Antenna Density Function (ADF)

• ADF is a continuous representation of antenna placement that reformulates discrete indexing as a density over a normalized aperture, enabling functional optimization.
• It leverages variational analysis and gradient-based numerical methods to maximize achievable rate in point-to-point near-field channels under spherical-wave propagation.
• Closed-form and flexible array designs demonstrate that concentrating antenna density at aperture edges can significantly boost performance compared to uniform spacing.

Searching arXiv for the specified paper to ground the article and citation. Antenna Density Function (ADF) is a continuous representation of antenna placement introduced for near-field communication with massive movable antennas. In the formulation of "Near-Field Communication with Massive Movable Antennas: A Functional Perspective" (Liu et al., 2 Aug 2025), ADF recasts discrete antenna-index assignment into a density over a normalized aperture coordinate, enabling the original placement problem to be reformulated as a continuous functional optimization. The framework is developed for maximizing achievable rate in a point-to-point near-field channel, and it combines a continuous antenna position function, variational analysis, gradient-based numerical optimization, and closed-form characterization in the near-field line-of-sight (LoS) setting. A central conclusion is that edge antenna density plays a critical role in enhancing achievable rate, especially under spherical-wave propagation (Liu et al., 2 Aug 2025).

1. Formal definition and geometric interpretation

The ADF framework begins with an array of $M$ movable antennas whose discrete index $m \in \{1,\dots,M\}$ is mapped to physical positions through an Antenna Position Function (APF),

$$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$

Here $p$ is the normalized coordinate along the aperture of length $A_T=(M-1)d$ (Liu et al., 2 Aug 2025).

Instead of optimizing directly over the discrete APF, the framework introduces the Antenna Density Function $w(p)$, defined by

$$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$

Equivalently,

$$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$

so that $w(p)$ describes “antennas per unit length” (Liu et al., 2 Aug 2025).

The normalization constraint is

$\int_{-1}^{1} w(p)\,dp = f^{-1}(1)-f^{-1}(-1)=M-1.$

Because $m \in \{1,\dots,M\}$0 and integrates to $m \in \{1,\dots,M\}$1, it equivalently parametrizes all monotonically increasing APFs. In the paper’s formulation, this change of variables linearizes the geometry and converts a discrete placement problem into a continuous functional optimization problem (Liu et al., 2 Aug 2025).

A plausible implication is that ADF functions as a measure-theoretic surrogate for antenna indexing: instead of enumerating antenna coordinates individually, one optimizes a density that captures the aggregate spatial allocation of antennas across the aperture.

2. Continuous reformulation of antenna placement

Under an arbitrary spatial channel response $m \in \{1,\dots,M\}$2, the downlink Gram matrix is written as

$m \in \{1,\dots,M\}$3

By extending the antenna index to a continuous variable and changing variables through $m \in \{1,\dots,M\}$4, the formulation becomes

$m \in \{1,\dots,M\}$5

where $m \in \{1,\dots,M\}$6 is the linear interpolation of antenna coordinates (Liu et al., 2 Aug 2025).

With SNR $m \in \{1,\dots,M\}$7 and isotropic transmit covariance $m \in \{1,\dots,M\}$8, the achievable rate becomes the functional

$m \in \{1,\dots,M\}$9

subject to

$$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$0

This yields a continuous optimization problem:

$$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$1

subject to the positivity and normalization constraints (Liu et al., 2 Aug 2025).

The significance of this reformulation lies in the replacement of the original combinatorial placement problem $$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$2 by the continuous problem $$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$3. In the terminology of the paper, achievable rate is optimized with respect to the ADF rather than over discrete antenna coordinates. This suggests that the framework is especially suited to massive-antenna regimes, where direct combinatorial search over placements becomes difficult.

3. Variational characterization of the optimal ADF

The optimality condition for ADF is derived through functional analysis and variational methods. The Lagrangian is

$$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$4

Considering perturbations $$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$5 with $$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$6, the functional derivative is given by

$$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$7

where

$$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$8

The first-order optimality condition is then

$$p=f(m), \qquad p\in[-1,1], \qquad f(1)=-1,\; f(M)=+1,\; f(m+1)>f(m).$$9

constant for all $p$0, together with positivity and normalization constraints (Liu et al., 2 Aug 2025).

In this formulation, the optimal ADF equalizes the functional derivative of achievable rate across the aperture. The paper identifies this as the Euler–Lagrange condition for the continuous placement problem. The condition is important because it provides both an analytical characterization of optimality and the basis for numerical solution methods under general channel conditions.

A plausible interpretation is that the optimal ADF distributes antenna density so that no infinitesimal reallocation of density can increase the achievable rate to first order.

4. Gradient-based numerical solution

For general channel conditions, the paper proposes an iterative gradient-ascent method on the ADF. The normalized coordinate $p$1 is discretized on $p$2 grid points, and the initial density is chosen uniformly:

$p$3

At iteration $p$4, the procedure is:

1. Build $p$5 using the kernel integral.
2. Compute $p$6 and the gradient $p$7.
3. Update

$p$8

1. Project by clipping

$p$9

then renormalize so that $A_T=(M-1)d$0.

The algorithm stops when $A_T=(M-1)d$1 is small. Its complexity is stated as $A_T=(M-1)d$2, linear in $A_T=(M-1)d$3 when $A_T=(M-1)d$4 (Liu et al., 2 Aug 2025).

The numerical method serves as the general-purpose solver in the framework. The paper positions it as applicable beyond the analytically tractable LoS case, including realistic mixed LoS/NLoS channels. The use of a projection step reflects the fact that admissible ADFs must remain nonnegative and properly normalized throughout optimization.

5. Closed-form ADF in the near-field LoS regime

For the near-field LoS scenario, the paper derives a closed-form optimal ADF. Under the spherical-wave model,

$A_T=(M-1)d$5

and with Fresnel and denominator approximations, the Gram matrix takes the Toeplitz form

$A_T=(M-1)d$6

where $A_T=(M-1)d$7, $A_T=(M-1)d$8, and

$A_T=(M-1)d$9

By classical Toeplitz-symbol asymptotics (Fisher–Hartwig), the asymptotic log-determinant is maximized when the generating function exhibits controlled pole-type singularities at $w(p)$0 of maximal admissible order $w(p)$1 (Liu et al., 2 Aug 2025).

The corresponding optimal ADF is

$w(p)$2

with $w(p)$3 chosen so that $w(p)$4. In the extreme-order case $w(p)$5, the ADF becomes “U-shaped” and blows up at $w(p)$6, corresponding to the aperture edges (Liu et al., 2 Aug 2025).

The paper explicitly interprets this edge concentration as reflecting the need to sample high spatial-frequency content of spherical waves near the aperture rim. For moderate $w(p)$7, the form can be simplified by dropping the offset term, yielding

$w(p)$8

This closed-form LoS solution is one of the central analytical contributions of the framework. It establishes that nonuniform antenna density, rather than uniform spacing, is rate-optimal in the asymptotic near-field setting considered.

6. Discretization, deployment, and practical implementation

Once an optimal continuous ADF is obtained, the discrete antenna positions $w(p)$9 are recovered through the cumulative ADF (CADF),

$$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$0

and the inversion rule

$$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$1

For the simplified form $$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$2, the paper states that closed-form expressions can be obtained via inverse incomplete Beta-functions (Liu et al., 2 Aug 2025).

A practical issue arises because extreme edge clustering, especially for $$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$3, may violate minimum-spacing requirements and cause mutual coupling. To mitigate this, the authors propose a “flexible array” substrate whose physical curve $$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$4 is designed so that the density of projected antenna indices onto $$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$5 follows the target ADF. The construction solves

$$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$6

where $$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$7 is the curve length. This produces a bent or folded array whose Euclidean spacing remains uniform while its projection matches the desired density (Liu et al., 2 Aug 2025).

The implementation strategy links the continuous optimization theory to realizable antenna geometries. The paper further notes that uniform circular arrays emerge as a promising geometry for balancing performance and deployment feasibility in near-field communications (Liu et al., 2 Aug 2025). This suggests that practical embodiments of ADF-optimized placement need not rely solely on straight apertures.

7. Simulation results and interpretive context

The reported simulations consider a representative point-to-point setup with $$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$8 receive elements, SNR $$w(p)=\lim_{\Delta p \to 0}\frac{f^{-1}(p+\Delta p)-f^{-1}(p)}{\Delta p} =\frac{d}{dp}[f^{-1}(p)]>0, \qquad p\in[-1,1].$$9, and varying $$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$0 and $$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$1 in the near-field. The compared schemes are: Uniform Linear Array (ULA) with $$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$2, closed-form ADF with $$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$3 for $$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$4, gradient-based ADF via Algorithm 1, and a baseline greedy Antenna-Selection (AS) scheme on a dense candidate grid (Liu et al., 2 Aug 2025).

The simulation findings reported in the paper are summarized below.

Scheme or observation	Reported result
Closed-form ADF with $$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$5	$$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$6–$$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$7 rate gain over ULA for $$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$8 up to $$f^{-1}(p)=\int_{-1}^{p} w(u)\,du + (\text{constant}),$$9
Variational Algorithm 1	Further boosts performance in realistic mixed LoS/NLoS channels
Algorithm 1 convergence	Converges in $w(p)$0 iterations with per-iteration cost linear in $w(p)$1
Flexible “bent” arrays	Recover essentially the same performance while maintaining practical minimum spacing

These results support the paper’s broader claim that the ADF framework turns a combinatorial massive-antenna placement problem into a tractable continuous functional optimization with analytical structure in the near-field LoS case and fast numerical solvers in more general settings (Liu et al., 2 Aug 2025).

A common misconception would be to interpret ADF merely as a heuristic nonuniform spacing rule. In the framework under discussion, ADF is instead the primary optimization variable in a functional-rate maximization problem. Another possible misconception is that edge concentration necessarily implies impractical physical clustering; the flexible-array construction is introduced precisely to mitigate mutual coupling and minimum-spacing violations while preserving the projected density profile.