Ferkans — Interactive Telecom Tutor

What Controls the Rank of a Reflective Channel?

Sections 21.1 and 21.2 analyzed the scalar link budget. To understand multiuser and multistream operation, we need the singular-value structure of the cascaded channel $\mathbf{H}_{\text{eff}} = \mathbf{H}_{\text{RIS-Rx}}\, \boldsymbol{\Phi}\, \mathbf{H}_{\text{Tx-RIS}}$ . Two questions shape the analysis: (i) what is the rank of $\mathbf{H}_{\text{eff}}$ , and (ii) how do its dominant singular values depend on $N_a$ , $N_{\text{RIS}}$ , and $\boldsymbol{\phi}$ ?

The answer that underlies the whole chapter is that the passive RIS cannot create rank. A diagonal matrix cannot raise the rank of the product beyond the rank of the narrowest factor. In the array-fed setting, the narrowest factor is almost always the active feed: $\mathbf{H}_{\text{eff}}$ has rank at most $\min(N_r, N_a)$ , no matter how many RIS elements we throw in. But the value of the non-zero singular values does grow with $N_{\text{RIS}}$ — the passive aperture concentrates energy onto the $N_a$ available modes. This distinction between rank and gain is the key to understanding both the benefits and the fundamental limits of the architecture.

Definition:
Effective Cascaded MIMO Channel

Consider an array-fed RIS transmitter with $N_a$ active elements feeding an $N_{\text{RIS}}$ -element passive RIS, and an $N_r$ -antenna receiver. The effective cascaded channel is

$\mathbf{H}_{\text{eff}}(\boldsymbol{\phi}) = \underbrace{\mathbf{H}_{\text{RIS-Rx}}}_{N_r \times N_{\text{RIS}}}\, \underbrace{\text{diag}(\boldsymbol{\phi})}_{N_{\text{RIS}} \times N_{\text{RIS}}}\, \underbrace{\mathbf{G}_f}_{N_{\text{RIS}} \times N_a} \quad \in \mathbb{C}^{N_r \times N_a},$

where $\boldsymbol{\phi}$ is the vector of unit-modulus RIS reflection coefficients. Writing $\text{diag}(\boldsymbol{\phi}) = \mathbf{D}(\boldsymbol{\phi})$ , a useful identity expresses $\mathbf{H}_{\text{eff}}$ as a linear function of $\boldsymbol{\phi}$ :

$\mathbf{H}_{\text{eff}}(\boldsymbol{\phi}) = \sum_{n=1}^{N_{\text{RIS}}} \phi_n\, \mathbf{r}_n\, \mathbf{g}_n^T,$

where $\mathbf{r}_n = [\mathbf{H}_{\text{RIS-Rx}}]_{:,n}$ and $\mathbf{g}_n^T = [\mathbf{G}_f]_{n,:}$ are the $n$ -th column/row of the two factor matrices. Thus $\mathbf{H}_{\text{eff}}$ is a linear combination of $N_{\text{RIS}}$ rank-one matrices with unit-modulus coefficients, which gives the design problem an elegant structure.

Theorem: Rank Upper Bound of the Effective Channel

For any RIS phase profile $\boldsymbol{\phi}$ ,

$\text{rank}(\mathbf{H}_{\text{eff}}(\boldsymbol{\phi})) \leq \min(N_r,\, N_a,\, N_{\text{RIS}}).$

In particular, when $N_a \leq \min(N_r, N_{\text{RIS}})$ , the effective rank is at most $N_a$ regardless of how many RIS elements are used.

Multiplying a full-rank matrix by a diagonal matrix cannot increase its rank. The active-feed coupling $\mathbf{G}_f$ is $N_{\text{RIS}} \times N_a$ , so its rank is at most $N_a$ . Left-multiplying by $\text{diag}(\boldsymbol{\phi})$ cannot help, and pre-multiplying by $\mathbf{H}_{\text{RIS-Rx}}$ only caps rank at $N_r$ . A passive RIS can redirect energy between the $N_a$ available modes but cannot create new spatial degrees of freedom.

Proof

Rank of a matrix product is bounded by the narrowest factor

For any matrices $\mathbf{A}, \mathbf{B}, \mathbf{C}$ of compatible sizes, $\text{rank}(\mathbf{ABC}) \leq \min\{\text{rank}\,\mathbf{A}, \text{rank}\,\mathbf{B}, \text{rank}\,\mathbf{C}\}$ .

Diagonal-phase matrix has full rank

$\text{diag}(\boldsymbol{\phi})$ is invertible because every diagonal entry $e^{j\phi_n}$ has unit modulus. So $\text{rank}(\text{diag}(\boldsymbol{\phi})) = N_{\text{RIS}}$ , which is the maximum possible.

Combine the three bounds

$\text{rank}(\mathbf{H}_{\text{eff}}) \leq \min\{N_r, N_{\text{RIS}}, N_a\}$ . When $N_a$ is the smallest, it dominates the bound. $\blacksquare$

Theorem: Dominant Singular Values Scale with $N_{ ext{RIS}}$

Let the forward coupling be normalized so that $\mathbf{G}_f^H \mathbf{G}_f = (N_{\text{RIS}}/N_a)\, \mathbf{I}_{N_a}$ (reactive-near-field approximation for a well-matched feed) and let $\mathbf{H}_{\text{RIS-Rx}}$ have i.i.d. zero-mean entries with variance $\sigma_r^2$ . For the optimal RIS phase profile that co-phases the dominant right singular vector of $\mathbf{H}_{\text{RIS-Rx}}$ with the corresponding column of $\mathbf{G}_f$ ,

$\sigma_1(\mathbf{H}_{\text{eff}}) \asymp \sigma_r\, \sqrt{\frac{N_{\text{RIS}}}{N_a}}\, \sqrt{N_{\text{RIS}} N_a} = \sigma_r\, N_{\text{RIS}}.$

The k-th singular value satisfies $\sigma_k(\mathbf{H}_{\text{eff}}) \asymp \sigma_r \sqrt{N_{\text{RIS}} N_a}/\sqrt{N_a} = \sigma_r N_{\text{RIS}}$ for $k \leq N_a$ (all are of the same order) and zero for $k > N_a$ .

Two mechanisms compound. First, each of the $N_a$ non-zero singular values of $\mathbf{H}_{\text{eff}}$ inherits an amplitude proportional to $\sqrt{N_{\text{RIS}}}$ from the aperture-gain side of the RIS reflection. Second, the RIS phase profile co-phases one specific direction, promoting that direction's amplitude by another factor of $\sqrt{N_{\text{RIS}}}$ . The resulting gain is quadratic in $N_{\text{RIS}}$ in power — exactly the passive-RIS aperture law — but confined to the $N_a$ -dimensional column space.

Proof

Factor the cascaded channel

Write $\mathbf{H}_{\text{eff}}(\boldsymbol{\phi}) = \mathbf{H}_{\text{RIS-Rx}} \mathbf{D}(\boldsymbol{\phi}) \mathbf{G}_f$ . The singular values of the product are bounded by $\sigma_k(\mathbf{H}_{\text{RIS-Rx}}) \cdot \sigma_k(\mathbf{D}) \cdot \sigma_k(\mathbf{G}_f)$ via the Horn inequality, but equality for the dominant mode is achievable by co-phasing.

Co-phasing the top mode

Let $\mathbf{v}_1 \in \mathbb{C}^{N_{\text{RIS}}}$ be the right dominant singular vector of $\mathbf{H}_{\text{RIS-Rx}}$ and $\mathbf{u}_1$ the corresponding column of $\mathbf{G}_f$ (by the reactive-near-field structure, $\mathbf{G}_f$ has a DFT-like set of columns). Setting $\phi_n = \arg(\mathbf{v}_1[n]^*\, \mathbf{u}_1[n]^*)$ makes the product coherent on mode 1, delivering amplitude $|\mathbf{v}_1|^H |\mathbf{u}_1|$ , which for random i.i.d. vectors of length $N_{\text{RIS}}$ grows as $N_{\text{RIS}}/\sqrt{N_a}$ .

Apply the $\mathbf{G}_f$ normalization

With the stated normalization, the dominant singular value becomes $\sigma_r N_{\text{RIS}}$ . The other $N_a - 1$ non-zero singular values share the remaining energy of the cascaded channel, all of the same order. $\blacksquare$

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Cascaded channel (RIS)

The end-to-end MIMO channel of an RIS-aided link, written as $\mathbf{H}_{\text{eff}} = \mathbf{H}_{\text{RIS-Rx}}\, \text{diag}(\boldsymbol{\phi})\, \mathbf{G}_f$ . The cascaded channel is bilinear in the RIS phase profile $\boldsymbol{\phi}$ and the active-feed coupling, has rank at most $\min(N_r, N_a, N_{\text{RIS}})$ , and is the central object of channel estimation, precoding, and capacity analysis for array-fed RIS architectures.

Aperture gain

The directivity boost of an aperture antenna of effective area $A_{\text{eff}}$ at wavelength $\lambda$ , given by $G_{\text{ap}} = 4\pi A_{\text{eff}}/\lambda^{2}$ . For a half-wavelength-spaced array of $N$ elements, $G_{\text{ap}} \approx N$ . For an array-fed RIS, the transmit aperture gain is set by $N_{\text{RIS}}$ even though only $N_a \ll N_{\text{RIS}}$ RF chains exist.

Eigenmode

A right singular vector of the channel matrix $\mathbf{H}$ , viewed as a "spatial mode" along which information can be transmitted at the gain given by the corresponding singular value $\sigma_k$ . SVD-based transmission decomposes the MIMO channel into parallel scalar eigenmode subchannels.

Related: Svd, Spatial Multiplexing

Key Takeaway

Rank is set by $N_a$ ; gain is set by $N_{\text{RIS}}$ . The array-fed RIS can support at most $N_a$ parallel streams, no matter how big the RIS is — that is the cost of moving all RF chains to the feed. But each of those streams enjoys an array gain proportional to $N_{\text{RIS}}$ , which is what makes the architecture worthwhile. To support $K$ users, pick $N_a \geq K$ (small); to boost SINR, pick $N_{\text{RIS}}$ large. The two design parameters are almost decoupled.

Singular-Value Spectrum of $\mathbf{H}_{\text{eff}}$

Plot the ordered singular values of $\mathbf{H}_{\text{eff}}(\boldsymbol{\phi})$ for a random Rayleigh reflected channel. See the sharp cutoff at $k = N_a$ predicted by Theorem TRank Upper Bound of the Effective Channel and watch the dominant singular values grow with $N_{\text{RIS}}$ as predicted by Theorem $N_{ ext{RIS}}$ $N_{e x t R I S}$ " data-ref-type="theorem">TDominant Singular Values Scale with $N_{ ext{RIS}}$ .

Parameters

N_{\text{RIS}}

256

N_a

8

N_r

16

\boldsymbol{\phi}

policy

Example: Rank and Gain for a Sample Configuration

Consider an array-fed RIS with $N_{\text{RIS}} = 1024$ , $N_a = 8$ , $N_r = 16$ . Assume $\mathbf{H}_{\text{RIS-Rx}}$ is i.i.d. $\mathcal{CN}(0, 1)$ . Compute (i) the rank of $\mathbf{H}_{\text{eff}}$ , (ii) the expected dominant singular value under the optimal phase profile, and (iii) the ratio $\sigma_1/\sigma_{N_a}$ (the condition number of the non-zero block).

Solution

Rank

By Theorem TRank Upper Bound of the Effective Channel, $\text{rank}(\mathbf{H}_{\text{eff}}) \leq \min(16, 8, 1024) = 8$ .

Dominant singular value

$\sigma_1 \asymp N_{\text{RIS}} = 1024$ (in the normalization where $\sigma_r = 1$ ). In dB, the dominant-mode power is $20 \log_{10}(1024) \approx 60$ dB over the baseline per-element channel.

Condition number

Because all $N_a = 8$ non-zero singular values scale as $N_{\text{RIS}}$ , the condition number is $\mathcal{O}(1)$ — not $\mathcal{O}(N_{\text{RIS}})$ . The architecture delivers eight comparably strong spatial modes, which is exactly what is needed for multiuser multiplexing. $\blacksquare$

Historical Note: Lens Arrays and Continuous-Aperture Radar

1950s–present

Before "reflectarray" and long before "RIS," the radar community built lens arrays — discrete or quasi-continuous dielectric lenses illuminated by a compact feed. The Luneburg lens (1944) is the classical example, and dielectric-lens antennas were widely used in X-band radars and Earth-station receivers through the 1970s. What metasurface technology added was electronic reconfigurability: the phase profile that a Luneburg lens imposes geometrically, a programmable reflectarray imposes electrically. Caire's array-fed RIS is in a direct line of descent from this tradition, and the eigenmode analysis of this section mirrors the "focal-plane array" analysis that mm-wave radar engineers have done for decades.

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Common Mistake: Thinking the RIS Provides Extra Spatial DoF

Mistake:

A common misreading is that $N_{\text{RIS}}$ effective antennas yield $N_{\text{RIS}}$ spatial degrees of freedom — so a 1024-element RIS supports 1024 simultaneous streams.

Correction:

The RIS is a diagonal transformation. It has $N_{\text{RIS}}$ knobs (the phases) but only $N_a$ modes because its rank is bounded by that of the active feed it illuminates. The DoF available for spatial multiplexing is $\min(N_a, N_r)$ , not $N_{\text{RIS}}$ . What the RIS does provide is per-mode array gain scaling with $N_{\text{RIS}}$ , which boosts SINR without unlocking extra streams. This distinction is subtle but central: confusing it produces massive overestimates of achievable sum rate and explains several early enthusiastic RIS papers.

Quick Check

An array-fed RIS has $N_{\text{RIS}} = 4096$ , $N_a = 4$ , $N_r = 64$ . How many simultaneous independent data streams can it support?

4096

64

4

$\sqrt{N_a N_{\text{RIS}}} = 128$