Ferkans — Interactive Telecom Tutor

The Algebraic Heart of Array-Fed RIS

The near-field BS-RIS channel $\mathbf{H}_1$ has a rich SVD structure. Each singular value corresponds to an "eigenbeam" that the active array can excite and the RIS can focus coherently. The number of usable eigenbeams is essentially the number of independent streams the array-fed RIS can carry. Section 11.2 develops the eigenmode analysis and its consequences for multiplexing capacity.

Definition:
Eigenmode Decomposition of $\mathbf{H}_1$

The SVD of the BS-RIS channel is

$\mathbf{H}_1 = \mathbf{U}_1 \boldsymbol{\Sigma}_1 \mathbf{V}_1^H, \quad \mathbf{U}_1 \in \mathbb{C}^{N \times r}, \mathbf{V}_1 \in \mathbb{C}^{N_t \times r},$

where $r = \text{rank}(\mathbf{H}_1)$ and $\boldsymbol{\Sigma}_1 = \text{diag}(\sigma_1, \sigma_2, \ldots, \sigma_r)$ with $\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_r > 0$ .

$\mathbf{v}_k$ (right singular vector) is an active-array beam direction that maximally couples into the $k$ -th eigenmode.
$\mathbf{u}_k$ (left singular vector) is the RIS-side signature of the $k$ -th eigenmode — a specific phase profile across the RIS elements.
$\sigma_k$ is the coupling gain of the $k$ -th eigenmode.

In the near-field regime, $r \approx N_t$ (full rank in the smaller dimension), giving $N_t$ parallel eigenmodes.

Theorem: Eigenbeams Map to Users

Suppose the array-fed RIS has $\text{rank}(\mathbf{H}_1) = r$ . Then the RIS can form $r$ orthogonal eigenbeams, one per eigenmode of $\mathbf{H}_1$ . For $K \leq r$ users in known directions, the joint active+passive beamforming can deliver each user a dedicated eigenbeam with:

No inter-beam interference at the ideal solution (eigenmodes are orthogonal by SVD).
Per-user array gain: $|\sigma_k|^2$ from eigenmode coupling.
Per-user aperture gain: $N$ from coherent RIS focusing on user $k$ 's direction.

Total multiplexing capacity: $K$ parallel streams of rate $\log_2(1 + P_t \sigma_k^2 N / (\sigma^2 K))$ at the per-user level. Sum rate: $\sum_k \log_2(1 + \cdots)$ , which can exceed that of a conventional massive-MIMO system at the same $N_t$ .

Each eigenmode of $\mathbf{H}_1$ produces a different RIS-side signature $\mathbf{u}_k$ . The RIS can "steer" the $k$ -th eigenmode to a specific user via its phase-shift matrix $\boldsymbol{\Phi}$ . With $K$ users, we want to assign each user to one of the $r$ eigenmodes (one beam per user). When $r \geq K$ , this assignment is possible without interference crosstalk.

Proof

Active precoder aligned with eigenmodes

Choose active precoder $\mathbf{W} = \mathbf{V}_1^\star \text{diag}(p_k)$ where $p_k$ is power-allocation to eigenmode $k$ . Then $\mathbf{H}_1 \mathbf{W} = \mathbf{U}_1 \text{diag}(\sigma_k p_k) = \sum_k \sigma_k p_k \mathbf{u}_k \mathbf{e}_k^T$ . Each eigenmode appears as a distinct RIS-side excitation.

RIS phases focus per-user

Choose $\boldsymbol{\Phi}$ to focus eigenmode $k$ toward user $k$ 's direction: $\boldsymbol{\phi} = \sum_k \text{diag}(\mathbf{u}_k^*) \text{matched-filter}(\mathbf{h}_{k,2})$ . Since the $\mathbf{u}_k$ are orthogonal (left singular vectors), the focused beams are nearly orthogonal.

Per-user rate

User $k$ receives signal through the $k$ -th eigenmode with per-user SNR $\propto \sigma_k^2 N$ . Sum rate sums over $K = r$ eigenmodes. $\blacksquare$

Capacity of Array-Fed RIS is Near Massive-MIMO

Under ideal eigenmode alignment, the array-fed RIS capacity is

$C_{\text{AF-RIS}} = \sum_{k=1}^{\min(N_t, K)} \log_2\!\left(1 + \sigma_k^2 \cdot \frac{P_t\,N}{K \sigma^2}\right).$

Compare with a fully-digital massive-MIMO BS of equivalent active-antenna count $N_t$ (without RIS):

$C_{\text{mMIMO}} = \sum_{k=1}^{\min(N_t, K)} \log_2\!\left(1 + \lambda_k^2 \cdot \frac{P_t}{K \sigma^2}\right),$

where $\lambda_k$ are singular values of the BS-UE direct channel. The RIS adds the factor $N$ to each eigenmode's SNR — effectively doing the work of $\sqrt{N}$ additional antennas per eigenmode. At $N = 256$ : the "effective array size" is $N_t + 16 = 24$ antennas' worth of aperture. At $N = 1024$ : 32 antennas' worth.

Practical outcome: $N_t = 8$ + $N = 256$ delivers performance comparable to a $\sim 30$ -antenna fully digital array, at dramatically lower cost. This is the architectural case for array-fed RIS.

Eigenmode Spectrum of $\mathbf{H}_1$ vs. Geometry

Sweep the array-RIS distance $d_{\text{AR}}$ and plot the singular value spectrum of $\mathbf{H}_1$ . Near-field distances give many significant eigenvalues; as distance grows, the spectrum collapses to a few dominant values (far-field). The transition is the design sweet spot for array-fed RIS.

Parameters

Active array antennas

N_t

16

RIS elements

N

256

Array-RIS distance / wavelength50

Array aperture / wavelength10

Example: Eigenmode Count for a 28 GHz Design

$N_t = 8$ active antennas in a $10\lambda$ -aperture ULA, $N = 256$ RIS elements in a $50\lambda$ -aperture UPA, $d_{\text{AR}} = 3\lambda$ (about 3 cm at 28 GHz). Compute the effective number of eigenmodes.

Solution

Fraunhofer distance

$D_{\text{max}} = \max(10, 50) \cdot \lambda = 50\lambda$ . $d_F^{\text{AR}} = 2 D_{\text{max}}^2 / \lambda = 2 \cdot 2500\,\lambda = 5000\lambda$ . $d_{\text{AR}} = 3\lambda \ll d_F^{\text{AR}}$ ✓ (near-field).

Near-field DoF

$N_{\text{eff}} \sim D_A \cdot D_R / (d_{\text{AR}} \lambda) = 10 \cdot 50 / (3 \cdot 1) \approx 167$ modes. Since $N_t = 8 \ll N_{\text{eff}}$ , the bottleneck is the active array. Effective eigenmode count: $r = N_t = 8$ .

Streams

The array-fed RIS supports up to $r = 8$ parallel streams at this geometry. Compared to a far-field alternative (where $d_{\text{AR}} > d_F$ would force $r \sim 1$ rank-1), this is a $\sim 8\times$ improvement in multi-user multiplexing.

Practical

With $K = 8$ users, each served on their own eigenmode, the array-fed RIS provides 8 parallel streams at full $N$ -fold aperture gain — i.e., 8 users each seeing $\sim 256$ coherent elements. $\blacksquare$

Eigenmode Allocation for Array-Fed RIS

Complexity:

O(N N_t^{2})

for SVD +

O(NK)

for assignments

Input:

\mathbf{H}_1

, user channels

\{\mathbf{h}_{k,2}\}_{k=1}^{K}

, total power

P_t

.

Output:

\mathbf{W}, \boldsymbol{\Phi}

.

1. Eigenmode setup:

\mathbf{H}_1 = \mathbf{U}_1 \boldsymbol{\Sigma}_1 \mathbf{V}_1^H

via SVD.

Keep

r = \min(N_t, N)

dominant eigenmodes.

2. User-eigenmode assignment: match each user to one eigenmode (e.g., by strength in

\mathbf{u}_k^H \mathbf{h}_{k,2}

).

3. Active precoder:

\mathbf{W} = \mathbf{V}_1 \text{diag}(p_k) \text{diag}(\sqrt{P_k})

,

where

P_k

is water-filling power to eigenmode

k

.

4. RIS phases: for each user

k

, compute matched-filter phase

\phi_n^{(k)} = e^{-j\arg[(\mathbf{u}_k)_n^* (\mathbf{h}_{k,2})_n]}

.

Combine via weighted sum (weighted by power allocation).

5. return

\mathbf{W}, \boldsymbol{\Phi}

.

The algorithm is near-closed-form thanks to the eigenmode structure. No AO outer loop is needed once $\mathbf{H}_1$ 's SVD is computed — a major computational simplification over general RIS optimization. This is the central algorithmic payoff of the array-fed RIS architecture.

Common Mistake: In the Far-Field, Rank Collapses

Mistake:

"Far from the RIS, $\mathbf{H}_1$ is still a proper channel matrix; we should be fine."

Correction:

In the far-field, all active-array elements "see" the same spatial direction toward the RIS: $\mathbf{H}_1$ becomes a rank-1 outer product $\mathbf{a}_{\text{RIS}}(\Omega) \mathbf{a}_{\text{BS}}^H(\Omega)$ . Only ONE eigenmode exists; only ONE user can be served independently. Multi-user multiplexing through a single RIS in far-field requires rich per-user scattering on the RIS-UE side — which mmWave doesn't provide. Array-fed RIS is specifically designed around near-field geometry; far-field usage negates its raison d'être. Verify $d_{\text{AR}} \ll d_F^{\text{AR}}$ at design time.

Eigenmode Analysis of the BS-RIS Channel