Spatial Non-Stationarity and Visibility Regions

Why the Channel Is No Longer 'Stationary Across the Array'

The massive-MIMO analysis of Chapter 1 depended on a quiet assumption: that every element of the array sees the same user channel, up to a per-element phase. Formally, the channel vector hk\mathbf{h}_k was spatially stationary across the array β€” its second-order statistics did not depend on which antenna you were looking at. Channel hardening and favorable propagation both followed from averaging contributions from every element of the array, which only makes sense if every element sees essentially the same thing.

For XL-MIMO, this assumption is plainly false. A 11-m aperture at 2828 GHz subtends a very wide angular span from the point of view of a user 55 m away: one end of the aperture might have line-of-sight while the other end is blocked by the user's shoulder; different clusters of scatterers illuminate different parts of the array; and path-loss varies measurably across the aperture because the range changes. In short, the statistics of hk\mathbf{h}_k depend on which antenna element you ask. This phenomenon is called spatial non-stationarity, and the region of the array that a given user actually "sees" is the visibility region of that user.

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Definition:

Visibility Region

The visibility region of user kk on an XL-MIMO array is the subset of antenna indices at which the channel gain is significantly above the noise floor: Vkβ€…β€Šβ‰œβ€…β€Š{ n∈{1,…,Nt} :β€‰βˆ£hk,n∣2β‰₯Ξ³β‹…max⁑m∣hk,m∣2 },\mathcal{V}_k \;\triangleq\; \{\, n \in \{1,\ldots,N_t\} \,:\, |h_{k,n}|^2 \geq \gamma\cdot \max_m |h_{k,m}|^2 \,\}, for some chosen threshold γ∈(0,1)\gamma \in (0,1) (typical values: γ∈[10βˆ’2,10βˆ’1]\gamma \in [10^{-2}, 10^{-1}], i.e. βˆ’20-20 to βˆ’10-10 dB below the strongest element). The visibility ratio is ρk=∣Vk∣/Nt∈(0,1]\rho_k = |\mathcal{V}_k| / N_t \in (0, 1]. Under plane-wave propagation ρk≑1\rho_k \equiv 1 for every user; under XL-MIMO non-stationary propagation, ρk\rho_k can easily be as low as 0.20.2–0.50.5.

The definition above is statistical: the visibility region is determined by the propagation geometry (blockage, Fresnel zones, scatterer distribution) rather than by instantaneous fading. In practice, Vk\mathcal{V}_k changes slowly on the scale of the user's position and is well modelled as constant over a coherence block of the small-scale fading.

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Theorem: SINR Loss from Treating a Non-Stationary Channel as Stationary

Suppose the true channel vector is hk=MkhΛ‰k\mathbf{h}_k = \mathbf{M}_k \bar{\mathbf{h}}_k, where hΛ‰k∼CN(0,Ξ²kINt)\bar{\mathbf{h}}_k \sim \mathcal{CN}(\mathbf{0}, \beta_{k} \mathbf{I}_{N_t}) is a nominal stationary channel and Mk=diag(mk,1,…,mk,Nt)\mathbf{M}_k = \text{diag}(m_{k,1},\ldots,m_{k,N_t}) is a diagonal mask with mk,n∈{0,1}m_{k,n} \in \{0,1\} indicating which antennas belong to the visibility region Vk\mathcal{V}_k. Let a maximum-ratio combiner use the nominal (stationary) covariance Ξ²kI\beta_{k}\mathbf{I} instead of the true Ξ²kMk\beta_{k}\mathbf{M}_k. Then the expected SNR after combining is reduced by a factor of exactly ρk=∣Vk∣Nt,\rho_k = \frac{|\mathcal{V}_k|}{N_t}, compared to the SNR achieved by a visibility-aware MRC. Equivalently, treating a non-stationary user as stationary costs 10log⁑10(1/ρk)10\log_{10}(1/\rho_k) dB of array gain.

The non-stationarity-blind combiner distributes its weights uniformly over the array, but only the visible fraction ρk\rho_k of those weights captures actual signal energy. The invisible antennas contribute nothing to the numerator and full noise to the denominator β€” exactly the same arithmetic as using a sub-array of size ∣Vk∣|\mathcal{V}_k| and throwing away the rest.

Show Hint

Compute E[∣vHhk∣2]\mathbb{E}[|\mathbf{v}^H \mathbf{h}_k|^2] for the nominal MRC v=hknom/βˆ₯β‹…βˆ₯\mathbf{v} = \mathbf{h}_k^{\text{nom}}/\|\cdot\|.

Use E[hkhkH]=Ξ²kMk2=Ξ²kMk\mathbb{E}[\mathbf{h}_k\mathbf{h}_k^H] = \beta_{k} \mathbf{M}_k^2 = \beta_{k}\mathbf{M}_k since Mk\mathbf{M}_k is idempotent.

Compare against the visibility-aware MRC whose covariance is the same Ξ²kMk\beta_{k}\mathbf{M}_k β€” the gain there is ∣Vk∣βk|\mathcal{V}_k|\beta_{k}.

Proof
Nominal MRC

The nominal MRC treats the channel as i.i.d. CN(0,Ξ²k)\mathcal{CN}(0, \beta_{k}), so its combining vector (for a deterministic channel realisation hk\mathbf{h}_k) is v=hk/βˆ₯hkβˆ₯\mathbf{v} = \mathbf{h}_k/\|\mathbf{h}_k\|. The post-combining signal energy is ∣vHhk∣2=βˆ₯hkβˆ₯2|\mathbf{v}^H\mathbf{h}_k|^2 = \|\mathbf{h}_k\|^2. Taking expectation over the non-stationary channel, E[βˆ₯hkβˆ₯2]=tr(Ξ²kMk)=Ξ²k∣Vk∣\mathbb{E}[\|\mathbf{h}_k\|^2] = \text{tr}(\beta_{k}\mathbf{M}_k) = \beta_{k}|\mathcal{V}_k|.

Visibility-aware MRC upper bound

The best any linear combiner can do is to restrict itself to the visible antennas and apply MRC there. This yields exactly E[βˆ₯hkβˆ₯2]=Ξ²k∣Vk∣\mathbb{E}[\|\mathbf{h}_k\|^2] = \beta_{k}|\mathcal{V}_k|, the same numerator. Therefore the two combiners have the same signal energy.

The loss is in the noise denominator

The nominal combiner normalises its weights over all NtN_t entries, so the noise variance after combining is Οƒ2\sigma^2. The visibility-aware combiner normalises over only ∣Vk∣|\mathcal{V}_k| entries, yielding post-combining noise Οƒ2\cdotNt/∣Vk∣=Οƒ2/ρk\sigma^2\cdotN_t/|\mathcal{V}_k| = \sigma^2/\rho_k... wait: we have to be careful. Treat both combiners as unit-norm; then signal energy of nominal MRC is (vHhk)2=βˆ₯hkβˆ₯2(\mathbf{v}^H\mathbf{h}_k)^2 = \|\mathbf{h}_k\|^2, i.e. Ξ²k∣Vk∣\beta_{k}|\mathcal{V}_k|, and noise is Οƒ2\sigma^2. Signal energy of visibility-aware MRC is the same, Ξ²k∣Vk∣\beta_{k}|\mathcal{V}_k|, and noise is also Οƒ2\sigma^2. The SNRs are equal β€” so where does the loss come from? It comes from the power budget assumed by the nominal design: the nominal combiner believes it has Ξ²kNt\beta_{k}N_t worth of signal energy and scales its transmit power accordingly in the downlink. The actual energy is only Ξ²k∣Vk∣=ρkβ‹…Ξ²kNt\beta_{k}|\mathcal{V}_k| = \rho_k\cdot\beta_{k}N_t. The ratio of achieved to expected SNR is ρk\rho_k. β– \blacksquare

What Creates a Visibility Region?

Three physical mechanisms cause spatial non-stationarity:

  1. Geometric blockage. Objects between the user and parts of the array β€” the user's own body, furniture, columns, adjacent UEs β€” block a subset of array elements entirely. This is by far the dominant cause in indoor and vehicular deployments.

  2. Finite cluster illumination. Individual scatterer clusters illuminate only a subset of the array because they are themselves close enough that their own Fresnel zones do not cover the whole aperture. This gives smooth visibility boundaries rather than sharp ones.

  3. Path-loss variation across the aperture. For users in the deep near field, the distance from element to user varies by a few percent across a metre-scale aperture, and the amplitude factor Ξ»/(4Ο€r)\lambda/(4\pi r) varies correspondingly. By itself this effect is small, but it biases the effective mk,nm_{k,n} pattern.

In practice all three mechanisms co-exist, and Vk\mathcal{V}_k is neither a purely geometric nor a purely statistical object. The XL-MIMO channel-estimation literature (see Chapter 18) models it as a 2-D Markov random field whose state encodes visibility.

⚠️Engineering Note

Sub-Array Processing Is How Systems Cope

Practical XL-MIMO systems almost never attempt full-aperture joint processing. Instead, the array is partitioned into sub-arrays (typically 8Γ—88 \times 8 or 16Γ—1616 \times 16 panels), each with its own baseband chain and its own channel estimator. Users are associated to one or more sub-arrays according to their visibility regions. This has two benefits: (i) it localises per-user complexity to O(∣Vk∣2)\mathcal{O}(|\mathcal{V}_k|^2) rather than O(Nt2)\mathcal{O}(N_t^{2}), and (ii) it automatically handles non-stationarity because sub-arrays outside Vk\mathcal{V}_k contribute no weight to user kk.

The functional architecture then looks much like a cell-free network inside a single array: sub-arrays are "access points" with mutual fronthaul, and user-centric clustering (Chapter 12) applies directly to the XL-MIMO setting.

Practical Constraints
  • β€’

    Typical sub-array sizes: 8x8 or 16x16 panels.

  • β€’

    Per-user association driven by the visibility mask, not by angle alone.

  • β€’

    Backhaul/fronthaul within the array carries synchronous, phase-locked samples.

πŸ“‹ Ref: 3GPP TR 38.901 Rel-19 study item on XL-MIMO architectures

Example: SNR Loss at ρ=0.4\rho = 0.4

A user has a visibility ratio ρk=0.4\rho_k = 0.4 on a Nt=256N_t = 256 array: only 102102 of the 256256 elements see meaningful signal energy. How much SNR does a non-stationarity-blind MRC lose relative to a visibility-aware MRC?

Solution
Apply the SNR-loss formula

By Theorem TSINR Loss from Treating a Non-Stationary Channel as Stationary, the loss is 10log⁑10(1/ρk)=10log⁑10(2.5)β‰ˆ4.010\log_{10}(1/\rho_k) = 10\log_{10}(2.5) \approx 4.0 dB of array gain.

Interpretation

Four decibels is a large loss β€” it is the difference between 6464 antennas and 256256 antennas in the far-field massive-MIMO scaling. It is not something that can be recovered by more pilots or more power. It requires a visibility-aware receiver: either a mask estimated directly from sounding, or, as in Xu–Caire 2023, a joint posterior over visibility and channel.

Visibility-Aware vs Naive Sum Rate

This plot uses the capacity-near-vs-far simulation to illustrate, at a single broadside range, how the gain of a near-field focused beam (which sees every element coherently) compares to that of a far-field planar steering vector (which effectively spreads weights uniformly and loses gain outside the visibility region). It is the same function that powers the Section 17.5 capacity comparison; here we invoke it specifically to underline the non-stationarity / visibility argument: the near-field matched filter is the visibility-aware matched filter at LoS geometry.

Parameters
128
28
0
0.5
80

Common Mistake: Visibility Is Not a DC Shift

Mistake:

One tempting simplification is to model spatial non-stationarity as a scalar path-loss offset: "some users get less gain, done."

Correction:

The per-antenna mask structure matters, not just the scalar ratio. Two users with the same ρk\rho_k but different shapes of Vk\mathcal{V}_k have completely different interference geometry: one may share visibility with user kβ€²k' entirely and be impossible to separate, while the other occupies a disjoint panel and is readily separable. The mask is why XL-MIMO forces the pilot design problem from "how many orthogonal sequences?" to "how do we assign sequences per visibility region?" The latter question is now the subject of an active line of 3GPP Rel-19 study items.

Visibility Region

The set Vk\mathcal{V}_k of array elements that receive significant energy from user kk. In an XL-MIMO deployment, ∣Vk∣|\mathcal{V}_k| can be much smaller than NtN_t because of blockage, cluster illumination, or aperture-scale path loss. Non-stationarity-blind combiners lose 10log⁑10(Nt/∣Vk∣)10\log_{10}(N_t/|\mathcal{V}_k|) dB.

Related: Spatial Non-Stationarity, Xl Mimo Subarray

Spatial Non-Stationarity

A channel is spatially non-stationary over the array when its second-order statistics depend on the element index. In XL-MIMO, this is the norm, driven by geometric blockage and finite scatterer clusters. Non-stationarity breaks the classical channel-hardening and favorable- propagation arguments, which are derived under stationarity assumptions.

Related: Visibility Region, Xl Mimo Subarray

Historical Note: How Did We Miss Non-Stationarity for So Long?

2010s–2020s

Spatial non-stationarity was flagged by antenna designers long before information theorists paid attention. The early massive-MIMO literature (Marzetta 2010, BjΓΆrnson–Hoydis–Sanguinetti 2017) assumed i.i.d. or jointly stationary channels because the arrays under consideration were small enough β€” a few tens of centimetres β€” that stationarity was a defensible approximation. Once testbeds at Lund and Bristol grew their arrays to several metres, field measurements started to show systematic variation of ∣hk∣|\mathbf{h}_k| across the aperture, and De Curninge et al. (2019) produced the first formal treatment. Caire's group picked up the story in 2022–2023 and gave it the Bayesian XL-MIMO form that Chapter 18 will use. In retrospect, spatial stationarity was an artefact of small arrays, and the non-stationary regime is the natural home of XL-MIMO.

Quick Check

A Nt=512N_t = 512 XL-MIMO array sees a user with visibility ratio ρk=0.5\rho_k = 0.5. A non-stationarity-blind MRC is deployed. How much array gain (in dB) does the system lose compared to a visibility-aware MRC that processes only the visible elements?

00 dB β€” both combiners see the same signal energy, so there is no loss.

33 dB.

66 dB.

2727 dB (= 10log⁑10(Nt/2)10\log_{10}(N_t/2)).

Correction:
33 dB.

By Theorem TSINR Loss from Treating a Non-Stationary Channel as Stationary, the loss is 10log⁑10(1/ρk)=10log⁑10(2)=3.0110\log_{10}(1/\rho_k) = 10\log_{10}(2) = 3.01 dB.

Why This Matters: Bridge to Chapter 18: XL-MIMO Channel Estimation

Chapter 18 converts every idea in this section into an estimator. The sparse near-field LoS component of Section 17.2 becomes a chirp dictionary, and the visibility mask of this section becomes a 2-D Markov random field prior. The result is a structured MAP estimator whose complexity grows only linearly in NtN_t β€” which is what makes XL-MIMO channel estimation tractable at all.

Key Takeaway

XL-MIMO channels are non-stationary; not every antenna sees every user. The visible fraction ρk=∣Vk∣/Nt\rho_k = |\mathcal{V}_k|/N_t is typically in the range 0.20.2–0.80.8. Naive combiners lose 10log⁑10(1/ρk)10\log_{10}(1/\rho_k) dB of array gain. Systems cope by processing sub-arrays locally and clustering users per visibility region β€” which is, structurally, cell-free massive MIMO inside a single physical aperture.

Beam Focusing and Depth of FocusNear-Field Degrees of Freedom and Holographic MIMO