Ferkans — Interactive Telecom Tutor

One Model to Rule Them All

We now present the cascaded channel model in full generality: multi-antenna BS, multi-element RIS, multi-antenna UE. The same model covers single-user and multi-user scenarios (with appropriate stacking), single-RIS and multi-RIS (with appropriate summation), narrowband and wideband (per subcarrier). Every optimization problem in this book — rate maximization, phase shift design, channel estimation, localization — starts from the formulas in this section. Nail them down now, and the rest of the book reads as a series of optimization tricks applied to these formulas.

Definition:
Multi-Antenna Cascaded Channel

Consider a BS with $N_t$ antennas, an RIS with $N$ elements, and a UE with $N_r$ antennas. Define:

$\mathbf{H}_1 \in \mathbb{C}^{N \times N_t}$ : BS-to-RIS channel. Rows index RIS elements; columns index BS antennas.
$\mathbf{H}_2 \in \mathbb{C}^{N_r \times N}$ : RIS-to-UE channel. Rows index UE antennas; columns index RIS elements.
$\mathbf{H}_d \in \mathbb{C}^{N_r \times N_t}$ : direct BS-to-UE channel (possibly zero).
$\boldsymbol{\Phi} = \text{diag}(e^{j\theta_1}, \ldots, e^{j\theta_N})$ : the RIS phase-shift matrix.

The effective end-to-end channel is the $N_r \times N_t$ matrix

$\boxed{\;\mathbf{H}_{\text{eff}} \;=\; \mathbf{H}_d \;+\; \mathbf{H}_2\, \boldsymbol{\Phi}\, \mathbf{H}_1\;}$

The received signal under BS transmit signal $\mathbf{x} \in \mathbb{C}^{N_t}$ with $\mathbb{E}[\mathbf{x}\mathbf{x}^H] = P_t \mathbf{Q}$ (covariance normalized by total power) is

$\mathbf{y} = \mathbf{H}_{\text{eff}}\, \mathbf{x} + \mathbf{w}, \qquad \mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I}_{N_r}).$

Every dimension in $\mathbf{H}_{\text{eff}}$ has an interpretation: rows are the UE's spatial dimensions (which antenna receives the signal) and columns are the BS's (which BS antenna transmits the signal). The RIS is "in the middle" — it does not add dimensions, it modulates the dimensions it has.

,

Dimensions of the Cascaded Channel — Block diagram showing the dimensions of $\mathbf{H}_1, \boldsymbol{\Phi}, \mathbf{H}_2$ and their product $\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1$ . The inner dimension $N$ is where the RIS acts; the outer dimensions $N_t$ and $N_r$ are inherited from the BS and UE hardware.

Theorem: Rank of the Cascaded Channel

For the cascaded product $\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1$ ,

$\text{rank}(\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1) \leq \min\!\left(\text{rank}(\mathbf{H}_1),\ N,\ \text{rank}(\mathbf{H}_2)\right).$

In particular, in a pure LoS scenario where both $\mathbf{H}_1$ and $\mathbf{H}_2$ are rank-1 outer products of steering vectors, the cascaded product is also rank-1, regardless of $N$ . A high-rank (rich-scattering) link is needed on both sides to get a multi-stream RIS channel.

The cascade can at most propagate as many independent streams as the narrowest channel admits. If the BS–RIS path has only a few independent directions, adding a high-rank RIS–UE path doesn't help — the bottleneck is upstream.

Proof

Rank of a product

$\text{rank}(\mathbf{AB}) \leq \min(\text{rank}(\mathbf{A}), \text{rank}(\mathbf{B}))$ is standard linear algebra.

Apply to the cascade

Setting $\mathbf{A} = \mathbf{H}_2 \boldsymbol{\Phi}, \mathbf{B} = \mathbf{H}_1$ : $\text{rank}(\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1) \leq \min(\text{rank}(\mathbf{H}_2 \boldsymbol{\Phi}), \text{rank}(\mathbf{H}_1))$ . Since $\boldsymbol{\Phi}$ is invertible ( $|\phi_n| = 1 \neq 0$ ), $\text{rank}(\mathbf{H}_2 \boldsymbol{\Phi}) = \text{rank}(\mathbf{H}_2)$ . Hence the bound.

Keyhole implication

LoS with planar wavefronts yields rank-1 matrices: $\mathbf{H}_1 = \mathbf{a}_{\text{RIS}} \mathbf{a}_{\text{BS}}^H \cdot h_{1,\text{gain}}$ , similarly for $\mathbf{H}_2$ . Product is rank-1. This is the RIS version of the well-known keyhole channel — a cascade of two rank-1 channels always gives a rank-1 end-to-end channel, which limits multi-antenna multiplexing gain. $\blacksquare$

Key Takeaway

The cascaded channel is linear in $\boldsymbol{\phi}$ and in $\mathbf{W}$ separately — but not jointly. The product $\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1$ is linear in $\boldsymbol{\phi}$ for any fixed $\mathbf{H}_1, \mathbf{H}_2$ , and the resulting $\mathbf{H}_{\text{eff}} \mathbf{W}$ is linear in $\mathbf{W}$ for any fixed $\boldsymbol{\Phi}$ . This bi-linear structure is what makes alternating optimization (Chapter 5) such a natural algorithm: fix one, optimize the other.

Theorem: Vectorized Cascaded Channel

Using the identity $\text{diag}(\boldsymbol{\phi})\mathbf{A} = \text{diag}(\boldsymbol{\phi}) \mathbf{A}$ and the vec/Kronecker identities, the vectorized effective channel is

$\text{vec}(\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1) = \big(\mathbf{H}_1^T \odot_{\text{col}} \mathbf{H}_2\big) \boldsymbol{\phi},$

where $\odot_{\text{col}}$ denotes the column-wise Khatri–Rao product: the $n$ -th column of $\mathbf{H}_2$ times the $n$ -th row of $\mathbf{H}_1^T$ , vectorized. Concretely, $\mathbf{H}_1^T \odot_{\text{col}} \mathbf{H}_2$ is the $N_t\,N_r \times N$ matrix whose $n$ -th column is $(\mathbf{H}_1)_{n,:}^T \otimes (\mathbf{H}_2)_{:,n} \in \mathbb{C}^{N_tN_r}$ .

The product $\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1$ can be rewritten using the vec-diag identity so that $\boldsymbol{\phi}$ appears as a linear factor of a known matrix. This is the form preferred by optimization algorithms.

Proof

Write the product as a sum

$\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1 = \sum_n \phi_n (\mathbf{H}_2)_{:,n}\,(\mathbf{H}_1)_{n,:}$ , a sum of rank-1 matrices weighted by $\phi_n$ .

Vectorize

$\text{vec}\big(\sum_n \phi_n (\mathbf{H}_2)_{:,n} (\mathbf{H}_1)_{n,:}\big) = \sum_n \phi_n \text{vec}\big((\mathbf{H}_2)_{:,n} (\mathbf{H}_1)_{n,:}\big) = \sum_n \phi_n \big((\mathbf{H}_1)_{n,:}^T \otimes (\mathbf{H}_2)_{:,n}\big)$ .

Matrix-vector form

The sum is a matrix-vector product: columns of the matrix are the $\text{vec}$ -Kronecker factors, vector is $\boldsymbol{\phi}$ . This is precisely the Khatri–Rao structure stated. $\blacksquare$

Example: MISO-RIS in Vectorized Form

For a single-antenna UE, $\mathbf{H}_2 \to \mathbf{h}_2^T \in \mathbb{C}^{1 \times N}$ and $\mathbf{H}_{\text{eff}} = \mathbf{h}_d^T + \mathbf{h}_2^T \boldsymbol{\Phi} \mathbf{H}_1 \in \mathbb{C}^{1 \times N_t}$ . Rewrite this using the Khatri-Rao/vec identity.

Solution

Transpose setup

$\mathbf{h}_2^T \boldsymbol{\Phi} \mathbf{H}_1 = (\mathbf{h}_2 \odot \mathbf{1})^T \text{diag}(\boldsymbol{\phi}) \mathbf{H}_1$ — but the cleanest form uses $\boldsymbol{\phi}^T$ : $\mathbf{h}_2^T \boldsymbol{\Phi} \mathbf{H}_1 = \boldsymbol{\phi}^T (\mathbf{h}_2 \odot \mathbf{H}_1 \text{ along rows})$ .

Apply the vector identity

Introducing $\mathbf{G} = \text{diag}(\mathbf{h}_2) \mathbf{H}_1 \in \mathbb{C}^{N \times N_t}$ , we have $\mathbf{h}_2^T \boldsymbol{\Phi} \mathbf{H}_1 = \boldsymbol{\phi}^T \mathbf{G}$ . For a specific beamformer $\mathbf{v}$ , the end-to-end scalar gain is $\mathbf{h}_{\text{eff}}^T \mathbf{v} = \mathbf{h}_d^T \mathbf{v} + \boldsymbol{\phi}^T \mathbf{G} \mathbf{v}$ — linear in $\boldsymbol{\phi}$ , a convenient form for optimization.

Multi-User Stacking

For $K$ users each with a single antenna: stack their $\mathbf{h}_2^{(k)}$ vectors into a matrix $\mathbf{H}_2^{\text{MU}} \in \mathbb{C}^{K \times N}$ (one row per user). The cascaded channel becomes the $K \times N_t$ matrix $\mathbf{H}_{\text{eff}}^{\text{MU}} = \mathbf{H}_d^{\text{MU}} + \mathbf{H}_2^{\text{MU}} \boldsymbol{\Phi} \mathbf{H}_1$ , and multi-user precoding uses this matrix the same way it uses any multi-user channel. Chapter 7 develops the multi-user algorithms; the model is an immediate extension of what we have here.

Singular Value Spectrum of the Cascaded Channel

Set the rank of $\mathbf{H}_1$ (by choosing how many scattering clusters feed the BS–RIS link) and of $\mathbf{H}_2$ . Observe the singular values of the product $\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1$ : the number of significant values is limited by the minimum of the two factor ranks. Only when both sides have sufficient scattering can the cascade support multiple streams.

Parameters

BS antennas

N_t

8

RIS elements

N

64

UE antennas

N_r

4

Rank of

\mathbf{H}_1

4

Rank of

\mathbf{H}_2

4

Common Mistake: Getting the Conjugation Right in Multi-Antenna Cascades

Mistake:

Using $\boldsymbol{\Phi}^H$ instead of $\boldsymbol{\Phi}$ when simplifying the multi-antenna cascade.

Correction:

$\boldsymbol{\Phi}$ is a diagonal matrix of (non-conjugated) phase shifts $e^{j\theta_n}$ . The cascade is always written as $\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1$ — no Hermitian on $\boldsymbol{\Phi}$ . Misplacing the Hermitian inside the cascade flips the sign of every phase shift, sending every subsequent optimization in the wrong direction. This is a common source of bugs in RIS codebases — always sanity-check with the single-antenna identity of TDiagonal Factorization of the Cascaded Channel.

The Cascaded Channel, Dimensions, and Linearity

One Model to Rule Them All

Definition: Multi-Antenna Cascaded Channel

Dimensions of the Cascaded Channel

Theorem: Rank of the Cascaded Channel

Rank of a product

Apply to the cascade

Keyhole implication

Key Takeaway

Theorem: Vectorized Cascaded Channel

Write the product as a sum

Vectorize

Matrix-vector form

Example: MISO-RIS in Vectorized Form

Transpose setup

Apply the vector identity

Multi-User Stacking

Singular Value Spectrum of the Cascaded Channel

Parameters

Common Mistake: Getting the Conjugation Right in Multi-Antenna Cascades

Definition:
Multi-Antenna Cascaded Channel