Ferkans — Interactive Telecom Tutor

From Massive MIMO to Extremely Large Arrays

Massive MIMO (Chapter 18) operates in the far field, where the wavefront arriving at the array is well approximated as a plane wave. Beamforming then amounts to steering a beam in a given direction (angle). As arrays grow to hundreds or thousands of elements at FR3/sub-THz frequencies — so-called XL-MIMO or extremely large aperture arrays (ELAA) — two qualitative shifts occur:

Near-field operation: The Rayleigh distance grows with the aperture squared, placing many users inside the near field where the spherical wavefront must be modelled. Beamforming becomes beamfocusing: energy is concentrated at a specific 3D point $(r, \theta, \phi)$ rather than just a direction $(\theta, \phi)$ .
Spatial non-stationarity: Different parts of the array "see" different sets of scatterers because the array aperture spans a significant fraction of the propagation environment. The familiar assumption that all array elements share the same large-scale fading breaks down.

These two phenomena fundamentally change MIMO signal processing and motivate new channel models, codebook designs, and scheduling algorithms for 6G.

Definition:
Rayleigh Distance (Near-Field Boundary)

The Rayleigh distance $d_R$ marks the boundary between the radiative near field (Fresnel region) and the far field (Fraunhofer region) of an antenna or array with physical aperture $D$ :

$d_R = \frac{2D^2}{\lambda}$

where $\lambda = c/f$ is the wavelength. For distances $d < d_R$ , the spherical curvature of the wavefront across the array aperture introduces phase errors exceeding $\pi/8$ relative to the plane-wave approximation. In this regime:

The channel between a point source at $(r_0, \theta_0)$ and the $n$ -th array element depends on both range $r_0$ and angle $\theta_0$ (not just angle).
Conventional far-field array response vectors $\mathbf{a}(\theta) = [1, e^{j\pi\sin\theta}, \ldots]^T$ must be replaced by near-field response vectors $\mathbf{a}(r, \theta)$ that incorporate element-specific distances.
Beamfocusing at a point $(r_0, \theta_0)$ achieves higher gain at the target location than far-field beamsteering, at the expense of a tighter focal spot and reduced gain at other ranges.

The Rayleigh distance is also called the Fraunhofer distance $d_F$ . For a ULA with $N$ elements and half-wavelength spacing, $D = (N-1)\lambda/2 \approx N\lambda/2$ , giving $d_R \approx N^2\lambda/2 = N^2 c/(2f)$ . Doubling the number of elements quadruples $d_R$ .

,

Near-Field vs Far-Field Geometry — (Left) In the far field ( $d \gg d_R$ ), the wavefront is approximately planar and the array response depends only on angle $\theta$ . (Right) In the near field ( $d < d_R$ ), the spherical wavefront creates element-dependent path lengths, enabling beamfocusing at a specific point $(r_0, \theta_0)$ .

Example: Rayleigh Distance Scaling Across 6G Bands

Compute the Rayleigh distance for the following three array configurations, representative of 6G deployments:

(a) FR3 at 10 GHz, $N = 256$ elements, half-wavelength spacing.

(b) mmWave at 28 GHz, $N = 512$ elements, half-wavelength spacing.

(c) Sub-THz at 140 GHz, $N = 1024$ elements, half-wavelength spacing.

For each case, determine whether a user at $d = 20$ m is in the near field or far field.

Solution

FR3, 10 GHz, $N = 256$

$\lambda = 30$ mm, $D = 256 \times 30/2 = 3840$ mm $= 3.84$ m.

$d_R = \frac{2 \times 3.84^2}{0.030} = \frac{29.49}{0.030} = 983\;\text{m}$

At $d = 20$ m: deep near field ( $d \ll d_R$ ). Note the enormous aperture ( $3.84$ m) — this would be a large wall-mounted panel.

mmWave, 28 GHz, $N = 512$

$\lambda = 10.7$ mm, $D = 512 \times 10.7/2 = 2739$ mm $= 2.74$ m.

$d_R = \frac{2 \times 2.74^2}{0.0107} = \frac{15.01}{0.0107} = 1403\;\text{m}$

At $d = 20$ m: deep near field.

Sub-THz, 140 GHz, $N = 1024$

$\lambda = 2.14$ mm, $D = 1024 \times 2.14/2 = 1096$ mm $= 1.10$ m.

$d_R = \frac{2 \times 1.10^2}{0.00214} = \frac{2.42}{0.00214} = 1131\;\text{m}$

At $d = 20$ m: deep near field.

All three configurations place a 20 m user firmly in the near field — underscoring that near-field operation is the norm, not the exception, for XL-MIMO at any 6G frequency. $\blacksquare$

Beamfocusing vs Beamsteering

In the far field, the beamforming gain at a target user is determined solely by the angular alignment. A user at the same angle but a different range receives the same beam gain. In the near field, beamfocusing concentrates energy at a specific point in 3D space:

The beamforming vector for focusing at $(r_0, \theta_0)$ is the conjugate of the near-field steering vector: $\mathbf{w}(r_0, \theta_0) = \frac{1}{\sqrt{N}} \left[ e^{j2\pi d_1/\lambda}, \ldots, e^{j2\pi d_N/\lambda} \right]^T$ where $d_n$ is the distance from the $n$ -th element to the focal point.
A user at the same angle but a different range $r \neq r_0$ experiences a focus gain loss that depends on $|r - r_0|/\Delta r$ , where $\Delta r$ is the depth of focus.
This range selectivity creates a new spatial dimension for user scheduling: two users at the same angle but different ranges can be simultaneously served with separate focused beams — impossible in far-field MIMO.

The interactive plot below compares beamfocusing and beamsteering gain as a function of user range.

Beamfocusing vs Beamsteering Gain

Compare the received power as a function of range for near-field beamfocusing (matched to a specific focal point) and far-field beamsteering (matched only to angle). The vertical dashed line marks the Rayleigh distance $d_R = 2D^2/\lambda$ . Observe how beamfocusing provides higher gain at the target range but drops off more steeply away from the focus point.

Parameters

Number of elements

N

64

Frequency (GHz)28

Focus range

r_0

(m)10

Definition:
Spatial Non-Stationarity in XL-MIMO

In conventional massive MIMO, all $N$ array elements share the same set of scattering clusters — the channel is spatially stationary across the array. In XL-MIMO, the array aperture $D$ may span several metres, and different parts of the array observe different propagation environments. This phenomenon is called spatial non-stationarity.

Formally, let $\mathcal{S} = \{s_1, \ldots, s_L\}$ be the set of scattering clusters. For each cluster $s_\ell$ , define the visibility region $\mathcal{V}_\ell \subseteq \{1, \ldots, N\}$ as the subset of array elements that can "see" cluster $s_\ell$ . The channel vector for user $k$ becomes:

$h_n^{(k)} = \sum_{\ell=1}^{L} \mathbf{1}[n \in \mathcal{V}_\ell^{(k)}] \, \alpha_\ell^{(k)} \, e^{-j2\pi d_{n,\ell}/\lambda}$

where $\mathbf{1}[\cdot]$ is the indicator function and $\alpha_\ell^{(k)}$ is the complex gain of cluster $\ell$ .

Consequences for signal processing:

MR/ZF/MMSE precoding must account for per-element visibility; a global covariance matrix is no longer Toeplitz.
Channel estimation: pilot signals may need to be processed over sub-arrays (segments of the XL array) rather than the full array.
User scheduling can exploit the fact that two users with non-overlapping visibility regions cause zero inter-user interference, even without precoding.

XL-MIMO Spatial Non-Stationarity

Visualise the visibility map (which array elements see which scatterers) and the resulting channel power variation across the array for a single user. Adjust the number of array elements, user position, and number of scattering clusters to observe how spatial non-stationarity becomes more pronounced with larger arrays.

Parameters

Number of elements

N

128

User distance (m)20

Number of scatterers6

Near-Field Beamfocusing Transition

Watch the beam pattern evolve as the focal range sweeps from deep near-field (tight 3D focus) through the Rayleigh distance to the far field (broad angular beam). The beam gain profile narrows and sharpens at closer focal ranges.

32-element ULA at 30 GHz. As

r_0

increases past

d_R

, the focused beam widens to become indistinguishable from far-field beamsteering.

Near-Field Beamfocusing Range Sweep

Watch the beam pattern evolve as the focal range sweeps from 1 m (deep near field) through the Rayleigh distance to the far field. Observe the transition from tight 3D focusing (narrow depth of focus) to broad angular beamsteering (range-independent).

Parameters

Number of elements

N

64

Frequency (GHz)28

Focus range

r_0

(m)10

Near-Field Beamfocusing for XL-MIMO

Input: Array element positions

\{(x_n, y_n, z_n)\}_{n=1}^{N}

,

focal point

(r_0, \theta_0, \phi_0)

in spherical coordinates,

carrier frequency

f

Output: Beamfocusing weight vector

\mathbf{w} \in \mathbb{C}^{N}

1. Convert focal point to Cartesian:

\mathbf{p}_0 = (r_0\sin\theta_0\cos\phi_0,\; r_0\sin\theta_0\sin\phi_0,\; r_0\cos\theta_0)

2. Compute wavelength:

\lambda = c / f

3. for

n = 1, 2, \ldots, N

do

4.

\quad

Compute exact distance:

d_n = \|\mathbf{p}_0 - (x_n, y_n, z_n)\|_2

5.

\quad

Compute phase:

\psi_n = 2\pi d_n / \lambda

6.

\quad

Set weight (conjugate matching):

w_n = \frac{1}{\sqrt{N}} e^{-j\psi_n}

7. end for

8. Return

\mathbf{w} = [w_1, \ldots, w_N]^T

Complexity:

\mathcal{O}(N)

— one square root and one complex

exponential per element. Compare with far-field beamsteering,

which uses the same

\mathcal{O}(N)

complexity but with a linear

(not quadratic) phase progression.

Note: In practice, the focal point is estimated from uplink

pilot signals using near-field channel estimation algorithms

(e.g., polar-domain MUSIC or compressed sensing on a range-angle

dictionary).

Open Research Directions for Near-Field XL-MIMO

Near-field XL-MIMO is one of the most active 6G research areas. Key open problems include:

Near-field channel estimation: The search space is 2D (range $\times$ angle) rather than 1D (angle only), increasing pilot overhead. Polar-domain sparsity and parametric estimation methods are being developed to reduce this overhead.
Codebook design: Far-field DFT codebooks are suboptimal in the near field. Polar-domain codebooks that jointly quantise range and angle are needed for beam management.
Wideband near-field effects: At sub-THz bandwidths ( $> 10$ GHz), the beam focus point becomes frequency-dependent — a phenomenon called beam squint in range, distinct from the angular beam squint studied in Chapter 27.
Channel modelling: Measurement campaigns for XL-MIMO at FR3 and sub-THz frequencies are still scarce. Standardised near-field, spatially non-stationary channel models are needed for system-level evaluation.
Hybrid architectures for near-field: Analog beamforming with phase shifters can only approximate the element-dependent phases needed for beamfocusing. True-time-delay (TTD) elements or sub-array-based hybrid architectures are promising but add hardware complexity.

Theorem: Spatial Degrees of Freedom in the Near Field

For a continuous linear aperture of length $D$ communicating with a point source at distance $r$ in the near field ( $r < d_R$ ), the number of effective spatial degrees of freedom (DoF) scales as:

$\text{DoF} \approx \frac{D^2}{2\lambda r}$

This is a factor of $D/(2r)$ larger than the far-field DoF (which is 1 for a point source). Equivalently, the angular resolution improves with decreasing range in the near field, enabling range-domain multiplexing — serving users at the same angle but different ranges on separate spatial streams.

In the far-field limit ( $r \gg d_R$ ), $\text{DoF} \to 1$ (beamsteering provides only angular selectivity).

In the near field, the array "sees" the source from a wider angular extent (the subtended angle $\approx D/r$ increases as range decreases). More angular extent means more resolvable directions, hence more DoF. This is why near-field XL-MIMO can support more simultaneous users than conventional massive MIMO.

Proof

Subtended angle and resolution

A source at range $r$ subtends an angle $\Delta\theta \approx D/r$ at the aperture. The angular resolution of the aperture is $\delta\theta \approx \lambda/D$ . The number of resolvable directions is $\text{DoF} = \Delta\theta/\delta\theta = D^2/(\lambda r)$ . Including the factor-of-2 from the Nyquist spatial sampling rate: $\text{DoF} \approx D^2/(2\lambda r)$ . $\blacksquare$

Theorem: Beamfocusing Gain over Beamsteering

For a ULA with $N$ elements and half-wavelength spacing, the beamfocusing gain at the focal point $(r_0, \theta_0)$ relative to far-field beamsteering (matched only to $\theta_0$ ) is:

$G_{\text{focus}} / G_{\text{steer}} = 1 + \frac{N^2 \lambda^{2}}{32 r_0^2}$

For $r_0 \ll d_R = N^2\lambda/2$ , this ratio approaches $N/16$ , representing a significant additional gain from range matching. The depth of focus (range resolution) is approximately:

$\Delta r \approx \frac{8 r_0^2}{N^2 \lambda}$

which shrinks quadratically with range, enabling tighter focusing at closer distances.

Beamfocusing exploits the spherical wavefront curvature to concentrate energy at a specific range. The additional gain comes from constructive interference across all $N$ elements being optimised for the exact distance, not just the angle.

Proof

Phase error analysis

Far-field beamsteering applies linear phase progression $\psi_n^{\text{FF}} = n \pi \sin\theta_0$ . The true near-field phase includes a quadratic term $\propto n^2/(r_0)$ . The mismatch between FF and NF phases causes a gain loss that equals $G_{\text{steer}}/G_{\text{focus}}$ when the source is at range $r_0$ . Computing the array factor ratio yields the stated expression. $\blacksquare$

🎓CommIT Contribution(2024)

2D Markov Prior for Visibility Region Detection in XL-MIMO

Y. Xu, G. Caire — IEEE Transactions on Signal Processing

Proposed a 2D Markov random field prior for modelling the spatial non-stationarity pattern (visibility regions) in XL-MIMO channels. The key insight is that visibility regions exhibit spatial smoothness — neighbouring array elements are likely to share the same visibility state. By embedding this prior into a Bayesian channel estimation framework, the method achieves near-oracle performance in estimating both the channel coefficients and the visibility mask, with significantly reduced pilot overhead compared to per-element estimation. This work directly addresses the spatial non-stationarity challenge described in this section.

xl-mimonear-fieldspatial-non-stationaritychannel-estimation

Key Takeaway

Near-field operation is the norm for XL-MIMO at 6G frequencies — a 256-element array at 10 GHz has $d_R \approx 1$ km. Beamfocusing enables range-domain user separation impossible in far-field MIMO, but demands new channel models, codebooks, and estimation algorithms that operate in the 2D (range $\times$ angle) domain.

Rayleigh Distance

The boundary between the near field and far field of an antenna array: $d_R = 2D^2/\lambda$ . Users at $d < d_R$ experience spherical wavefronts; users at $d > d_R$ see approximately planar wavefronts.

Beamfocusing

Near-field beamforming that concentrates energy at a specific 3D point $(r, \theta, \phi)$ by matching the spherical-wave phase profile across the array, rather than steering to a direction only (far-field beamsteering).

XL-MIMO (Extremely Large MIMO)

MIMO systems with apertures large enough that near-field effects, spatial non-stationarity, and per-element visibility regions become significant. Typical configurations: 256 -- 4096 elements at FR3/sub-THz frequencies.

Quick Check

A ULA with $N = 128$ elements and half-wavelength spacing operates at $f = 30$ GHz ( $\lambda = 10$ mm). What is the approximate Rayleigh distance?

$d_R \approx 8.2$ m

$d_R \approx 82$ m

$d_R \approx 820$ m

$d_R \approx 0.82$ m

Correction:

d_R \approx 82

m

Correct. $D = 128 \times 10/2 = 640$ mm $= 0.64$ m. $d_R = 2D^2/\lambda = 2 \times 0.64^2 / 0.01 = 81.9$ m $\approx 82$ m. Any user within 82 m of this array is in the near field — which encompasses most indoor and many outdoor urban scenarios.

Near-Field Communications and XL-MIMO

From Massive MIMO to Extremely Large Arrays

Definition: Rayleigh Distance (Near-Field Boundary)

Near-Field vs Far-Field Geometry

Example: Rayleigh Distance Scaling Across 6G Bands

FR3, 10 GHz, $N = 256$

mmWave, 28 GHz, $N = 512$

Sub-THz, 140 GHz, $N = 1024$

Beamfocusing vs Beamsteering

Beamfocusing vs Beamsteering Gain

Parameters

Definition: Spatial Non-Stationarity in XL-MIMO

XL-MIMO Spatial Non-Stationarity

Parameters

Near-Field Beamfocusing Transition

Near-Field Beamfocusing Range Sweep

Parameters

Near-Field Beamfocusing for XL-MIMO

Open Research Directions for Near-Field XL-MIMO

Theorem: Spatial Degrees of Freedom in the Near Field

Subtended angle and resolution

Theorem: Beamfocusing Gain over Beamsteering

Phase error analysis

2D Markov Prior for Visibility Region Detection in XL-MIMO

Key Takeaway

Rayleigh Distance

Beamfocusing

XL-MIMO (Extremely Large MIMO)

Quick Check

Definition:
Rayleigh Distance (Near-Field Boundary)

Definition:
Spatial Non-Stationarity in XL-MIMO