complete the proof.Source: Appendix B · pp. 53–54

APPENDIX C

• Appendix C obtains the LMMSE estimate of qk from yk using the mean and covariance matrices from Lemma 1 and the Woodbury matrix identity.It expands the resulting linear expression to recover equation (17).Source: Appendix C · pp. 54–55
• The estimation error ek has zero mean, and Appendix C expresses its NMSE as the trace of its MSE matrix divided by the trace of qk's covariance.The supplied passage gives the resulting expression as M(ak5 + ak6), though the surrounding equation formatting is incomplete.Source: Appendix C · pp. 55–56

APPENDIX D

• The coefficients ek1, ek2, and ek3 are nonnegative and at most 1.The appendix establishes nonnegativity by inspection and shows the numerator of the expanded ek2 is smaller than its denominator.Source: Appendix D · p. 56
• As τp→∞ or N→∞, ek1, ek2, and ek3 approach 1; as τp→0, all three approach 0.These limits follow from ak3→0 and ak4→1 in the first regime, and ak3, ak4→0 in the second.Source: Appendix D · p. 57

A. Signal Term and Noise Term

• The appendix derives the signal and noise expectations using LMMSE orthogonality, zero-mean terms, and independence among the channel components and noise.The noise calculation selects non-zero terms in the expansion and uses the stated identities and lemmas.Source: A. Signal Term and Noise Term · p. 57
• For Ak=IM and τ→∞, the estimate q̂k becomes the perfect estimate qk, and the appendix specializes the signal expectation to that case.The derivation substitutes these conditions into (148).Source: A. Signal Term and Noise Term · p. 58

B. Interference Term

• The interference derivation addresses dependent user channels qk and qi, which share the RIS–BS channel, and an LMMSE error ek that is uncorrelated with but dependent on estimate q̂k.The passage attributes the latter dependence to the cascaded channel being non-Gaussian.Source: B. Interference Term · pp. 58–59
• Equation (160) is derived by calculating expectations of 16 modulus-square terms and other cross-terms; the former are evaluated one by one using Lemmas 7 and 10 and independence between ˜H2, ˜hk, and ˜hi.Of the 16 × 15 cross-terms, only a few are non-zero; Iki (Φ) is obtained by combining (155)–(159), (161)–(176), and (179)–(182) to (154).Source: B. Interference Term · pp. 60, 64, 66
• Only 8 cross-terms have non-zero expectation in the interference expansion; the interference term is then obtained by combining the derived expectations.The derivation uses independence and zero-mean properties of the channel components to identify the non-zero terms.Source: B. Interference Term · pp. 64, 66

C. Signal Leakage

• The signal-leakage derivation expands an expectation and removes terms with zero expectation using zero-mean properties and independence.The stated independence is between the cascaded channel, direct channel, and noise.Source: C. Signal Leakage · p. 66
• Two cross-terms vanish: one by Lemma 8 and another because odd-order central moments of a zero-mean Gaussian variable are zero [78, Eq. (12)].Source: C. Signal Leakage · p. 66
• The second and third terms in (184) are derived, with their results given in (150) and (151), respectively.Source: C. Signal Leakage · p. 67
• For the fourth term in (184), the derivation uses the Wishart distribution property (126); the fifth term is also evaluated, with its result given in (151).Source: C. Signal Leakage · p. 67
• The sixth and seventh terms in (184) are derived, with a result for the seventh term given in (153).Source: C. Signal Leakage · p. 67
• The first term in (184) is expanded into 16 modulus-square terms and the remaining cross-terms.The derivation then considers the modulus-square terms and cross-terms in turn.Source: C. Signal Leakage · p. 68

2 H2Φ˜hk

• The calculation completes the 16 modulus-square terms before turning to the expectation of the remaining cross-terms.Source: §2 · p. 73
• Of 16 × 15 cross-terms, only 20 are nonzero; independence and zero-mean properties reduce them to 10 terms.Source: §2 · p. 73
• The derivation omits the real operator because the 10 cross-term results have only real parts.Source: §2 · p. 73

APPENDIX F

• For N > 1, 0 ≤ |fk(Φ)| ≤ N, with equality when the RIS phase shifts are aligned as specified.Source: Appendix F · p. 77
• The minimum of |fk(Φ)| is zero for both even and odd N.Source: Appendix F · p. 77
• When RIS phases maximize |fk(Φ)|, |fi(Φ)| remains bounded as N → ∞ for a user with a different AoA, shown rigorously under ULA and deduced for USPA.Source: Appendix F · pp. 77–78

APPENDIX G

• Appendix G derives the aggregated-channel and observation statistics, including their covariances, and uses them to form the LMMSE channel estimate.Source: Appendix G · pp. 78–79

APPENDIX H

• Appendix H derives noise and EMI terms using orthogonality and channel, noise, and EMI independence.Source: Appendix H · pp. 79–80
• Appendix H derives interference and signal-leakage terms by evaluating component expectations; the detailed calculations are omitted or follow the interference procedure.Source: Appendix H · pp. 80–81

APPENDIX I

• Appendix I applies the chain rule to derive a trace gradient with respect to RIS phase elements and proves an identity under unitary-matrix conditions.Source: Appendix I · p. 81

APPENDIX J

• Appendix J estimates the direct channel and cascaded channel in two pilot phases, requiring 1 and N pilots, respectively, for N+1 total.Source: Appendix J · p. 82
• Appendix J addresses imperfect-CSI phase optimization with a low-complexity method maximizing desired-signal power, alternating MRC and RIS phase updates until convergence.Source: Appendix J · pp. 82–83
• Appendix J’s instantaneous-CSI average rate includes pilot overhead and becomes zero when N+1 > τ_c, leaving no symbols for data.Source: Appendix J · p. 83
• Appendix J also presents a rate obtained by replacing the instantaneous scheme’s pilot-loss factor with the two-timescale factor, but states that this rate is not achievable.Source: Appendix J · p. 83

APPENDIX K EXPRESSIONS FOR GRADIENT VECTORS

• Theorem 6 states the gradient of f(θ) with respect to θ.The displayed expression is partially garbled in the supplied text.Source: Appendix K · p. 84
• Theorem 7 gives the gradient of fc(θ) with respect to θ, and the appendix separately identifies the gradient of signal.The supplied passages do not preserve the full expressions clearly.Source: Appendix K · p. 85
• The appendix also provides gradients for EMI and signal leakage.The EMI expression is fragmented; the signal-leakage gradient is introduced but its expression is not legible in the supplied text.Source: Appendix K · pp. 85, 88