Precoding-Induced Correlation-Averaging

• Precoding-Induced Correlation-Averaging is a mechanism that leverages spatial covariance eigenmodes to distribute energy across a user’s transmission sphere in near-field communications.
• It averages fast spatial fluctuations by spreading precoded energy over statistically persistent modes, ensuring stable received power and effective interference suppression.
• The approach employs convex optimization and eigenspace shaping to achieve robust beamforming in the presence of mobility-induced uncertainties and channel variations.

Precoding-induced correlation-averaging denotes a precoding mechanism in which transmission is deliberately distributed over statistically persistent modes so that fast spatial or channel fluctuations are smoothed rather than tracked pointwise. In near-field communications, the term refers specifically to using the spatial covariance of a user’s channel over an uncertain three-dimensional support set—a sphere around the nominal user position—to spread precoded energy across dominant covariance eigenmodes, maintain received power over a target sphere, and suppress interference over non-target spheres. In this form, the concept is central to sphere precoding for robust near-field links with mobility (Luo et al., 9 Mar 2025).

1. Near-field channel, one-sphere modeling, and mobility-aware covariance

The near-field setting replaces planar-wave steering with spherical-wave propagation. For an array element at position $p_n$ and a user at $r$, the channel coefficient is

$$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$

The corresponding robust formulation in sphere precoding is built on a one-sphere channel model, which extends the one-ring model by placing scatterers uniformly on a sphere of radius $R_s$ around the user position $q_k$. A scatterer at spherical coordinates $(r,\theta,\phi)$ has position

$$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$

with element-to-scatterer distance $$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$. The static-user covariance is then

$$[R_k]_{m,n}=\frac{1}{2\pi^2R_s}\int_0^{R_s}\int_0^{2\pi}\int_0^\pi e^{-j\frac{2\pi}{\lambda}[d_{m,k}(r,\theta,\phi)-d_{n,k}(r,\theta,\phi)]} \,dr\,d\theta\,d\phi.$$

This covariance is a one-sphere analogue of the one-ring covariance, but with exact spherical-wave phase differences (Luo et al., 9 Mar 2025).

Mobility expands the user support from a point neighborhood to a transmission zone. If user $k$ moves by $r$0 during a slot, the user-plus-scatterer region is

$r$1

Averaging the channel over this region yields the mobility-aware covariance approximation

$r$2

Equivalently, with support set $r$3 and distribution $r$4,

$r$5

The eigendecomposition

$r$6

leads to the Karhunen–Loève representation

$r$7

so the columns of $r$8 are spatial modes supported over the user’s transmission sphere, and $r$9 specifies their power.

2. Eigenspace shaping and the averaging mechanism

In sphere precoding, the beamformer for user $$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$0 is parameterized as

$$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$1

This confines precoding to the subspace in which the near-field channel has energy across the user’s transmission sphere. The central averaging step is the equal projection principle,

$$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$2

which enforces nearly equal projection of the beamformer onto selected singular vectors of $$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$3. Under the channel representation above, this makes

$$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$4

less sensitive to the exact user location inside $$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$5, because transmit energy is spread across the principal modes that remain active over the whole sphere rather than concentrated at a single focal point (Luo et al., 9 Mar 2025).

The same mechanism yields interference suppression through subspace rejection. For a non-target user $$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$6, the constraint

$$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$7

forces $$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$8 to be nearly orthogonal to the subspace where user $$h_n(r)=\alpha_n \frac{e^{-j\frac{2\pi}{\lambda}\|r-p_n\|}}{\|r-p_n\|}, \qquad h(r)=[h_1(r),\ldots,h_N(r)]^T.$$9 has channel energy across its own support set $R_s$0. As a result, $R_s$1 remains small throughout the non-target sphere. The resulting beampattern is therefore not a point focus but a volumetric structure: a three-dimensional focal region that covers the target transmission sphere and null regions that cover the spheres of other users.

This mechanism is the precise content of correlation-averaging in the near-field setting. Instantaneous phase variations due to path-length changes are not eliminated; they are mixed across covariance modes so that their effect is averaged over the spatial distribution induced by mobility and local scattering. The design objective is thus robust sphere shaping rather than peak gain at a single coordinate.

3. Robust SINR formulation and convex relaxation

The robustness target is probabilistic. For user $R_s$2 at position $R_s$3, the near-field SINR is

$R_s$4

The original design problem maximizes the minimum SINR satisfaction probability over users subject to power constraints:

$R_s$5

In the paper’s unit-norm-per-column formulation this appears as

$R_s$6

Because this objective is probabilistic and non-convex, sphere precoding replaces it by deterministic covariance-structure constraints that encode correlation-averaged target power and non-target suppression (Luo et al., 9 Mar 2025).

For each user $R_s$7, the relaxed problem is

$R_s$8

subject to

$R_s$9

This is a convex second-order cone program. The relaxation approximates the original satisfaction-probability objective by requiring balanced energy over the dominant sphere modes of the target user and low projection onto the sphere modes of the other users. In implementation, one first computes $q_k$0 and its eigendecomposition $q_k$1 for all users, then solves the per-user SOCP sequentially for $q_k$2, and stacks the resulting $q_k$3 as the columns of $q_k$4. The simulations use MOSEK via CVX.

The algorithmic cost is dominated by covariance decomposition and convex optimization. Dense SVD has per-user complexity $q_k$5, typically reducible by exploiting structure or low rank, while standard interior-point SOCP solvers scale roughly as $q_k$6 per user for moderate $q_k$7. An optional refinement is to iterate over users after the first pass to improve multiuser interference coupling.

4. Performance regimes, baselines, and trade-offs

The reported simulations consider a $q_k$8 UPA at $q_k$9 GHz with antenna spacing $(r,\theta,\phi)$0 and noise variance $(r,\theta,\phi)$1. The scatterer sphere radius is $(r,\theta,\phi)$2 m, the transmission sphere radius is $(r,\theta,\phi)$3, the covariance is estimated from $(r,\theta,\phi)$4 channel samples per sphere, and each user has $(r,\theta,\phi)$5 scatterers. Baselines include conjugate beamforming based on LoS, dominant-eigenvector beamforming using $(r,\theta,\phi)$6, equal projection without interference constraints, zero-forcing on the initial LoS channel, and a fractional programming robust method (Luo et al., 9 Mar 2025).

Under mobility of approximately $(r,\theta,\phi)$7 m, sphere precoding produces beampatterns that focus around the target sphere and place nulls around non-target spheres, whereas zero-forcing over-focuses at the initial positions and loses both power and null depth as users move. At approximately $(r,\theta,\phi)$8 m, sphere precoding achieves about $(r,\theta,\phi)$9 of channel samples with $$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$0 dB, while zero-forcing and the fractional programming method achieve about $$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$1. Average-SINR curves versus mobility further show that methods ignoring inter-user interference degrade faster, and that equal projection alone can be worse than conjugate beamforming or dominant-eigenvector transmission because it spreads power without null control.

These results expose the central trade-off. Equal projection across dominant covariance modes sacrifices peak focal gain at a single point in exchange for stable received power across the target sphere. Likewise, enforcing nulls across the other users’ spheres reduces the feasible set for $$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$2 and can reduce maximum rate in static scenarios, but it improves robustness under mobility. The method therefore trades pointwise optimality for volumetric reliability.

The reported sensitivity trends are consistent with this interpretation. Correlation-averaging becomes more valuable as array size grows, because larger apertures tighten near-field focal spots and make conventional beamforming more sensitive to position errors. Gains are also larger at mmWave and above, where smaller wavelengths increase phase sensitivity. Moderate transmission-sphere radii preserve a compact dominant eigenspace and make averaging effective, whereas very large spheres diffuse the eigenstructure and reduce both beamforming gain and robustness. If $$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$3 is non-uniform rather than uniform over the sphere, weighting the projection toward higher-probability modes can further improve robustness. This suggests that the mechanism is fundamentally statistical: it is strongest when uncertainty is structured enough to admit a stable low-dimensional covariance description.

5. Practical construction, operational guidance, and limitations

Operationally, the method begins by constructing each support set $$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$4 from mobility or sensing information: the radius is $$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$5, combining user movement with local scatter extent. The covariance $$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$6 can then be estimated analytically when geometry is known or numerically from samples over $$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$7; the practical guideline in the simulations is to use at least about $$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$8 samples so that the covariance captures the sphere structure with sufficient fidelity. Once $$s_k(r,\theta,\phi)=q_k+\begin{bmatrix} r\sin\phi\cos\theta\ r\sin\phi\sin\theta\ r\cos\phi \end{bmatrix},$$9 is obtained, it defines the target subspace, while the design constants $$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$0 and $$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$1 control, respectively, the strength of equal projection and the tolerance for leakage into non-target subspaces (Luo et al., 9 Mar 2025).

Several implementation rules follow directly from the formulation. Including several dominant modes is preferred to single-mode beamforming when robustness across $$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$2 is required. The beamformer should be normalized so that $$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$3. The leakage threshold $$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$4 must be small enough to suppress interference but not so small that feasibility is over-constrained. Since both mobility and sensing evolve over time, $$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$5 and the full precoder $$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$6 should be recomputed when updated position or scatter information arrives. The relevant update rate is determined by user speed and sensing latency.

The limitations are structural. The one-sphere model assumes uniformly distributed scatterers and spherical support, while actual environments may be non-uniform, anisotropic, or clustered. Covariance-estimation errors caused by position, velocity, or sensing inaccuracies perturb $$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$7; the sphere geometry provides some tolerance because a solid sphere is less brittle than a point target, but large errors still degrade performance. The sequential per-user SOCP does not globally optimize joint multiuser coupling. Mutual coupling, calibration errors, and RF nonidealities are not modeled, even though they can alter the effective covariance seen by the array. Extremely large support spheres weaken the dominant eigenspace and diminish the benefit of correlation-averaging. Finally, the unit-norm per-stream constraint is simple; more realistic sum-power or per-antenna constraints can be incorporated, but only with careful solver configuration.

6. Related formulations in other precoding domains

The same general idea appears in several other communication settings, although the object being averaged differs. In OFDM over fading channels, a fixed-size normalized Hadamard precoder combined with time-frequency interleaving redistributes each data symbol across multiple independently faded samples. After clipped-modified zero forcing and deprecoding, the effective noise covariance is

$$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$8

so every output component sees the same noise variance, equal to the average of the per-subcarrier equalizer-output variances. The paper’s central moment result,

$$d_{n,k}(r,\theta,\phi)=\|p_n-s_k(r,\theta,\phi)\|_2$$9

formalizes correlation-averaging as variance reduction of instantaneous noise power after mixing (Ortin et al., 24 Jan 2025).

In tri-polarized near-field holographic MIMO surfaces, two-layer precoding creates an effective correlation matrix

$$[R_k]_{m,n}=\frac{1}{2\pi^2R_s}\int_0^{R_s}\int_0^{2\pi}\int_0^\pi e^{-j\frac{2\pi}{\lambda}[d_{m,k}(r,\theta,\phi)-d_{n,k}(r,\theta,\phi)]} \,dr\,d\theta\,d\phi.$$0

with the first layer suppressing cross-polarization interference and the second layer block-diagonalizing inter-user interference within each polarization. The resulting effective channel is block-diagonal across polarizations and more nearly block-diagonal across users, with reduced off-diagonal correlation coefficients and reduced eigenvalue spread. In that setting, the averaging is not over a mobility sphere but over polarization and user subspaces (Wei et al., 2022).

In large-system secrecy precoding for correlated MISO broadcast channels, regularized channel inversion leads to deterministic equivalents in which transmit-side correlation enters only through spectral averages of the eigenvalue distribution of the correlation matrix. Quantities such as

$$[R_k]_{m,n}=\frac{1}{2\pi^2R_s}\int_0^{R_s}\int_0^{2\pi}\int_0^\pi e^{-j\frac{2\pi}{\lambda}[d_{m,k}(r,\theta,\phi)-d_{n,k}(r,\theta,\phi)]} \,dr\,d\theta\,d\phi.$$1

replace dependence on the detailed eigenvectors of the covariance. Here correlation-averaging is resolvent-based spectral smoothing rather than spatial sphere shaping, but the underlying principle is again that precoding converts fine-grained correlation structure into a smaller set of stable statistics (Geraci et al., 2013).

These formulations suggest that precoding-induced correlation-averaging is best understood not as a single algorithm but as a recurring design pattern. In near-field sphere precoding it averages spatial correlation over user-position uncertainty; in OFDM it averages unequal noise amplification across interleaved tones; in polarized holographic MIMO it averages correlation by block-diagonalizing coupled subspaces; and in large-system correlated broadcast channels it averages correlation spectrally through deterministic equivalents. The common element is the replacement of fragile pointwise design by mode-domain design over a statistical support set.