Ferkans — Interactive Telecom Tutor

The Fundamental Challenge: Estimating Channels Through Passive Elements

In conventional MIMO systems, channel estimation is performed by transmitting known pilot symbols and measuring the received signal. An RIS, however, is purely passive — it has no RF chains, no ADCs, and no baseband processing. It can neither transmit pilots nor receive and process them. This creates a fundamental asymmetry: the BS-RIS channel $\mathbf{G}$ and the RIS-user channel $\mathbf{h}_r$ cannot be estimated separately through standard pilot-based methods.

What can be observed at the receiver is only the cascaded channel $\mathbf{h}_r^H \boldsymbol{\Theta} \mathbf{G}$ , which is a function of the RIS configuration $\boldsymbol{\Theta}$ . By varying $\boldsymbol{\Theta}$ across multiple pilot slots and observing the corresponding received signals, the receiver can extract information about the individual channels. The key question is: how many pilot slots are needed, and how should $\boldsymbol{\Theta}$ be varied to enable efficient estimation?

Definition:
RIS Channel Estimation Problem

Consider $T$ pilot slots during which the BS transmits known pilots $\{x_t\}_{t=1}^T$ and the RIS applies configurations $\{\boldsymbol{\Theta}_t\}_{t=1}^T$ . The received signal at pilot slot $t$ is:

$y_t = (\mathbf{h}_d^H + \mathbf{h}_r^H \boldsymbol{\Theta}_t \mathbf{G}) \mathbf{w}_t x_t + n_t, \quad t = 1, \ldots, T$

For a single BS antenna ( $M = 1$ ) and denoting the cascaded channel coefficients $v_n = h_{r,n}^* g_n$ (where $g_n$ is the $n$ -th element of the BS-RIS channel vector), the model simplifies to:

$y_t = h_d x_t + \boldsymbol{\phi}_t^T \mathbf{v} \, x_t + n_t$

where $\boldsymbol{\phi}_t = [\phi_{t,1}, \ldots, \phi_{t,N}]^T$ is the RIS phase vector at slot $t$ and $\mathbf{v} = [v_1, \ldots, v_N]^T$ .

Stacking all $T$ observations (with $x_t = 1$ for simplicity):

$\mathbf{y} = \boldsymbol{\Phi} \begin{bmatrix} h_d \\ \mathbf{v} \end{bmatrix} + \mathbf{n}$

where $\boldsymbol{\Phi} \in \mathbb{C}^{T \times (N+1)}$ has rows $[1, \boldsymbol{\phi}_t^T]$ . The channel estimation problem is to recover the $(N+1)$ -dimensional vector $[h_d; \mathbf{v}]$ from $T$ noisy observations.

Identifiability requirement: The matrix $\boldsymbol{\Phi}$ must have rank $N + 1$ , which requires $T \geq N + 1$ pilot slots. This is the fundamental training overhead of RIS channel estimation.

For MIMO BS with $M$ antennas, the cascaded channel has $NM$ unknowns (plus $M$ for the direct channel), requiring $T \geq NM + M$ pilot slots in the worst case. This overhead can be prohibitive for large $N$ and motivates the structured estimation approaches described below.

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Theorem: Training Overhead Lower Bound for RIS Channel Estimation

For an RIS-assisted system with $N$ reflecting elements, $M$ BS antennas, and a single-antenna user, any unbiased estimator of the cascaded channel $\mathbf{v}_n = h_{r,n}^* \mathbf{g}_n \in \mathbb{C}^M$ for all $n = 1, \ldots, N$ requires at least

$T \geq N + 1$

pilot training slots (for $M = 1$ ), or more generally

$T \geq \left\lceil \frac{NM + M}{M} \right\rceil = N + 1$

pilot time slots when the BS can transmit $M$ orthogonal pilots per slot. The minimum number of total pilot symbols is $NM + M = M(N + 1)$ .

Furthermore, the Cram'{e}r-Rao lower bound (CRLB) for the mean squared error of the cascaded channel estimate is:

$\mathrm{MSE} \geq \frac{(N+1)\sigma^2}{TP}$

when $\boldsymbol{\Phi}$ is a $(N+1) \times (N+1)$ unitary matrix (e.g., a DFT matrix) and $P$ is the pilot transmit power.

The cascaded channel has $N$ unknown complex coefficients (one per RIS element), plus the direct channel. Each distinct RIS configuration provides one independent linear measurement of these unknowns. Therefore, at least $N + 1$ distinct configurations are needed. The DFT-based design achieves the CRLB because the DFT matrix is unitary, providing maximally spread measurements.

Proof

Degrees of freedom argument

The unknown parameter vector $\boldsymbol{\eta} = [h_d, v_1, \ldots, v_N]^T \in \mathbb{C}^{N+1}$ has $N + 1$ complex degrees of freedom. The observation at each pilot slot provides one complex-valued measurement $y_t = \boldsymbol{\phi}_t^T \boldsymbol{\eta} + n_t$ . For the system $\mathbf{y} = \boldsymbol{\Phi} \boldsymbol{\eta} + \mathbf{n}$ to have a unique solution, we need $\mathrm{rank}(\boldsymbol{\Phi}) = N + 1$ , which requires $T \geq N + 1$ .

CRLB derivation

The Fisher information matrix for the linear model $\mathbf{y} = \boldsymbol{\Phi}\boldsymbol{\eta} + \mathbf{n}$ with $\mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I})$ is:

$\mathbf{J} = \frac{P}{\sigma^2} \boldsymbol{\Phi}^H \boldsymbol{\Phi}$

The CRLB on total MSE is:

$\mathrm{MSE} \geq \mathrm{tr}(\mathbf{J}^{-1}) = \frac{\sigma^2}{P} \mathrm{tr}((\boldsymbol{\Phi}^H \boldsymbol{\Phi})^{-1})$

When $\boldsymbol{\Phi}$ is a scaled unitary matrix with $\boldsymbol{\Phi}^H \boldsymbol{\Phi} = T \mathbf{I}$ (achievable with DFT-based phase configurations), this gives:

$\mathrm{MSE} \geq \frac{(N+1)\sigma^2}{TP}$

which is minimised (for fixed $T$ ) by the unitary design. $\blacksquare$

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Grouped Element Channel Estimation

Input: Number of elements

N

, group size

G

, pilot power

P

, noise variance

\sigma^2

Output: Estimated cascaded channel

\hat{\mathbf{v}}

1. Set

\bar{N} = \lceil N/G \rceil

(number of groups)

2. Design phase configurations:

- Construct

(\bar{N}+1) \times (\bar{N}+1)

DFT matrix

\mathbf{F}

- For slot

t = 1, \ldots, \bar{N}+1

:

- Set group

k

's phase:

\bar{\phi}_{t,k} = F_{t,k}

for

k = 1, \ldots, \bar{N}

- All elements in group

k

use phase

\bar{\phi}_{t,k}

:

\phi_{t,n} = \bar{\phi}_{t,\lceil n/G \rceil}

for

n = 1, \ldots, N

3. Collect observations: transmit pilots with each configuration

y_t = h_d + \sum_{k=1}^{\bar{N}} \bar{\phi}_{t,k} \bar{v}_k + n_t

where

\bar{v}_k = \sum_{n \in \mathcal{G}_k} v_n

is the grouped channel

4. Estimate grouped channel:

\hat{\bar{\boldsymbol{\eta}}} = (\bar{\boldsymbol{\Phi}}^H \bar{\boldsymbol{\Phi}})^{-1} \bar{\boldsymbol{\Phi}}^H \mathbf{y}

(least squares estimate of

[h_d, \bar{v}_1, \ldots, \bar{v}_{\bar{N}}]^T

)

5. Reconstruct element-level channel: Assign

\hat{v}_n = \hat{\bar{v}}_{\lceil n/G \rceil} / G

for

n = 1, \ldots, N

(uniform allocation within each group)

6. Return

\hat{\mathbf{v}} = [\hat{v}_1, \ldots, \hat{v}_N]^T

Complexity:

O(\bar{N}^2) = O(N^2/G^2)

for least squares.

Training overhead:

\bar{N} + 1 = \lceil N/G \rceil + 1

pilot slots.

Advanced Channel Estimation Strategies

Beyond the basic ON/OFF and grouped estimation protocols, several advanced strategies exploit channel structure:

1. Codebook-based estimation with DFT patterns. The RIS cycles through a codebook of $T$ phase configurations, typically drawn from a DFT matrix. If $T = N + 1$ , the full cascaded channel can be recovered. For $T < N + 1$ , the system operates in a compressed regime.

2. Compressed sensing / sparse recovery. In mmWave/sub-THz bands, both the BS-RIS and RIS-user channels exhibit angular sparsity: only a few dominant paths exist. If the cascaded channel $\mathbf{v}$ is $S$ -sparse in the angular domain (i.e., $\mathbf{v} = \mathbf{A}_{\mathrm{dict}} \mathbf{s}$ where $\mathbf{s}$ has only $S \ll N$ nonzero entries), then only $T = O(S \log(N/S))$ pilot slots suffice. Standard algorithms (OMP, LASSO, AMP) can recover $\mathbf{s}$ .

3. ON/OFF protocol (Mishra and Johansson 2019). In round $t$ , only element $t$ is turned ON (reflecting) while all others are OFF (absorbing). This provides $y_t = h_d + v_t + n_t$ , directly revealing $v_t$ (after subtracting the known $h_d$ ). Simple but requires $N$ rounds and wastes the potential array gain during estimation.

4. Two-timescale estimation. The BS-RIS channel $\mathbf{G}$ is quasi-static (both nodes are fixed), while the RIS-user channel $\mathbf{h}_r$ varies with user mobility. Estimate $\mathbf{G}$ infrequently (slow timescale) and track $\mathbf{h}_r$ at each coherence interval (fast timescale), reducing per-interval overhead from $O(NM)$ to $O(N)$ .

Quick Check

An RIS has $N = 256$ elements and the BS has $M = 1$ antenna. Using a DFT-based estimation protocol with no sparsity exploitation, how many pilot slots are required for full cascaded channel estimation?

$N = 256$ pilot slots

$N + 1 = 257$ pilot slots

$2N = 512$ pilot slots (real and imaginary parts)

$N \log N$ pilot slots for a DFT-based scheme

Correction:

N + 1 = 257

pilot slots

The cascaded channel has $N$ unknown complex coefficients $\{v_n\}$ plus the direct channel $h_d$ , totalling $N + 1 = 257$ unknowns. Each pilot slot with a distinct RIS configuration provides one linear equation. A DFT-based design with $T = N + 1 = 257$ rows forms a unitary measurement matrix, achieving the CRLB. Compressed sensing can reduce this to $O(S \log N)$ if the channel is $S$ -sparse, but without exploiting structure, 257 is the minimum.

Channel Estimation for RIS