Ferkans — Interactive Telecom Tutor

When Is the RIS a Diagonal Matrix?

The diagonal model $\boldsymbol{\Phi} = \text{diag}(\phi_1, \ldots, \phi_N)$ assumes element $n$ reflects only the wave incident on element $n$ , with coefficient $\phi_n$ . In a real RIS, adjacent elements are coupled electromagnetically — the current induced on element $n$ radiates not just backward but also laterally, driving currents on its neighbours. The true relationship between incident and reflected fields is then given by a full matrix $\boldsymbol{\Phi}_{\text{full}} \in \mathbb{C}^{N \times N}$ with non-zero off-diagonal entries. When does the diagonal approximation hold, and what do we lose when it doesn't?

Definition:
Mutual Coupling and the Full Response Matrix

Let $\mathbf{i} \in \mathbb{C}^N$ be the vector of currents induced on the $N$ RIS elements by an incident field $\mathbf{e} \in \mathbb{C}^N$ . In the presence of mutual coupling, the governing relation is

$\big(\mathbf{Z}_s + \mathbf{Z}\big)\,\mathbf{i} = \mathbf{e},$

where $\mathbf{Z} \in \mathbb{C}^{N \times N}$ is the mutual impedance matrix (off-diagonal entries $Z_{nm}$ encode coupling between elements $n$ and $m$ ) and $\mathbf{Z}_s = \text{diag}(z_1, \ldots, z_N)$ are the per-element tunable load impedances. The reflected field is a linear function of the currents — in matrix form, the full RIS response is

$\boldsymbol{\Phi}_{\text{full}} = \mathbf{A}\,\big(\mathbf{Z}_s + \mathbf{Z}\big)^{-1}\,\mathbf{B}$

for geometry-dependent matrices $\mathbf{A}, \mathbf{B}$ . The diagonal model sets $\mathbf{Z}$ to be diagonal itself; in that approximation $\boldsymbol{\Phi}_{\text{full}} \to \text{diag}(\phi_n)$ with $\phi_n = f(z_n)$ for a per-element function $f$ .

Mutual coupling is always present at some level; the question is how much off-diagonal mass $\mathbf{Z}$ has, and whether ignoring it causes a significant error in the predicted beam pattern.

Theorem: When the Diagonal Model Is Accurate

Assume that the mutual-impedance matrix satisfies the decay $|Z_{nm}| \leq C / d_{nm}^\nu$ for some $C > 0, \nu \geq 1$ . Under half-wavelength element spacing, the relative spectral error of the diagonal approximation satisfies

$\frac{\|\boldsymbol{\Phi}_{\text{full}} - \text{diag}(\phi_n)\|_F}{\|\text{diag}(\phi_n)\|_F} = \mathcal{O}\!\left(\frac{1}{d^\nu_{\min}}\right) = \mathcal{O}(1) \text{ at } \lambda/2,$

which is small but not negligible. At $\lambda/4$ , the error grows by the factor $2^\nu \geq 2$ and the diagonal approximation should not be trusted for beam-shaping.

Mutual coupling decays rapidly with element spacing: the off-diagonal entry $|Z_{nm}|$ is typically $\propto 1/d_{nm}^\nu$ with $\nu \in [1, 3]$ depending on element geometry. At half-wavelength spacing ( $\lambda/2$ ), nearest-neighbour coupling is typically $\sim -10\text{ dB}$ of the self impedance; next-nearest, $\sim -20\text{ dB}$ . For communication-rate purposes, this off-diagonal mass usually perturbs beam patterns by $< 1\text{ dB}$ — tolerable. At tighter spacings ( $\lambda/4$ or less), coupling grows and the diagonal model becomes unreliable.

,

The Practical Rule of Thumb

For most deployed RIS panels with half-wavelength element spacing and resonant patches:

Beam pattern: the diagonal model is accurate to within $1\text{ dB}$ of measured main-lobe gain and within $3\text{ dB}$ of first sidelobes.
Coherent-sum SNR: the diagonal model is accurate to within $1\text{ dB}$ for $N \leq 1024$ at half-wavelength spacing.
Polarization and cross-polarization: not captured by the scalar diagonal model; requires a full tensor treatment.

For the optimization theory in the rest of this book, we stay with the diagonal model. For high-accuracy system-level simulation or calibration of deployed panels, upgrade to $\boldsymbol{\Phi}_{\text{full}}$ .

Beam Pattern: Diagonal vs. Full-Coupling Model

Computed main-beam and sidelobe pattern of a 16x16 RIS panel under the diagonal model (dashed) and the full coupling model (solid). Differences appear mostly in the sidelobe structure; main-beam direction and peak gain match closely at half-wavelength spacing.

Historical Note: Why Mutual Coupling Is an Array-Processing Legacy

1950s–2020s

Mutual coupling is not a new problem — antenna engineers have argued about it since the 1950s, when arrays first became large enough for coupling to matter. The "method of moments" (MoM) analysis of antenna arrays, introduced by Harrington in 1967, remains the gold standard for computing mutual-impedance matrices. RIS revives these techniques because the close element spacing that makes coherent-sum beamforming effective also makes mutual coupling non-trivial. A beneficial cross-pollination: the RIS community is re-learning antenna engineering, and the antenna-engineering community is updating its tools to handle the tunable-impedance context.

Common Mistake: Closer Spacing Is Not Always Better

Mistake:

"Packing more RIS elements into the same area (tighter spacing) increases $N$ and hence the $N^2$ coherent gain."

Correction:

Only under the diagonal model. With proper mutual-coupling treatment, reducing spacing below $\lambda/2$ causes off-diagonal impedance growth, which reduces the reachable phase range per element and introduces amplitude variation. Measurement shows that $\lambda/4$ -spaced arrays typically underperform $\lambda/2$ -spaced arrays of the same physical size, despite having $4\times$ more elements. The half-wavelength grid is a genuine design sweet spot.

Example: First-Order Coupling Correction

Consider a 1D RIS array of $N = 8$ elements at $\lambda/2$ spacing. Model the mutual-impedance matrix as tridiagonal with diagonal $Z_{nn} = 1$ and off-diagonal $Z_{n,n\pm 1} = -0.3\,e^{j\pi/4}$ ; all other entries zero. Under all-ones load $\mathbf{Z}_s = \mathbf{I}$ , compute the spectral error of the diagonal approximation.

Solution

Form the system

$(\mathbf{Z}_s + \mathbf{Z}) = (1 + 1)\mathbf{I} + \text{off-diag} = 2\mathbf{I} + \mathbf{T}$ , where $\mathbf{T}$ has tridiagonal $\pm 1$ -band with coefficient $-0.3 e^{j\pi/4}$ .

Diagonal approximation

Diag: $(\mathbf{Z}_s + \mathbf{Z})^{-1} \approx (1/2) \mathbf{I}$ . Full: solve $(2\mathbf{I} + \mathbf{T})^{-1}$ numerically. The difference is the off-diagonal part divided by $\approx 4$ , i.e., $\sim 0.075\,e^{j\pi/4}$ on adjacent off-diagonals.

Frobenius norm error

$\|\text{off-diag}\|_F \approx \sqrt{2 \cdot 7} \cdot 0.075 = 0.28$ , while $\|\text{diag}\|_F = \sqrt{8 \cdot 0.25} \approx 1.41$ . Ratio $\approx 0.20$ , or $-14\text{ dB}$ . The diagonal approximation is $\sim 1.5\text{ dB}$ off in the coherent-sum gain in this illustrative case.

🔧Engineering Note

Handling Mutual Coupling in Practice

Strategies for dealing with mutual coupling:

Design spacing. Stick to $\lambda/2$ unless there is a hard physical constraint.
Per-element calibration. Measure each element's response with neighbours in known states, build an empirical correction matrix that approximates $\boldsymbol{\Phi}_{\text{full}}^{-1} \text{diag}(\phi_n)$ .
Full-wave simulation for design. Use MoM or FDTD to compute the mutual-impedance matrix before deployment; design the element geometry to minimize off-diagonal coupling.
Coupling-aware optimization. Treat $\boldsymbol{\Phi}_{\text{full}}$ as the optimization variable (more complex but exact). Reserved for research deployments where the last $1\text{ dB}$ matters.

Practical Constraints

•
Typical neighbour coupling at $\lambda/2$ : $-10\text{ dB}$ to $-15\text{ dB}$ of self impedance.
•
Computational cost of full $\boldsymbol{\Phi}_{\text{full}}$ simulation grows as $O(N^3)$ .

Mutual Coupling and Diagonal-Model Validity