Ferkans — Interactive Telecom Tutor

Real RIS Channels Are Not Pure LoS

At sub-6 GHz or in cluttered indoor mmWave, both hops of the RIS link see multipath: reflections off walls, scattering from furniture, weak blockage. The channel is a superposition of a strong LoS ray and many weaker non-LoS (NLoS) components. This section models that structure: Ricean fading per hop, spatial correlation across RIS elements, and the cascaded rank / keyhole implications for end-to-end performance.

Definition:
Ricean Fading on Both Hops

Each hop of the RIS link is modelled as a Ricean-fading channel with K-factor $K_1, K_2$ :

$\mathbf{H}_1 = \sqrt{\frac{K_1}{K_1 + 1}}\, \mathbf{H}_1^{\text{LoS}} + \sqrt{\frac{1}{K_1 + 1}}\, \mathbf{H}_1^{\text{NLoS}},$

where $\mathbf{H}_1^{\text{LoS}}$ is the rank-1 deterministic LoS component (Section 3.2) and $\mathbf{H}_1^{\text{NLoS}}$ is a zero-mean complex Gaussian matrix capturing multipath scattering. Similarly for $\mathbf{H}_2$ . Large $K$ (pure LoS) collapses to the deterministic rank-1 model; $K = 0$ gives pure Rayleigh fading.

Keyhole Channel

A cascaded channel of the form $\mathbf{H}_{\text{eff}} = \mathbf{h}_2 \alpha \mathbf{h}_1^H$ where $\mathbf{h}_1, \mathbf{h}_2$ are vectors. Keyhole channels have rank 1 — the signal must pass through a "keyhole" that limits the spatial degrees of freedom. LoS-on-both-hops RIS channels are exactly keyhole channels; LoS-plus- multipath RIS channels are approximately keyhole at high K-factor.

Theorem: Cascaded Rayleigh: The Double-Rayleigh Distribution

For a single-antenna BS ( $N_t = 1$ ) and single-antenna UE ( $N_r = 1$ ) with independent Rayleigh fading on both hops: $\mathbf{h}_1 \sim \mathcal{CN}(\mathbf{0}, \sigma_1^2 \mathbf{I}_N), \mathbf{h}_2 \sim \mathcal{CN}(\mathbf{0}, \sigma_2^2 \mathbf{I}_N)$ and a random-phase RIS, the cascaded gain is $z = \mathbf{h}_2^H \boldsymbol{\Phi} \mathbf{h}_1 = \sum_n e^{j\theta_n} h_{1,n} h_{2,n}^*$ . Its magnitude squared has mean

$\mathbb{E}|z|^2 = N \sigma_1^2 \sigma_2^2,$

but tail behaviour heavier than Rayleigh — the PDF is related to a modified Bessel function $K_0$ . The implication: outage probability for cascaded Rayleigh RIS is considerably higher than for a direct Rayleigh channel of the same mean SNR.

When both hops are Rayleigh-faded and independent, the cascaded end-to-end scalar gain is the product of two complex Gaussians. This is not itself Gaussian — it has heavier tails, smaller mean squared magnitude, and leads to worse outage behaviour than a single Rayleigh fade of the same average power.

Proof

Scalar gain

Let $x_n = h_{1,n} h_{2,n}^*$ . For independent $h_{1,n} \sim \mathcal{CN}(0, \sigma_1^2)$ and $h_{2,n}^* \sim \mathcal{CN}(0, \sigma_2^2)$ , $x_n$ is a product of two independent complex Gaussians — the "double-Rayleigh" or "complex double-Gaussian" distribution.

Moments

$\mathbb{E}[x_n] = \mathbb{E}[h_{1,n}] \mathbb{E}[h_{2,n}^*] = 0$ . $\mathbb{E}|x_n|^2 = \mathbb{E}|h_{1,n}|^2 \mathbb{E}|h_{2,n}|^2 = \sigma_1^2 \sigma_2^2$ . By independence across $n$ and random phases, $\mathbb{E}|z|^2 = \sum_n |e^{j\theta_n}|^2 \mathbb{E}|x_n|^2 = N \sigma_1^2 \sigma_2^2$ .

Distribution of $|z|$

For large $N$ , by the CLT, $z \to \mathcal{CN}(0, N \sigma_1^2 \sigma_2^2)$ — which is Rayleigh in magnitude. For moderate $N$ , the distribution retains the heavier-tailed "double-Rayleigh" character of individual terms, causing higher outage probability. See Simon & Alouini (2005) for the analytic form of the double-Rayleigh PDF. $\blacksquare$

CLT Rescues Large- $N$ RIS

The heavier-tailed double-Rayleigh distribution is a real concern — at small $N$ . For $N = 4$ or $N = 8$ , the fat tails of cascaded Rayleigh lead to noticeable outage penalties. But for practical RIS sizes ( $N \geq 256$ ), the CLT kicks in: the sum of $N$ independent double-Rayleigh terms is, to excellent approximation, Gaussian with the stated mean and variance. Small- $N$ RIS has unfriendly fading; large- $N$ RIS behaves like a well-behaved Rayleigh channel with scale $\sqrt{N \sigma_1^2 \sigma_2^2}$ . Another argument, if one were needed, for designing with $N$ in the hundreds.

Definition:
Spatial Correlation of an RIS in a Scattering Field

When the RIS is immersed in a rich-scattering field (e.g., the NLoS component of $\mathbf{H}_1$ under large $K$ or a near-resonance regime), adjacent elements receive correlated signals. The spatial correlation matrix of the incident field at the RIS is

$[\mathbf{R}_{\text{RIS}}]_{n,m} = \text{sinc}\!\left(\frac{2 \|\mathbf{r}_n - \mathbf{r}_m\|}{\lambda}\right),$

where $\mathbf{r}_n$ is the position of element $n$ . For $\lambda/2$ -spaced elements, adjacent-element correlation is $\text{sinc}(1) \approx 0$ — uncorrelated! This is the 3D isotropic-scattering assumption. At tighter spacings or in structured scattering environments, correlation is nonzero and reduces the effective number of independent channel realizations.

The half-wavelength spacing of a typical RIS is not arbitrary: it is the spacing that makes i.i.d. scattering approximately valid. Tighter spacing buys more elements but trades away independence; the effective degrees of freedom of the RIS as a correlated array are upper-bounded by the rank of $\mathbf{R}_{\text{RIS}}$ .

Theorem: Effective Degrees of Freedom via Correlation Rank

The effective number of independent RIS elements in a scattering field is

$N_{\text{eff}} = \left(\sum_k \lambda_k\right)^2 / \sum_k \lambda_k^2 = \frac{\text{tr}(\mathbf{R}_{\text{RIS}})^2}{\|\mathbf{R}_{\text{RIS}}\|_F^2},$

where $\{\lambda_k\}$ are the eigenvalues of $\mathbf{R}_{\text{RIS}}$ . For i.i.d. elements ( $\mathbf{R}_{\text{RIS}} = \mathbf{I}$ ), $N_{\text{eff}} = N$ . For fully correlated ( $\mathbf{R}_{\text{RIS}} = \mathbf{1}\mathbf{1}^T$ ), $N_{\text{eff}} = 1$ . Real RIS panels at half-wavelength spacing typically have $N_{\text{eff}} / N > 0.9$ — correlation is mild.

Proof

Participation ratio

The quantity $\big(\sum_k \lambda_k\big)^2 / \sum_k \lambda_k^2$ is the participation ratio of the eigenvalue distribution. It equals $N$ when eigenvalues are uniform (i.i.d. case) and equals 1 when one eigenvalue dominates (fully correlated).

Equivalent formula

Using $\sum_k \lambda_k = \text{tr}(\mathbf{R}_{\text{RIS}})$ and $\sum_k \lambda_k^2 = \|\mathbf{R}_{\text{RIS}}\|_F^2 = \text{tr}(\mathbf{R}_{\text{RIS}}^2)$ . $\blacksquare$

Spatial Correlation Map and Effective DoF

Set the element spacing and the scattering richness (effective angle-spread). The heatmap shows $|\mathbf{R}_{\text{RIS}}|$ ; the text shows $N_{\text{eff}}$ . Half-wavelength spacing gives near-diagonal $\mathbf{R}$ ; tighter spacing makes off-diagonals significant.

Parameters

RIS elements

N

64

Spacing (in units of

\lambda

)0.5

Angular spread of scattering (deg)30

Example: Outage Probability for a Random-Phase RIS

A single-antenna BS and UE communicate through an $N = 16$ RIS under cascaded Rayleigh fading, $\sigma_1 = \sigma_2 = 1$ , $P_t/\sigma^2 = 10\text{ dB}$ , random RIS phases. What is the outage probability at target SNR $\text{SNR}_{T} = 0\text{ dB}$ ? Compare to a direct Rayleigh link of the same average SNR.

Solution

Average SNR

$\bar{\text{SNR}} = (P_t/\sigma^2) \cdot N \sigma_1^2 \sigma_2^2 = 10 \cdot 16 = 160 \approx 22\text{ dB}$ .

Outage under CLT approximation

By CLT at $N = 16$ , $|z|^2 \approx \text{Exp}(\text{mean} = 16)$ . $\Pr[\text{SNR} < 0\text{ dB}] = \Pr[|z|^2 < 0.1] = 1 - e^{-0.1/16} \approx 0.0062$ — a modest outage of $0.6\%$ .

Direct Rayleigh comparison

A direct Rayleigh link with the same mean SNR (22 dB): $\Pr[\text{SNR} < 0\text{ dB}] = 1 - e^{-0.1/160} \approx 6.25 \cdot 10^{-4}$ — an outage of $0.06\%$ .

Interpretation

At $N = 16$ the cascaded channel's fatter tails give $\sim 10\times$ worse outage than a direct link of the same mean SNR — the keyhole penalty. At $N = 256$ the CLT makes the cascaded channel effectively as well-behaved as direct Rayleigh.

Historical Note: The Keyhole Discovery

2001–2005

Keyhole channels were observed experimentally in the early 2000s (Chizhik et al., 2002) in MIMO measurements across hallways — the effective signal path was constrained by a door or other narrow aperture, and the MIMO capacity dropped below the i.i.d. prediction despite rich scattering on either side. The theoretical understanding matured with Gesbert et al. (2002) and Almers et al. (2005), who showed that cascaded narrow-aperture scenarios universally yield rank-1 end-to-end channels. The RIS keyhole (rank-1 under LoS on both hops) is the same phenomenon, now a deliberate design choice rather than an accident of geometry.

Common Mistake: Don't Ignore Correlation at Tight Spacings

Mistake:

"Assume independent elements regardless of spacing — the CLT works for any i.i.d. distribution."

Correction:

The CLT requires independence, which presupposes enough physical spacing for the scattering field to decorrelate. At sub-wavelength spacing, adjacent elements see almost identical fields and are strongly correlated. Treating them as independent overestimates the effective $N$ and the coherent gain. Always compute $N_{\text{eff}}$ from the correlation matrix when designing for tight-packed RIS; the "free" $N^2$ scaling does not survive below $\lambda/2$ spacing.

Fading, Correlation, and the Keyhole Effect