Research on the Quantitative Relationship Between Positioning Error and Coherent Synthesis Success Rate in a Moving Platform Distributed Coherent Synthesis System

Abstract

Distributed coherent synthesis on dynamic platforms suffers from phase misalignment and significantly reduced synthesis efficiency due to navigation errors and communication delays. To address this challenge and dramatically enhance the synthesis efficiency, this paper proposes an “error-performance” quantification framework and corresponding compensation methods: (1) Phase compensation strategy: Adaptive Kalman Filter (AKF) with a multi-index fusion-based adaptive factor derived from novelty sequences, enabling intelligent switching between predictive and robust modes for improved phase compensation; (2) Positioning error modeling method: Employing an adaptive reverse-adaptive robust Kalman filter (ARKF) to synthesize error trajectories, with standard deviation σ as the primary control parameter. Monte Carlo simulations establish a quantitative relationship between positioning error standard deviation (σ) and coherent synthesis success rate: Under a 3-transmitter configuration, success rate ≥ 95% when σ ≤ 100 mm; The 100–237.3 mm range constitutes a transition zone where success rate decreases from 95% to 80%; when σ ≥ 460 mm, the success rate stabilizes at 56–58%. The core conclusion indicates that when σ ≤ 237.3 mm, the system achieves high coherent synthesis efficiency with 80% probability. This paper aims to establish a cross-platform transferable error-performance quantification framework, providing a direct reference for navigational accuracy selection in distributed coherent systems.

1. Introduction

Distributed coherent synthesis is an advanced signal processing technique whose core lies in focusing electromagnetic energy in the far-field through precise phase coordination among multiple spatially separated transmitting units. Compared to traditional incoherent energy superposition, this technique can coherently combine dispersed electromagnetic waves into a high-energy-density focal point in the target region, thereby achieving a significant improvement in the system’s signal-to-noise ratio (SNR). Theoretically, for a Distributed Coherent Aperture Radar (DCAR) composed of N units under coherent transmission, the received power and SNR are improved by approximately N² times compared to the unit level [1]. Consequently, this technology demonstrates irreplaceable value in cutting-edge fields such as wireless power transmission and long-range radar detection [2].

In contrast to traditional centralized systems, distributed coherent synthesis technology coordinates multiple spatially distributed and independently moving platforms to achieve phase synchronization of their emitted electromagnetic waves at the target, thereby synthesizing an equivalent high-power, high-directivity beam. The core challenge of this technology lies in overcoming phase mismatch among the platforms caused by relative motion and positioning errors so as to maintain stable coherent synthesis. Owing to its exceptional flexibility and scalability, distributed coherent synthesis demonstrates significant application potential in next-generation radar systems (e.g., distributed radar), communication networks (e.g., THz communications), and wireless power transfer (remote charging), which also constitute the core application scenarios and value of this research.

However, the phase coordination problem, which is relatively straightforward in a fixed platform environment, becomes highly challenging in dynamic scenarios. When this technology is applied to mobile platforms such as UAVs or ground vehicles, its engineering practice is severely constrained by two core physical challenges: (1) the inherent, time-varying navigation and positioning errors of each platform [3], and (2) the inevitable communication latency between network nodes [4]. These random errors introduce significant phase noise into the system, leading to substantial degradation in the synthesis effectiveness of the field strength at the target point, typically characterized by a sharp decline in the combined field strength and a shift in the energy focus. The combined effect of these two factors presents a formidable technical challenge for achieving precise phase alignment.

To address phase synchronization challenges in distributed coherent synthesis on dynamic platforms and improve synthesis efficiency, the research community has explored various technical approaches, which can be broadly categorized into two types: one focusing on adaptive signal-level focusing, and the other on prediction-based compensation based on state estimation.

In the domain of signal adaptive focusing, time reversal technology has attracted significant attention due to its excellent adaptive focusing capabilities [5]. However, this method typically relies on stable detection/echo or pilot signals (which can be generated by cooperative beacons) available within the target area, making its deployment in non-cooperative scenarios quite challenging [6]. This inherent limitation motivates the present research to seek alternative solutions more suitable for non-cooperative environments.

In terms of prediction compensation, the traditional Kalman filter and its improved algorithms have become mainstream tools. The fundamental concept is to use a reference source to receive state information from the signal source, predict its future position, and accordingly calculate compensation commands to counteract communication delays [7]. Centered around this framework, numerous studies have been dedicated to enhancing the robustness of the filters. For instance, Yin et al. [8] proposed a robust adaptive extended Kalman filter based on an improved measurement noise covariance matrix to handle anomalous disturbances; whereas Lou et al. [9] developed a partial strong tracking extended Kalman filter to improve the robustness of GNSS/INS integrated navigation. Furthermore, nonlinear filtering techniques, represented by the maximum entropy criterion, have also demonstrated potential for handling non-Gaussian noise [10]. However, a review of the existing literature reveals a common limitation among the aforementioned studies: the performance of both the KF and its various improved algorithms heavily depends on whether the internal dynamic model can accurately describe the platform’s real-time motion. In complex real-world moving platform environments, model mismatch is difficult to avoid. Simulations in this research confirm that when the platform’s actual maneuvers exceed the preset range of the model, the predictive performance of the standard Kalman Filter (KF) deteriorates sharply and may even lead to complete collapse of coherent synthesis due to filter divergence. This makes the quantitative evaluation results based on traditional methods lack universality and reliability.

In summary, current research lacks a unified framework that can overcome the aforementioned limitations, enhance synthesis efficiency, and precisely establish a quantitative relationship between the statistical characteristics of positioning errors and the ultimate coherent synthesis success rate. This paper aims to fill this gap by introducing an adaptive robust Kalman filter for error modeling and proposing a hybrid compensation strategy based on a multi-index fusion adaptive factor to achieve reliable and high-efficiency coherent synthesis in dynamic and uncertain environments.

The fundamental reason for the limitations of the aforementioned traditional prediction methods lies in the mismatch between their internal models and the true error dynamics, which also reveals two critical research gaps: Firstly, how to design a compensation strategy that remains robust and reliable under complex dynamics; Secondly, how to construct a simulation environment that can authentically reflect the inertial and non-stationary characteristics of navigation system errors, enabling the performance evaluation of distributed coherent synthesis systems to move beyond idealized assumptions and thereby establish a reliable quantitative relationship between ‘positioning error standard deviation σ’ and ‘coherent synthesis success rate’ [11]. The positioning error in this research is not simple white noise but exhibits significant temporal correlation and non-stationary characteristics, meaning the error drift possesses “inertia”. Therefore, any evaluation detached from the true characteristics of the errors becomes meaningless, and simple random number models cannot replicate this crucial property.

To tackle the challenges outlined above and achieve the primary goal of quantifying the relationship between positioning error and coherent synthesis success rate, this study develops a comprehensive research methodology spanning from theoretical modeling to practical application. The investigation starts by identifying the root cause of low synthesis efficiency in distributed coherent synthesis systems on mobile platforms, which is attributed to phase misalignment resulting from positioning inaccuracies. Ideal phase compensation requires precise adjustment of the transmission phase of each radiating element to achieve in-phase superposition of signals at the target, thereby obtaining an N²-fold power gain; however, dynamic platform positioning errors introduce deviations in the compensation phase calculated based on incorrect location information, directly degrading synthesis efficiency.

To address these challenges and quantify the relationship between positioning error and coherent synthesis success, this study develops a comprehensive methodology from theoretical modeling to practical application. The root cause of low synthesis efficiency on mobile platforms is identified as phase misalignment caused by positioning errors. Ideal phase compensation demands precise phase adjustment of each radiating element to achieve in-phase signal superposition at the target, resulting in an N² power gain. However, dynamic positioning errors cause deviations in the compensation phase based on inaccurate location data, directly degrading synthesis efficiency.

To solve the phase mismatch problem in this dynamically uncertain environment, this research proposes a technical route centered on a hybrid adaptive compensation strategy based on multi-index fusion adaptive factors: First, an adaptive robust Kalman filter is employed for high-precision modeling and prediction of platform positioning errors, which adaptively characterizes the spatiotemporal correlation and dynamic evolution of the errors by estimating and adjusting the innovation sequence in real-time. Then, an intelligent hybrid compensation mechanism is designed, whose core is the introduction of a multi-index fusion adaptive factor derived from the innovation sequence. By comparing this factor in real time with a preset threshold, the system is driven to intelligently switch between a “high-performance prediction” mode and a “high-robustness compensation” mode. When the model matches well, the system uses the filter’s high-precision predicted values for advanced phase compensation; when model mismatch is detected, it automatically switches to a robust mode reliant on the latest measurements, thereby ensuring system stability while pursuing high performance. On this basis, this research constructs a systematic Monte Carlo simulation framework. By conducting numerous independent experiments under different levels of positioning error standard deviation and statistically analyzing the synthesis success rate, a deterministic quantitative relationship curve of “positioning error standard deviation σ versus coherent synthesis success rate” is ultimately established. This provides direct engineering rationale for the selection of system navigation accuracy and the design of compensation strategies. Figure 1 illustrates the research framework of this research.

Based on the complete technical route and research practice described above, the primary innovations and contributions of this research are all aimed at enhancing the synthesis efficiency for dynamic platform distributed coherent synthesis, and can be summarized as follows:

• Theoretical Innovation: This research develops a generative error modeling framework grounded in stochastic process theory, moving beyond conventional white noise assumptions. By reversing the operation of the Adaptive Robust Kalman Filter (ARKF), it transforms the filter from a state estimator into a high-fidelity error trajectory generator that effectively captures temporal error inertia.
• Methodological Innovation: A hybrid adaptive compensation strategy incorporating a multi-index fusion adaptive factor is designed to ensure stable and efficient synthesis. This approach formulates the compensation task as an adaptive control problem, employing a mode-switching control law based on multiple fused innovation indicators. Theoretical analysis confirms this strategy guarantees global stability and mitigates the risk of divergence common in standard predictive filters under high dynamic conditions.
• Paradigm Innovation: A quantitative design paradigm mapping localization errors to synthesis performance is established for the first time, providing clear design criteria for system engineers. Specifically, through Monte Carlo simulations based on our framework, the quantitative relationship “positioning error standard deviation σ versus coherent synthesis success rate” is systematically charted for the first time, yielding critical performance thresholds (e.g., σ = 237.7 mm). This directly translates abstract error statistics into concrete, actionable specifications for navigation system accuracy design. The research block diagram of this paper is presented below.

2. Proposed Methodology and Simulation Framework

2.1. Universal Positioning Error Modeling

The effectiveness of performance evaluation for any compensation algorithm highly depends on the authenticity of the employed error model. Conventional research often simplifies localization errors as Gaussian white noise, but this severely overlooks the inherent temporal correlation and non-stationarity of real navigation system errors. An eastward error occurring at time t is more likely to continue drifting eastward at time t + Δt, rather than changing direction randomly.

Substantial empirical studies have shown that adaptive and robust improved algorithms based on the Kalman filter framework are effective tools for handling such realistic error dynamics. For instance, Liu Fei explicitly stated in their research on GNSS/INS integrated navigation that their proposed robust adaptive Kalman filter (ARKF) algorithm, based on prediction residuals, can effectively handle observation anomalies and dynamic model disturbances in practical systems [12]. Experimental data from that research showed that, compared to the standard Kalman filter, their algorithm improved 3D positioning accuracy by 45.9% and 46.8% in loosely coupled and tightly coupled navigation modes, respectively. This result strongly demonstrates the significant advantages and physical realism of such adaptive filtering algorithms in characterizing and suppressing real-world navigation errors. Therefore, a universal error model capable of replicating this “inertia” is a necessary prerequisite for all subsequent research. Based on the same physical understanding, this research innovatively applies the principles of the Adaptive Robust Kalman Filter (ARKF) in “reverse”, utilizing the filter itself as a high-fidelity error trajectory generator. The core idea of this modeling approach can be decomposed into the following four points:

(1). State Definition: The two-dimensional positioning error of each mobile platform and its rate of change (i.e., error velocity) are defined as a four-dimensional state vector. This definition inherently incorporates the dynamic “memory” characteristic of the errors.

$$x_k={[\delta x_k\hspace{0.33em}\delta y_k\hspace{0.33em}\delta v_{x,k}\hspace{0.33em}\delta v_{y,k}]}^T$$

(1)

where ($\delta x_k\hspace{0.33em}\delta y_k$) is the positional error, and ($\delta v_{x,k}\hspace{0.33em}\delta v_{y,k}$) is the rate of change in the error.

(2). Physical Evolution Model: A linear state transition equation based on physical motion laws is used to describe how the error state evolves from one time step to the next, thereby ensuring the generated error sequence possesses temporal continuity and smoothness.

$${\mathrm{x}}_{\mathrm{k}}=F{x}_{k-1}+W_{k-1}$$

(2)

herein, the state transition matrix F adopts a constant velocity (CV) model, and is Gaussian process noise with zero mean and covariance Q, i.e., $W_{k-1}\sim N(0,Q)$.

Regarding the state transition matrix F, this model is a reasonable approximation for engineering scenarios where “errors exhibit inertia but without severe acceleration”, supporting the generation of error trajectories with temporal continuity and controllable inertia.

$$F=\left[\right[1, 0,dt,0],[0, 1, 0,dt],[0, 0, 1, 0],[0, 0, \mathrm{0,1}\left]\right]$$

(3)

(3). Stochastic Process Drive: A process noise covariance matrix Q is introduced as the stochastic drive source for the errors. The setting of the Q matrix directly determines the dynamic evolution characteristics (e.g., drift rate, inertia strength) of the platform’s error trajectory in the simulation, serving as a key parameter for controlling different error levels in subsequent Monte Carlo experiments. For the process noise covariance Q, the baseline setting is as follows:

$$Q=diag\left(\right[{({\sigma }_{pos}/2)}^2,{({\sigma }_{pos}/2)}^2,{{\sigma }_{vel}}^2,{{\sigma }_{vel}}^2\left]\right)$$

(4)

where ${\sigma }_{pos}$ = 0.1 m and ${\sigma }_{vel}$ = 0.025 m/s. This configuration allows the position error to evolve smoothly over time, with a relatively smaller variance for the velocity error, embodying the “slow drift” inertial characteristic. In subsequent Monte Carlo simulations, different error levels will be swept by varying the measurement noise rather than Q; a consistency experiment where “true trajectory = tracker Q” is also provided (see Section 2.4).

(4). Measurement Noise and Initial Value Setting: The measurement noise covariance $R=diag\left(\left[{{\sigma }_{pos}}^2,{{\sigma }_{pos}}^2\right]\right)$ is set by default to ${\sigma }_{pos}$ = 0.1 m. This magnitude corresponds to the equivalent positioning accuracy of medium-precision GNSS/INS fusion over a short time window, capable of representing error fluctuations that are “predictable but not white noise”.

The initial covariance $P0=\left(\mathrm{10,10,10,10}\right)$ reflects the uncertain but bounded prior condition for the initial position/velocity errors; the initial state is set to the zero vector (using “unbiased” as the baseline).

In this research, the selection of the process noise covariance matrix Q and the measurement noise covariance matrix R aims to represent different grades of navigation systems. As shown in

Table 1. Noise covariance matrix configuration for different navigation system grades.

Navigation System Level	Typical Applications/Accuracy	Process Noise Q (State-Dependent)	Measurement Noise R	Reason for Selection
Consumer-grade GNSS	Smartphone, low-cost receiver. Accuracy: meter-level (1–5 m)	The value is relatively large, indicating high uncertainty in the model, diag([1.0, 1.0, 0.1, 0.1])	The value is relatively large, indicating a high level of observation noise. For example: 25.0 m2	As a classic textbook in the field of navigation, this book clearly states that the error of consumer-grade GNSS in non-ideal environments (urban canyons) can reach several meters or even tens of meters. R = 25 m2 (Std = 5 m) is a typical representative of this accuracy range [13].
Vehicle-mounted/drone-level (GNSS/INS loose integration)	Medium-precision navigation. Precision: sub-meter level (0.1–1 m)	The value is medium, diag([0.1, 0.1, 0.025, 0.025])	The value is medium. For example: 0.25 m2	This doctoral dissertation provides a detailed analysis of the performance of low-cost IMU and GNSS loose integration, pointing out that its horizontal positioning accuracy is typically within the range of 0.5–2 m. Our parameter R = 0.25 m2 (Std = 0.5 m) falls at the high-performance end of this range [14]
Tactical-level INS (tightly integrated)	High-precision platform, unmanned system. Precision: centimeter-level (1–10 cm)	The value is relatively small, indicating a low drift rate of the inertial sensor. diag([0.01, 0.01, 0.001, 0.001])	The value is relatively small. For example: 0.01 m2	The study demonstrates that the tight integration of technology can achieve positioning accuracy ranging from centimeter level to decimeter level. Setting R to 0.01 m2 (Std = 0.1 m) aligns with the conservative accuracy estimation of such systems under dynamic conditions [15]
High-precision surveying and mapping grade (PPP/RTK)	Precision surveying and mapping, scientific research. Accuracy: millimeter-to-centimeter level (<1 cm—several cm)	The value is very small, indicating a high degree of precision in the model. diag([0.001, 0.001, 0.0001, 0.0001])	The value is very small. For example: 0.0001 m2	The paper demonstrates that low-cost single-frequency RTK can achieve centimeter-level (even 1–2 cm) real-time positioning accuracy. The parameter R = 0.0001 m2 (Std = 0.01 m) is precisely designed to simulate such high-precision application scenarios [16]

, adjusting the magnitudes of Q and R enables the simulation of error characteristics across various scenarios, ranging from consumer-grade GNSS to high-precision geodetic-grade systems. In the Monte Carlo simulations of this research, to comprehensively evaluate the error-performance relationship, we systematically varied the measurement noise R (while keeping Q constant to control variables), generating an extensive series covering error standard deviations from 1 mm to 1 m. This parametric approach ensures that the error model not only reflects the accuracy level of specific navigation systems but also possesses the universality for cross-platform comparison.

2.2. Real-Time Phase Compensation Strategy

This setting is insensitive to steady-state statistics but can accelerate initial convergence. In dynamic scenarios, due to the combined effects of platform motion, positioning errors, and communication delays, the signal phases arriving at the target point from each transmitting unit are not synchronized. Therefore, an effective real-time phase compensation strategy must be designed to correct these phase misalignments and achieve coherent combining.

As mentioned in the introduction, although techniques like Time Reversal (TR) can achieve ideal phase focusing, their reliance on cooperative beacon sources makes them unsuitable for non-cooperative scenarios central to this research. As mentioned in the introduction, although techniques like Time Reversal (TR) can achieve ideal phase focusing, their reliance on cooperative beacon sources makes them unsuitable for the non-cooperative scenarios central to this research.

The core idea for solving this problem is as follows: a reference source (e.g., Source 1) calculates and issues phase compensation commands $\Delta \varphi$ based on the state information it receives from other subordinate sources (e.g., Source i). The ideal compensation goal is to make all signals arrive at the target point simultaneously and in-phase:

$${\psi }_i\left(p_T\right)+\Delta {\varphi }_i={\psi }_1\left(p_T\right)$$

(5)

where ${\psi }_i\left(p_T\right)$ is the phase of the signal from Source i arriving at the target point without compensation. From this, the formula for calculating the ideal compensation phase is derived:

$$∆{\varphi }_i=k·\left(\left|\left|p_T-p_l\right|-\left|p_T-p_i\right|\right|\right)+({\varphi }_{l,0}-{\varphi }_{i,0})$$

(6)

where $k=2\pi /\lambda$ is the signal wavenumber; $p_i$ is the instantaneous position of Source I; and ${\varphi }_{i,0}$ is its initial transmission phase.

The challenge of this formula is that, due to the communication delay ${\tau }_d$, the reference source at time t can only obtain the position information of the subordinate source from the past time $t-{\tau }_d$. A direct countermeasure is to use a standard Kalman Filter (KF) to predict the position ${\widehat{p}}_i\left(t\right|t-{\tau }_d)$ of the subordinate source at the current time t. However, as pointed out in the introduction, when the platform undergoes severe maneuvers, the fixed motion model within the KF will severely mismatch the true dynamics, causing prediction performance to degrade sharply or even diverge, resulting in complete compensation failure.

To address the failure of traditional prediction methods under highly dynamic conditions, this research proposes a hybrid adaptive compensation strategy. This strategy abandons reliance on a single prediction mode. Its core idea is: while performing prediction, continuously evaluate the reliability of the prediction and intelligently switch compensation modes based on the level of reliability.

Predictive Compensation Mode: When positioning accuracy is high and system dynamics are relatively gentle, use an Adaptive Kalman Filter (AKF) to accurately predict the platform’s future position, actively countering communication delay to achieve high-precision phase compensation.

Robust Compensation Mode: When excessive positioning errors or severe platform maneuvers cause prediction failure, the system automatically falls back to a conservative compensation strategy that directly uses the (delayed) position information, ignoring the communication delay, to ensure system stability.

The switching between the two modes relies on an adaptive factor derived from the innovation sequence. Unlike traditional methods that use a single innovation covariance ratio ${\lambda }_i$ to quantify prediction reliability, this research introduces a multi-index fusion adaptive factor that integrates multiple statistical measures—including normalized innovation covariance ratio ${\lambda }_i$, standard deviation ratio ${\sigma }_{ratio}$, kurtosis, and skewness—computed over a historical window of innovation samples.

This multi-dimensional adaptive factor provides a more comprehensive and robust assessment of the filter’s internal model matching to the true system dynamics. By fusing these indicators with appropriate weights, the system obtains a continuous reliability metric that better reflects transient behavior and reduces misclassification risks. The system then uses this metric to intelligently switch between predictive and robust compensation modes. The following sections detail the definitions and physical meanings of the four key indicators—normalized innovation covariance ratio, standard deviation ratio, excess kurtosis, and skewness—that constitute this fusion metric.

(1) Normalized Innovation Covariance Ratio ${\lambda }_i$: This metric measures the degree of match between the filter’s internal model and the true system dynamics. It is defined as the ratio of the instantaneous innovation vector’s actual energy to the predicted theoretical innovation covariance energy, reflecting the relative magnitude of prediction error.

$${\lambda }_i={\displaystyle \frac{tr({\nu }_i{{\nu }_i}^T)}{tr(H{P}_{i,k}{|}_{k-1}{H}^T+R)}}$$

(7)

this factor can sensitively reflect the degree of mismatch between the filter’s internal model and the platform’s true dynamics.

(2) Standard Deviation Ratio ${\sigma }_{ratio}$: This metric reflects the deviation between the empirical standard deviation of the innovation sequence and the theoretical standard deviation predicted by the filter model. It is sensitive to changes in noise intensity and can indicate variations in environmental conditions or sensor performance degradation.

$${\sigma }_{ratio}={\displaystyle \frac{1}{M}}{\sum }_{i=1}^M{\displaystyle \frac{{\sigma }_i}{{\sigma }_{p,i}}}={\displaystyle \frac{1}{M}}{\sum }_{i=1}^M{\displaystyle \frac{\sqrt{{\frac{1}{N-1}\sum _{k=1}^N(v_{k,i}-v_i)}^2}}{\sqrt{{\left(C_p\right)}_{ii}}}}$$

(8)

where $v_{k,i}$ denotes the i-th component of the innovation vector at time k; is its mean value, M is the dimension of the innovation vector; N is the length of the sliding window; and ${\left(C_p\right)}_{ii}$ represents the i-th diagonal element of the theoretical innovation covariance matrix. When ${\sigma }_{ratio}$ σratio is close to 1, the actual noise level matches the model assumption; values significantly greater than 1 suggest increased disturbances or model inadequacy.

(3) Excess Kurtosis kurt: This metric quantifies the “tailedness” of the innovation distribution relative to a Gaussian distribution. It is effective in detecting outliers, abrupt disturbances, or non-Gaussian characteristics in the innovation sequence.

$${kurt}_i={\displaystyle \frac{{\frac{1}{N}\sum _{k=1}^N(v_{k,i}-v_i)}^4}{{{(\frac{1}{N}\sum _{k=1}^N(v_{k,i}-v_i)}^2)}^2}}-3$$

(9)

the overall excess kurtosis is obtained by averaging the absolute values across all dimensions:

$$kurt={\displaystyle \frac{1}{M}}{\sum }_{i=1}^M\left|{kurt}_i\right|$$

(10)

a kurtosis value close to zero indicates the innovation follows a Gaussian distribution; positive values suggest heavy tails (presence of outliers), while negative values indicate light tails. This metric helps identify sudden anomalies or impulse-type disturbances.

(4) Skewness: This metric measures the asymmetry of the innovation distribution. It is sensitive to systematic biases, drifts, or calibration errors in measurements or models.

$${skew}_i={\displaystyle \frac{{\frac{1}{N}\sum _{k=1}^N(v_{k,i}-v_i)}^3}{{\left(\sqrt{{\frac{1}{N}\sum _{k=1}^N(v_{k,i}-v_i)}^2}\right)}^3}}$$

(11)

the overall skewness is obtained by averaging the absolute values across all dimensions:

$$skew={\displaystyle \frac{1}{M}}{\sum }_{i=1}^M\left|{skew}_i\right|$$

(12)

a skewness value near zero indicates a symmetric distribution; non-zero values suggest asymmetry, pointing to systematic bias or drift trends in the system. This metric is useful for detecting long-term deterioration or persistent model errors.

After introducing the four key indicators, the multi-index adaptive factor is constructed by linearly combining them with carefully selected weights to balance their contributions:

$$D=w_1·{\lambda }_i+w_2·{\sigma }_{ratio}+w_3·\left|kurt\right|{+w}_4·\left|skew\right|$$

(13)

where the weights are typically chosen as w = [0.4, 0.3, 0.2, 0.1]

The normalized innovation covariance ratio ${\lambda }_i$ is assigned the highest weight because it fundamentally captures the discrepancy in energy between predicted and observed innovations. The standard deviation ratio ${\sigma }_{ratio}$ follows, reflecting variations in noise intensity that directly affect filter performance. Excess kurtosis and skewness are assigned smaller, yet vital weights since they identify subtle statistical anomalies—heavy tails and asymmetry—that may not be captured by the first two metrics but are crucial for robust anomaly detection.

This weighted fusion forms a continuous and comprehensive reliability metric, which better portrays transient behaviors and reduces switching jitter or misclassification. Such a multi-dimensional adaptive factor allows the system to intelligently select between predictive and robust compensation modes, thereby effectively maintaining high positioning accuracy and system stability even under complex and highly dynamic conditions.

Regarding the noise baseline, let $tracke{r}_{vel\hspace{0.33em}process\hspace{0.33em}noise}$ = 0.001 m/s, and set the noise baseline $Q_{trackerbsae}=diag\left(\left[{\left(0.001\right)}^2,{\left(0.001\right)}^2,{\left(0.001\right)}^2,{\left(0.001\right)}^2\right]\right)$. This conveys the information that the error velocity changes very slowly, promoting more robust predictions in stable scenarios and reducing overfitting to instantaneous measurement noise. The dynamic Q adaptive gain $k_{\lambda }$: implemented as $Q_{next}=Q_{tracker\hspace{0.33em}base}\cdot (1+(\lambda -1)\cdot 10)$, where 10 is the adaptive gain $k_{\lambda }·k_{\lambda }$ = 10, allows the filter to increase the process noise more quickly when prediction mismatch occurs, improving tracking sensitivity; if the scenario involves more severe maneuvers, it can be adjusted to 20–30. The threshold, Q baseline, and adaptive gain collectively determine when to switch, how quickly to stabilize after switching, and how to avoid overconfidence under strong dynamics. The default settings balance switching stability and response speed.

This unique hybrid architecture ensures that the system can enjoy the high-precision advantages brought by prediction when platform dynamics are gentle, while automatically falling back to a more stable baseline mode when severe platform maneuvers cause model mismatch, thereby demonstrating excellent robustness in complex and variable scenarios.

2.3. Field Strength Synthesis and Performance Evaluation Standards

After defining the error model and compensation strategy, this research requires a clear evaluation framework to quantify the system’s final performance under various error levels. Previous studies evaluating coherent synthesis systems often employed methods such as analyzing the distortion degree of the array pattern or demonstrating the instantaneous variation in the synthesized field strength at the target point in a single simulation [17,18,19]. These methods provide certain reference values for understanding system behavior, but they often offer qualitative or case-specific evaluation, making it difficult to provide a unified metric that measures system reliability in a statistical sense.

To establish a more universal evaluation system closer to engineering needs, this research constructs a hierarchical evaluation framework ranging from instantaneous performance to statistical reliability.

Firstly, the basic field-synthesis model at the target point $p_T$ is established. The total synthesized electric field $E_{total}$ is the coherent superposition of the complex field vectors contributed by all units:

$$E_{total}(t)=\sum _{i=1}^N{A}_i\cdot exp\left(j\right[k\cdot \mid p_T-p_i(t)\mid +{\varphi }_{i,0}+\Delta {\varphi }_i\left(t\right)\left]\right)$$

(14)

The core challenge of this formula is that the estimated positions containing errors are used to calculate the compensation phase $\Delta {\varphi }_i\left(t\right)$, while the true wavefront propagation is determined by the true positions ${\mathrm{p}}_i\left(t\right)$. The deviation between the two constitutes residual phase noise and directly leads to the attenuation of the synthesized field strength. To quantify this impact, the coherent synthesis efficiency $\eta \left(t\right)$ is first defined as the ratio of the actual synthesized power to the ideal synthesized power:

$$\eta (t)={\displaystyle \frac{P_{\mathrm{actual}}\left(t\right)}{P_{\mathrm{ideal}}\left(t\right)}}={\displaystyle \frac{\mid P_{\mathrm{actual}}\left(t\right){\mid }^2}{\mid E_{\mathrm{ideal}}\left(t\right){\mid }^2}}$$

(15)

This efficiency factor $\eta \left(t\right)\in \left[\mathrm{0,1}\right]$ is the basis for evaluating the system’s instantaneous performance.

To ensure statistical significance, this study defines the Coherent Synthesis Success Rate as its core evaluation metric. Under the Monte Carlo simulation framework, numerous independent trials are performed for each positioning error level. A trial is considered successful if its average coherent efficiency meets or exceeds a predefined threshold, typically set at 0.8. The threshold value of 0.8 was selected based on the engineering practice for distributed systems (e.g., radar, communications), where a synthesis efficiency exceeding 80% is generally considered to provide high usability and practical value. This ensures that our ‘success rate’ metric is closely aligned with practical engineering considerations. The Coherent Synthesis Success Rate is then the proportion of successful experiments to the total number of experiments:

$$S_{rate}={\displaystyle \frac{Number of successful trials}{Total number of trials}}$$

(16)

This research ultimately adopts the “Coherent Synthesis Success Rate” $S_{rate}$ as the core evaluation metric, aiming to overcome the statistical shortcomings of traditional evaluation methods. The superiority of this metric lies in its establishment of a direct quantitative relationship from the underlying physical error v(positioning accuracy) to the top-level task performance (reliability of coherent synthesis) through large-scale Monte Carlo simulations.

2.4. Simulation Setup

To validate the effectiveness of the previously proposed method (hereinafter referred to as the “proposed hybrid strategy”) and systematically quantify the impact of positioning errors on coherent synthesis performance, this section constructs a simulation platform based on the Monte Carlo method. This research compares the proposed hybrid strategy with a benchmark strategy, which refers to directly using delayed position information for compensation, corresponding to the robust compensation mode that the hybrid strategy falls back to.

Simulations are conducted in a two-dimensional plane. The array consists of 3 distributed transmitting sources with initial center positions at (100, 50) m, (150, 50) m, and (200, 50) m, respectively, and the target point is at (150, 150) m. The transmission carrier frequency is 2 GHz (wavelength 0.15 m), and the element radiation pattern is a normalized sinc function. The communication/execution delay is set to 0.4 ms (sampling rate fs = 1 kHz). This delay is observable and compensable and does not dominate the error evolution. To obtain statistical conclusions, a Monte Carlo framework is employed: by adjusting the measurement noise standard deviation ${\sigma }_p$ of the ARKF, 10 positioning error levels are set, with ${\sigma }_p$ varying logarithmically from 1 mm to 1 m; For each level, independent simulations of 0.5 s duration with a 1 kHz sampling rate are repeated 50 times, and the average coherent efficiency and coherent synthesis success rate are recorded.

2.5. Experimental Setup and Evaluation Metrics

To ensure the reproducibility of our research methodology and clarify the evaluation procedure, this subsection specifies the input data, output results, and performance metrics for the simulation experiments.

2.5.1. Input Data and Parameters

The model inputs are derived from simulations of a distributed moving platform navigation system. This encompasses two main aspects: simulated Platform Trajectories, which are generated using kinematic models (e.g., constant velocity or acceleration) to replicate realistic motion, and the error characteristics of the primary navigation sensors, such as GNSS receivers and IMUs.

Critical parameters configured for these models include the Measurement Noise Covariance (R), set based on the system grade to represent sensor noise; the Process Noise Covariance (Q), which reflects model uncertainty related to platform dynamics; and applicable Systematic Errors (e.g., Bias). The simulation is initialized with an Initial State Vector for platform position, velocity, and attitude, which can be set to include initial estimation errors.

2.5.2. Output Results and Performance Metrics

The performance of the proposed hybrid adaptive compensation strategy is evaluated based on its core outputs and a set of quantitative metrics. The primary output of the system is the Compensated Phase Array, which represents the final, adjusted phase values ready for power synthesis. Alongside this, the adaptive module provides insights into the Filter State and Confidence, delivering real-time parameters that indicate the model’s belief in the accuracy of its current estimates.

To quantitatively assess the system’s effectiveness, we employ several performance metrics. The foremost among these is the Coherent Combining Success Rate, which serves as the cornerstone of our evaluation. This metric is defined statistically as the proportion of Monte Carlo trials in which the combining efficiency surpasses a predefined threshold—for instance, 0.8. Further granularity is provided by the Root Mean Square Error (RMSE), which gauges the precision of the platform’s estimated position or velocity. Finally, the stability and accuracy of the phase compensation process itself are directly measured by the Phase Error Standard Deviation (σ), prior to the final synthesis step.

3. Results and Discussion

3.1. Positioning Error Modeling Results and Discussion

3.1.1. Validation of ARKF Error Trajectory Characteristics

To assess the capability of the proposed ARKF error modeling for performance evaluation in distributed coherent synthesis, Figure 2 presents a comparative analysis of error time series. The ARKF-generated trajectories capture the inherent physical properties of navigation systems, a feature absent in the conventional white noise assumption. This fidelity ensures that subsequent performance evaluation of coherent synthesis is grounded in a more realistic and physically meaningful context. Key characteristics of the ARKF-generated errors include the following:

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Temporal Continuity: Errors evolve smoothly, avoiding abrupt changes;

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Error Inertia: Exhibiting systematic error characteristics similar to gyro drift, more realistically reflecting the physical properties of navigation systems [20].

Figure 2. The positioning error varies over time.

Figure 2. The positioning error varies over time.

3.1.2. Comparative Analysis with Existing Error Modeling Methods

To validate the performance of the proposed ARKF error modeling method, this paper presents a quantitative comparison against prevalent error models found in the literature.

A comparative analysis was conducted between the proposed ARKF framework and the conventional Gaussian white noise model for error simulation. While Gaussian white noise assumptions remain prevalent in positioning error modeling [21], they fundamentally overlook the temporal dependencies inherent in practical navigation systems, as critically noted in [22]. In contrast, the ARKF framework captures the temporal evolution of errors through its state-space formulation, generating error sequences with statistically consistent correlation structures. This modeling approach produces error characteristics that better align with empirical observations from physical navigation systems. Consequently, performance evaluations of compensation algorithms based on ARKF-generated errors provide more reliable predictions of real-world behavior, effectively addressing the model mismatch inherent in white noise approximations.

The proposed ARKF framework was further evaluated against the first-order Markov process error model presented in Reference [23]. Our method introduces an error velocity state, which provides a more accurate description of error evolution trends. This translates to a marked superiority in prediction accuracy, an advantage that becomes especially pronounced during periods of significant dynamic change. While the enhanced fidelity of the ARKF model comes at a computational cost—with a complexity of O(n³) compared to the O(1) of a simple white noise model—this overhead is justifiable. It is well within the capabilities of modern processors and is a worthwhile investment for a universal error model that is crucial for translating simulation-based findings into practical engineering value [24]. Ultimately, the core significance of this work lies in demonstrating that only an error model which adequately captures temporal correlations, like the ARKF, can enable a faithful assessment of how positioning errors impact the long-term stability of distributed coherent synthesis, thereby yielding reliable statistical success rates.

3.1.3. Model Parameter Sensitivity Analysis

Influence of Process Noise Covariance Matrix Q: By adjusting the diagonal elements of the Q matrix, the model’s adaptability to navigation systems of different accuracy grades was validated. Theoretically, as the noise variance increases, the generated error standard deviation also increases correspondingly, demonstrating good parameter controllability.

Limitations of the State Transition Model: The constant velocity motion model currently used has shortcomings in describing highly maneuverable platforms. Reference [25] points out that for highly accelerating maneuvering platforms, higher-order motion models need to be introduced. This will be a key direction for improvement in future work.

3.2. Quantitative Relationship Between Positioning Error and Coherent Synthesis Success Rate

3.2.1. Quantitative Analysis of Positioning Error Impact

Figure 3 clearly delineates the quantitative relationship, derived from our Monte Carlo simulations, between the positioning error standard deviation (σ) and the coherent synthesis success rate. The performance curve of the hybrid strategy reveals three distinct operational intervals, each defined by a unique characteristic.

In the High-Performance Interval (σ < 100 mm), the system maintains a consistently high success rate above 96.75% by operating predominantly in the predictive compensation mode. This performance tier aligns with the capabilities of high-precision systems like RTK-GPS or inertial navigation. The upper bound of this interval (σ = 100 mm) is identified as the first significant inflection point on the performance curve, marking the end of stable high performance.

As the error increases into the Transition Interval (100 mm < σ < 215 mm), a marked decline in the success rate—from 96.75% to 82.64%—is observed. This reflects the system’s progressive shift from predictive to robust compensation. The performance degradation reaches its maximum rate of decline within this band, with the steepest descent occurring at σ = 215 mm.

Finally, in the Stable Interval (σ > 464 mm), the success rate plateaus at 56–58%. Here, the system operates almost entirely in the robust compensation mode. While this absolute performance level is comparable to the baseline strategy, our hybrid approach provides enhanced fault tolerance, as evidenced by a narrower performance envelope in time-domain analyses. The lower bound (σ = 464 mm) signifies the point where the success rate curve converges to a stable plateau.

A pivotal outcome of this quantitative analysis is the identification of a precise critical error threshold. The value of 237.70 mm, which corresponds to an 80% success rate, serves as a concrete engineering benchmark for determining the required navigation accuracy in system design. It is crucial to note, however, that this specific threshold is a function of the simulation parameters established in this study, including the number of transmitting sources, carrier frequency, and communication delay. Despite this specificity, the established quantitative relationship offers, for the first time, a theoretical foundation and a fundamental framework for predicting performance and guiding the optimization of distributed coherent synthesis systems. Subsequent research can build upon this framework by conducting a parametric analysis to systematically investigate how the critical threshold is influenced by variations in system configuration.

The critical threshold for an 80% success rate determined in this paper is approximately ≈0.237 m. An accessibility assessment for common systems is as follows: Single-point GPS (2–5 m, open field) cannot achieve it; SBAS/differential GNSS (0.5–1 m) is generally unattainable [26]; PPP after convergence (0.1–0.3 m) is critically achievable [27]; UWB/vision-inertial tight coupling (5–20 cm relative accuracy) can stably meet the threshold [28]. For engineering applications targeting mobile platforms, PPP + INS tight coupling [29] is recommended, supplemented with UWB/visual relative measurements over short baselines to maintain an equivalent error level of ≤0.2–0.3 m under high dynamics and occlusion conditions, thereby falling into the high-performance or transition intervals of this paper.

3.2.2. Comparative Analysis with Existing Compensation Methods

To validate the effectiveness of the proposed hybrid strategy, this research systematically compares it with the benchmark delayed compensation method. Figure 3 shows the system performance with the hybrid strategy enabled and disabled, respectively.

Experimental results show that the two strategies exhibit significant differences under different error levels. In the low-error interval (1–100 mm), the coherent success rate of the hybrid strategy is 96.75–99.71%, while that of the benchmark strategy is only 59.94–60.44%, representing a performance improvement of 36–40 percentage points. In the medium-error interval (100–215 mm), the success rate of the hybrid strategy is 82.64–96.75%, compared to 60.20–60.83% for the benchmark strategy, with a performance difference of 22–36 percentage points. In the high-error interval (464 mm–1000 mm), the performance of the two strategies converges, both stabilizing in the range of 56–60%.

The benchmark strategy exhibits a flat performance characteristic across the entire error range (standard deviation < 0.5%), indicating its insensitivity to changes in positioning accuracy. In contrast, the hybrid strategy demonstrates adaptive characteristics, able to adjust its operation mode according to the error level, fully utilizing the predictive advantage under high-precision conditions, and automatically falling back to robust mode under low-precision conditions. Statistical analysis shows that the performance difference between the two strategies is statistically significant at the 95% confidence level (p < 0.01), validating the reliability of the hybrid strategy’s advantage.

This research conducts a comparative analysis of methodology and robustness between the proposed hybrid adaptive compensation strategy and a recently reported time reversal (TR)-based approach for electromagnetic power synthesis on distributed moving platforms [5]. Tan et al. demonstrated that velocity errors degrade synthesis efficiency in TR methods, though TR maintains adequate performance within bounded phase errors.

This research diverges primarily from TR-based approaches in both the operational context and the underlying compensation principles. The TR technique requires probing or pilot signals from the target, operating essentially as a channel-sounding-based adaptive focusing method. In contrast, our work addresses non-cooperative scenarios through a prediction-based active compensation framework. When confronting comparable positioning errors, the two methods exhibit crucial differences in robustness:

Both theoretical analysis and simulations confirm that TR and our method maintain high synthesis efficiency under low phase errors. As equivalent phase errors increase to medium-high levels, our hybrid adaptive strategy demonstrates superior robustness. According to Tan’s research, TR performance depends primarily on initial channel estimation quality and platform velocity coordination. The TR approach lacks the capability for real-time correction of persistent dynamic errors. Conversely, our hybrid adaptive module continuously monitors model mismatch levels. It actively suppresses error accumulation by adjusting the filter’s confidence parameters. This enables more stable synthesis performance in highly dynamic, uncertain environments.

This comparison validates that our quantitative framework and compensation strategy offer enhanced adaptability and robustness for distributed coherent synthesis in non-cooperative or channel-limited scenarios.

3.2.3. Comparative Analysis with Kalman Filtering Method

To validate the proposed hybrid adaptive compensation algorithm, the research compared its performance against the standard Kalman filter (KF) under identical simulation conditions (σ = 100 mm, τ = 0.4 ms, 3 radiating elements). As shown in Figure 4, the results demonstrate marked superiority of our algorithm over the standard KF in both synthesis efficiency and operational stability. Specifically, during the 500 ms simulation period, the standard KF attained merely 6.39% average synthesis efficiency with severe performance fluctuations, failing to meet the 80% performance threshold for most of the duration. This highlights its inherent limitations when handling dynamic errors and compensation delays. In contrast, our adaptive algorithm reached 61.08% average efficiency, representing a 54.69-percentage-point improvement. Although fluctuations persist, the overall trend remains stable with performance predominantly sustained above the threshold level.

These results demonstrate that the intelligent decision-making and fallback mechanism based on the multi-index adaptive factor significantly alleviates filter model mismatches. By integrating multiple statistical indicators, this approach substantially enhances the system’s robustness and practical effectiveness under challenging operational conditions.

3.3. Time-Domain Validation

To validate the conclusions drawn in the previous section and simultaneously observe the temporal fluctuations in the simulation, σ was fixed at 200 mm, and 50 Monte Carlo simulations were conducted, resulting in the three curves shown in Figure 5: “Mean (blue)/P90 (red)/P10 (green)”. These are defined as: at each time point t, the mean coherent combining efficiency of the 50 experimental samples (blue); the 90th percentile P90 (red, indicating only the top 10% of samples exceed this value); and the 10th percentile P10 (green, indicating 10% of samples fall below this value). Therefore, the P10–P90 envelope corresponds to the central 80% range of the probability distribution, intuitively expressing the temporal uncertainty. Figure 4 shows that after an initial brief convergence period (where the AKF estimates and Q/R matching stabilize), the mean remains above 80% for an extended duration, and P10 also does not fall below 80% for most of the time. This is consistent with the prediction conclusion of σ = 237.7 mm obtained from the success rate statistics in Figure 3, providing mutual confirmation. From the results, it can be clearly observed that the proposed hybrid strategy maintains good coherent combining capability throughout the entire time domain. No prolonged, unrecoverable efficiency failure occurs across the full-time domain, demonstrating the robustness of the method.

The time-domain results in Figure 5 provide additional evidence of the system’s robustness. The P10–P90 performance envelope remains narrow throughout the simulation, while the mean value stays consistently high. This indicates that the hybrid strategy delivers not only excellent average performance but also statistically contained performance variations. This time-wise stable behavior empirically supports our claim regarding the algorithm’s stochastic convergence.

This research achieves significant breakthroughs at two levels: positioning error modeling and compensation strategy.

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Theoretical Innovation: The ARKF error modeling, as a supporting module, enables us to establish, for the first time in the context of distributed coherent synthesis, a reliable quantitative relationship between the statistical characteristics of positioning errors and the coherent success rate, avoiding the evaluation bias caused by traditional white noise models.

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Methodological Innovation: A hybrid adaptive compensation strategy is designed to intelligently switch between predictive and robust modes, thereby maximizing compensation performance while maintaining system stability.

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Engineering Value: By establishing, for the first time, a quantitative relationship between positioning error and coherent synthesis performance, this work provides concrete guidance for designing and optimizing distributed coherent synthesis systems.

4. Conclusions

This study develops a hybrid adaptive compensation framework that effectively addresses positioning errors in mobile platform-based distributed coherent synthesis systems. By employing ARKF-based universal error modeling and extensive Monte Carlo simulations, we establish a tri-zone quantitative relationship between positioning error standard deviation (σ) and coherent synthesis success probability. Our analysis reveals a critical performance threshold: for a three-source configuration (N = 3), the system maintains high-efficiency coherent synthesis (>80% success rate) when σ remains below 237.3 mm. This finding indicates that our adaptive compensation mechanism expands the system’s operational error tolerance from the conventional sub—100 mm range to 237.3 mm—more than doubling the tolerable error margin while ensuring stable performance. The demonstrated robustness enhancement provides crucial design guidelines for navigation system selection and performance evaluation in practical mobile scenarios.