Ferkans — Interactive Telecom Tutor

Why Calibration Is Mandatory

The commanded phase $\theta_n$ and the physically realized phase $\tilde\theta_n$ differ for two reasons: (i) the PIN-diode / varactor has a nonlinear bias-voltage-to-phase relationship that is element-dependent, (ii) fabrication tolerances shift the resonance of each element by a random amount. Without calibration, the commanded coherent-combining pattern is decoherent at the RIS aperture, losing most of the $N^2$ gain.

Definition:
RIS Calibration

RIS calibration is the process of measuring the mapping $\tilde\theta_n = g_n(V_n)$ between the bias voltage $V_n$ applied to element $n$ and the actually realized phase shift $\tilde\theta_n$ . The output is a lookup table (LUT) $V_n(\theta)$ per element. At deployment, commanding $\theta_n^\star$ applies $V_n(\theta_n^\star)$ , yielding a realized phase with residual error $\delta_n = \tilde\theta_n - \theta_n^\star$ .

Theorem: Coherent Combining Loss from Phase Errors

Under independent phase errors $\delta_n \sim \mathcal{N}(0, \sigma_\phi^2)$ , the expected RIS beamforming gain is $\mathbb{E}[G_{\text{BF}}] = N + N(N-1) e^{-\sigma_\phi^2}$ which for $\sigma_\phi^2 \ll 1$ reduces to $G_{\text{BF}} \approx N^2(1 - \sigma_\phi^2)$ . For perfect calibration ( $\sigma_\phi = 0$ ), $G_{\text{BF}} = N^2$ ; for uniform phases ( $\sigma_\phi \to \infty$ ), $G_{\text{BF}} \to N$ .

Proof

Sum-of-phasors structure

The signal amplitude at the UE is a sum of $N$ phasors: $A = \sum_{n=1}^N e^{j\delta_n}$ . The power is $|A|^2 = \sum_n e^{j\delta_n} \sum_m e^{-j\delta_m} = N + 2\sum_{n < m} \cos(\delta_n - \delta_m)$ .

Expectation

By independence, $\mathbb{E}[\cos(\delta_n - \delta_m)] = \mathbb{E}[\cos \delta_n] \mathbb{E}[\cos \delta_m] - \mathbb{E}[\sin \delta_n] \mathbb{E}[\sin \delta_m]$ . For $\delta_n \sim \mathcal{N}(0, \sigma_\phi^2)$ : $\mathbb{E}[\cos \delta_n] = e^{-\sigma_\phi^2/2}$ , $\mathbb{E}[\sin \delta_n] = 0$ , giving $\mathbb{E}[\cos(\delta_n - \delta_m)] = e^{-\sigma_\phi^2}$ .

Gain formula

$\mathbb{E}[|A|^2] = N + 2 \cdot \binom{N}{2} e^{-\sigma_\phi^2} = N + N(N-1) e^{-\sigma_\phi^2}$ . At $\sigma_\phi = 0$ : $N + N(N-1) = N^2$ . $\checkmark$ At $\sigma_\phi \to \infty$ : $N + 0 = N$ . $\checkmark$ For small $\sigma_\phi$ : Taylor expansion gives $N^2(1 - \sigma_\phi^2) + O(\sigma_\phi^4)$ . $\blacksquare$

Example: Calibration Accuracy Budget

A 28 GHz $N = 1024$ RIS panel must achieve at least 55 dB beamforming gain relative to metallic reference (target $\eta_{\text{cal}} \geq 0.8$ ). What per-element phase error standard deviation is allowed?

Solution

Budget

$\eta_{\text{cal}} = e^{-\sigma_\phi^2} \geq 0.8$ $\Rightarrow \sigma_\phi^2 \leq 0.223$ rad² $\Rightarrow \sigma_\phi \leq 0.47$ rad $= 27°$ .

Engineering interpretation

$27°$ RMS phase error per element is achievable with 2-bit quantization + careful calibration. For 1-bit, the quantization alone gives $\sigma_\phi \approx 52°$ — too high. This is why 2-bit is the industry sweet spot. $\blacksquare$

Per-Element Phase Calibration

Complexity: O(N · K) VNA sweeps

Input: VNA, phase reference, V_range = [V_min, V_max]

Output: LUT V_n(θ) for each element n = 1..N

1. for n = 1, ..., N:

2. Set all other elements to reference bias V_ref

3. for k = 1, ..., K:

4. V_k = V_min + (V_max - V_min) · k/K

5. Set bias of element n to V_k

6. Measure S21 (transmission coefficient) → φ_nk

7. end for

8. Fit cubic spline φ_n = g_n(V) to the measured points

9. Invert to get V_n(θ) over θ ∈ [0, 2π)

10. end for

11. return LUT array [V_n(θ)] for n=1..N

🚨Critical Engineering Note

Thermal Drift and Re-Calibration Frequency

PIN-diode capacitance varies with temperature: $\Delta C \approx 2.5 \times 10^{-3} \cdot \Delta T / C$ . Over a $\Delta T = 10°$ C swing (typical indoor day-to-night), the phase shifts by $\sim 15°$ — enough to wipe out coherent combining on large panels ( $N > 256$ ). Re-calibration cadence: every 30 min for outdoor deployments, every 4 h for indoor. Commercial systems implement automatic thermal-compensation via temperature sensors on the panel and real-time LUT adjustment.

Beamforming Gain vs. Phase-Error Variance (Theorem 3.1 Curve)

Plot the formula $G_{\text{BF}} = N + N(N-1) e^{-\sigma_\phi^2}$ as a function of $\sigma_\phi$ for several values of $N$ . The curve transitions from $N^2$ (coherent) to $N$ (incoherent). The transition region is where calibration effort pays off.

Parameters

Max

N

1024

Max phase error RMS (deg)90

Common Mistake: Don't Calibrate Elements in Parallel

Mistake:

Setting many elements simultaneously during calibration to speed it up.

Correction:

Elements couple electromagnetically (edge-mode coupling at $\lambda/2$ spacing). Calibrating element $n$ with its neighbors at random bias biases the measurement. Always calibrate one element at a time with all others at a reference bias, ensuring the coupling error is common-mode and cancels when applied to all elements at deployment. Parallel calibration sounds appealing but yields a 20+ dB performance loss.

Factory vs. Field Calibration

Factory calibration (anechoic-chamber VNA) yields the best accuracy but requires panel removal for refresh. Field calibration using the BS-RIS-UE link as the reference path is less accurate but continuous — errors appear as residual phase noise in the deployed system. Commercial systems use both: factory cal at manufacture, field cal online to track thermal drift.

Calibration of Phase Errors