Antenna Position Function (APF)

• Antenna Position Function (APF) is a mapping that converts discrete antenna indices into continuous spatial coordinates or response maps, capturing key geometric and electromagnetic characteristics.
• It provides a unifying framework for representing antenna positions, field responses, and scalar performance metrics across applications such as movable arrays, AirComp, and antenna pointing.
• Key design insights include edge-enhanced density functions and symmetric configurations, which improve resolution, control phase responses, and optimize overall system performance.

Antenna Position Function (APF) denotes a position-encoding or position-induced response map whose precise meaning depends on the antenna problem under study. In its most explicit recent usage, APF is the monotone map $f(m)$ that assigns antenna index $m$ to a normalized aperture coordinate $p\in[-1,1]$, thereby converting discrete array placement into a continuous functional design problem (Liu et al., 2 Aug 2025). In closely related movable-antenna, sensing, AirComp, relaying, and antenna-pointing literature, the term is often not used explicitly; instead, the nearest equivalents are the antenna position vector (APV), the position-dependent field or channel response $h_k(\mathbf r)$, or a scalar performance map such as received signal strength, computation mean square error (CMSE), or achievable rate (Cheng et al., 2023, Li et al., 2024, Pi et al., 2023). Taken together, these works suggest that APF is best understood as a family of mathematically equivalent representations that connect antenna geometry to electromagnetic response, channel realization, and system-level objective.

1. Terminological scope and formal definitions

In "Near-Field Communication with Massive Movable Antennas: A Functional Perspective" (Liu et al., 2 Aug 2025), the APF is defined directly as a scalar-valued map

$f:\mathcal M\to[-1,1],$

with $\mathcal M=\{1,\dots,M\}$, where $p=f(m)$ is a normalized antenna position coordinate on the aperture. The paper states that $f(m)$ is a monotonically increasing function bounded by $-1\le f(m)\le 1$, with boundary conditions $f(1)=-1$ and $m$0. For a conventional ULA, the APF is

$m$1

The same paper then extends the APF domain from the discrete index set to the interval $m$2, so that continuous functional analysis can be applied (Liu et al., 2 Aug 2025).

Most other papers use different terminology. "Unlocking the Potential of Movable Antennas: General and Practical Antenna Position Optimization" states that it does not explicitly define the term APF and instead works with antenna position optimization (APO), antenna position vector (APV), sampling-point selection, point-wise CSI or channel maps, and learned mappings from partial measurements to optimal MA positions (Mei et al., 10 Jun 2026). "Multi-Beam Forming with Movable-Antenna Array" likewise uses the antenna position vector

$m$3

and notes that, in APF language, a natural interpretation is the discrete map $m$4 from antenna index to spatial coordinate (Ma et al., 2023).

A third meaning appears in antenna pointing. "Extremum Seeking Control for Antenna Pointing via Symmetric Product Approximation" does not use the term APF explicitly, but it defines the unknown received-power map $m$5, with $m$6, and its equivalent minimization form

$m$7

The details identify $m$8 and $m$9 as exactly the kind of object requested under the APF label: an analytically unknown scalar function of antenna orientation whose value is the received signal strength or equivalent cost (Wang et al., 24 Feb 2025).

2. APF as an index-to-position map

The most literal APF formulation treats antenna placement itself as a function of antenna label. In the near-field functional formulation, once $p\in[-1,1]$0 is specified, the $p\in[-1,1]$1-th antenna location is obtained through the line parameterization

$p\in[-1,1]$2

so the APF does not directly map into arbitrary $p\in[-1,1]$3-D or $p\in[-1,1]$4-D space; it maps to a scalar aperture coordinate, and that scalar then parameterizes a prescribed geometric support (Liu et al., 2 Aug 2025).

In one-dimensional movable arrays, the APF-equivalent object is usually the ordered APV. "Multi-Beam Forming with Movable-Antenna Array" imposes

$p\in[-1,1]$5

so the geometry is completely described by the discrete index-to-position map $p\in[-1,1]$6 over a bounded line segment (Ma et al., 2023). "Movable Antenna Enhanced Wireless Sensing Via Antenna Position Optimization" uses the same pattern for sensing arrays, with $p\in[-1,1]$7, ordered positions, and minimum inter-antenna spacing $p\in[-1,1]$8 (Ma et al., 2024). In both cases, the APF is discrete, ordered, and constrained by aperture and coupling-avoidance conditions.

In two-dimensional formulations, the APF-equivalent object becomes a stacked APV rather than a scalar coordinate function. "Multiuser Communications with Movable-Antenna Base Station Via Antenna Position Optimization" defines

$p\in[-1,1]$9

with $h_k(\mathbf r)$0 (Pi et al., 2023). "Movable Antenna Enhanced Integrated Sensing and Communication Via Antenna Position Optimization" similarly defines a planar APV

$h_k(\mathbf r)$1

and then separates it into horizontal and vertical APVs $h_k(\mathbf r)$2 and $h_k(\mathbf r)$3 for alternating optimization (Ma et al., 13 Jan 2025). This suggests that beyond one dimension, APF is often operationally realized as a vector-valued position representation rather than as a single scalar function.

3. APF as a position-to-field or position-to-channel mapping

A second major meaning of APF is the map from antenna position to field response or channel coefficient. In AirComp with movable antennas, the canonical example is

$h_k(\mathbf r)$4

which maps a receive location $h_k(\mathbf r)$5 to the complex channel coefficient from sensor $h_k(\mathbf r)$6 to that location (Cheng et al., 2023). The effective receive channel vector then becomes

$h_k(\mathbf r)$7

The same paper states that antenna positioning reshapes the receive channel to improve AirComp distortion, and explicitly notes that, different from the FPA-based case, the MSE depends on MA positions (Cheng et al., 2023).

Field-response formulations make this dependence more explicit. In uplink multiuser communications, the receive field-response vector is

$h_k(\mathbf r)$8

with

$h_k(\mathbf r)$9

and the user channel vector is

$f:\mathcal M\to[-1,1],$0

(Pi et al., 2023). "Over-the-Air Computation via 2D Movable Antenna Array" uses the same compositional structure, writing

$f:\mathcal M\to[-1,1],$1

and emphasizes that under the far-field condition local movement changes only the phase of each multipath component, while amplitudes and AoAs remain unchanged (Li et al., 2024).

A related formulation appears in derivative-free MA optimization. "A Derivative-Free Position Optimization Approach for Movable Antenna Multi-User Communication Systems" writes

$f:\mathcal M\to[-1,1],$2

where $f:\mathcal M\to[-1,1],$3 is the full stacked position vector (Zeng et al., 25 May 2025). The paper explicitly interprets this as the input to a black-box optimization problem, rather than as a globally reconstructed spatial map. In relaying, the same position-to-channel logic is split across two stages: $f:\mathcal M\to[-1,1],$4 so receive-stage and transmit-stage antenna positions define two distinct APF-like mappings (Li et al., 14 Jan 2025).

Taken together, these formulations indicate that APF frequently denotes a deterministic geometry-to-response map whose output is a steering vector, field-response vector, channel coefficient, or channel matrix. The dominant mechanism is phase control through position-dependent inner products such as $f:\mathcal M\to[-1,1],$5, $f:\mathcal M\to[-1,1],$6, or $f:\mathcal M\to[-1,1],$7.

4. APF as a scalar objective or utility landscape

A third meaning of APF is a scalar-valued function over antenna orientation or position whose extremizer is the design target. In antenna pointing without angular measurements, the APF is the received-power map

$f:\mathcal M\to[-1,1],$8

and the equivalent minimization cost

$f:\mathcal M\to[-1,1],$9

The assumptions imposed on $\mathcal M=\{1,\dots,M\}$0 are that it is smooth and positive everywhere, has a unique global minimizer $\mathcal M=\{1,\dots,M\}$1, is separable as $\mathcal M=\{1,\dots,M\}$2, and has globally bounded Hessian (Wang et al., 24 Feb 2025). Under this interpretation, APF is an unknown scalar field over orientation, and antenna control becomes extremum seeking on that field.

In movable-antenna communication systems, equivalent scalar APFs are often performance functions induced by the position-dependent channel. In AirComp with a $\mathcal M=\{1,\dots,M\}$3-D movable receive array, the position-dependent objective is

$\mathcal M=\{1,\dots,M\}$4

so the full APV $\mathcal M=\{1,\dots,M\}$5 is mapped to a single scalar distortion measure (Li et al., 2024). In multiuser uplink communications, the paper defines

$\mathcal M=\{1,\dots,M\}$6

as the maximum min-user achievable rate under a given APV, after inner-loop optimization of receive combining and user powers (Pi et al., 2023). In derivative-free MA optimization, the paper defines single-user and multi-user scalar objectives

$\mathcal M=\{1,\dots,M\}$7

and

$\mathcal M=\{1,\dots,M\}$8

respectively (Zeng et al., 25 May 2025).

This suggests a broad but technically precise usage: APF may denote either the primitive position-to-field map or the composite position-to-performance map obtained after combining, beamforming, power control, or control-law elimination. The distinction matters because some papers optimize the primitive map directly, while others optimize only the scalar utility induced by it.

5. Functional reformulations and optimization methodologies

The explicit APF literature introduces a functional reformulation from discrete coordinates to continuous density. In the near-field functional approach, the inverse derivative of the APF,

$\mathcal M=\{1,\dots,M\}$9

is defined as the antenna density function (ADF), with

$p=f(m)$0

The achievable-rate functional is then written as

$p=f(m)$1

and optimized over positive densities $p=f(m)$2 subject to the integral constraint (Liu et al., 2 Aug 2025). The paper derives the functional derivative

$p=f(m)$3

and proposes variational gradient ascent with nonnegative clipping and normalization, after which the APF is recovered by integrating the ADF and inverting the cumulative map (Liu et al., 2 Aug 2025).

In discrete APV optimization, alternating optimization and successive convex approximation recur. "Multi-Beam Forming with Movable-Antenna Array" alternates between antenna weight vector and antenna position vector optimization, and solves convex QCQPs built from SCA lower and upper bounds of the beam gain (Ma et al., 2023). "Movable Antenna Enhanced Wireless Sensing Via Antenna Position Optimization" uses AO-SCA on horizontal and vertical coordinates, with convex subproblems derived from the variance and covariance structure of the CRB denominators (Ma et al., 2024). "Movable Antenna Enhanced Multi-Region Beam Coverage" first builds a continuous multi-notch-filter-inspired profile $p=f(m)$4, then uses sequential update and Gibbs sampling to choose a discrete subset of positions that best sample that profile (Wang et al., 30 Dec 2025).

Swarm-based and derivative-free methods address cases where analytical structure is weak or full channel maps are impractical. In multiuser communications and AirComp, PSO is used in an outer loop over APV, with inner-loop updates of receive combining, power, or transmit coefficients (Pi et al., 2023, Li et al., 2024). "A Derivative-Free Position Optimization Approach for Movable Antenna Multi-User Communication Systems" instead treats the objective as a black box and uses the one-point zeroth-order estimator

$p=f(m)$5

within a ZO-AdaMM update, thereby avoiding global CSI reconstruction (Zeng et al., 25 May 2025).

A broader methodological taxonomy is given in the APO survey-style paper, which separates continuous APO, discrete APO, and partial-CSI APO. The listed methods include SCA-based and filter-based continuous APO; graph-theoretic, branch-and-bound, and sequential update with Gibbs sampling for discrete APO; and ZO-AdaMM, RACOS, supervised learning, and unsupervised learning for partial-CSI APO (Mei et al., 10 Jun 2026). This suggests that APF optimization now spans functional analysis, nonconvex continuous optimization, discrete combinatorial selection, and learned position-decision mappings.

6. Structural results and recurring design principles

Several papers derive explicit APF or APV structures rather than only numerical solutions. In near-field line-of-sight communication, the optimal ADF is edge-enhanced. The smooth-case solution is

$p=f(m)$6

while the singular-case asymptotically optimal density has the form

$p=f(m)$7

which is U-shaped and places higher density near $p=f(m)$8 (Liu et al., 2 Aug 2025). The paper interprets this as evidence that edge antennas better capture higher spatial frequencies induced by spherical-wave curvature.

In one-dimensional wireless sensing, the CRB-only optimum is the closed-form edge-clustered APV

$p=f(m)$9

which maximizes the position variance and therefore minimizes

$f(m)$0

(Ma et al., 2024). In a circular $f(m)$1-D movement region, the same paper shows that optimum designs satisfy symmetry conditions such as $f(m)$2, $f(m)$3, and $f(m)$4, so boundary placement and geometric balance become the key principles (Ma et al., 2024).

Robust sensing introduces an opposing tendency. "Movable Antennas for Robust Wireless Sensing via Joint Cramér-Rao Bound and Sidelobe Minimization" states that minimizing the CRB prefers a narrower mainlobe, where antennas are concentrated near the two edges of the one-dimensional movement region, whereas minimizing the maximum sidelobe level favors a wider mainlobe, where antennas are distributed more densely near the center (Ma et al., 22 Jun 2026). Its continuous MSL-oriented density is symmetric and contains both edge impulses and a center-peaked smooth component. This suggests that APF design can be SNR-regime dependent: high-SNR operation favors edge-heavy high-resolution configurations, while low-SNR robustness favors more center-dense low-sidelobe configurations.

Integrated sensing and communication yields a related boundary-loading result. In receive-MA-only ISAC, the sensing objective reduces to

$f(m)$5

and the paper gives a closed-form optimal receive APV consisting of two edge clusters. It further shows that employing MAs over conventional FPAs can achieve a sensing performance gain upper-bounded by $f(m)$6 dB (Chen et al., 20 Feb 2025). A plausible implication is that APF design is frequently governed by centered second moments of position, but the optimal balance between edge loading, central loading, and symmetry depends on whether the dominant figure of merit is resolution, ambiguity suppression, communication conditioning, or a joint trade-off.

Overall, the APF concept has evolved from a narrow index-to-position map into a broader analytical framework for geometry-aware design. It can denote a monotone aperture parameterization, a position-to-field or position-to-channel map, or a composite scalar utility over orientation or placement. Across near-field communications, movable-array beamforming, AirComp, relaying, sensing, ISAC, and antenna pointing, the common thread is that APF formalizes how controlled antenna geometry reshapes phase, response, and ultimately performance.