Ferkans — Interactive Telecom Tutor

Net Throughput = Rate × (1 − Overhead) × CSI-Gain

We have seen how to estimate $\mathbf{G}$ with different overheads. The next question, genuinely practical: given a fixed coherence budget $T$ , what is the optimal $\tau_p$ ? Too few pilots leaves $\mathbf{G}$ poorly known, eroding the coherent beamforming gain. Too many pilots leaves little time for data. The optimum balances estimation error against pilot cost, and its solution — a function of SNR, $N$ , and $T$ — determines the realizable throughput of any RIS deployment.

Theorem: SNR Loss from Imperfect CSI

Suppose the DFT-codebook estimator produces $\hat{\mathbf{G}}$ with normalized error $\epsilon_{\text{CSI}}^2 = \|\tilde{\mathbf{G}}\|_F^2 / \|\mathbf{G}\|_F^2$ . The beamformer $\boldsymbol{\Phi}^\star$ designed from $\hat{\mathbf{G}}$ (matched filter on the estimate) achieves expected SNR

$\mathbb{E}[\text{SNR}] \;=\; \text{SNR}^{\text{ideal}} \cdot (1 - \epsilon_{\text{CSI}}^2),$

where $\text{SNR}^{\text{ideal}} = P_t N^2 / \sigma^2$ is the perfect-CSI coherent SNR. For small CSI error, the coherent-gain penalty is approximately $\epsilon_{\text{CSI}}^2 \cdot N^2$ — the $N^2$ gain is eroded by the CSI quality.

If we use the channel estimate $\hat{\mathbf{G}}$ to design the RIS phases, the actual received signal sees the true $\mathbf{G}$ , not the estimate. The mismatch is quantified by the CSI error $\tilde{\mathbf{G}} = \mathbf{G} - \hat{\mathbf{G}}$ . The mean-coherent beamforming gain scales as $(N - \text{error penalty})^2$ . For small error, the penalty is linear in error, and the effective $N$ becomes $N_{\text{eff}} = N (1 - \epsilon_{\text{CSI}}^2)$ .

Proof

Beamformer misalignment

The matched-filter beamformer on $\hat{\mathbf{G}}$ points in direction $\hat{\mathbf{v}} = \hat{\mathbf{G}}/\|\hat{\mathbf{G}}\|$ . The true channel $\mathbf{G}$ decomposes as $\mathbf{G} = \hat{\mathbf{G}} + \tilde{\mathbf{G}}$ with $\tilde{\mathbf{G}}$ zero-mean and orthogonal to $\hat{\mathbf{G}}$ in expectation (the LS estimator is unbiased).

Inner product

$|\mathbf{G}^H \hat{\mathbf{v}}|^2 = |\hat{\mathbf{G}}^H \hat{\mathbf{v}}|^2 + |\tilde{\mathbf{G}}^H \hat{\mathbf{v}}|^2 - 2 \Re\{...\}$ , where the cross-term vanishes in expectation.

Expected SNR

Taking expectations, $\mathbb{E}[|\mathbf{G}^H \hat{\mathbf{v}}|^2] = \mathbb{E}[|\hat{\mathbf{G}}|^2] (1 - \epsilon_{\text{CSI}}^2) = \|\mathbf{G}\|^2 (1 - \epsilon_{\text{CSI}}^2)$ , yielding the stated formula. $\blacksquare$

Theorem: Optimal Pilot Length Under Pilot-Power Budget

Under the DFT-codebook protocol with per-slot pilot energy $P_t T_s$ and coherence block $T$ , the effective throughput is

$R_{\text{eff}}(\tau_p) = \left(1 - \frac{\tau_p}{T}\right) \log_2\!\left(1 + \text{SNR}^{\text{ideal}} \left(1 - \frac{N_t \sigma^2}{\tau_p P_t}\right)\right).$

The optimal $\tau_p^\star$ satisfies (approximately, at high SNR)

$\tau_p^\star \approx \sqrt{\frac{N_t\, \sigma^2\, T}{P_t\, \ln 2}},$

i.e., grows as $\sqrt{T / \text{SNR}}$ , not linearly in $N$ — good news for large- $N$ deployments where CSI error is the first-order concern rather than pilot count.

Given a fixed total energy (pilot power times pilot time $=$ constant), increasing $\tau_p$ reduces per-element MSE at a $1/\tau_p$ rate but reduces data time at a $1 - \tau_p/T$ rate. Differentiating the product reveals an interior optimum.

Proof

Write R_eff

$R_{\text{eff}} = (1 - \tau_p/T) \log_2(1 + \text{SNR}_{\text{eff}})$ , where $\text{SNR}_{\text{eff}} = \text{SNR}^{\text{ideal}} (1 - N_t\sigma^2/(\tau_p P_t))$ from Thm. 4.6 plus DFT-MSE from Thm. 4.4.

Differentiate

$dR_{\text{eff}}/d\tau_p = 0$ gives a transcendental equation. In the high-SNR limit, $\log(1 + x) \approx \log x$ , and the optimum satisfies $1/\tau_p \cdot (1/T) \cdot \log(x) = (1/T) \cdot d(\log(1 - c/\tau_p))/d\tau_p$ , leading to the stated scaling.

Operational interpretation

The square-root scaling is the familiar asymptotic from training-based MIMO (Hassibi and Hochwald 2003 for massive MIMO). The RIS problem inherits this structure; only the constants differ. $\blacksquare$

Effective Throughput vs. Pilot Length

Plot $R_{\text{eff}}(\tau_p)$ for DFT codebook and compare with ideal-CSI throughput. The interior optimum shows the pilot-vs-data tradeoff explicitly. Increase $T$ to see the optimum $\tau_p$ grow; increase SNR and the optimum decreases.

Parameters

RIS elements

N

256

BS antennas

N_t

8

Pilot SNR (dB)10

Coherence length

T

500

Example: Optimal Pilot Length for a Mid-Band Deployment

Deploy $N = 256$ RIS at 3.5 GHz with $T_c = 500$ symbols (roughly $5\text{ ms}$ at $10\,\mu\text{s}$ symbol duration, appropriate for pedestrian mobility). BS antennas $N_t = 8$ , pilot SNR = $10\text{ dB}$ . Find the optimal $\tau_p$ and the resulting fraction of coherent gain retained.

Solution

Apply the formula

$\tau_p^\star \approx \sqrt{N_t\,\sigma^2\,T/(P_t\ln 2)} = \sqrt{8 \cdot (1/10) \cdot 500 / \ln 2} \approx \sqrt{577} \approx 24$ .

CSI error at optimum

$\epsilon_{\text{CSI}}^2 = N_t\sigma^2/(\tau_p^\star P_t) = 8/(24 \cdot 10) = 0.033$ , retaining $(1 - 0.033) = 96.7\%$ of the coherent gain.

Overhead

$\tau_p/T = 24/500 = 4.8\%$ — a modest overhead for almost perfect CSI. Compare with the naive $\tau_p = N = 256$ (51% overhead): at the cost of only $3\%$ of coherent gain, we recover nearly half the coherence block for data.

Operational meaning

Under the square-root scaling, large- $N$ RIS deployments are CSI-feasible at modest cost. The $\tau_p \propto \sqrt{T}$ dependence scales favorably: the coherence block is also a function of deployment (mobile vs. fixed), and slower mobility gives more pilot budget.

Key Takeaway

Optimal pilot length scales as $\sqrt{T/\text{SNR}}$ , not linearly with $N$ . For reasonable operating conditions ( $\text{SNR} \geq 0\text{ dB}$ , $T \geq 500$ ), the optimal pilot fraction is $< 10\%$ even for $N = 256$ . The $\mathcal{O}(N)$ pilot overhead of the naive DFT-codebook protocol is a worst-case figure; the real-world optimum is much better once pilots are not required to hit the CRB exactly.

Common Mistake: Don't Evaluate RIS Gains With Perfect CSI Only

Mistake:

A paper reports a $40\text{ dB}$ SNR improvement from $N = 1024$ coherent RIS beamforming — under perfect CSI.

Correction:

Under perfect CSI, the $N^2$ gain is real. Under imperfect CSI with realistic pilot overhead and finite pilot SNR, the effective gain is $N^2 (1 - \epsilon_{\text{CSI}}^2)(1 - \tau_p/T)$ , often $10$ – $15\text{ dB}$ smaller than the perfect-CSI figure. Any RIS paper claiming dB gains without specifying the CSI assumption should be read with deep suspicion. Always report effective throughput $R_{\text{eff}}$ under a realistic pilot protocol.

Quick Check

For $N = 256$ -element RIS with pilot SNR $= 10\text{ dB}$ , $N_t = 8$ BS antennas, and $T = 500$ symbol coherence block, the optimal pilot length $\tau_p^\star$ under DFT codebook is approximately:

$\sim 256$ (= N)

$\sim 60$ - $80$

$\sim 20$ - $25$

$\sim 5$

Correction:

\sim 20

-

25

From the formula $\tau_p^\star \approx \sqrt{N_t\,\sigma^2\,T/(P_t\,\ln 2)} = \sqrt{8 \cdot 0.1 \cdot 500/\ln 2} \approx 24$ .

Cascaded Channel

The two-hop composite channel $\mathbf{G} = \text{diag}(\mathbf{h}_2^*)\mathbf{H}_1 \in \mathbb{C}^{N \times N_t}$ between the BS and the RIS elements, which includes the effect of the RIS-UE channel $\mathbf{h}_2$ . This is the estimable object; $\mathbf{H}_1$ and $\mathbf{h}_2$ are not separately identifiable from passive-RIS pilot observations.

DFT Codebook

A pilot-design scheme in which the RIS applies phase shifts corresponding to the columns of the $N \times N$ DFT matrix, cycling through $N$ orthogonal configurations across $N$ pilot slots. Achieves the minimum MSE for a given total pilot energy and is the practical default for RIS channel estimation.

🚨Critical Engineering Note

CSI Budget in Practice

Practical CSI guidelines for real RIS deployments:

Start from the coherence time. Mobile pedestrians at $1\,\text{m/s}$ at 3.5 GHz have coherence time $\sim 30\,\text{ms}$ → $T \sim 3000$ symbols at $10\,\mu\text{s}$ symbols. Fixed users have $T \sim 100\,\text{ms}$ or more.
Scale pilot overhead by $\sqrt{T}$ . Optimal $\tau_p$ grows with the square root of the coherence budget — large $N$ deployments are most profitable for slowly-varying users.
Use compressed sensing for mobile scenarios. For $T < 500$ , CS brings $\tau_p$ below the $\sqrt{T}$ square-root scaling by exploiting angular sparsity.
Reserve a retraining budget. RIS channel statistics drift slowly (temperature, hardware aging); plan periodic long-pilot sessions (every few seconds) to track the slow component, interleaved with fast CS-based updates on the fast component.

Practical Constraints

•
Fixed-wireless access (FWA): $T \sim 10^5$ symbols, $\tau_p \sim 30$ symbols (0.03% overhead).
•
Pedestrian mobility: $T \sim 3000$ , $\tau_p \sim 20$ (0.7% overhead).
•
Vehicular mobility: $T \sim 200$ , $\tau_p \geq 10$ with CS; overhead $> 5\%$ likely.

The Overhead-Accuracy Tradeoff