Ferkans — Interactive Telecom Tutor

The mmWave Scenario Demands LoS

The most compelling RIS deployments are at mmWave and sub-THz, where the direct BS–UE path is often blocked by obstacles. The RIS is mounted with a clear view to both the BS and the candidate UE area — meaning the BS–RIS and RIS–UE links themselves are line-of-sight. Under LoS, the channel matrices $\mathbf{H}_1, \mathbf{H}_2$ have a clean, deterministic, rank-1 structure that is easy to analyze and well worth the time to write down carefully.

Definition:
LoS Channel Between Two Arrays

Consider two uniform planar arrays (UPAs) in the far-field: array A of size $N_A$ at position $\mathbf{p}_A$ , and array B of size $N_B$ at $\mathbf{p}_B$ . Let $\mathbf{a}_A(\Omega_A) \in \mathbb{C}^{N_A}$ be the steering vector of array A at direction $\Omega_A$ (azimuth and elevation), and similarly $\mathbf{a}_B(\Omega_B)$ . The LoS channel from A to B is

$\mathbf{H}_{\text{LoS}} = \alpha_\text{LoS}\, \mathbf{a}_B(\Omega_B)\, \mathbf{a}_A^H(\Omega_A),$

a rank-1 outer product, where $\alpha_\text{LoS}$ is the complex path gain. Here $\Omega_A = \Omega_A^{(AB)}$ is the angle of departure from A in the direction of B, and $\Omega_B = \Omega_B^{(BA)}$ the angle of arrival at B from the direction of A.

Theorem: The LoS Cascaded Channel Is Rank-1

Under pure LoS conditions on both hops:

$\mathbf{H}_1 = \alpha_1\, \mathbf{a}_{\text{RIS}}(\Omega_{\text{RIS}}^{\text{BS}})\, \mathbf{a}_{\text{BS}}^H(\Omega_{\text{BS}}^{\text{RIS}}), \qquad \mathbf{H}_2 = \alpha_2\, \mathbf{a}_{\text{UE}}(\Omega_{\text{UE}}^{\text{RIS}})\, \mathbf{a}_{\text{RIS}}^H(\Omega_{\text{RIS}}^{\text{UE}}),$

the cascaded channel is

$\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1 = \alpha_1 \alpha_2\, \big[\mathbf{a}_{\text{RIS}}^H(\Omega_{\text{RIS}}^{\text{UE}})\, \boldsymbol{\Phi}\, \mathbf{a}_{\text{RIS}}(\Omega_{\text{RIS}}^{\text{BS}})\big] \mathbf{a}_{\text{UE}}(\Omega_{\text{UE}}^{\text{RIS}})\, \mathbf{a}_{\text{BS}}^H(\Omega_{\text{BS}}^{\text{RIS}}).$

This is the product of a scalar (the bracketed quantity in $\boldsymbol{\Phi}$ ) and a rank-1 outer product of UE- and BS-side steering vectors. The RIS's role reduces to tuning the scalar via $\boldsymbol{\phi}$ .

Proof

Substitute the rank-1 forms

$\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1 = \alpha_2 \mathbf{a}_{\text{UE}} \mathbf{a}_{\text{RIS}}^H(\Omega_{\text{RIS}}^{\text{UE}}) \boldsymbol{\Phi} \cdot \alpha_1 \mathbf{a}_{\text{RIS}}(\Omega_{\text{RIS}}^{\text{BS}}) \mathbf{a}_{\text{BS}}^H$ .

Extract the scalar

$\mathbf{a}_{\text{RIS}}^H(\Omega_{\text{RIS}}^{\text{UE}}) \boldsymbol{\Phi} \mathbf{a}_{\text{RIS}}(\Omega_{\text{RIS}}^{\text{BS}})$ is a $1 \times 1$ complex scalar. Move it to the front: $\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1 = \alpha_1 \alpha_2\, s(\boldsymbol{\phi})\, \mathbf{a}_{\text{UE}}\mathbf{a}_{\text{BS}}^H$ , with $s(\boldsymbol{\phi}) = \mathbf{a}_{\text{RIS}}^H(\Omega_{\text{RIS}}^{\text{UE}}) \boldsymbol{\Phi} \mathbf{a}_{\text{RIS}}(\Omega_{\text{RIS}}^{\text{BS}}) = \sum_n \phi_n a_{\text{RIS},n}(\Omega_{\text{RIS}}^{\text{BS}}) a_{\text{RIS},n}^*(\Omega_{\text{RIS}}^{\text{UE}})$ .

Optimal $\boldsymbol{\phi}$

Maximizing $|s(\boldsymbol{\phi})|$ under $|\phi_n| = 1$ is again a matched-filter problem (Section 1.3). The optimum is $\phi_n^\star = e^{j\arg(a_{\text{RIS},n}(\Omega_{\text{RIS}}^{\text{UE}}) a_{\text{RIS},n}^*(\Omega_{\text{RIS}}^{\text{BS}}))}$ , $|s^\star| = N$ , giving the $N^2$ SNR scaling. $\blacksquare$

Example: Closed-Form Phases for a UPA

An $N_1 \times N_2$ UPA RIS at the origin communicates with a BS at direction $(\theta_1, \phi_1)$ and a UE at $(\theta_2, \phi_2)$ . Half-wavelength element spacing, broadside reference. Give the optimal phase profile $\theta_n^\star$ for the coherent RIS link.

Solution

Element indexing

Label element $(n_1, n_2) \in \{0, \ldots, N_1 - 1\} \times \{0, \ldots, N_2 - 1\}$ . Steering-vector phase from direction $(\theta, \phi)$ : $\psi_{n_1, n_2}(\theta, \phi) = \pi[n_1 \sin\theta\cos\phi + n_2 \sin\theta\sin\phi]$ .

Optimal RIS phase

From Theorem 3.2's optimum, $\theta_{n_1,n_2}^\star = \psi_{n_1,n_2}(\theta_2, \phi_2) - \psi_{n_1,n_2}(\theta_1, \phi_1) = \pi[n_1(\sin\theta_2\cos\phi_2 - \sin\theta_1\cos\phi_1) + n_2(\sin\theta_2\sin\phi_2 - \sin\theta_1\sin\phi_1)]$ .

Interpretation

The optimal phase profile is a linear function of the element index — a 2D sawtooth pattern. This is exactly the phased-array beamsteering phase, extended to two endpoints (one for the incoming beam from the BS, one for the outgoing beam to the UE). The RIS is implementing a generalized reflection along the line $(\Omega_{\text{RIS}}^{\text{BS}}, \Omega_{\text{RIS}}^{\text{UE}})$ .

Keyhole: The LoS Rank-1 Curse

Pure LoS on both hops means the cascaded channel is rank-1, regardless of $N_t, N_r$ . Consequences:

Multiplexing is impossible: the channel supports a single spatial stream only. No matter how many BS or UE antennas, $\min(\text{rank}(\mathbf{H}_{\text{eff}}), N_t, N_r) = 1$ .
Multi-user MU-MIMO is severely limited: all users share the same rank-1 cascaded direction $\mathbf{a}_{\text{BS}}^H$ from the BS side, so spatial multiplexing through the RIS works only if users differ in their UE-side directions $\mathbf{a}_{\text{UE}}^{(k)}$ . Even then, the scalar $s(\boldsymbol{\phi})$ is shared — it takes a separate per-user analysis (Chapter 7).

The practical escape: add scatterers. Even one scatterer per hop introduces a second rank direction. At mmWave this happens naturally from reflections off nearby walls and furniture; chapter 3 of MIMO book's discussion of sparse mmWave scatterers applies directly.

RIS Beampattern: Incident and Reflected Beams

Set the angle-of-arrival from the BS and the desired angle-of-departure toward the UE. The plot shows the resulting RIS beam pattern as a function of azimuth, with the desired-UE direction marked. Notice that the RIS can redirect the main lobe anywhere in the forward half-space, subject to gain falloff at steep angles.

Parameters

Horizontal elements

N_1

16

Vertical elements

N_2

16

BS angle-of-arrival

\theta_1

(deg)-20

UE angle-of-departure

\theta_2

(deg)30

RIS Beam Steering via Phase Profile

Animation of the RIS beam rotating as the desired UE direction

\theta_2

sweeps from

-45^\circ

to

+45^\circ

. The phase profile (shown in color on the RIS surface) is a linear slope whose gradient matches the desired angle; the resulting reflected beam tracks smoothly.

Theorem: SNR at the Optimum for Two LoS Hops

For the pure LoS cascaded channel of Theorem 3.2, under a unit-norm BS beamformer $\mathbf{v}^\star = \mathbf{a}_{\text{BS}}/\|\mathbf{a}_{\text{BS}}\| = \mathbf{a}_{\text{BS}}/\sqrt{N_t}$ (matched filter at BS) and a unit-norm UE combiner $\mathbf{v}_{\text{UE}}^\star = \mathbf{a}_{\text{UE}}/\sqrt{N_r}$ , the maximum received SNR over all unit-modulus $\boldsymbol{\Phi}$ is

$\text{SNR}^\star_{\text{LoS}} = \frac{P_t}{\sigma^2}\, N_t\, N_r\, N^2\, |\alpha_1|^2 |\alpha_2|^2.$

The three scaling factors are: $N_t$ from BS beamforming gain, $N_r$ from UE aperture gain, and $N^2$ from coherent RIS combining.

Proof

Plug in beamformer

$\mathbf{a}_{\text{BS}}^H \mathbf{v}^\star = \|\mathbf{a}_{\text{BS}}\|^2/\sqrt{N_t} = N_t/\sqrt{N_t} = \sqrt{N_t}$ . Similarly $\mathbf{v}_{\text{UE}}^{\star H} \mathbf{a}_{\text{UE}} = \sqrt{N_r}$ .

Plug into the cascaded form

Using Theorem 3.2, the scalar gain at the UE combiner is $\alpha_1 \alpha_2 s(\boldsymbol{\phi}) \sqrt{N_r N_t}$ , where $s(\boldsymbol{\phi})$ is the RIS phase sum.

Optimize over $\boldsymbol{\phi}$

$|s^\star| = N$ by matched filtering. Squaring and dividing by noise: $\text{SNR}^\star = |\alpha_1 \alpha_2 N \sqrt{N_rN_t}|^2 P_t/\sigma^2 = N_tN_r N^2 |\alpha_1|^2 |\alpha_2|^2 P_t/\sigma^2$ . $\blacksquare$

Quick Check

In a pure LoS RIS deployment with $N_t = 8, N_r = 4, N = 256$ , the maximum number of independent spatial streams the cascaded channel can carry is:

1

4

8

256

Correction:

1

Pure LoS on both hops gives rank-1 $\mathbf{H}_1$ and $\mathbf{H}_2$ , so the cascaded product is rank-1 regardless of $N_t, N_r, N$ . This is the keyhole phenomenon.

⚠️Engineering Note

LoS Geometry Determines RIS Angle Range

RIS beam-steering is only effective in the forward half-space — the surface cannot reflect back toward directions behind it. Practical consequences:

RIS orientation matters. Deploy the RIS so that the BS and the target UE area both lie within ~ $60^\circ$ of broadside.
Element gain falls with angle. A typical patch antenna has $\cos^p \theta$ gain with $p \approx 1$ – $2$ . At $60^\circ$ from broadside, gain drops by $3$ – $6\text{ dB}$ .
Wrap-around deployment. For coverage in all directions, consider multiple RIS panels on orthogonal facets (Chapter 12).

Practical Constraints

•
Practical steering range: $\pm 60^\circ$ from broadside per face.
•
Per-element directivity: $\sim 5\text{ dB}$ at broadside, $\sim 0\text{ dB}$ at $60^\circ$ .
•
Panel orientation tolerance: $\pm 5^\circ$ before beam distortion is noticeable.

LoS RIS Channel: Steering Vectors and Rank