Cross-Antenna Dynamic Freezing Constraints

• Cross-Antenna Dynamic Freezing Constraints are a design principle that fixes slow or structural antenna variables while allowing fast, adaptive updates for enhanced system performance.
• They enable directional modulation by preserving even-symmetric average radiation and dynamically reorienting an odd-symmetric differential component to control the information beam.
• The approach operates on a multi-timescale framework in varied architectures, balancing hardware limitations with agile digital optimization to reduce actuation overhead.

Searching arXiv for the specified papers to ground the article and verify metadata. {"queries":[{"query":"id:(Huang et al., 19 Jun 2026)"},{"query":"id:(Khalili et al., 24 Mar 2025)"},{"query":"id:(Zhu et al., 6 May 2025)"},{"query":"id:(Zheng et al., 8 Jan 2026)"},{"query":"id:(Wang et al., 30 Apr 2026)"}]} Cross-Antenna Dynamic Freezing Constraints are a class of coupled antenna-domain constraints that keep a designated cross-antenna quantity invariant over a prescribed set of antennas, antenna groups, or time intervals while allowing complementary variables to adapt. In the compact cross-structured dynamic antenna, they preserve an even-symmetric average radiation pattern while an odd-symmetric differential component is reoriented to rotate the information-recoverable sector used for directional modulation (Huang et al., 19 Jun 2026). In movable-antenna and cross-linked array systems, they freeze antenna positions or shared row/column/panel degrees of freedom over a scanning period or long statistical interval while beamforming, receive combining, transmit power, sensing waveform, or snapshot duration remain adaptive (Khalili et al., 24 Mar 2025, Zhu et al., 6 May 2025, Zheng et al., 8 Jan 2026). In monostatic ISAC with antenna flexibility, they appear as exact multiplicative Tx/Rx/Off constraints that force inactive antennas to have zero transmit or receive coefficients and then progressively harden relaxed assignments into binary frozen decisions (Wang et al., 30 Apr 2026).

1. Canonical structure of dynamic freezing

Across the cited works, the frozen variable is always a slow, shared, or structurally coupled degree of freedom, whereas the adaptive variable is electronic, algorithmic, or symbol/snapshot dependent. This suggests a common abstraction: a system partitions its design variables into a constrained subset that must remain invariant over some antenna index set or time interval, and a complementary subset that is re-optimized under that invariant structure.

Context	Frozen quantity	Adaptive quantity
Cross-structured dynamic antenna	Even-symmetric average radiation $AF_{\text{avg}}$	Odd-symmetric differential term $AF_\Delta$
MA-enabled ISAC	Antenna positions $\mathbf{B}(n)=\mathbf{B}$	Beamforming, sensing covariance, durations
CL-MA / CL-RA	Shared APVs or shared row/column/panel angles	Combining, powers, beamformers
Non-uniform monostatic ISAC	Hard Tx/Rx/Off antenna modes after hardening	Precoding, combining, WMMSE variables

The relevant invariance mechanism depends on the architecture. In the dynamic antenna, it is enforced by complementary excitation states whose even-symmetric port sums are identical and whose odd-symmetric port differences flip sign. In movable-antenna ISAC, it is enforced by per-period equality constraints such as $x_i(n)=x_i$ and $\mathbf{B}(n)=\mathbf{B}$. In cross-linked movable and rotatable arrays, it is enforced by row-column coupling, such as $\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$ or $\alpha_{m,n}=\alpha_m$, $\beta_{m,n}=\beta_n$. In antenna-flexible monostatic ISAC, it is enforced by diagonal mode-selection matrices $\mathbf{A}_T$ and $\mathbf{A}_R$ that zero inactive transmit or receive entries.

A plausible implication is that “dynamic freezing” is not a single hardware technique but a multi-timescale design principle. The hardware or geometry variables with high reconfiguration cost, strong coupling, or strict physical symmetry are constrained to remain fixed over a longer interval, while lower-latency digital variables are repeatedly updated within that frozen scaffold.

2. Dynamic freezing in cross-structured directional modulation

In "A Compact Cross-Structured Dynamic Antenna for Reconfigurable Directional Modulation" (Huang et al., 19 Jun 2026), Cross-Antenna Dynamic Freezing Constraints refer to the structural symmetries and excitation rules that keep the antenna’s broad omnidirectional radiation “frozen” while a complementary, odd-symmetric differential component is dynamically reoriented to create and rotate a narrow information-recoverable sector. The antenna uses four identical printed meander-line monopoles arranged as a planar cross on a single-layer Rogers RO4350B substrate with $AF_\Delta$0, $AF_\Delta$1, thickness $AF_\Delta$2 mm, and operates at 5.05 GHz with a footprint of $AF_\Delta$3. The element phase-center coordinates are $AF_\Delta$4, $AF_\Delta$5, $AF_\Delta$6, and $AF_\Delta$7.

The switching architecture realizes two complementary excitation states per mode. A single RF chain is split into four paths by a broadband power splitter, each branch includes attenuators and phase shifters for amplitude ratio $AF_\Delta$8 and phase calibration, and two MCU-controlled DPDT switches select the two complementary states. Opposite pairs $AF_\Delta$9–$\mathbf{B}(n)=\mathbf{B}$0 and $\mathbf{B}(n)=\mathbf{B}$1–$\mathbf{B}(n)=\mathbf{B}$2 are used for $\mathbf{B}(n)=\mathbf{B}$3 and $\mathbf{B}(n)=\mathbf{B}$4; diagonal groups $\mathbf{B}(n)=\mathbf{B}$5 and $\mathbf{B}(n)=\mathbf{B}$6 are used for $\mathbf{B}(n)=\mathbf{B}$7 and $\mathbf{B}(n)=\mathbf{B}$8. The supported information-beam azimuths are therefore $\mathbf{B}(n)=\mathbf{B}$9, $x_i(n)=x_i$0, $x_i(n)=x_i$1, and $x_i(n)=x_i$2.

The freezing mechanism is explicit in the array-factor decomposition. With

$x_i(n)=x_i$3

the state-dependent array factor is

$x_i(n)=x_i$4

Under two-state switching,

$x_i(n)=x_i$5

with

$x_i(n)=x_i$6

The key constraint is that the even-symmetric sums of opposite ports are identical in both complementary states, while the odd-symmetric differences flip sign. As a result, $x_i(n)=x_i$7 is largely invariant under switching and preserves broad omnidirectional coverage, whereas $x_i(n)=x_i$8 rotates the angle-dependent magnitude and phase distortion that determines the recoverable information sector.

Representative E-plane forms make the mechanism transparent. For $x_i(n)=x_i$9, with $\mathbf{B}(n)=\mathbf{B}$0 and $\mathbf{B}(n)=\mathbf{B}$1, one obtains $\mathbf{B}(n)=\mathbf{B}$2 and $\mathbf{B}(n)=\mathbf{B}$3. For $\mathbf{B}(n)=\mathbf{B}$4, $\mathbf{B}(n)=\mathbf{B}$5 and $\mathbf{B}(n)=\mathbf{B}$6. For the two diagonal modes, $\mathbf{B}(n)=\mathbf{B}$7, while $\mathbf{B}(n)=\mathbf{B}$8 becomes either $\mathbf{B}(n)=\mathbf{B}$9 or $\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$0, depending on the selected diagonal port grouping. The average power under switching,

$\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$1

therefore separates omnidirectional frozen coverage from angularly selective modulation distortion.

The design constraints that preserve freezing are concrete. The active ports in each state must follow the calibrated amplitude ratio $\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$2; experiments use $\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$3 dB in many results. Opposite ports must realize a $\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$4 phase inversion between complementary states. Inter-port coupling must remain lower than reflections, and the ground length should remain below approximately $\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$5 to preserve near-broadside average radiation. Measured S-parameters confirm that all four ports are reasonably matched at 5.05 GHz and that inter-port coupling is lower than the reflection responses.

The communication effect is directional modulation rather than conventional power-beam steering. The received field is expressed as $\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$6, where the even-symmetric $\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$7 varies slowly and the odd-symmetric $\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$8 produces angle-dependent magnitude and phase distortion. Information beamwidth is defined by the span where $\mathbf{p}_{m,n}=[x_m,y_n]^{\mathrm T}$9 for both simulated and measured 16-QAM and 256-QAM. In the E-plane, low BER is confined to the intended sectors; off-beam angles show large magnitude and phase errors and high BER or unrecoverable constellations, even with received SNR $\alpha_{m,n}=\alpha_m$0 dB in valid samples. In the H-plane, low BER is maintained over nearly the full angular span with received SNR $\alpha_{m,n}=\alpha_m$1 dB, confirming that the omnidirectional average remains recoverable. The realized-gain patterns of the two complementary states in each mode are nearly identical, around $\alpha_{m,n}=\alpha_m$2 dBi, which validates that the effect is phase-pattern reconfiguration rather than power redistribution.

3. Two-timescale freezing in movable-antenna ISAC

In "Movable Antenna Enabled ISAC: Tackling Slow Antenna Movement, Dynamic RCS, and Imperfect CSI via Two-timescale Optimization" (Khalili et al., 24 Mar 2025), dynamic freezing is formulated as a per-period cross-antenna position constraint. A DFRC-BS employs $\alpha_{m,n}=\alpha_m$3 movable antenna elements, each occupying one of $\alpha_{m,n}=\alpha_m$4 discrete grid locations in a 2D transmitter area. A sector of width $\alpha_{m,n}=\alpha_m$5 degrees is scanned in $\alpha_{m,n}=\alpha_m$6 snapshots over a scanning period $\alpha_{m,n}=\alpha_m$7 while serving $\alpha_{m,n}=\alpha_m$8 single-antenna users. Because mechanical repositioning is slow, on the order of milliseconds, the antenna positions are adjusted only once per scanning period, while beamforming vectors and snapshot durations are adapted in every snapshot.

The formal freezing constraint is

$\alpha_{m,n}=\alpha_m$9

or, in the discrete-grid representation,

$\beta_{m,n}=\beta_n$0

with block-diagonal selection matrix $\beta_{m,n}=\beta_n$1 constructed from the binary vectors $\beta_{m,n}=\beta_n$2. This fixed $\beta_{m,n}=\beta_n$3 is then used in all per-snapshot communication and sensing expressions. The transmit signal in snapshot $\beta_{m,n}=\beta_n$4 is

$\beta_{m,n}=\beta_n$5

with covariance

$\beta_{m,n}=\beta_n$6

and instantaneous power

$\beta_{m,n}=\beta_n$7

The robust communication model uses bounded CSI uncertainty, and the sensing model incorporates dynamic RCS variation through a chance constraint. The nominal user-$\beta_{m,n}=\beta_n$8 SINR depends on $\beta_{m,n}=\beta_n$9, and the robust communication QoS is enforced over the uncertainty set $\mathbf{A}_T$0 via an S-procedure-based LMI. The sensing beampattern is matched to a binary mask $\mathbf{A}_T$1 through an MSE constraint, and the sensing output SNR

$\mathbf{A}_T$2

is required to satisfy $\mathbf{A}_T$3. Because $\mathbf{A}_T$4, this chance constraint yields a deterministic equivalent lower bound on the realized beamforming gain. The resulting optimization problem jointly handles binary position selection, minimum spacing, robust communication QoS, sensing beampattern matching, sensing chance constraints, and timing constraints.

The solution is a hierarchical alternative-optimization procedure. The outer loop updates the discrete positions $\mathbf{A}_T$5 on the slow timescale. The inner loop optimizes, on the fast timescale, the communication beamforming, sensing covariance, and snapshot durations. The beamforming/covariance block uses SDP relaxation, the robust SINR LMI via the S-procedure, and a linear sensing chance bound. The duration/slack block uses a difference-of-convex identity for $\mathbf{A}_T$6 and successive convex approximation. The outer position block transforms binary spacing constraints via auxiliary binaries and Glover linearization, while the bilinear couplings $\mathbf{A}_T$7 and $\mathbf{A}_T$8 are handled with Schur-complement LMIs and DC penalties.

The numerical setup uses $\mathbf{A}_T$9 movable antennas, $\mathbf{A}_R$0 users, $\mathbf{A}_R$1 snapshots, carrier 5 GHz with $\mathbf{A}_R$2 m, $\mathbf{A}_R$3 m, grid step $\mathbf{A}_R$4 m, $\mathbf{A}_R$5 dBm, $\mathbf{A}_R$6 bps/Hz, $\mathbf{A}_R$7 dB, and $\mathbf{A}_R$8 ms. The TTS frozen-position design significantly reduces transmit power relative to fixed-position and antenna-selection baselines and nearly matches the upper bound with per-snapshot repositioning. Tightening the chance-constraint parameter $\mathbf{A}_R$9 increases power, coarsening the grid degrades beamforming accuracy and increases power, and larger transmitter area sizes reduce power by expanding the feasible position set. Because per-snapshot repositioning would require $AF_\Delta$00 position updates per period, whereas TTS freezing requires only one, the framework saves approximately $AF_\Delta$01 per period; with $AF_\Delta$02 and 1–3 ms per move, the saving is multi-millisecond.

4. Cross-linked position and rotation freezing

In "Multiuser Communications Aided by Cross-Linked Movable Antenna Array: Architecture and Optimization" (Zhu et al., 6 May 2025), dynamic freezing arises from collective movement constraints and from a statistical two-timescale policy. The CL-MA uses $AF_\Delta$03 vertical sliding tracks and $AF_\Delta$04 horizontal sliding tracks; each antenna lies at the cross-point of one vertical and one horizontal track. A motor moves each vertical track horizontally and each horizontal track vertically, so the number of motors scales as $AF_\Delta$05 rather than $AF_\Delta$06. The antenna position vectors are

$AF_\Delta$07

and each antenna position is

$AF_\Delta$08

All elements in a column share the same $AF_\Delta$09, and all elements in a row share the same $AF_\Delta$10. The feasible region is $AF_\Delta$11 with spacing constraints $AF_\Delta$12 and $AF_\Delta$13.

The uplink channel is built from horizontal and vertical field-response matrices and a path-response vector $AF_\Delta$14. Under ZF combining,

$AF_\Delta$15

and the APV-only power objective becomes

$AF_\Delta$16

A global lower bound on the per-user power is

$AF_\Delta$17

For single-path channels, this bound is achievable when

$AF_\Delta$18

and the optimal APVs admit the structured form $AF_\Delta$19, $AF_\Delta$20. For multipath channels, the paper uses a discrete APV optimization algorithm with sequential elimination and successive refinement. The statistical channel-based design then freezes $AF_\Delta$21 and $AF_\Delta$22 over a long period and updates only $AF_\Delta$23 and $AF_\Delta$24 in real time. In simulations with a $AF_\Delta$25 CL-MA, carrier 30 GHz, region $AF_\Delta$26, $AF_\Delta$27, and Monte Carlo size $AF_\Delta$28, the instantaneous optimization closely approaches the global lower bound, the statistical design is about 2 dB worse, and element-wise MA offers at most a 1 dB additional gain. For $AF_\Delta$29, CL-MA saves more than 30 dB relative to UPA, and the objective typically evolves through about 75 sequential-elimination iterations and about 25 successive-refinement iterations.

In "Wireless Communication with Cross-Linked Rotatable Antenna Array: Architecture Design and Rotation Optimization" (Zheng et al., 8 Jan 2026), the frozen quantity is orientation rather than position. In the element-level CL-RA architecture, the BS hosts an $AF_\Delta$30 rotatable array on horizontal and vertical rotation tracks. All elements in row $AF_\Delta$31 share the elevation angle $AF_\Delta$32, and all elements in column $AF_\Delta$33 share the azimuth angle $AF_\Delta$34, so the orientation of antenna $AF_\Delta$35 is

$AF_\Delta$36

with equality constraints

$AF_\Delta$37

This reduces the degrees of freedom from $AF_\Delta$38 to $AF_\Delta$39. The rotation matrices are $AF_\Delta$40, and the rotated boresight is

$AF_\Delta$41

A panel-level variant freezes all elements within a panel to a shared orientation and imposes additional inter-panel reflection-avoidance and CPU-blockage constraints. Sum-rate maximization is solved by alternating between MMSE receive beamforming and a feasible-direction method for the shared rotation angles; discrete angle selection is handled by a genetic algorithm.

The single-user LoS ULA case shows that CL rotation can achieve the same performance as fully flexible orientation while using $AF_\Delta$42 motors instead of $AF_\Delta$43. In the multiuser case, simulations at 3.5 GHz with $AF_\Delta$44, $AF_\Delta$45, $AF_\Delta$46, $AF_\Delta$47, and $AF_\Delta$48 show that the CL element-level scheme surpasses the CL panel-level scheme by about 25% and improves sum-rate by about 128% over fixed-direction antennas. Element-level CL also exceeds random orientation by about 79% at $AF_\Delta$49 dBm. With discrete angles, the GA-based design achieves an 84% gain over fixed orientation and about 15% over nearest-projection for $AF_\Delta$50, while the continuous search remains about 32% better than GA for $AF_\Delta$51. The alternating optimization converges in approximately 10 iterations.

Taken together, the CL-MA and CL-RA results show two distinct forms of cross-antenna freezing: collective translational freezing through shared APVs, and collective rotational freezing through shared row/column or panel angles. In both cases, the constraint is not merely hardware-economical; it is also central to the mathematical optimization, because it changes the feasible set from element-wise independent control to a lower-dimensional but still highly effective manifold.

5. Antenna-mode freezing in non-uniform monostatic ISAC

In "Harnessing the Freedom of Non-Uniformity in Monostatic ISAC with Antenna Flexibility" (Wang et al., 30 Apr 2026), dynamic freezing is neither geometric nor kinematic. It is a mode-assignment constraint over a pool of candidate antennas. Each antenna can be assigned to transmit, receive, or inactive mode through binary vectors

$AF_\Delta$52

with per-antenna exclusivity

$AF_\Delta$53

and activation budget

$AF_\Delta$54

The corresponding diagonal operators are $AF_\Delta$55 and $AF_\Delta$56.

The exact freezing effect is multiplicative. The transmitted ISAC signal is

$AF_\Delta$57

so if $AF_\Delta$58, the $AF_\Delta$59th entry of every transmit precoder is forced to zero. Similarly, the sensing combiner always appears through $AF_\Delta$60, so if $AF_\Delta$61, the $AF_\Delta$62th receive coefficient is forced to zero. The communication SINR, sensing SINR, and total transmit covariance all inherit this exact coupling. No big-$AF_\Delta$63 constants are used.

The sum-rate maximization problem is converted to a WMMSE form with equalizers $AF_\Delta$64, weights $AF_\Delta$65, and MSEs $AF_\Delta$66. The communication-side WMMSE objective is minimized jointly with the sensing SINR constraint and the power constraint. The sensing combiner admits a closed-form Rayleigh-quotient solution,

$AF_\Delta$67

The precoding subproblems become convex QCQPs or SCA-based QCQPs once the current assignment variables are fixed.

Binary assignments are relaxed to $AF_\Delta$68, and the paper uses the concave penalty

$AF_\Delta$69

which is nonnegative on $AF_\Delta$70 and vanishes only at binary points. The objective is augmented by $AF_\Delta$71, and SCA linearizes the concave quadratic terms. The Tx-assignment subproblem is expressed in quadratic form

$AF_\Delta$72

with additional quadratic power and sensing constraints. The Rx-assignment subproblem similarly uses quadratic sensing expressions in $AF_\Delta$73. Progressive hardening then converts stable relaxed assignments into frozen binary sets through thresholds $AF_\Delta$74, producing hard decisions $AF_\Delta$75, $AF_\Delta$76, or $AF_\Delta$77 for Tx, Rx, or Off.

The numerical results show that the non-uniform array consistently outperforms uniform-array baselines. For $AF_\Delta$78, the sum-rate improvement over UPA-opt is about 30%, whereas for $AF_\Delta$79, it is about 6.2%. The proposed scheme with $AF_\Delta$80 is within 2.4% of UPA-opt with $AF_\Delta$81, corresponding to approximately 51% fewer active antennas. Increasing the antenna spacing from $AF_\Delta$82 to $AF_\Delta$83 increases sum-rate under fixed $AF_\Delta$84, with gains around 10.5% for $AF_\Delta$85. As the sensing threshold $AF_\Delta$86 increases, sum-rate decreases for all schemes, but the dynamically frozen non-uniform design remains best. Here, freezing is not “no use” of antennas in a static sense; it is an optimization-driven decision to bind a subset of antennas to Tx, Rx, or Off roles and then exploit that non-uniform effective geometry jointly with precoding and combining.

6. Cross-cutting design principles, trade-offs, and scope

Taken together, these works suggest a common design doctrine: freeze the slow or structurally coupled antenna variables, and adapt the fast electronic variables around them. In the cross-structured dynamic antenna, the frozen object is the omnidirectional even-symmetric average radiation, preserved through geometry symmetry, complementary states, amplitude ratio calibration, $AF_\Delta$87 phase inversion, low mutual coupling, and a ground length below approximately $AF_\Delta$88 (Huang et al., 19 Jun 2026). In MA-enabled ISAC, the frozen object is the mechanical placement matrix over a scanning period, because movement is slow and costly relative to electronic beamformer updates (Khalili et al., 24 Mar 2025). In CL-MA and CL-RA, shared APVs or shared row/column/panel orientations reduce actuators from element-wise control to collective motion, while statistical freezing further reduces update overhead (Zhu et al., 6 May 2025, Zheng et al., 8 Jan 2026). In monostatic ISAC with antenna flexibility, freezing is a sparsity-inducing mode-selection mechanism that couples geometry, sensing, and communication through exact diagonal masking (Wang et al., 30 Apr 2026).

The principal trade-off is between degrees of freedom and implementation cost. Freezing reduces search dimensionality, actuation overhead, and calibration complexity, but it can also restrict the feasible set. The cited papers quantify this differently. Increasing the excitation ratio $AF_\Delta$89 in the dynamic antenna strengthens $AF_\Delta$90 and shrinks the information beamwidth. Tightening the sensing chance constraint or coarsening the position grid in movable-antenna ISAC increases required transmit power. Statistical APV freezing in CL-MA incurs about 2 dB loss relative to instantaneous re-optimization but drastically lowers movement overhead. CL-RA panel-level freezing is more restrictive than element-level freezing because physical panel constraints shrink feasible orientations. In non-uniform monostatic ISAC, aggressive hardening improves implementability but changes the balance among sum-rate, sensing SINR, and activation budget.

The literature also distinguishes freezing from complete stasis. In every case, some key variables remain adaptive. The dynamic antenna reorients the odd-symmetric differential component while preserving the average. The TTS movable-antenna framework freezes positions but adapts beamforming, sensing covariance, and snapshot durations every snapshot. Statistical CL-MA freezes APVs over long intervals but updates combining and powers with instantaneous channels. CL-RA freezes shared row/column or panel angles but still optimizes them jointly with MMSE receive filters. The non-uniform ISAC design freezes antenna modes only after relaxed assignments stabilize, while WMMSE variables and precoders continue to evolve during the iterations.

A plausible implication is that Cross-Antenna Dynamic Freezing Constraints define a broader multi-timescale methodology for antenna-flexible systems. The movable-antenna ISAC paper explicitly notes that the same principle extends to other systems with slow hardware knobs, including reconfigurable metasurfaces and fluid antennas (Khalili et al., 24 Mar 2025). Within the cited corpus, the unifying requirement is always the same: the frozen quantity must be chosen so that the retained adaptive variables can still enforce the target objective—directional modulation, robust communication, sensing reliability, actuator reduction, or antenna-limited sum-rate—without being overwhelmed by the loss of instantaneous element-wise freedom.