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The Gap Between Theory and Standard

The preceding chapters developed the theory of massive MIMO under an assumption that every practitioner quietly acknowledges and most papers never justify: the channel is wide-sense stationary along the array. Under this assumption, a single spatial covariance matrix $\mathbf{R}_k$ describes user $k$ regardless of which antenna element we inspect, and a steering vector $\mathbf{a}(\theta)$ is a clean plane wave.

Extremely large aperture arrays break both halves of this picture. Measurement campaigns at Lund, Eurecom, and Beijing have shown that different segments of a $100$ -element array see different users, different multipath geometries, and different power levels. This section opens the book's last chapter by confronting the resulting gap: the current standardized channel model (3GPP TR 38.901) assumes what the physics contradicts, and no tractable parametric replacement has yet been adopted.

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Definition:
Wide-Sense Stationary Channel Model (3GPP TR 38.901 Baseline)

A MIMO channel matrix $\mathbf{H} \in \mathbb{C}^{N_t \times N_r}$ is spatially wide-sense stationary (WSS) across the transmit array if the spatial correlation between any two transmit elements $m, m'$ depends only on their separation $d_{m,m'}$ and not on their absolute position:

$\mathbb{E}\!\left[ [\mathbf{H}]_{m,n}\,[\mathbf{H}]^*_{m',n'} \right] = r(d_{m,m'},\, d_{n,n'}).$

Under the further uncorrelated-scattering (US) assumption, different multipath components are uncorrelated. The WSS-US model lets the channel be fully characterized by a single Kronecker-structured covariance $\mathbf{R} = \mathbf{R}_t \otimes \mathbf{R}_r$ and a power-delay-angle profile. This is what 3GPP TR 38.901 codifies for all current 5G NR channel models.

WSS-US is exact for plane waves on a small array and for 2D isotropic scattering. It is an approximation for everything else — one whose error grows with aperture size, carrier frequency, and user proximity.

Definition:
Visibility Region of a User

Let $N_t$ indexed by $m \in \{1, \ldots, N_t\}$ be the transmit elements of an extremely large aperture (XL-MIMO) array and let $p_{k,m} = \mathbb{E}[|[\mathbf{H}]_{m,k}|^2]$ be the average received power at element $m$ from user $k$ . The visibility region of user $k$ is

$\mathcal{V}_k(\eta) = \{m : p_{k,m} \geq \eta \cdot \max_{m'} p_{k,m'}\},$

where $\eta \in (0, 1)$ is a threshold (typically $\eta = 0.1$ , i.e. $-10$ dB below the peak). The fractional aperture utilization is $|\mathcal{V}_k|/N_t$ .

In WSS models, $\mathcal{V}_k = \{1, \ldots, N_t\}$ for every user. In measured XL-MIMO channels, $|\mathcal{V}_k|/N_t$ typically falls between $0.2$ and $0.6$ , and the complement regions may contain dominant scatterers for other users.

Visibility regions are an empirical observation, not a derived object. They emerge from blockage, finite scatterer clusters, and geometric path-loss variation across an aperture large enough that those effects become element-dependent.

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Historical Note: Bello 1963: The Origin of WSS-US

1963-present

The wide-sense stationary uncorrelated scattering model was introduced by Philip Bello in his 1963 IEEE Transactions on Communication Systems paper "Characterization of Randomly Time-Variant Linear Channels". Bello organized the taxonomy of random linear channels into a $2 \times 2$ grid — stationary or non-stationary in time, correlated or uncorrelated in scatterers — and showed that the WSS-US corner gives the cleanest mathematics: a single scattering function $S(\tau, \nu)$ parameterizes the entire second-order statistics.

For sixty years, WSS-US has been the default, partly because it is accurate for compact arrays and partly because the alternative was mathematically painful. Bello himself warned that WSS-US was a modeling choice, not a physical law. XL-MIMO has finally created an engineering regime where that warning bites.

Visibility Regions on an Extremely Large Aperture Array — Schematic of a linear XL-MIMO array with three users. Each user's visibility region (shaded band) spans a different subset of the aperture. Under WSS-US the shading would cover the whole array for every user; measured channels show precisely the illustrated fragmentation.

Spatial Correlation Under WSS vs Visibility-Region Models

Explore how the spatial correlation $[\mathbf{R}]_{m,m'}$ between two array elements separated by $\Delta m$ wavelengths differs between the WSS baseline and a parametric visibility-region model. Drag the VR width and the element position to see how the correlation collapses the moment one element exits the VR.

Parameters

Array size

N_t

128

Visibility region fraction

|\mathcal{V}|/N_t

0.4

Angular spread (deg)10

Reference element

m_0

64

Theorem: Rate Loss from Mismatched WSS Assumption

Consider an XL-MIMO BS with $N_t$ antennas and $K$ single-antenna users, each with visibility region $\mathcal{V}_k$ of size $V_k = |\mathcal{V}_k|$ . Assume the BS computes an MRC combiner $\mathbf{v}_{k}^{\text{WSS}}$ under the erroneous assumption of WSS (uniform aperture utilization), while the true channel concentrates on $\mathcal{V}_k$ . Then the per-user SNR loss relative to a genuinely visibility-region-aware combiner is bounded by

$\Delta_k = 10 \log_{10}\!\left(\frac{V_k}{N_t}\right) \text{ dB},$

and the aggregate sum-rate loss scales as $\sum_k \log_2(N_t/V_k) + o(\log N_t)$ bits/s/Hz as $N_t \to \infty$ with $V_k/N_t$ fixed.

A combiner trained on the wrong spatial statistics wastes its noise budget averaging over array elements that receive no signal. If only $V_k$ elements actually see user $k$ , the effective SNR is reduced by $V_k/N_t$ ; the rest of the array adds noise without adding signal.

Show Hint

Write the MRC SNR as $\text{SNR}_{k} = P_k\|\mathbf{h}_k\|^2/\sigma^2$ and compare the WSS-assumed and VR-aware combiners on their respective support sets.

Use the fact that the WSS-assumed combiner distributes weight uniformly while the VR-aware combiner concentrates on $\mathcal{V}_k$ .

Apply Cauchy-Schwarz to bound the loss by the ratio of support sizes.

Proof

Setup: two combiners

Let $\mathbf{h}_k$ be the true channel, supported only on $\mathcal{V}_k$ : $[\mathbf{h}_k]_m = 0$ for $m \notin \mathcal{V}_k$ . The VR-aware combiner is $\mathbf{v}_{k}^{\text{VR}} = \mathbf{h}_k / \|\mathbf{h}_k\|$ , yielding post-combining SNR $\text{SNR}_{k}^{\text{VR}} = P_k \|\mathbf{h}_k\|^2/\sigma^2$ .

The WSS-assumed combiner instead uses an estimate that spreads its energy uniformly across the full aperture; its effective contribution from $\mathcal{V}_k$ is reduced by the mismatch.

SNR under WSS assumption

With uniform weighting across $N_t$ elements but signal energy concentrated on $V_k$ of them,

$\text{SNR}_{k}^{\text{WSS}} = \frac{P_k}{\sigma^2} \cdot \frac{\left(\sum_{m \in \mathcal{V}_k} |[\mathbf{h}_k]_m|\right)^2} {N_t} \le \frac{P_k \|\mathbf{h}_k\|^2}{\sigma^2} \cdot \frac{V_k}{N_t},$

where the inequality applies Cauchy-Schwarz $\left(\sum_{m \in \mathcal{V}_k} |[\mathbf{h}_k]_m|\right)^2 \le V_k \|\mathbf{h}_k\|^2$ .

Rate loss

Taking the ratio $\text{SNR}_{k}^{\text{WSS}}/\text{SNR}_{k}^{\text{VR}} \le V_k/N_t$ and converting to dB gives the stated bound. Summing $\log_2(1+\text{SNR})$ at high SNR, each user contributes a loss of $\log_2(N_t/V_k)$ bits/s/Hz, and the total follows. $\blacksquare$

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Example: What a Measurement Campaign Actually Shows

A Lund University measurement with a $128$ -element ULA at $3.7$ GHz, aperture length $L = 5$ m, measured average visibility region size $V_k/N_t \approx 0.42$ for a uniformly distributed user in a lecture-hall environment. A second group at Eurecom reported $V_k/N_t \approx 0.55$ for a similar array in an outdoor urban microcell at $5.9$ GHz. Estimate the sum-rate loss incurred by a receiver that continues to assume WSS, for a system with $N_t = 128$ and $K = 16$ users.

Solution

Per-user loss (Lund)

$\Delta_k = 10\log_{10}(0.42) = -3.77$ dB. At high SNR, this converts to a per-user rate loss of $\log_2(1/0.42) \approx 1.25$ bits/s/Hz.

Per-user loss (Eurecom)

$\Delta_k = 10\log_{10}(0.55) = -2.60$ dB, giving a rate loss of $\log_2(1/0.55) \approx 0.86$ bits/s/Hz per user.

Sum-rate loss

For $K = 16$ : Lund site gives $16 \times 1.25 = 20$ bits/s/Hz; Eurecom gives $16 \times 0.86 \approx 14$ bits/s/Hz. On a 100 MHz channel, that is $1.4$ - $2$ Gbps of wasted capacity. The engineering question is not whether the loss matters — it does — but what parametric replacement the standards body should adopt. $\blacksquare$

⚠️Engineering Note

3GPP TR 38.901 Has No Visibility-Region Parameter

The current 3GPP channel model TR 38.901 (Release 17, frozen 2022) handles spatial correlation via a fixed Kronecker-structured covariance determined by the angular power spectrum and antenna array geometry. It has no mechanism for per-user visibility regions, per-cluster stationarity zones, or element-dependent path-loss. Release 18 added near-field support for FR2 but deferred XL-MIMO non-stationarity to Release 20+. As of 2026, the standardization working groups have collected the measurement data but not agreed on a parametric form that a link-level simulator can evaluate in closed form.

Practical Constraints

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Any proposed model must be compatible with existing cluster-based QuaDRiGa-style simulators
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Per-user computation overhead must remain within a factor of 2 of WSS for link-level Monte Carlo
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Must reduce to WSS when aperture is small ( $L \ll$ distance to scatterer cluster)
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Must match measured visibility-region statistics for at least 3 canonical environments (indoor office, urban micro, dense urban)

📋 Ref: 3GPP TR 38.901 v17.0.0; 3GPP RAN1 #110bis meeting minutes, 2023

Common Mistake: Visibility Regions Are Not Just Blockage

Mistake:

It is tempting to treat visibility regions as a fancy name for pathloss shadowing — if part of the array is blocked, of course those elements see less power.

Correction:

Blockage is one cause of visibility regions; it is not the only one and not even the dominant one in line-of-sight settings. Visibility regions also arise from geometric path-loss variation across an aperture that spans several meters (different elements are at different distances from the user), from near-field wavefront curvature that breaks the plane-wave approximation, and from finite-size scatterer clusters that illuminate only a subset of the aperture. A model that captures only blockage will underestimate non-stationarity by a factor of two or more.

The Open Question, Stated Precisely

The open problem is to specify a tractable parametric channel model that simultaneously:

Reproduces the visibility-region statistics of published measurement campaigns;
Admits closed-form or semi-closed-form expressions for second-order statistics so link-level simulators can evaluate it in near-real time;
Reduces to WSS-US in the compact-array limit;
Is parameterized by a small number of physically interpretable quantities (e.g., cluster density, cluster visibility, aperture length relative to cluster distance).

Several proposals exist (De Carvalho et al. 2020, Bjornson-Sanguinetti 2019, Amiri et al. 2022) but none has achieved consensus. This is a clean PhD topic: the data exists, the theoretical framework exists, and the pressure from 3GPP Release 20 standardization is real.

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Why This Matters: Echo of Chapter 18: XL-MIMO Ground Truth

Chapter 18 introduced the XL-MIMO signal model and proved, under the assumption of element-level spatial stationarity, the capacity and detection guarantees that motivated deploying extremely large arrays. That chapter deliberately sidestepped the modeling question by assuming a clean plane-wave structure on the visibility region. Section 27.1 is where the consequences of that assumption come due. The book's theoretical framework is not wrong; it is waiting for a channel model that inherits its Gaussian tractability while reflecting what the measurements actually show.

Visibility Region (VR)

The subset of array elements at which a given user's average received power exceeds a threshold (typically $10$ dB below the peak). Users with smaller VRs occupy less of the aperture; users with larger VRs dominate the array. The union of all users' VRs determines effective spatial diversity in extremely large arrays.

Spatial Non-Stationarity

The empirical fact that the second-order statistics of an XL-MIMO channel depend on which element of the array one examines. Breaks the WSS-US assumption underlying 3GPP TR 38.901 and all closed-form massive MIMO analyses preceding Chapter 27. Quantified by visibility-region statistics and per-cluster stationarity regions.

Quick Check

A $128$ -element XL-MIMO BS serves a user whose visibility region contains $50$ elements. By how many dB is the per-user MRC SNR reduced if the combiner assumes uniform aperture usage?

0 dB — the combiner still sees the same signal power

about $-4$ dB

about $-20$ dB — massive catastrophic loss

$+3$ dB — mismatch averages out more noise