Ferkans — Interactive Telecom Tutor

All Elements On, All the Time

ON/OFF wastes $(N-1)/N$ of the RIS aperture during every pilot slot — a consequence of activating only one element at a time. The DFT codebook flips this around: during every pilot slot, all $N$ elements are active, with orthogonal phase patterns across slots. The orthogonality lets the BS extract each element's cascaded channel by a simple inverse DFT, while exploiting the full $N$ -fold reflection aperture at every pilot slot. The result is a $N$ -fold improvement in estimation SNR relative to ON/OFF, at the same pilot overhead $\tau_p = N$ .

Definition:
DFT-Codebook Pilot Design

Let $\mathbf{F} \in \mathbb{C}^{N \times N}$ be the normalized $N$ -point DFT matrix, $[\mathbf{F}]_{n,k} = \frac{1}{\sqrt{N}} e^{-j 2\pi n k / N}$ .

In the DFT-codebook protocol, the $t$ -th RIS configuration is $\boldsymbol{\phi}^{(t)} = \sqrt{N}\, \mathbf{f}_t$ , where $\mathbf{f}_t$ is the $t$ -th column of $\mathbf{F}$ . That is, $\phi_n^{(t)} = e^{-j 2\pi n t / N}$ for all $n, t$ .

Every element has unit modulus in every slot; across slots, the phase pattern forms a DFT. The stacked configuration matrix $\boldsymbol{\Phi}^{\text{stack}} = \sqrt{N}\,\mathbf{F}$ is unitary (up to scaling): $(\boldsymbol{\Phi}^{\text{stack}})^H \boldsymbol{\Phi}^{\text{stack}} = N \mathbf{I}$ .

The DFT is the unique choice of orthogonal codebook with unit-modulus entries that can be realized by any RIS hardware — every element only needs to apply one of $N$ fixed phases. Crucially, no amplitude variation is required (which would be impossible for a passive RIS). Other unitary codebooks exist (e.g., Hadamard when $N$ is a power of 2), but DFT is the workhorse choice.

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Theorem: DFT Codebook Estimator and MSE

Under the DFT-codebook pilot design with unit-power pilots ( $|x| = 1$ ) and noise $\mathbf{W} \sim \mathcal{CN}$ with per-entry variance $\sigma^2$ , the least-squares estimator

$\hat{\mathbf{G}}^H = \frac{1}{\sqrt{N}}\, \mathbf{Y}\, \mathbf{F}^H$

is unbiased with per-element MSE

$\mathbb{E}\big[\|\hat{\mathbf{g}}_n - \mathbf{g}_n\|^2\big] = \frac{N_t\, \sigma^2}{N\, P_t}.$

Compared with the ON/OFF MSE of $2N_t\,\sigma^2/P_t$ , the DFT estimator improves per-element MSE by a factor of $2N$ — at the same pilot length $\tau_p = N$ .

Stacking pilots, the received data is $\mathbf{Y} = \mathbf{G}^H \boldsymbol{\Phi}^{\text{stack}} + \mathbf{W}$ . Multiplying on the right by the inverse codebook $\mathbf{F}^H / \sqrt{N}$ recovers $\mathbf{G}^H$ without amplification of noise — the DFT is unitary, so $\|\mathbf{W} \mathbf{F}^H / \sqrt{N}\|_F^2 = \|\mathbf{W}\|_F^2 / N$ is the same as before. The signal, however, has been coherently combined across $N$ slots.

Proof

Stacked signal model

With $\boldsymbol{\Phi}^{\text{stack}} = \sqrt{N}\mathbf{F}$ , the stacked observation is $\mathbf{Y} = \mathbf{G}^H \sqrt{N} \mathbf{F} + \mathbf{W} \in \mathbb{C}^{N_t \times N}$ .

Apply inverse DFT

$\hat{\mathbf{G}}^H = \frac{1}{\sqrt{N}} \mathbf{Y}\mathbf{F}^H = \mathbf{G}^H \mathbf{F}\mathbf{F}^H + \frac{1}{\sqrt{N}}\mathbf{W}\mathbf{F}^H = \mathbf{G}^H + \frac{1}{\sqrt{N}}\mathbf{W}\mathbf{F}^H$ , using $\mathbf{F}\mathbf{F}^H = \mathbf{I}$ (since $\mathbf{F}$ is unitary).

Variance after projection

The effective noise $\tilde{\mathbf{W}} = \mathbf{W}\mathbf{F}^H/\sqrt{N}$ is still i.i.d. Gaussian with per-entry variance $\sigma^2/N$ (unitary rotation preserves covariance after the $1/\sqrt{N}$ normalization: $\mathbb{E}[\tilde{w}_{ij} \tilde{w}_{ij}^*] = \sigma^2/N$ ).

Per-element MSE

The $n$ -th row of $\hat{\mathbf{G}}^H$ has MSE $N_t \cdot \sigma^2/N / P_t \cdot P_t = N_t\sigma^2/(N P_t)$ . $\blacksquare$

Key Takeaway

DFT codebook beats ON/OFF by a factor of $2N$ in per-element MSE, at identical pilot overhead. The mechanism is that DFT keeps all $N$ elements active every slot, exploiting coherent reflection for estimation rather than just for data. This is the single biggest practical reason DFT codebooks are the default for RIS estimation in the literature — and why the ON/OFF protocol is taught primarily as pedagogical baseline.

Estimation MSE: ON/OFF vs. DFT Codebook

Compare the per-element MSE achieved by the ON/OFF protocol and the DFT-codebook estimator as a function of pilot SNR and $N$ . The DFT codebook is uniformly better by a factor of $2N$ (log axis makes the separation very clean).

Parameters

RIS elements

N

128

BS antennas

N_t

8

Min pilot SNR (dB)0

Max pilot SNR (dB)30

Example: DFT Codebook Recovery by Hand: $N = 4$

An $N = 4$ RIS is probed with the $4$ -point DFT codebook. The BS has a single antenna ( $N_t = 1$ ). The noiseless received samples are $\mathbf{y} = [y_1, y_2, y_3, y_4]^T = \mathbf{G}^H \sqrt{4} \mathbf{F}$ , where $\mathbf{G}^H = [g_1^*, g_2^*, g_3^*, g_4^*]$ is the row of cascaded channels. Recover $\mathbf{G}^H$ from $\mathbf{y}$ .

Solution

Write out the DFT

The $4 \times 4$ DFT matrix has entries $[\mathbf{F}]_{n,k} = e^{-j\pi n k/2}/2$ . Explicitly, $\mathbf{F}_{:,k} = \frac{1}{2}[1, e^{-j\pi k/2}, e^{-j\pi k}, e^{-j3\pi k/2}]^T$ . So $\sqrt{4}\mathbf{F} = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -j & -1 & j \\ 1 & -1 & 1 & -1 \\ 1 & j & -1 & -j \end{pmatrix}$ .

Identify the codebook columns

Slot 1: $\phi^{(1)} = [1, 1, 1, 1]^T$ — all elements in phase. Slot 2: $\phi^{(2)} = [1, -j, -1, j]^T$ — progressive $90^\circ$ rotation. Slot 3, 4: similar with larger rotation.

Recover $\mathbf{G}^H$ by inverse DFT

$\hat{\mathbf{G}}^H = \mathbf{y}^T \mathbf{F}^H / \sqrt{4}$ . Each entry: $\hat{g}_n^* = \frac{1}{4} \sum_t y_t e^{j\pi n (t-1) / 2}$ . For example, $\hat{g}_1^* = (y_1 + y_2 + y_3 + y_4)/4$ — the average of all slot observations. $\hat{g}_2^*$ is a phase-rotated average, picking up the $2\pi/4$ component. The $N$ -fold improvement over ON/OFF is evident: every slot contributes to every $\hat{g}_n$ estimate. $\blacksquare$

Common Mistake: DFT Requires Pilot Synchronization

Mistake:

"Just apply the DFT inverse to any received pilot vector — it works for any unitary codebook, not just the DFT specifically."

Correction:

The reconstruction $\hat{\mathbf{G}}^H = \mathbf{Y}\mathbf{F}^H/\sqrt{N}$ assumes that the BS knows exactly which RIS configuration was applied in each pilot slot. If slot timing slips by even one symbol — e.g., slot 2 gets misaligned with $\boldsymbol{\phi}^{(2)}$ — the estimator applies the wrong inverse codebook column, producing a systematic bias that does not vanish with more pilots. Synchronization between BS controller and RIS is a real engineering concern. Most deployments use a dedicated control link with known latency.

Historical Note: From Hadamard to DFT

2019–2020

The first RIS estimation papers (Mishra and Johansson, 2019; Jensen and de Carvalho, 2020) used the Hadamard codebook — an orthogonal $\pm 1$ matrix valid for $N = 2^k$ . It has the same $N$ -fold improvement over ON/OFF and only requires $\pm 1$ phase states, easily realized by a 1-bit PIN-diode element. The Hadamard transform was the natural choice for the earliest 1-bit RIS prototypes.

The DFT codebook (Zheng and Zhang, 2020) generalizes to any $N$ and is optimal in the sense that its eigenvectors align with steering-vector structure of plane-wave channels — meaning the DFT pilot design implicitly probes angular directions. For RIS hardware with at least 2-bit phase control, DFT is the preferred choice; for 1-bit hardware, Hadamard is equally good. Both give the same $\mathcal{O}(N)$ pilot-length lower bound.

🔧Engineering Note

DFT in Deployed Panels

Modern RIS prototypes and pre-commercial panels use DFT-based pilot sequences as the default. Practical notes:

The DFT requires continuous or at least fine-resolution phase shifts (for $N = 256$ , phase step of $\sim 1.4^\circ$ ). Low-resolution ( $B = 2$ -bit) hardware approximates DFT by rounding to nearest grid phase; the resulting non-orthogonality adds $\sim 1\text{ dB}$ estimation MSE penalty.
Pilots are interleaved with the control link: the BS tells the RIS controller to cycle through the DFT columns, one per pilot symbol.
For high-rate updates, the codebook can be pre-programmed in RIS firmware and triggered by a single synchronization signal.

Practical Constraints

•
Control-link bandwidth for $\tau_p = N$ updates at $\sim 10\,\text{ms}$ coherence: $B N / 0.01 = 100 B N$ bps ( $\sim 300$ kbps at $B=3$ , $N=1024$ ).
•
Practical DFT approximation error with $B = 3$ -bit: $\sim 0.3\text{ dB}$ MSE penalty.
•
Synchronization jitter tolerance: $\pm 1$ symbol; beyond this, orthogonality breaks.

DFT Codebook Estimation