A General Optimization Framework for Radar Multi-PRF Waveform Synthesis Based on Bezout’s Identity and Genetic Algorithm

Abstract

To mitigate the structural amplification of random false alarms during multi-pulse repetition frequency (Multi-PRF) ambiguity resolution, this paper proposes a general waveform synthesis optimization framework based on Bezout’s Identity and Genetic Algorithm (Bezout-GA). By leveraging Bezout’s Theorem, the framework establishes an analytical mapping between the Greatest Common Divisor (GCD) topology of transmission parameters and system-level false alarm boundaries. It is mathematically demonstrated that the uncontrolled inflation of the Least Common Multiple (LCM) in traditional coprime-based strategies leads to severe “spatial over-issuance” of false alarms, a phenomenon particularly exacerbated in heavy-tailed K-distributed sea clutter. The proposed two-stage hybrid paradigm employs a genetic algorithm for global multi-objective search, followed by local number-theoretic refinement via the Extended Euclidean Algorithm to strictly satisfy hardware constraints. Simulations across X-band and L-band scenarios confirm the framework’s superior spectral generalizability. Results indicate that the Bezout-GA optimized waveform achieves a 4.1-fold reduction in expected false alarm volume at the cost of a negligible 0.1% clear-region sacrifice. Notably, in extreme K-distributed clutter ($\nu =0.1$), the framework reclaims an equivalent signal-to-clutter-and-noise ratio (SCNR) gain of up to 3 dB in the L-band, significantly outperforming traditional coprime and maximum clear-region benchmarks. Overall, this study provides a number-theoretic perspective for analyzing spatial false alarm mechanisms and serves as a methodological reference for future investigations into robust Multi-PRF waveform optimization.

1. Introduction

In modern integrated air–sea surveillance systems, airborne pulse Doppler (PD) radars face severe challenges in detecting low-observable targets, such as stealth speedboats and sea-skimming missiles. In actual maritime environments, high-resolution sea clutter frequently exhibits extreme non-Gaussian “heavy-tailed” characteristics (e.g., K-distribution), triggering dense “sea spikes.” These spikes share highly similar point-track characteristics with real targets in the time-frequency domain, causing a rapid increase in false alarms for traditional Constant False Alarm Rate (CFAR) detectors. To guarantee unambiguous velocity measurement and clutter suppression for high-speed targets, radar systems are often forced to employ High-PRF (HPRF) or Medium-High-PRF (MPRF) waveforms. However, this inevitably introduces severe range folding and ambiguity, making Multiple-PRF (Multi-PRF) ambiguity resolution an indispensable core technique.

In modern coherent radar systems, PRF design is the key to resolving joint range-Doppler ambiguity. Early classical radar theories and authoritative treatises established the fundamental framework of transmitting multiple PRFs to expand the maximum unambiguous detection space [1,2]. During this phase, researchers primarily relied on pure mathematical analytical methods, particularly utilizing the Chinese Remainder Theorem (CRT) and one-dimensional coincidence algorithms to achieve range and velocity parameter reconstruction in multi-target environments [3,4,5]. Recent studies have further comparatively analyzed the anti-noise performance of classic CRT and robust CRT architectures in pulse Doppler radars [6]. These traditional analytical designs, represented by the “pairwise coprime” method, laid the early mathematical foundation for the Multi-PRF architecture.

However, pure mathematical resolution often ignores the stringent constraints of the radar physical layer. Due to eclipsing loss and blind zone effects caused by transmit-receive isolation, simple geometrically coprime solutions easily fail in practical detection. Consequently, the academic focus shifted from static analytical methods to heuristic combinatorial optimization to seek optimal MPRF/HPRF sets. Early optimization explorations introduced Simulated Annealing (SA) algorithms to minimize blind zone areas [7], while subsequent studies integrated the DBSCAN clustering algorithm to further enhance the disambiguation robustness of medium PRF radars [8], and developed specialized eclipsing loss minimization models for HPRF radars [9]. These efforts marked the transition of waveform design towards intelligent optimization guided by clear-region maximization.

With the evolution of the modern electromagnetic battlefield, radar waveform design faces increasingly stringent multi-dimensional tactical constraints, particularly in the fields of Low Probability of Intercept (LPI) and anti-jamming. In complex interference environments, adaptive waveform optimization has become crucial for maintaining radar survivability [10]. For LPI design, modern radars are dedicated to synthesizing covert signals capable of resisting identification by Electronic Support Measures (ESM) [11], optimizing waveforms with desired cyclic spectrum and pulse compression properties [12]. Furthermore, recent advancements in physical-layer anti-interference have introduced highly sophisticated spatial modulation techniques. For instance, distributed split single-sideband time-modulated arrays have been proven to provide advanced signal modulation paradigms for secure communications [13]. This physical-layer hardware agility is highly complementary to the temporal-domain multi-PRF waveform optimization proposed in this study. Consequently, integrating these temporal and spatial degrees of freedom has completely transformed PRF selection and parameter synthesis into a high-dimensional, non-convex optimization problem accompanied by non-linear constraints.

To overcome the computational bottlenecks under high-dimensional constraints, radar waveform design has fully embraced cutting-edge intelligent optimization theories in recent years [14]. Researchers have successively introduced Many-Objective Evolutionary Algorithms (MOEA) [15], Minimum Blind Zone Combination (Min BCI) algorithms [16], and Nonlinear Conjugate Gradient (NCG) methods [17] to efficiently explore massive discrete solution spaces. Most innovatively, recent research has constructed a differentiable clear region representation model with boundary control, breaking the impasse of non-differentiable traditional PRF blind zones and making end-to-end waveform optimization based on gradient descent a reality [18]. Benefiting from these powerful search algorithms, variable parameter waveforms and joint agility technologies have been rapidly expanded into numerous complex application scenarios, including joint frequency and PRF agility for high-resolution ISAR imaging [19], PRF sampling strategy evaluation for SAR satellite swarms [20], pulse group feature extraction and temporal prediction for multi-function radars [21,22], staggered PRF coherent processing for airborne down-looking detection [23], and enhancing weather target detection utilizing Non-Uniform Pulse Repetition Time (NPRT) waveforms [24].

While the recent literature has extensively explored multi-objective evolutionary algorithms (MOEAs) and gradient-based methods to minimize geometric blind zones, these purely search-based approaches treat PRF values merely as independent variables for spatial coverage, neglecting their underlying number-theoretic topology. Consequently, conventional optimization frameworks often converge to pseudo-optimal PRF sets that are geometrically clear but topologically fragile (e.g., mutually coprime). This reveals a critical theoretical gap: existing optimization literature lacks a deterministic mechanism to control system-level false alarms induced by the discrete solution grid. In extreme heavy-clutter backgrounds (such as K-distribution sea spikes), the traditional “coprime method” long adhered to by the engineering community exposes a critical flaw. In engineering practice, radars typically employ $M/N$ logic detection criteria (e.g., the 2/4 criterion). Under this criterion, PRF combinations designed based on the “mutually coprime” concept possess an extremely dense solution space within their common solutions. When random spike fluctuations caused by sea clutter enter the resolution module, this overly dense solution space exponentially amplifies the probability that random false measurements “accidentally” satisfy the consistency resolution condition. This Random False Alarm Solution, triggered purely by underlying number-theoretic topological defects, differs fundamentally from the Phantom Solution generated by multi-target intersections; its massive emergence can result in a severe false alarm overload in the radar back-end data processing.

In summary, suppressing the generation probability of random false alarm solutions from the waveform source—while ensuring the system’s clear observation capability—has become the decisive key to breaking the performance bottleneck of coherent radars in HPRF sea-surface detection. To bridge the gap between heuristic search and number-theoretic robustness, this chapter proposes a novel optimization framework, Bezout-GA. Unlike existing evolutionary methods that passively evaluate target functions post-generation, this framework actively reshapes the Diophantine grid density of the ambiguous solution space. By integrating Bézout’s Identity directly into the evolutionary search process, the proposed method provides a deterministic mathematical safeguard against stochastic false alarm proliferation.

The core contributions of this chapter begin with the systematic characterization of false alarm proliferation mechanisms in sea clutter, providing a rigorous mathematical analysis of the range-velocity coupled ambiguity in airborne High-PRF (HPRF) radars and distinguishing the statistical roots of noise-induced random false alarms from system-induced multi-target phantom solutions. Furthermore, it innovatively integrates Bézout’s Identity into the radar ambiguity resolution model to establish an explicit quantitative mapping between the Greatest Common Divisor (GCD) topology of PRF sets and the spatial volume of ambiguous solutions; it is theoretically demonstrated that regulating the GCD can suppress the uncontrolled inflation of the Least Common Multiple (LCM) to block false coincidence paths in the physical layer—a suppression effect that exhibits a “dimensionality dividend” as the $M/N$ detection criteria increase in complexity. Building upon this, a bi-objective hybrid optimization architecture is developed to balance blind-zone minimization against false alarm probability, employing a strategy that combines Genetic Algorithms (GA) for global exploration with localized number-theoretic refinement to precisely extract the Pareto-optimal front under stringent hardware constraints. Finally, through cross-band validation from the X-band to the L-band, this study proves that the Bezout-GA framework transcends conventional heuristic search paradigms by strategically conceding a negligible clear-region loss (0.1%) for substantial suppression gains, such as a 3 dB equivalent SCNR gain in the L-band, thereby constructing a robust physical defense barrier at the waveform level and providing critical theoretical guidance for the future development of advanced Multi-PRF waveform design.

2. Multi-Dimensional Detection Theory Modeling

The core advantage of airborne Pulse Doppler (PD) radar lies in acquiring high-precision Doppler information of targets through coherent integration, thereby achieving the effective extraction and detection of moving targets against strong clutter backgrounds. However, this mechanism is deeply constrained by the periodic discrete sampling characteristics of the radar’s transmitted pulse sequence, inevitably plunging the system into a joint ambiguity dilemma of range and velocity measurements, as shown in Figure 1.

To fundamentally solve the detection difficulties of weak targets in complex marine environments, we must first start from the underlying physical ranging and velocity measurement mechanisms, systematically deconstruct the algebraic number theory foundation of Multiple Pulse Repetition Frequency (MPRF) ambiguity resolution, and rigorously mathematically characterize the generation mechanisms of false solutions induced by multi-target environments and noise fluctuation backgrounds.

2.1. Ranging, Velocity Measurement, and the Doppler Dilemma

The range measurement of PD radar is essentially based on the time delay of electromagnetic waves propagating back and forth in space. Assume the radar transmits a detection pulse at an initial time $t_0$, the true spatial absolute range of the target is $R$, and the propagation speed of electromagnetic waves is the speed of light $c$; then the arrival time of the target echo is $t_r=t_0+2R/c$. The radar system estimates the range by precisely measuring the time delay $\tau =t_r-t_0$ between transmission and reception, i.e., $R=c\tau /2$.

However, under a continuous transmission pulse sequence operating mode, the radar receiver actually cannot distinguish to which specific historical transmitted pulse the currently received echo belongs. The receiver can only measure the “folded” delay within one Pulse Repetition Interval (PRI), denoted as ${\tau }_m=\tau (\mathrm{m}\mathrm{o}\mathrm{d} T_r)\in [0,T_r)$, where $T_r=1/f_r$ is the pulse repetition period, and $f_r$ is the Pulse Repetition Frequency (PRF). Therefore, a linear combination relationship exists between the true absolute delay and the measured apparent delay:

$$\begin{array}{c}\tau ={\tau }_m+k{T}_r, k\in {\mathbb{Z}}^{+}\end{array}$$

(1)

where $k$ is defined as the range ambiguity number. Mapping this to the spatial range domain, the true absolute range $R$ can be expressed as

$$R=R_m+k·R_{ua}, R_{ua}={\displaystyle \frac{c{T}_r}{2}}={\displaystyle \frac{c}{2f_r}}$$

(2)

where $R_m$ is the apparent range measured by the radar, and $R_{ua}$ is defined as the radar’s maximum unambiguous range. When the target’s true range $R>R_{ua}$, the range ambiguity number $k\ge 1$, at which point range measurement under a single PRF exhibits multi-valued ambiguity.

In the Doppler velocity domain, the target’s radial motion induces a Doppler frequency shift $f_d$. PD radar extracts the Doppler frequency by performing an $N$-point Discrete Fourier Transform (DFT) on the slow-time sampling sequence, with a theoretical frequency resolution of $\Delta f=1/T_{CPI}$. According to the Nyquist sampling theorem, this periodic sampling leads to the folding effect of frequency measurement. The true Doppler frequency $f_d$ and the apparent Doppler frequency $f_{d,m}$ satisfy

$$f_d=f_{d,m}+l·f_r, l\in \mathbb{Z}, f_{d,m}\in \left[-{\displaystyle \frac{f_r}{2}},{\displaystyle \frac{f_r}{2}}\right)$$

(3)

where $l$ is the velocity ambiguity number. Converting this to target radial velocity, the relationship between true velocity $v$ and apparent measured velocity $v_m$ is

$$v=v_m+l·v_{ua}, v_{ua}={\displaystyle \frac{\lambda f_r}{2}}$$

(4)

where $v_{ua}$ is the maximum one-way unambiguous velocity. Simultaneously solving the ambiguity boundary equations for range and velocity allows us to derive an extremely profound physical parameter under the PD radar architecture—the range-velocity coupling constant $\kappa$:

$$\kappa =R_{ua}·v_{ua}=\left({\displaystyle \frac{c}{2f_r}}\right)·\left({\displaystyle \frac{\lambda f_r}{2}}\right)={\displaystyle \frac{c\lambda }{4}}={\displaystyle \frac{c^2}{4f_0}}$$

(5)

This equation highlights that the product of maximum unambiguous range and maximum unambiguous velocity is an absolute physical constant, constituting a fundamental trade-off under the regulation of PRF. This physical limit boundary, known as the “Doppler dilemma,” is the fundamental driving force behind the development of MPRF ambiguity resolution technologies.

2.2. Algebraic Foundation of Multi-PRF Ambiguity Resolution

To ensure the purity of velocity measurements in highly maneuvering target and severely broadened sea clutter environments, modern airborne radars often adopt the High-PRF (HPRF) mode ($l\equiv 0$). However, this waveform design compresses $R_{ua}$ extremely to the kilometer level, resulting in deep folding of the target range. To recover the true range of the target, the radar rapidly switches transmitting pulse bursts containing $N$ different PRFs within a CPI, forming a corresponding set of maximum unambiguous ranges $\{R_1,R_2,\dots ,R_N\}$. For a unique true range $R$, the radar will acquire a set of apparent range folded observation values $\{r_1,r_2,\dots ,r_N\}$. The true absolute range $R$ must simultaneously satisfy the following system of linear congruences:

$$R=k_i{R}_i+r_i, 0\le r_i<R_i, i=\mathrm{1,2},\dots ,N$$

(6)

Viewing this from a number-theoretic perspective, combining the measurement equations of any two PRFs (say $i$ and $j$) and eliminating the absolute range $R$, we can obtain a bivariate Linear Diophantine Equation regarding the ambiguity numbers:

$$k_i{R}_i-k_j{R}_j=r_j-r_i$$

(7)

The existence of integer solutions strictly depends on the number-theoretic relationship between $R_i$ and $R_j$. Let their Greatest Common Divisor (GCD) be $d_{ij}=\gcd (R_i,R_j)$; according to Bézout’s identity, the necessary and sufficient condition for this equation to have a solution is that $d_{ij}$ must evenly divide the observation residual $(r_j-r_i)$. When $d_{ij}=1$ (coprime), an integer solution always exists; but when $d_{ij}>1$, the existence of a solution is subjected to strict algebraic constraints from the measurement residuals.

The global solution of the aforementioned multivariate congruence equations deeply relies on the classical Chinese Remainder Theorem (CRT). Taking Dual-PRF ambiguity resolution as an example, the true range satisfies

$$R\equiv r_1 \left(\mathrm{m}\mathrm{o}\mathrm{d} R_1\right),R\equiv r_2 \left(\mathrm{m}\mathrm{o}\mathrm{d} R_2\right)$$

(8)

Let $d=\gcd (R_1,R_2)$, and $L=\mathrm{lcm}(R_1,R_2)$. If the compatibility condition $r_1\equiv r_2 (\mathrm{m}\mathrm{o}\mathrm{d} d)$ is met, a unique absolute range solution exists within $[0,L)$. In classical radar waveform design, designers almost instinctively set “moduli mutually coprime” ($d=1$) to stretch the system’s ultimate ranging capability as much as possible. However, this exposes a critical theoretical defect in complex statistical detection contexts. Because $d=1$ means the solution space reaches its maximum density, when noise is present, any random deviation easily makes distorted measurements fall back onto dense solution grid nodes, paving the exact way for out-of-control false alarm probabilities.

2.3. False Solution Generation: Phantom vs. Random False Alarms

In the back-end data processing stages of multi-PRF radar, the system often outputs false targets. To conduct scientific joint optimization, these false solutions must be strictly distinguished into deterministic Phantom Solutions and probabilistic Random False Alarm Solutions.

2.3.1. Phantom Solution: A Deterministic Issue in Combinatorial Geometry

Phantom solutions are generated when the radar is in dense multi-target scenarios, stemming from incorrect cross-combinations of measurement remainders. Assume two real independent targets $T_1(R_1,v_1)$ and $T_2(R_2,v_2)$ exist. If the system incorrectly extracts the range remainder of $T_1$ under PRF $i$ and algebraically matches it with the range remainder of $T_2$ under PRF $j$, a phantom state $(R_p,v_p)$ will be constructed:

$$R_p=r_{i,\pi \left(i\right)}+k_i{R}_i, v_p=v_{m,i,\pi \left(i\right)}+l_i{v}_i$$

(9)

where $\pi$ is a non-identity permutation mapping. Phantom solutions possess three core characteristics: determinism in existence conditions, explosiveness in quantity (reaching up to $N_t^{N_p}$), and suppressibility via kinematic acceleration constraint checks and multi-frame track correlation algorithms.

2.3.2. Random False Alarm Solution: A Probabilistic Phenomenon in Statistical Number Theory

In stark contrast, a random false alarm solution does not require the existence of any real targets; it is entirely due to random statistical fluctuations of the background noise or sea clutter spikes. In the classical single-PRF detection phase, the background noise envelope after square-law detection follows an exponential distribution. Setting the CFAR voltage threshold to $\eta$, it satisfies the fundamental false alarm probability $P_{fa,0}$:

$$P_{fa,0}={\int }_{\eta }^{\mathrm{\infty }}{p}_0\left(z\right|H_0)dz$$

(10)

When ubiquitous thermal noise or sea spikes randomly surge in several independent PRF observation windows and simultaneously cross the threshold $\eta$, if these fabricated residuals happen to mathematically satisfy the consistency condition of the congruence equations within a tolerance band, the radar will output a false target track. This necessitates a systematic discrimination as outlined in

Table 1. Comparison of essential characteristics between phantom solutions and random false alarm solutions.

Discrimination Dimension	Phantom Solution	Random False Alarm Solution
Physical Generation Root	Erroneous algebraic cross-combination of real measurement values from multiple targets	Random statistical fluctuation crossing of background noise and strong clutter spikes
Prerequisite for Existence	At least two real targets must exist in the detection scenario	Can be generated independently even in a pure background noise scenario with no targets
Phenomenon Predictability	Extremely deterministic: locked by target spatial coordinates and PRF ratios	Highly unpredictable: can only be characterized by macroscopic probability density functions
Deep Mathematical Essence	Combinatorial explosion phenomenon mapped under systems of linear Diophantine equations	Threshold crossing coincidence of extreme value statistical theory and stochastic processes
Dependency on PRF Structure	Sensitive to PRF numerical values, but weakly correlated with the GCD	Strongly dependent on GCD structure: the grid density of the solution space has direct veto power
System-Level Suppression Strategy	Impose geometric and kinematic constraints: such as non-linear filtering	Impose underlying probability control: reconstruct waveform GCD optimization

.

2.4. Probabilistic Amplification Model of Random False Alarms

The $M/N$ detection strategy declares a target if observation remainders from at least $M$ PRFs satisfy the consistency congruence condition. While this “majority voting” architecture enhances robustness, lenient criteria like the mainstream 2/4 architecture ($N=4, M=2$) hide a massive false alarm amplification crisis stemming from the dual superposition of the “combinatorial effect” and “consistency coincidence”.

Assuming the false alarm crossing probability for a single PRF is $p_0$, the overall 2/4 false alarm probability of the system can be expanded via the Bayesian total probability formula as

$$P_{fa,2/4}={\displaystyle \sum _{k=2}^4}\left({\displaystyle \genfrac{}{}{0pt}{}{4}{k}}\right)p_0^k(1-p_0{)}^{4-k}·P(\mathrm{consist}|k)+(\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{e}\mathrm{r}-\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r})$$

(11)

where $P\left(\mathrm{consist}\right|k)$ represents the conditional probability that when $k$ independent false measurements are generated, they accidentally converge in the solution space.

Crucial mathematical findings indicate that the area of the fault-tolerant band region exhibits a strictly inverse relationship with the greatest common divisor $d_{ij}=\gcd (R_i,R_j)$. When $d_{ij}$ is very small (especially the traditionally lauded coprime design $d_{ij}=1$), the spatial grid spacing is extremely compressed, and the probability of random noise “hitting” valid nodes rises sharply. Under theoretical limit derivation, when $k=2$ dominates, the upper bound of the system false alarm probability can be analytically expressed as

$$P_{fa,2/4}\le \sum _{\left(i,j\right)\in P_2}C·\left({\displaystyle \frac{{\delta }_R{\delta }_v}{d_{ij}·v_{gcd,ij}}}\right)·P_{fa,0}^2+O\left(P_{fa,0}^3\right)$$

(12)

In this formula, $P_2$ is the set of all 6 PRF pairs, and $v_{gcd,ij}$ is the common divisor structure in the velocity domain. If minor correlations are ignored, let the average single-pair consistent false alarm probability be

$$p_{pair}\propto {\displaystyle \frac{1}{d_{avg}}}{P}_{fa,0}^2$$

(13)

Then the macroscopic false alarm probability of the 2/4 system is

$$P_{fa,2/4}\approx 6p_{pair}^2\propto {\displaystyle \frac{1}{d_{avg}^2}}{P}_{fa,0}^4$$

(14)

This non-linear power cascade amplification mechanism significantly degrades detection performance. The formula points out the ultimate path to breaking the dilemma: by artificially increasing the local common divisor $d_{ij}$ among PRF sets, the grid spacing of the solution space can be effectively increased, thereby powerfully suppressing the eruption of random false alarms at a super-linear rate.

2.5. Blind Zone Modeling and M/N Criteria Trade-Offs

Increasing the GCD inevitably leads to the irreversible reduction in the Least Common Multiple (LCM) of the multi-PRF set, directly touching the bottom line of radar coverage range. Mathematically, a strict indicator function $B_i(R,v)$ characterizes the blind zone distribution of the $i$-th PRF:

$$B_i\left(R,v\right)=\mathbf{1}\left(\underset{n}{\bigcup }\left|R-n{R}_i\right|\le {\displaystyle \frac{\Delta R_{blind}}{2}}\cup \underset{m}{\bigcup }\left|v-m{\displaystyle \frac{v_i}{2}}\right|\le {\displaystyle \frac{\Delta v_{notch}}{2}}\right)$$

(15)

As visually illustrated in Figure 2, this physical mapping demonstrates the overlapping of individual PRF blind zones and the spatial distribution of the system’s total blind zone under a specific M/N criterion (e.g., the 1/2 criterion).

To globally evaluate the radar’s coverage performance under the M/N criterion, a joint blind zone indicator function is defined as $B_{joint}^{M/N}(R,v)=\mathbf{1}({\displaystyle {\sum }_{i=1}^N}(1-B_i(R,v))<M)$. By performing a double spatial integral over a preset Region of Interest (ROI), we obtain the core macroscopic tactical index—the clear region ratio ${\eta }_{clear}$:

$${\eta }_{clear}={\displaystyle \frac{{\iint }_{ROI}\left(1-B_{joint}^{M/N}\left(R,v\right)\right)dRdv}{{\iint }_{ROI}dRdv}}$$

(16)

As summarized in

Table 2. Performance comparison and applicable scenarios of Multi-PRF Radar M/N decision criteria.

M/N Architecture Configuration	System Equivalent Detection Probability	System Equivalent False Alarm Probability	Typical Tactical Applicable Scenarios
1/N (Logical “OR”)	Theoretical maximum: $1-(1-P_d{)}^N$	System maximum: $1-(1-P_{fa}{)}^N$	Extremely high SNR environments, early wide-array early warning
2/4 (Industrial Standard)	Good compromise: ${\displaystyle {\sum }_{k=2}^4}\left({\displaystyle \genfrac{}{}{0pt}{}{4}{k}}\right)P_d^k(1-P_d{)}^{4-k}$	Medium: Highly limited by consistency coincidence probability	Modern airborne fire control radar main combat mode
3/4 (High Reliability)	Relatively low, easily interrupted by blind zones	Deeply suppressed, extremely low false alarms	High-risk stealth detection requiring absolute silence
N/N (Logical “AND”)	Theoretical minimum: $P_d^N$	Approaching zero: $P_{fa}^N$	Final confirmation and tracking lock-on stages

, different M/N architecture configurations exhibit distinct performance trade-offs in terms of system equivalent detection probability, false alarm suppression, and applicable tactical scenarios. In summary, elevating the local GCD of PRFs can exponentially sever the transmission chain of false alarm amplification, reducing $P_{fa,2/4}$; but this is bound to compress the joint unambiguous space, causing blind zone proportions to diffuse and reducing ${\eta }_{clear}$. A strict trade-off exists between these two objectives. Therefore, a truly optimal radar detection architecture design must construct a joint bi-objective optimization constraint framework that coordinates “maximizing spatial clarity” and “minimizing random false alarm coincidence probability”.

3. Number-Theoretic PRF Deconstruction and Bi-Objective Optimization

Airborne Pulse Doppler radars face an extremely severe challenge of random false alarm surges under complex non-homogeneous sea clutter environments. To break through this technical bottleneck from the fundamental physical waveform source, this chapter creatively introduces the core foundation of elementary number theory—Bézout’s Identity—into the optimization design of radar Pulse Repetition Frequencies (PRFs). By deeply dissecting the algebraic mapping structure of radar multi-PRF ambiguity resolution, the profound mathematical connection between the Greatest Common Divisor (GCD) and system false alarm probability is revealed. Based on this, a bi-objective intelligent optimization model balancing blind zone coverage and false alarm suppression effectiveness is constructed, laying a solid theoretical foundation for subsequent high-dimensional waveform optimization.

3.1. Bezout’s Identity and Solution Space Sparsification

Bézout’s Theorem is one of the most fundamental and profound theorems in elementary number theory characterizing the properties of integer linear combinations. Its classical algebraic expression is the following: assume there exist two integers $a,b\in \mathbb{Z}$ not both zero; if their greatest common divisor is $d=\gcd (a,b)$, then there must exist a set of coefficients $x,y\in \mathbb{Z}$ in the integer ring such that the linear Diophantine equation $ax+by=d$ holds. More crucially, the theorem explicitly states that $d$ is the smallest positive integer attainable in the set $\{ax+by:x,y\in \mathbb{Z}\}$. This existence proof usually relies on the back-substitution process of the Euclidean algorithm [25]. Although the Bezout coefficients $(x,y)$ satisfying the conditions are not unique in integer space, the minimum positive integer solution $d$ that the equation can represent possesses global absolute uniqueness. This algebraic uniqueness characteristic provides ample mathematical degrees of freedom for subsequently using GCD to regulate the radar ambiguity resolution physical space: the system can flexibly adjust coefficients to meet other multi-dimensional engineering constraints while maintaining the rigid premise of a constant $d$.

In modern airborne radar systems, a waveform burst comprising $N$ PRFs is typically utilized to achieve target range ambiguity resolution. Bézout’s Theorem can equally be rigorously extended to the $N$-dimensional integer domain: let $a_1,a_2,\dots ,a_N\in \mathbb{Z}$ and not be all zero, with their global greatest common divisor defined as $d=\gcd (a_1,a_2,\dots ,a_N)$; then there must exist an integer sequence $x_1,x_2,\dots ,x_N\in \mathbb{Z}$ satisfying ${\displaystyle {\sum }_{i=1}^N}{a}_i{x}_i=d$. This extended form under multiple integers directly reveals the number-theoretic topological structure of multi-PRF systems: the global common divisor $g$ and the pairwise local common divisors $d_{ij}=\gcd (a_i,a_j)$ are closely associated through a nested linear combination structure, forming the theoretical prototype of hierarchical optimization. At the engineering computation level, the Extended Euclidean Algorithm (XGCD) not only efficiently solves for the greatest common divisor but also synchronously outputs the Bezout coefficients during the recursive backtracking phase, with a time complexity of only $O\left(\mathrm{l}\mathrm{o}\mathrm{g}\right(\mathrm{m}\mathrm{i}\mathrm{n}(a,b)\left)\right)$. This extremely high computational efficiency provides fundamental computational power assurance for executing large-scale number-theoretic optimization within tens of thousands of genetic algorithm iteration cycles.

Mapping Bézout’s Theorem onto the physical context of radar ambiguity resolution can profoundly reveal the powerful constraining effect of GCD on the false solution space. We normalize the PRF sequences transmitted by the radar into a set of positive integer sequences $n_1,n_2,\dots ,n_N$ using the basic range gate width $t_0$ as the measurement benchmark. Extracting any pair of PRF components $n_i$ and $n_j$, all the integer linear combinations they can form create a discrete subgroup $L_{ij}=\{n_{i}x+n_{j}y:x,y\in \mathbb{Z}\}$ in algebraic space. According to the core corollary of Bézout’s Theorem, this set is strictly equivalent to $d_{ij}\mathbb{Z}$, i.e., an arithmetic progression with a step size of $d_{ij}=\gcd (n_i,n_j)$.

In the back-end data processing stage of the radar, due to the presence of receiver thermal noise and complex sea surface spike clutter, the system must tolerate a certain measurement error boundary. For an apparent residual drift purely induced by noise to be mistakenly recognized by the system as a legitimate ambiguity resolution result, its absolute offset in the Diophantine equation must strictly fall into the set $L_{ij}$. From this, it can be derived that the “relative physical density” of this solution set in local space is ${\rho }_{rel}\approx 1/d_{ij}$. This formula exposes a fundamental physical constraint in number-theoretic space: in traditional radar waveform design, to pursue the maximum unambiguous range as much as possible, systems universally adopt the “pairwise coprime” principle (forcibly setting $d_{ij}=1$). At this point, the relative density of the solution space reaches a peak of 1, meaning any minor noise fluctuation is highly likely to find a legal false target projection in the dense grid, thereby triggering a significant proliferation of the false alarm rate. Conversely, if we actively intervene in the number-theoretic structure of the waveform, making $d_{ij}>1$, the potential ambiguous solution space is forcibly “sparsified.” Noise must undergo a massive offset of exactly a multiple of $d_{ij}$ to slip through. This solution space sparsification effect realized through GCD control is precisely the core mathematical mechanism for linearly reducing or even super-linearly obliterating random false alarm solutions from the underlying probability.

To rigorously formalize the causal link between the number-theoretic structure of the PRF set and the system-level false-alarm behavior, one must consider the statistical mechanics of $M/N$ detection. A system-level false alarm occurs only when random clutter spikes or noise anomalies exceed the detection threshold in at least $M$ out of $N$ PRFs, aligning at the same unfolded range bin. The occurrence of these spatial alignments is fundamentally governed by the Diophantine equations formed by the PRF intervals. By purposefully elevating the pairwise greatest common divisors (GCDs) between individual PRFs, the algorithm fundamentally increases the step size of the ambiguity coincidence grid. This structural modification mathematically sparsifies the density of potential target-alignment nodes within the radar’s maximum observation window. As shown in Figure 3, this schematic representation visualizes the sparsified solution space, which effectively lowers the probability of random clutter spikes triggering the detection logic. Consequently, even under identical single-pulse false alarm rates, the sparsified coincidence grid significantly lowers the joint statistical probability of multiple random spikes aligning perfectly to trigger the $M/N$ logic. This establishes a direct, monotonically mapping relationship: structurally optimizing the local GCD distribution serves as a physical barrier that preemptively suppresses system-level multi-PRF false alarms.

Assuming that the noise- or clutter-induced threshold crossings across the $N=4$ diverse PRF channels are statistically independent, let $P_{fa,0}$ denote the basic constant false alarm probability of a single-pulse Doppler channel. In practical multi-PRF ambiguity resolution, two unwrapped measurements are considered consistent only if they fall within a predefined physical tolerance band $\Delta R$ (which accounts for target motion migration and system measurement errors). We define the normalized tolerance parameter as $\delta =\Delta R/R_{blind}$.

According to the topological properties of linear Diophantine equations, the probability that two independent false measurements accidentally align within this tolerance band $\delta$ on the unwrapped axis is strictly constrained by the solution grid density, which is proportional to $1/\mathrm{g}\mathrm{c}\mathrm{d}(n_i,n_j)$. Under the $2/4$ target decision criterion (i.e., if at least 2 out of 4 PRFs satisfy ambiguity resolution consistency, the target is declared present), we can apply the union bound (Boole’s inequality) to expand the overall probability of a random false alarm solution. The system’s false alarm space is bounded by the sum of all pairwise ($\left({\displaystyle \genfrac{}{}{0pt}{}{4}{2}}\right)$) and triplet ($\left({\displaystyle \genfrac{}{}{0pt}{}{4}{3}}\right)$) coincidence events. Thus, the explicit mathematical upper bound driven purely by noise can be rigorously derived as

$$P_{fa,solution}\le \left({\displaystyle \genfrac{}{}{0pt}{}{4}{2}}\right)\alpha {\displaystyle \frac{P_{fa,0}^2}{d_2\left(n\right)}}+\left({\displaystyle \genfrac{}{}{0pt}{}{4}{3}}\right)\beta {\displaystyle \frac{P_{fa,0}^3}{d_3\left(n\right)}}$$

(17)

In this rigorous formulation, the roles of the constants and variables are explicitly defined as follows:

• $d_2(n)={\left({\displaystyle \frac{1}{6}}\sum _{i<j}{\displaystyle \frac{1}{\mathrm{g}\mathrm{c}\mathrm{d}(n_i,n_j)}}\right)}^{-1}$ is the harmonic mean equivalent variable of the pairwise GCDs;
• $d_3(n)={\left({\displaystyle \frac{1}{4}}\sum _{i<j<k}{\displaystyle \frac{1}{\mathrm{g}\mathrm{c}\mathrm{d}(n_i,n_j,n_k)}}\right)}^{-1}$ represents the third-order harmonic GCD statistic, characterizing the grid sparsity for higher-order complex algebraic overlaps;
• $\alpha$ and $\beta$ are the constant tolerance overlap factors for pairwise and triplet coincidences, respectively. They are explicitly determined by the multi-dimensional integration of the measurement tolerance band $\delta$, and are deeply tied to the system’s range-Doppler resolution cells and coherent processing time.

This analytical upper bound transcends a mere empirical surrogate; it rigidly binds abstract false alarm probabilities with discrete number-theoretic parameters. It explicitly points out that: by maximizing the common divisor structural features ($d_2\left(n\right)$ and $d_3\left(n\right)$) within the PRF sets, the valid coincidence grid is forcefully sparsified, through which a non-linear precipitous reduction in the system false alarm rate can be achieved.

3.2. PRF Physical Constraints and GCD Optimization Boundaries

Although theoretically continuously increasing the greatest common divisor of PRF sets can suppress false alarms indefinitely, radar, as an electronic device constrained by rigorous physical laws and engineering limits, cannot have its waveform parameters selected to diverge infinitely. The end of GCD optimization is a zero-sum game with radar detection range (Least Common Multiple, LCM). According to the classical dual relationship $\gcd (a,b)·\mathrm{lcm}(a,b)=ab$ in elementary number theory, under the premise of constant hardware bandwidth, the increase in GCD inevitably leads to a sharp shrinkage of LCM, triggering catastrophic diffusion of detection blind zones. Therefore, the legitimate boundaries for GCD optimization must be precisely derived under the system’s multiple engineering physical constraints.

First, the microwave band characteristics and hardware timer precision of airborne radars constitute an insurmountable absolute bandwidth constraint: the normalized PRF integer $n_i$ must be strictly limited to the closed interval $[n_{min},n_{max}]$. In general radar system design, on the one hand, due to hardware and related factor constraints, the repetition period design is often selected as a multiple of the range gate width as the minimum unit; on the other hand, in the actual target position resolution process, all PRIs introduce the range gate as the unified minimum mathematical model unit, whereas when calculating velocity across different PRF dimensions, the minimum Doppler interval is determined by FD/FFT points and cannot be unified. Therefore, resolving range ambiguity is generally prioritized, followed by matching velocity ambiguity.

Furthermore, as the reciprocal of PRI, PRF is generally difficult to be evenly divided by a unit of 1 Hz, and in most cases are fractions, leading to a large maximum unambiguous velocity. Ambiguity is generally not easy to happen. But for typical X-band search radars, for instance, to cope with severe clutter broadening phenomena under high sea states, PRF usually needs to float within the medium-high PRF band. Under medium-high PRF scenarios, actual range ambiguities are much more severe than target velocity ambiguities. Thus, prioritizing the resolution of range dimension ambiguity is the focus. Meanwhile, to prevent high-power transmission components like Traveling-Wave Tubes (TWT) from burning out due to insufficient thermal dissipation, the product of pulse width $\tau$ and $f_{r,i}$ must strictly comply with the transmitter’s maximum duty cycle $D_{max}$ (e.g., typical value 10%), thereby deriving the upper bound constraint $n_i\le D_{max}/\left(\tau f_0\right)$. Moreover, to extract weak moving targets from intense noise floors and clutter, the number of coherent pulses assigned to a single PRF within the Coherent Processing Interval (CPI) must cross the minimum coherent integration threshold $N_{coh,min}$ (usually not lower than 64), thereby deriving the hard lower bound barrier $n_i\ge N·N_{coh,min}/\left(T_{CPI,total}{f}_0\right)$. Regarding the distribution of radar blind zone maps, we also wish to minimize complete blind velocity bands and blind range bands as much as possible, while considering the MTI main clutter suppression range under the radar architecture, all of which add supplementary constraints to the selection of waveform sets.

On the basis of satisfying the aforementioned underlying hardware and system design constraints, it is even more critical to strictly maintain the threshold of radar tactical requirements for the detection airspace under complex GCD regulation. For a detection system composed of $N$ PRFs, its joint maximum unambiguous range $R_{u,joint}$ and maximum unambiguous velocity $v_{u,joint}$ are determined by the Chinese Remainder Theorem. Let the global common divisor be $g=\gcd (n_1,\dots ,n_N)$, and introduce auxiliary variables $m_i=n_i/g$. Since $m_i$ tends to approach coprime states in later optimization stages, we can make the approximation $\mathrm{lcm}(m_1,\dots ,m_N)\approx {\displaystyle {\prod }_{i=1}^N}{m}_i$. Starting from this, via product expansion, the analytical expression for the joint unambiguous range can be approximately derived. The process is outlined below.

The maximum unambiguous range corresponding to a single PRI is

$$R_{u,i}={\displaystyle \frac{c·t_{r,i}}{2}}={\displaystyle \frac{c·n_i{t}_0}{2}}={\displaystyle \frac{c·n_i}{2f_0}}$$

(18)

The joint maximum unambiguous range (based on the Chinese Remainder Theorem) is

$$R_{u,joint}={\displaystyle \frac{c}{2}}·lcm\left(t_{r,1},\dots ,t_{r,N}\right)={\displaystyle \frac{c·t_0}{2}}·lcm\left(n_1,\dots ,n_N\right)$$

(19)

Let $g=\gcd (n_1,\dots ,n_N)$, let $n_i=g·m_i$, where $\gcd (m_1,\dots ,m_N)=1$, then

$$lcm\left(n_1,\dots ,n_N\right)=g·lcm\left(m_1,\dots ,m_N\right)$$

(20)

When each $m_i$ is nearly coprime, $\mathrm{lcm}(m_1,\dots ,m_N)\approx {\displaystyle {\prod }_{i=1}^N}{m}_i$, therefore,

$$R_{u,joint}\approx {\displaystyle \frac{c·t_0·g}{2}}·{\displaystyle \prod _{i=1}^N}{\displaystyle \frac{n_i}{g}}={\displaystyle \frac{c·t_0}{2}}·{\displaystyle \frac{{\displaystyle {\prod }_{i=1}^N}{n}_i}{g^{N-1}}}$$

(21)

Approximating ${\displaystyle {\prod }_{i=1}^N}{n}_i\approx n_{avg}^N$ using $n_{avg}$,

$$R_{u,joint}\approx {\displaystyle \frac{c·t_0·n_{avg}^N}{2·g^{N-1}}}$$

(22)

$$R_{u,joint}\approx {\displaystyle \frac{c·t_0·{\displaystyle {\prod }_{i=1}^N}{n}_i}{2g^{N-1}}}\approx {\displaystyle \frac{c·t_0·n_{avg}^N}{2g^{N-1}}}$$

(23)

where $n_{avg}$ is the equivalent average integer center of the PRF set, and $c$ is the speed of light. Notably, the approximation in (23) fundamentally assumes the auxiliary variables remain nearly coprime. If the global common divisor $g$ becomes excessively large, the true multi-variable LCM collapses. This would not only cause a rapid non-linear degradation of the actual unambiguous range to the power of $(N-1)$, but also introduce a non-negligible overestimation bias in this analytical model. To strictly preempt this theoretical vulnerability and guarantee the radar meets the basic tactical requirement range $R_{req}$, the proposed framework enforces a macroscopic coverage constraint. By requiring $R_{u,joint}\ge R_{req}$, an explicit upper bound constraint is imposed on $g$, which inherently forces the optimized PRF sets to operate safely within the valid, low-bias domain of this approximation:

$$g^{N-1}\le {\displaystyle \frac{c·t_0·n_{avg}^N}{2R_{req}}}$$

(24)

That is,

$$g\le {{\displaystyle \frac{n_{avg}}{c·t_0·\frac{n_{avg}^2}{R_{req}}}}}^{{\displaystyle \frac{1}{N-1}}}$$

(25)

In stark contrast to folding in the range domain, the unambiguous boundary in the velocity domain exhibits a high degree of positive duality. The joint maximum unambiguous velocity is

$$v_{u,joint}={\displaystyle \frac{\lambda }{4}}gcd\left(f_{r,1},\dots ,f_{r,N}\right)\approx {\displaystyle \frac{\lambda f_0·g^{N-1}}{4n_{avg}^N}}$$

(26)

Guaranteeing the velocity detection boundary $v_{req}$ requires the system common divisor not to be too low. Here, the lower bound constraint is derived. The maximum unambiguous velocity corresponding to a single PRF is

$$v_{u,i}={\displaystyle \frac{\lambda }{4t_{r,i}}}={\displaystyle \frac{\lambda }{4n_i{t}_0}}={\displaystyle \frac{\lambda f_0}{4n_i}}$$

(27)

The joint maximum unambiguous velocity is

$$v_{u,joint}={\displaystyle \frac{\lambda }{4}}·gcd\left(f_{r,1},\dots ,f_{r,N}\right)={\displaystyle \frac{\lambda f_0}{4}}·gcd\left({\displaystyle \frac{1}{n_1}},\dots ,{\displaystyle \frac{1}{n_N}}\right)$$

(28)

For positive integers, $\gcd ({\displaystyle \frac{1}{n_1}},\dots ,{\displaystyle \frac{1}{n_N}})={\displaystyle \frac{1}{\mathrm{lcm}(n_1,\dots ,n_N)}}$, therefore,

$$v_{u,joint}={\displaystyle \frac{\lambda f_0}{4·lcm\left(n_1,\dots ,n_N\right)}}={\displaystyle \frac{\lambda f_0·g^{N-1}}{4·{\displaystyle {\prod }_{i=1}^N}{n}_i}}$$

(29)

Approximating with $n_{avg}$,

$$v_{u,joint}\approx {\displaystyle \frac{\lambda f_0·g^{N-1}}{4·n_{avg}^N}}$$

(30)

Guaranteeing $v_{u,joint}\ge v_{req}$ requires

$$g^{N-1}\ge {\displaystyle \frac{4v_{req}·n_{avg}^N}{\lambda f_0}}$$

(31)

That is,

$$g\ge n_{avg}{\left({\displaystyle \frac{4v_{req}·n_{avg}}{\lambda f_0}}\right)}^{{\displaystyle \frac{1}{N-1}}}$$

(32)

$$g\ge {\left({\displaystyle \frac{4v_{req}·n_{avg}}{\lambda f_0}}\right)}^{{\displaystyle \frac{1}{N-1}}}$$

(33)

Integrating the dual tactical boundaries of ranging and velocity measurement, the viable physical optimization interval for the global common divisor $g$ is rigorously defined as

$$n_{avg}{\left({\displaystyle \frac{4v_{req}}{\lambda f_0{n}_{avg}}}\right)}^{{\displaystyle \frac{1}{N-1}}}\lesssim g\lesssim n_{avg}{\left({\displaystyle \frac{R_{req}}{c{t}_0{n}_{avg}^2}}\right)}^{{\displaystyle \frac{1}{N-1}}}$$

(34)

For typical modern airborne X-band parameters ($R_{req}=150$ km, $v_{req}=750$ m/s, $N=4$), the calculated values on both sides of this inequality typically span three orders of magnitude. This conclusively proves that: in the real physical world, this optimization interval is not only non-empty but also extremely generous, providing ample maneuver room to utilize intelligent optimization algorithms to deeply explore and reshape the number-theoretic topological structure of waveforms.

To strengthen the physical credibility of the aforementioned GCD boundary design, it is imperative to rigorously examine the validity range of the two core approximations utilized in deriving Equations (23) and (29): the near-coprime assumption ($lcm(m_1,\dots ,m_N)\approx {\displaystyle {\prod }_{i=1}^N}{m}_i$) and the average PRF surrogate (${\displaystyle {\prod }_{i=1}^N}{n}_i\approx n_{avg}^N$).

First, regarding the average PRF surrogate, in practical airborne pulse-Doppler radar designs, the PRF values within a single coherent processing burst are typically clustered tightly around a center frequency to maintain the stability of the transmitter’s duty cycle and average power. Let $n_i=n_{avg}(1+{\delta }_i)$, where ${\delta }_i$ represents the fractional deviation from the mean. In practical X-band HPRF operations, the maximum deviation is strictly bounded, typically $\left|{\delta }_i\right|\le 10\mathrm{\%}$. According to the AM-GM (Arithmetic Mean-Geometric Mean) inequality and Taylor expansion, the relative approximation error ${ϵ}_{avg}$ can be quantified as

$${ϵ}_{avg}=\left|{\displaystyle \frac{n_{avg}^N-{\displaystyle {\prod }_{i=1}^N}{n}_i}{n_{avg}^N}}\right|\approx \mathcal{O}({\sigma }_{\delta }^2)$$

where ${\sigma }_{\delta }^2$ is the variance of the fractional deviations. For a typical radar configuration with a $\pm 10\mathrm{\%}$ PRF stagger, the variance is minuscule, strictly bounding this surrogate error below $1.5\mathrm{\%}$.

Second, regarding the near-coprime assumption of the auxiliary variables $m_i$, it is deeply rooted in the inherent objective of multi-PRF radar design. To meet the macroscopic clear-region requirements, the optimization algorithm naturally penalizes sets with redundant common divisors among the $m_i$ terms, forcing them to be mutually coprime to maximize the usable LCM. Numerical checks over 10,000 sets of randomly generated HPRF parameters within the acceptable duty-cycle window show that the residual redundancy factor $\gamma =\prod m_i/lcm(m_1,\dots ,m_N)$ strictly equals 1 in over $94\mathrm{\%}$ of the cases, validating the near-coprime premise.

Most importantly, it must be emphasized that the approximations in Equations (23) and (29) are exclusively employed to theoretically deduce the macroscopic physical boundary constraints $[g_{min},g_{max}]$ prior to the optimization execution, ensuring the genetic algorithm is initialized within a valid, non-empty integer subspace. During the actual Bezout-GA evolutionary search, the fitness functions $f_1\left(n\right)$ and $f_2\left(n\right)$ strictly calculate the exact $LCM(n_1,\dots ,n_N)$ and perform exact discrete point-to-point blind-zone mapping without invoking any analytical approximations. Thus, the minor tolerance limits in the boundary estimations ($<1.5\mathrm{\%}$) do not introduce any bias or degradation into the final synthesized waveform parameters, thoroughly solidifying the rigor of the proposed framework.

3.3. Bi-Objective Fitness Function for Detection Effectiveness

Having established the dual bounding constraints of physics and number theory, the subsequent task is to construct an evaluation system capable of accurately guiding the optimization algorithm. Given the inherent trade-off between clear-region coverage and false-alarm suppression in HPRF waveform design, this paper formulates the problem strictly within a bi-objective evaluation framework. Rather than collapsing these multiple objectives into a single scalar fitness for population survival, the framework strictly maintains Pareto-based non-dominated sorting to preserve solution diversity, while utilizing dynamic weighted evaluations exclusively as heuristic guides for localized search operators.

• Optimization Objective 1: Minimization of Detection Blind Zone Coverage (Maximization of Clear Region Ratio)

The actual operational effectiveness of a radar highly depends on its blind zone distribution on the range-velocity two-dimensional plane. Setting the highest Doppler frequency shift and maximum range allowed for targets within the radar’s line of sight jointly delineates the Region of Interest (ROI). Under the 2/4 detection criterion, a local blind zone indicator function $B_{joint}^{2/4}(R,v;n)$ is defined; if clear echoes in a PRF burst lack at least 2 at a certain point on the 2D plane, that point is judged to be in a system-level blind zone, and the function is set to 1, otherwise 0. To quantify this coverage capability, the first fitness function $f_1\left(n\right)$ is defined as the blind zone coverage ratio (i.e., the complement of the clear region ratio ${\eta }_{clear}$):

$$f_1\left(\mathit{n}\right)=1-{\eta }_{clear}\left(\mathit{n}\right)={\displaystyle \frac{{\iint }_{ROI}{B}_{joint}^{\frac{2}{4}}\left(R,v;\mathit{n}\right)dRdv}{{\iint }_{ROI}dRdv}}$$

(35)

In the underlying computational engine of the algorithm, because the indicator function exhibits highly discontinuous jumps on the multi-dimensional manifold, analytical integration is impossible. Therefore, a Monte Carlo numerical integration engine based on ultra-large-scale sampling is introduced. By randomly sprinkling macro-inspection particles on the order of ${10}^5\sim {10}^6$ within the $R_{ROI}\times V_{ROI}$ domain and accurately counting the proportion of particles falling into clear regions, computing power is exchanged for extremely high fidelity in coverage evaluation.

• Optimization Objective 2: Minimization of Random False Alarm Solution Existence Probability

To fundamentally suppress consistency coincidences induced by clutter or noise, the second fitness function is directly anchored to the previously derived analytical upper bound of false alarm probability. To accelerate evaluation computations in high-frequency iterations and maximize the guiding force of Bézout’s Theorem on number-theoretic structures, the objective function $f_2\left(n\right)$ is defined as the dominant evaluation term of the system false alarm rate:

$$f_2\left(n\right)=P_{fa,sol}\left(n\right)=3\alpha {\displaystyle \frac{P_{fa,0}^2}{d_2\left(n\right)}}+\beta {\displaystyle \frac{P_{fa,0}^3}{d_3\left(n\right)}}$$

(36)

The minimization of this objective function effectively guides the optimization process, systematically driving the genetic population to reject suboptimal coprime configurations and converge toward number-theoretic structures characterized by elevated GCD distributions. It must be emphasized that the primary evolutionary search and survival selection throughout the framework remain genuinely Pareto-based, strictly utilizing non-dominated sorting and crowding distance mechanisms. The proposed method avoids collapsing the multi-objective problem into a single-objective scalar for population survival. However, to effectively navigate the trade-off between these two competing objectives during localized search operations, a dynamic weighted heuristic indicator, $H\left(t\right)$, is constructed. This indicator functions exclusively as a directional guide for the customized genetic operators and the local Bézout refinement phase, without altering the underlying Pareto ranks. Specifically, the time-varying weights adjust the heuristic focus: during the early exploration phase, a higher weight is assigned to $f_1$, directing the operators to rapidly identify viable configurations with extensive spatial coverage. Conversely, during the later exploitation phase, the weighting priority shifts toward $f_2$. In this stage, $H\left(t\right)$ determines which specific candidate solutions from the established Pareto front are prioritized for the local Bézout-guided directional perturbation. This mechanism ensures a systematic and rigorous number-theoretic refinement of the high-coverage candidates, thereby suppressing the false-alarm penalty while preserving the fundamental multi-objective integrity of the optimization framework.

4. Two-Stage Bezout-GA Hybrid Optimization Architecture

After clarifying the bi-objective fitness functions and multi-dimensional physical constraints, the core of the problem shifts to the solution domain. Radar PRF optimization belongs to a typical Mixed-Integer Non-Linear Programming (MINLP) problem; its solution space scale is as high as the order of ${10}^{16}$, and the fitness surface is covered with number-theoretic traps formed by irregular jumps of Diophantine equations. Traditional single heuristic genetic algorithms are extremely prone to premature population convergence in such problems or leaving fatal vulnerabilities in microscopic number-theoretic structures. To this end, this chapter innovatively proposes a two-stage hybrid optimization algorithm—Bezout-GA—that couples “large-scale global search” with “small-scale number-theoretic fine carving”.

4.1. Algorithm Architecture and Customized Genetic Operators

The design philosophy of the hybrid algorithm lies in: leveraging the excellent macroscopic parallel exploration capabilities of the genetic algorithm to lock onto high-quality candidate regions, and then utilizing the precise algebraic characterization capability of Bézout’s Theorem on GCD to patch microscopic number-theoretic fissures. As shown in

Table 3. Comparison of functional characteristics of Bezout-GA Two-Stage Hybrid Strategy.

Optimization Stage Division	Core Algorithm Driving Engine	Main Engineering Functional Positioning	Stage Key Feature Output
First Stage (Global Coarse Exploration)	Non-dominated Sorting Genetic Algorithm (NSGA-II Framework)	Rapidly explores within the enormously vast solution space to lock onto high-potential, high-coverage-rate, high-quality candidate regions	Generates rough Pareto approximate solution set $P_{GA}$
Second Stage (Local Fine Carving)	Number-theoretic guidance based on Bézout’s Theorem and Extended Euclidean Algorithm	Performs extremely refined number-theoretic vulnerability patching on the roughly explored solution set, achieving targeted localized enhancement of GCD structure	Outputs the final absolutely refined Pareto tactical waveform library front $P_{final}$

, the two-stage strategy achieves highly complementary advantages in functional positioning and output.

The overall architecture of the proposed Bezout-GA two-stage hybrid optimization strategy is illustrated in the flowchart of Figure 4. As shown in Figure 4, the algorithm logically transitions from an NSGA-II-based global search to a specialized Bezout refinement phase. In the global search of the first stage, the algorithm uses the NSGA-II framework as a base and deeply customizes it for the radar PRF problem [26]. Decision variables adopt a direct integer coding scheme, where each individual corresponds to a set of PRF sequences normalized to the range gate width. To prevent the loss of number-theoretic characteristic diversity during initialization, a “stratified constrained sampling” mechanism is built into the algorithm: first, randomly anchoring a global common divisor target $g_{target}$ within a reasonable physical interval, then generating chromosome genes on this basis to ensure the initial population inherently possesses a multi-dimensional common divisor skeleton from the start.

Regarding genetic evolution operators, to avoid destroying the non-coprime excellent genes that parent generations have painstakingly evolved, this paper originally designed the “GCD-Preserving Partially Mapped Crossover (GCD-PMX)” operator. After swapping cross segments, if the system detects that the critical pairwise common divisor $d_{ij}$ of the offspring has suffered a severe drop, the operator will automatically activate a gene repair routine, exhaustively searching for substitute values within the adjacent tiny mutation window $[n_i^{\prime }-\Delta ,n_i^{\prime }+\Delta ]$ to forcibly maintain and elevate the GCD. Meanwhile, the mutation operator is reconstructed as a “two-level joint strategy”: a large-step random global mutation is responsible for maintaining population diversity and preventing local deadlocks; while a low-probability triggered “local GCD-guided mutation” specifically picks out the PRF nodes with the worst GCD performance in an individual, executing minor directional translations along the direction of the Bezout coefficient vector calculated by the Extended Euclidean Algorithm. This directional number-theoretic perturbation, akin to gradient descent in physics, massively accelerates the population’s convergence toward highly robust regions. After strict culling via non-dominated sorting and crowding distance, after hundreds of generations, the algorithm will robustly output a set of Pareto coarse fronts with extremely high potential.

4.2. Execution Flow and Number-Theoretic Refinement

The second stage is a process of extreme squeezing and fine pruning of the coarse front $P_{GA}$ output by the GA. Due to the random nature of GA operators, the radar waveform combinations they output usually perform impeccably in macroscopic Doppler coverage; however, in hidden microscopic structures, two specific PRFs might accidentally form a coprime relationship ($d_{ij}=1$), thereby opening a “back door” for the infiltration of random false alarms.

The execution flow of the Bezout Theorem refinement phase is as follows: first, the system conducts a deep “number-theoretic physical examination” on every representative candidate solution in $P_{GA}$, calculating its complete $N\times N$ dimensional GCD adjacency matrix. Targeting the weakest node in the matrix that most severely drags down the overall false alarm performance (i.e., the PRF pair with the smallest local common divisor $(i^{\ast },j^{\ast })={\mathrm{argmin}}_{i<j}{d}_{ij}$), the algorithm immediately calls the XGCD engine to compute the precise Bezout coefficients $(x,y)$ representing that locality with extremely low computational overhead.

Subsequently, a targeted number-theoretic refinement procedure is executed. Leveraging the algebraic translation properties of Bézout’s identity, the algorithm constructs an optimization path along the coefficient vector and applies a directional perturbation with a minimal integer step size:

$$n_{i^{\ast }}^{\prime }=n_{i^{\ast }}+\delta ·{\displaystyle \frac{n_{j^{\ast }}}{d_{i^{\ast }{j}^{\ast }}}}, n_{j^{\ast }}^{\prime }=n_{j^{\ast }}-\delta ·{\displaystyle \frac{n_{i^{\ast }}}{d_{i^{\ast }{j}^{\ast }}}}$$

(37)

The primary analytical advantage of this operation is that it preserves the macroscopic linear combination span of the original PRF sequence, thereby maintaining the fundamental LCM-based detection coverage without significant degradation. Concurrently, it systematically resolves internal coprime vulnerabilities by increasing the local common divisor to structurally safer thresholds. Finally, the system re-evaluates the refined solution set, systematically excludes inferior solutions, and merges the remaining candidates to generate the final tactical configuration front, $P_{final}$. This Pareto front is then provided to the backend decision module, which selects the optimal configuration dictionary based on operational requirements, such as search-priority or track-priority modes.

Implementation Details for Reproducibility: To ensure the proposed hybrid framework is fully reproducible, the quantitative algorithmic settings and execution boundaries for the customized genetic operators and local Bézout refinement are explicitly specified as follows:

• Mutation Window ($\Delta$): In the GCD-preserving crossover and mutation repair routines, the algorithm exhaustively searches within a strictly bounded integer window $\Delta =\pm 3$ basic range bin units. This narrow window guarantees that the targeted GCD elevation does not violate the underlying transmitter duty cycle constraints.
• Refinement Step Size ($\delta$): During the XGCD-based local fine-tuning (as governed by Equation 45), the step size along the Bézout coefficient vector is initialized as $\delta \in \{-1,1\}$. Exploring exclusively adjacent integer steps ensures minimal disruption to the macroscopic LCM, thereby preserving the inherited geometric clear-region advantages.
• Acceptance Rules: A newly perturbed or refined offspring is accepted into the population repository only if it satisfies a strict constrained $ϵ$-dominance rule: it must either strictly Pareto-dominate its parent, or it must significantly improve the false-alarm penalty ($f_2$) while bounding any resulting degradation in the clear-region ratio ($f_1$) to a maximum tolerance of $0.5\mathrm{\%}$.
• Stopping Conditions: The localized Phase 2 refinement for a specific candidate solution terminates immediately if either of two conditions is met: (1) the step size $\delta$ reaches the maximum allowable boundary limit (${\delta }_{max}=\pm 5$); or (2) no acceptable solution satisfying the aforementioned acceptance rule is discovered after $T_{local}=5$ consecutive directional perturbations. Upon termination, the algorithm proceeds to evaluate the next candidate on the Pareto front.

4.3. Complexity Evaluation and Convergence Analysis

When evaluating the engineering usability of this hybrid architecture, computational complexity and memory consumption are two unavoidable hard metrics. The overall time complexity of the algorithm exhibits a significant phase superposition effect. During the GA global exploration stage, computing power is mainly consumed by the massive Monte Carlo integration engine, and its complexity is $O(G_{max}{N}_{pop}{N}_{MC}+N^2\mathrm{l}\mathrm{o}\mathrm{g}{n}_{max})$. Upon entering the Bezout refinement stage, since wide-area blind zone search is no longer needed, the computing core swiftly switches to underlying number-theoretic analysis, and the time complexity of this stage plummets to $O(N_{cand}{N}^2\mathrm{l}\mathrm{o}\mathrm{g}{n}_{max}+N_{local}·N\mathrm{l}\mathrm{o}\mathrm{g}{n}_{max})$, where $N_{local}$ is the number of attempts for minute perturbation.

Under typical HPRF airborne radar parameter configurations (e.g., maximum generations $G_{max}=300$, population size $N_{pop}=100$, integration test points $L_{test}={10}^4$, channels $N=8$), the overall execution of the Bezout-GA framework involves approximately $\mathcal{O}\left({10}^7\right)$ basic floating-point and bitwise operations. When deployed on a conventional general-purpose processing platform with C++ optimization, this hybrid algorithm can complete the offline reconstruction of highly complex waveform libraries within tens of seconds. In terms of spatial complexity, the peak memory occupation—encompassing the evolutionary population, fitness caching matrices, and the historical Pareto front archive—scales theoretically at the $\mathcal{O}(G_{max}·N_{pop})$ level. In practical simulation environments, the absolute physical memory footprint remains on the order of a few megabytes (MB). This extremely low-overhead spatial characteristic ensures that the proposed framework imposes no severe memory bottlenecks on resource-constrained airborne embedded computing units. Regarding practical engineering deployment, it must be objectively acknowledged that executing this complete evolutionary optimization strictly within a single Coherent Processing Interval (intra-CPI, typically a few milliseconds) remains computationally infeasible for standard airborne embedded platforms. However, the bounded computational complexity and the highly memory-efficient nature of the algorithm indicate substantial feasibility for cross-CPI or inter-dwell dynamic scheduling. In operational scenarios—such as mode transitions between volume search and precision track, or across consecutive antenna scanning cycles—the available time window extends significantly (e.g., hundreds of milliseconds to seconds). Within these extended operational timescales, leveraging modern parallel computing architectures (e.g., DSP or FPGA clusters) to execute this framework for near-real-time PRF reconfiguration is practically viable.

From a theoretical convergence perspective, the genetic algorithm mechanism, modeled via Markov chains, is mathematically proven to converge to the global Pareto boundary with a probability of 1 over infinite time. Concurrently, because the local directional translation in the Bézout stage is strictly governed by non-dominated acceptance criteria, this localized refinement preserves the integrity of the population’s evolutionary trajectory without degradation. Rather than disrupting the global search, it provides a robust acceleration in local convergence. Furthermore, comprehensive simulation evaluations demonstrate that, compared to classical heuristic algorithms relying primarily on standard computational search, the proposed Bezout-GA framework—integrating physical constraints with number-theoretic priors—outputs a waveform configuration set that achieves a measurable improvement in the comprehensive Hypervolume index. Ultimately, this approach endows the selected PRF sequences with explicit mathematical and physical interpretability, effectively validating its utility as a reliable, simulation-based PRF optimization method for resolving range-Doppler ambiguities.

5. Simulation and System-Level Performance Evaluation

To rigorously verify the algorithmic superiority and structural robustness of the proposed Bezout-GA framework, this section constructs a high-fidelity system-level simulation platform tailored to airborne High-PRF (HPRF) radar operational environments. Comprehensive evaluations are conducted against traditional baseline strategies—specifically the pairwise coprime and maximum clear region methods—to demonstrate the definitive advantages of our approach in suppressing spatial false alarms. To further validate the stability and generalizability of the proposed number-theoretic framework across diverse boundary conditions, this section incorporates a series of rigorous ablation studies, cross-band physical applicability tests (X-band and L-band), and multi-dimensional sensitivity analyses under higher-order $M/N$ detection criteria.

5.1. Simulation Setup and Algorithmic Characteristics

5.1.1. Scenario Configuration and Radar Parameters

The simulation system is set to operate in the X-band (carrier frequency $f_0=10$ GHz, wavelength $\lambda =0.03$ m). To conform to the practical demands of HPRF anti-clutter, $N=4$ sets of waveform sequences are configured and transmitted. Radar detection limits are calibrated as: detection range setting $R_{req}=20-300$ km, Doppler velocity interval of interest $v_{req}=\pm 1700$ m/s. Monte Carlo comparisons are based on K-distributed clutter and Gaussian-distributed noise scenarios. The system’s single-CFAR baseline false alarm defense line is set to $P_{fa,0}={10}^{-6}$, utilizing the 2/4 criterion, and the system duty cycle is set to 10%,.

Environmental configuration parameters: K-distributed sea clutter and Gaussian-distributed noise.

Based on this X-band baseline, higher-order M/N fault-tolerant criteria—specifically the 3/4 and 2/5 logics—are incorporated for comparative validation to assess the algorithmic robustness. Furthermore, to verify the cross-band applicability of the proposed framework, an L-band simulation scenario (carrier frequency 1.0 GHz) is additionally established, designed with an extended detection range of 40–700 km and evaluated under the baseline 2/4 criterion.

Genetic algorithm parameters: 300 generations, crossover probability 0.8, mutation probability 0.1, population size 1000.

To rigorously evaluate the performance and robustness of the proposed waveform optimization framework under realistic operational conditions, a comprehensive system-level simulation platform is established. The simulations are primarily conducted within an X-band High-PRF (HPRF) airborne radar context to counter severe sea clutter, and are subsequently extended to an L-band scenario to verify the cross-band generalizability of the algorithm.

In the baseline X-band scenario (carrier frequency $10 \mathrm{GHz}$), the radar is configured to transmit sets of $N=4$ waveform sequences, targeting a detection range of $20–300 \mathrm{km}$ and a broad Doppler velocity interval of $\pm 1700 \mathrm{m}/\mathrm{s}$. To strictly align with hardware limitations, the Pulse Repetition Interval (PRI) is constrained within $[30,50]\hspace{0.17em}\mathrm{\mu }\mathrm{s}$, with a system sampling rate of $5 \mathrm{MHz}$ and a maximum duty cycle of $10\mathrm{\%}$. The L-band scenario ($1.0 \mathrm{GHz}$) evaluates an extended detection range of $40–700 \mathrm{km}$, with its PRI specifically constrained to $[300,400]\hspace{0.17em}\mathrm{\mu }\mathrm{s}$.

The environmental models encompass heavy-tailed K-distributed sea clutter and Gaussian-distributed thermal noise. To assess the algorithmic robustness across varying decision logic stringencies, the system’s target detection performance is evaluated using the baseline $2/4$ criterion, alongside higher-order fault-tolerant criteria ($3/4$ and $2/5$ logics). The single-pulse Constant False Alarm Rate (CFAR) threshold is uniformly anchored at $P_{fa,0}={10}^{-6}$. The detailed systemic, environmental, and Genetic Algorithm (GA) parameters are systematically summarized in

Table 4. System-level simulation and algorithmic parameter configurations.

Parameter Category	Parameter Name	X-Band Scenario (Baseline)	L-Band Scenario (Extension)
Radar RF Specifications	Carrier Frequency ($f_0$)	$10 \mathrm{GHz}$	$1.0 \mathrm{GHz}$
	Wavelength ($\lambda$)	$0.03 \mathrm{m}$	$0.3 \mathrm{m}$
	Detection Range ($R_{req}$)	$20\sim 300 \mathrm{km}$	$40\sim 700 \mathrm{km}$
	Velocity Interval ($v_{req}$)	$\pm 1700 \mathrm{m}/\mathrm{s}$	$\pm 1700 \mathrm{m}/\mathrm{s}$
Hardware & Waveform Constraints	Number of PRFs ($N$)	$4$	$4$
	PRI Constraint Range	$[30,50]\hspace{0.17em}\mu \mathrm{s}$	$[300,400]\hspace{0.17em}\mu \mathrm{s}$
	System Sampling Rate ($f_s$)	$5 \mathrm{MHz}$	$5 \mathrm{MHz}$
	Maximum Duty Cycle ($D_{max}$)	$10\mathrm{\%}$	$10\mathrm{\%}$
Detection & Environment	Interference Models	K-distributed sea clutter & Gaussian thermal noise	K-distributed sea clutter & Gaussian thermal noise
	Target Decision Criteria	$2/4$, $3/4,2/5$	$2/4$
	Cell-Level False Alarm ($P_{fa,0}$)	${10}^{-6}$	${10}^{-6}$
Bezout-GA Configurations	Population Size ($N_{pop}$)	1000	1000
	Maximum Generations ($G_{max}$)	300	300
	Crossover Probability ($P_c$)	0.8	0.8
	Mutation Probability ($P_m$)	0.1	0.1

.

5.1.2. Controlled Experiment: Phantom Solutions and Random False-Alarm Solutions

To empirically validate the conceptual distinction between phantom solutions and random false-alarm solutions, and to demonstrate their fundamentally different physical origins, a controlled ablation experiment was conducted. We isolated the two failure mechanisms by mapping the 2D apparent range solution space under two extreme boundary scenarios: a noise-free, target-dense environment (Isolating Phantom Solutions), and a target-free, noisy environment (Isolating Random False Alarms). The traditional mutually coprime PRF set ($M_1=17,M_2=19$, GCD = 1) and the proposed Bezout-GA optimized PRF set ($M_1=16,M_2=20$, GCD = 4) were evaluated under identical target decision criteria. The visual results of the solution space topologies are presented in Figure 5.

Scenario A: Isolating Phantom Solutions (Target-Dense, Noise-Free) In this scenario, true targets with distinct range and velocity attributes (represented by blue triangles in Figure 5) were injected into the simulation. The background was set to be entirely noise-free, ensuring that any false output was strictly an algebraic phantom generated by the multi-target intersection of true echoes. As observed in both Figure 5a,b, the true target echoes inevitably generate valid phantoms (red crosses) across the diagonals. Both the traditional coprime set and the Bezout-GA optimized set generated these phantoms. This is physically expected: phantom solutions are algebraic inevitabilities driven by the Chinese Remainder Theorem (CRT) when true signal residues combine. Modifying the Greatest Common Divisor (GCD) topology does not eliminate true target returns; thus, mitigating phantom solutions inherently relies on downstream temporal tracking or filtering algorithms rather than physical-layer waveform design.

Scenario B: Isolating Random False Alarms (Target-Free, Heavy-Clutter) Conversely, in this scenario, random noise measurements (representing severe sea clutter spikes) were injected into the resolution modules. Here, any false output is strictly a random false-alarm solution generated by accidental noise coincidence. The results demonstrate a stark structural contrast:

As shown in Figure 5a, the traditional coprime PRF set creates an extremely dense Diophantine solution grid. Consequently, the noise rejection rate is 0%. Every random noise pairing (yellow stars) mathematically satisfies the consistency criteria, resulting in a significant proliferation of random coincidences and false alarms.

In sharp contrast, Figure 5b illustrates the Bezout-GA optimized set. By actively expanding the pairwise GCD (GCD = 4), the algorithm sparsifies the valid tolerance grid, leaving large topological gaps (invalid solution spaces). As a result, the noise rejection rate reaches 75%. The vast majority of random noise pairings (grey crosses) are mathematically rejected by the GCD filter at the physical layer, preventing them from forming valid false tracks.

Conclusion of the Experiment: By physically decoupling the simulation environments, this controlled experiment explicitly proves the theoretical assertions made in Section 3. The random false-alarm solution is fundamentally a topological vulnerability driven by grid density and measurement tolerance, which can be deterministically mitigated at the waveform source via Bezout-GA. Meanwhile, the phantom solution is an algebraic byproduct of multi-target density, which remains independent of the GCD structure. This isolation substantiates the targeted efficacy of the proposed number-theoretic optimization framework.

5.1.3. Characteristics of the Bi-Objective Hybrid Algorithm

To verify the effectiveness and convergence characteristics of the proposed Bezout-GA bi-objective hybrid optimization algorithm, this paper analyzes the fitness evolutionary trajectories of the algorithm at different evolutionary stages through simulations. The results are shown in Figure 6.

Simulation results in Figure 6a–c elucidate the trajectory of the bi-objective dynamic weight adjustment mechanism and its impact on sub-objective convergence. In the exploration phase ($t<100$), the system assigns a dominant initial weight ($w_1\approx 0.7$) to the blind zone coverage penalty, driving a rapid descent in the normalized cost $f_1$ to quickly secure viable baseline PRF sets. As evolution progresses into the exploitation phase ($100\le t<200$), the bi-objective weights gradually equilibrate. Here, the descent of $f_1$ slows as it approaches its physical limit, while the false alarm penalty $f_2$ initiates a steady downward trend. Upon entering the refinement phase ($t\ge 200$), the mechanism reverses: the false alarm suppression weight ($w_2$) takes precedence. Guided by this tilted weight and the intervention of the Bézout refinement strategy, $f_2$ achieves a precipitous secondary breakthrough. As depicted in Figure 6c, compared to a traditional static weight strategy (0.5/0.5), this dynamic adaptation successfully steers the heuristic indicator $H\left(t\right)$ away from premature stagnation, driving the Pareto front toward a deeper global optimum.

To further investigate the algorithmic mechanisms that prevent this stagnation, Figure 7 presents an ablation study comparing crossover operators through convergence trajectories. The trajectory of the Standard PMX operator exhibits prolonged horizontal plateaus—a classic symptom of falling into “coprime traps,” where standard combinatorial crossover destroys the latent number-theoretic structures required for further false alarm suppression. In stark contrast, the proposed GCD-Preserving PMX actively protects and inherits advantageous greatest common divisors during genetic transmission. Visually, this translates to significantly shorter plateau durations and a rapid, continuous trap-breaking capability. Ultimately, this structural preservation accelerates local convergence and yields a measurable improvement (approx. 10%) in the final system objective penalty.

In conclusion, coupling the GCD-preserving genetic operators with the dynamically weighted Bézout refinement effectively reconciles the zero-sum contradiction between blind zone coverage and false alarm suppression. This integrated structural design successfully endows the optimization framework with high efficiency and physical interpretability, realizing a substantial enhancement in overall radar detection performance.

5.2. Multi-Dimensional Verification of Waveform Optimization Performance

5.2.1. Typical Ablation Algorithm Comparison

After running the algorithm, three sets of waveforms output under four strategies are ultimately adopted: GA-Maximum clear region set: $[35.2,39.4,43.0,49.4]$ μs; Traditional coprime set: $[32.2,37.4,42.6,47.8]$ μs; Bezout-only set: $[36.0,40.0,44.0,48.0]$ μs; Optimal matched solution optimized by Bezout-GA: $[35.2,39.2,42.4,48.8]$ μs. Clear region ratios are: 0.97441, 0.97046, 0.96328 and 0.97387, respectively. The 2/4 clear region distributions for the three waveform sets are shown in Figure 8 below.

To comprehensively validate the structural advantages of the proposed framework, an ablation study was conducted evaluating four distinct PRF configurations: the GA-Max Clear Region set, the Traditional Coprime set, a purely number-theoretic “Bezout-only” baseline, and the proposed optimal Bezout-GA solution. Initial assessment of the 2D clear-region distributions (under the 2/4 fault-tolerant criterion, as illustrated in Figure 8) reveals highly comparable geometric clear-region ratios for the Max Clear Region, Traditional Coprime, and Bezout-GA sets (0.97441, 0.97046, and 0.97387, respectively). The Bezout-only baseline exhibits a marginally lower ratio of 0.96328, which, from a purely geometric standpoint, suggests only a minor and acceptable degradation in target visibility.

However, a rigorous topological analysis of the joint solution space (depicted in Figure 9) reveals a significant practical limitation of applying Bézout’s theorem in isolation. In the multi-PRF ambiguity resolution process, the boundary of the unambiguous joint solution space is strictly governed by the Least Common Multiple (LCM) of the pairwise PRFs. The Bezout-only strategy focuses on maximizing the Greatest Common Divisor (GCD) but fails to constrain the LCM. Consequently, the LCM values for the Bezout-only group fall significantly below the minimum system unambiguous range requirement ($N_{req}$, indicated by the dashed line in Figure 9). In practical pulse-Doppler radar operations, an LCM deficiency of this magnitude implies that the waveform cannot resolve range ambiguities up to the required tactical distance. This structural deficit in the solution space inevitably leads to significant spatial folding, generating numerous “phantom solutions” (ghost tracks) within the primary detection envelope. Because it violates the fundamental physical constraints required for radar ambiguity resolution, the Bezout-only configuration is deemed mathematically valid but practically inoperable. Therefore, it is consequently excluded from all subsequent system-level evaluations of false alarms and threshold performance.

With the invalid Bezout-only group excluded, comparison among the remaining viable sets underscores the superior regulatory capability of the Bezout-GA framework. The Traditional Coprime group strictly mandates absolute pairwise coprimality (GCD = 1) to maximize the theoretical unambiguous boundary. This approach causes its LCM values to increase substantially, far exceeding actual operational requirements. This results in significant “spatial over-issuance waste,” acting as a primary source for false alarms. While the Max Clear Region group performs slightly better, its LCM distribution remains disordered and exhibits significant spatial redundancy.

By contrast, the Bezout-GA optimal matched solution demonstrates precise number-theoretic regulation. The algorithm effectively compresses the redundant LCM space, fitting it compactly and safely above the required operational range boundary. By strategically conceding a negligible fraction of the geometric clear region (a reduction of roughly 0.1%, from 0.97453 to 0.97387), the proposed solution effectively trades this minor geometric sacrifice for a highly regularized and sparsified topological grid. This releases crucial algebraic degrees of freedom, laying a robust structural foundation capable of truncating the false alarm amplification chain at the physical waveform level.

As illustrated in Figure 10, the expected system false alarm volume ($E_{fa,sys}$) is strictly governed by the structural amplification factor of the waveform grid. Although the purely number-theoretic Bezout-only baseline mathematically yields the lowest amplification factor (10,540), its failure to satisfy the fundamental LCM range constraint renders it practically invalid for deployment. Among the physically viable configurations, the GA-Max Clear Region strategy primarily focuses on maximizing geometric target visibility. However, its lack of precise number-theoretic regulation leaves the spatial grid disordered, leading to a substantially higher amplification factor of 260,103. This inherent structural inefficiency geometrically amplifies weak underlying thermal noise, leading to a significant issue of “spatial false alarm over-issuance.” Conversely, the Bezout-GA optimized matched solution employs precise LCM clamping alongside directional GCD elevation to successfully reduce the amplification coefficient to 63,964. As explicitly demonstrated in the re-evaluation, under an identical cell-level false alarm probability, the proposed waveform achieves a substantial 4.1-fold reduction in the expected false alarm volume compared to the GA-Max Clear Region baseline. This inherent spatial suppression capability establishes a robust physical defense barrier at the waveform level, significantly outperforming the purely geometric optimization strategy without compromising the maximum unambiguous range requirement.

5.2.2. Clutter Intensity Sensitivity Analysis

As depicted in Figure 11, the spatial false alarm suppression achieved by optimizing the underlying waveform grid directly translates into equivalent hardware-level threshold gains at the receiver. While the purely number-theoretic “Bezout-only” baseline yields the highest theoretical gain, its failure to satisfy the fundamental unambiguous range constraint precludes it from practical engineering deployment. By maintaining a stringent system-level false alarm rate ($P_{fa,sys}={10}^{-6}$), the proposed Bezout-GA optimal matched waveform exploits its intrinsic false alarm mitigation properties to secure an equivalent Signal-to-Noise Ratio (SNR) advantage of approximately 0.4 dB against an ideal Gaussian white noise background. More importantly, under severe heavy-tailed K-distributed sea clutter ($\nu =0.1$), the traditional coprime configuration necessitates a drastically elevated detection threshold to counter spatial false alarm over-issuance. The Bezout-GA waveform circumvents this severe threshold penalty, thereby yielding a substantial equivalent Signal-to-Clutter Ratio (SCNR) enhancement exceeding 1.0 dB. Subsequent sensitivity analyses confirm that this topological regularization mechanism preserves robust performance gains across a diverse spectrum of clutter severities.

The systemic impact of this number-theoretic waveform refinement is clearly demonstrated by the system-level Receiver Operating Characteristic (ROC) curves in Figure 12, and is quantitatively summarized in

Table 5. Comprehensive structural and performance comparison of PRF waveform strategies (X-band Scenario).

Analytical Dimension	Traditional Coprime	GA-Max Clear Region	Bezout-Only (Ablation)	Bezout-GA Optimal (Proposed)
Output PRF Set (µs)	[32.2, 37.4, 42.6, 47.8]	[35.2, 39.4, 43.0, 49.4]	[36.0, 40.0, 44.0, 48.0]	[35.2, 39.2, 42.4, 48.8]
Normalized Integer Sequence **	[161, 187, 213, 239]	[176, 197, 215, 247]	[180, 200, 220, 240]	[176, 196, 212, 244]
Global GCD	1	1	20	4
Pairwise GCDs Set	[1, 1, 1, 1, 1, 1]	[1, 1, 1, 1, 1, 1]	[20, 20, 60, 20, 40, 20]	[4, 4, 4, 4, 4, 4]
Pairwise LCMs Range	[30,107, 50,907]	[34,672, 53,105]	[720, 2640]	[8624, 12,932]
LCM Range Constraint	Valid	Valid	Invalid (Fails min range)	Valid
Clear Region Ratio (2/4)	0.97441	0.97441	N/A	0.97387
False Alarm Amp. Factor	238,310	260,103	N/A (Discarded)	63,964 (Optimal)
Eq. Gain (Gaussian)	Baseline (0 dB)	+0 dB	N/A (Discarded)	+0.4 dB
Eq. Gain (K-Clutter, $v=0.1$)	Baseline (0 dB)	+0 dB	N/A (Discarded)	+1.0 dB

** Note: Normalized integers are derived using a basic time resolution unit of $t_0=0.2$ μs (defined by the 5 MHz sampling bandwidth) to enable rigorous Diophantine algebraic analysis.

. As a comprehensive structural and performance comparison of the PRF waveform strategies under the X-band scenario, Table 5 explicitly details the trade-offs between duty cycle, unambiguous range, and false alarm volume. Irrespective of the underlying clutter statistical distribution, the detection profile of the Bezout-GA optimal solution consistently exhibits a leftward shift in the ROC curves, delineating a superior physically valid performance envelope. This indicates that under identical, low-SCNR environmental conditions, the traditional coprime baseline experiences a severe degradation in detection probability due to its inherently stringent threshold requirements. Conversely, as supported by the performance metrics in Table 5, the proposed optimization framework facilitates the early interception and reliable tracking of low-observable targets without necessitating any hardware scaling in transmitter power or antenna aperture. Ultimately, this substantiates the effective translation of mathematical false alarm suppression into tangible system-level detection gains.

5.2.3. Robustness Validation Across Frequency Bands and Decision Criteria

Building upon the fundamental performance analysis in the X-band under the standard 2/4 detection logic, it is essential to further verify the generalizability and structural stability of the proposed framework. Accordingly, this section extends the evaluation to the L-band scenario and more complex $M/N$ decision criteria.

Simulation results demonstrate that the Bezout-GA framework maintains robust false alarm suppression across different frequency bands, successfully achieving cross-band validation from the X-band to the L-band. This confirms that the underlying number-theoretic mechanism—specifically the precise clamping of Least Common Multiple (LCM) boundaries and the directional elevation of the Greatest Common Divisor (GCD)—is independent of specific carrier frequency parameters, thereby establishing a universal physical defense barrier for radar hardware architectures operating across diverse spectral bands.

The specific performance analysis in the L-band reveals significant detection gains. As detailed in

Table 6. Performance comparison of PRF waveform strategies in L-band scenario.

Analytical Dimension	Traditional Coprime	GA-Max Clear Region	Bezout-Only (Ablation)	Bezout-GA Optimal (Proposed)
Output PRF Set (µs)	[301.9, 320.2, 356.9, 396.4]	[301.9, 320.2, 356.9, 396.6]	[360, 320, 360, 400]	[304, 328, 360, 392]
Normalized Integer Sequence **	[1507, 1659, 1811, 1963]	[1509, 1601, 1784, 1981]	[1500, 1600, 1800, 2000]	[1520, 1640, 1800, 1960]
Global GCD	1	1	100	40
Pairwise GCDs Set	[1, 1, 1, 1, 1, 1]	[1, 1, 1, 1, 1, 1]	[100, 300, 500, 200, 400, 200]	[40, 40, 40, 40, 40, 40]
Pairwise LCMs Range	[2,500,113, 3,554,993]	[2,415,909, 3,534,104]	[6000, 24,000]	[62,320, 88,200]
LCM Range Constraint	Valid	Valid	Invalid (Fails min range)	Valid
Clear Region Ratio (2/4)	0.9823	0.9825	N/A	0.9792
False Alarm Amp. Factor	18,003,590 (Severe)	17,659,163	N/A (Discarded)	447,560 (Optimal)
Eq. Gain (Gaussian)	Baseline (0 dB)	+0 dB	N/A (Discarded)	+1.2 dB
Eq. Gain (K-Clutter, $v=0.1$)	Baseline (0 dB)	+0.dB	N/A (Discarded)	+3.0 dB

** Note: Normalized integers are derived using a basic time resolution unit of $t_0=0.2$ μs (defined by the 5 MHz sampling bandwidth) to enable rigorous Diophantine algebraic analysis.

, the Bezout-GA optimal configuration (PRI set: [304, 328, 360, 392] μs) achieves an equivalent Signal-to-Noise Ratio (SNR) gain of +1.1 dB in Gaussian noise backgrounds while maintaining a high clear-region ratio of approximately 0.979. Crucially, in heavy-tailed K-distributed sea clutter, the optimized waveform yields an equivalent Signal-to-Clutter Ratio (SCNR) gain of up to +3 dB. The sensitivity curves in Figure 13 further substantiate that this performance edge remains stable even under stringent system false alarm requirements, effectively mitigating the severe threshold penalties faced by traditional strategies in lower frequency bands. Notably, the superiority of the Bezout-GA framework is particularly pronounced in the L-band. For low-frequency detection, unambiguous range constraints often lead to a denser distribution of ambiguity points within the joint solution space, where traditional coprime strategies are susceptible to “spatial false alarm over-issuance” due to uncontrolled LCM inflation. By precisely compressing redundant LCM space and releasing algebraic degrees of freedom to sparsify the topological grid, the proposed framework effectively suppresses the physical paths of false alarm amplification in low-frequency scenarios.

Furthermore, the evaluation is extended to higher-order decision criteria to assess topological robustness. Theoretically, within an $M/N$ detection logic, the parameter $M$ ($M\ge 3$) dictates the dimensionality of the joint solution space, which fundamentally governs the multi-dimensional spatial false alarm density distribution. Conversely, increasing the parameter $N$ merely expands the total number of available PRI combinations without altering the spatial dimensionality. To systematically investigate this, a 2/5 configuration is newly introduced and compared, as summarized in

Table 7. Performance comparison of PRF waveform strategies in 2/5 detection logic.

Analytical Dimension	Traditional Coprime	GA-Max Clear Region	Bezout-Only (Ablation)	Bezout-GA Optimal (Proposed)
Output PRF Set (µs)	[33.4, 36.2,39.4,42.2,45.4]	[30.4,33.8,39.4,42.6, 47.0]	[32,36.0,40.0,44.0,48.0]	[30.4,34.4,39.2,42.4,47.2]
Normalized Integer Sequence **	[167, 181, 197, 211, 227]	[152, 169, 197, 213, 235]	[160, 180, 200, 220, 240]	[152, 172, 196, 212, 236]
Global GCD	1	1	20	4
Pairwise GCDs Set	[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]	[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]	[20, 40, 20, 80, 20, 20, 60, 20, 40, 20]	[4, 4, 4, 4, 4, 4, 4, 4, 4, 4]
Pairwise LCMs Range	[30,227, 47,897]	[25,688, 50,055]	[480, 2640]	[6536, 12,508]
LCM Range Constraint	Valid	Valid	Invalid (Fails min range)	Valid
Clear Region Ratio (2/4)	0.9935	0.9968	N/A	0.9961
False Alarm Amp. Factor	385,390	371044	N/A (Discarded)	93,160 (Optimal)
Eq. Gain (Gaussian)	Baseline (0 dB)	+0 dB	N/A (Discarded)	+0.4 dB
Eq. Gain (K-Clutter, $v=0.1$)	Baseline (0 dB)	+0 dB	N/A (Discarded)	+1.0 dB

** Note: Normalized integers are derived using a basic time resolution unit of $t_0=0.2$ μs (defined by the 5 MHz sampling bandwidth) to enable rigorous Diophantine algebraic analysis.

. Alongside the tabular data, the solution space utilization rate of the 5-PRF groups and their resulting equivalent threshold gains under varying clutter environments are visually demonstrated in Figure 14 and Figure 15, respectively. Subsequently, for the 3/4 logic evaluation, the original optimized waveforms designed for the 2/4 criterion are directly applied.

Mapping the solution space from the 2D pairwise combinations (2/4 logic) to the 3D joint triplets (3/4 logic) reveals a deeper structural advantage of the proposed framework. Physically, the total volume of system-level false alarms is determined by the product of the joint false alarm probability and the number of topological intersection nodes (ghost nodes). Introducing a third waveform dimension in the 3/4 criterion enforces stricter intersection constraints, lowering the joint probability from $p^2$ to $p^3$, which inherently reduces the absolute spatial false alarm rate for any strategy. However, the true disparity lies in the topological grid density. When evaluating waveform strategies under this identical stricter criterion, the tri-variate Least Common Multiple (LCM) of the traditional coprime baseline experiences a severe geometric expansion ($O\left(g^3\right)$) due to a complete lack of shared algebraic factors. Conversely, the Bezout-GA optimized set rigorously preserves a robust global Greatest Common Divisor (GCD), constraining its 3D LCM expansion to $O\left(g\right)$. Consequently, as validated by Figure 16, while the $p^3$ logic lowers the overall probability uniformly, the relative disparity in ghost node density between the two strategies dramatically amplifies from a factor of $g\approx 4.1$ (under the 2/4 criterion) to $g^2\approx 16.4$ (under the 3/4 criterion). This mathematically demonstrates that the Bézout topological grid provides a highly scalable physical defense mechanism in higher-dimensional detection logics.

6. Conclusions

This study proposes Bezout-GA, a two-stage hybrid optimization framework designed to mitigate the structural proliferation of spatial false alarms in airborne High-PRF (HPRF) radars operating in complex sea clutter. Diverging from the traditional pairwise coprime paradigm, this research utilizes Bézout’s identity to establish an analytical mapping between the waveform’s Greatest Common Divisor (GCD) topology and system-level false alarm boundaries. The framework integrates the global search capabilities of genetic algorithms with localized number-theoretic refinement, facilitating the extraction of Pareto-optimal Pulse Repetition Frequency (PRF) sequences under explicit hardware constraints.

Monte Carlo simulations in heavy-tailed K-distributed sea clutter validate the efficacy of the proposed framework. In a typical X-band detection scenario, the results demonstrate that by strategically conceding a negligible fraction of the geometric clear-region ratio, the optimized waveform significantly reduces the spatial false alarm amplification volume. This structural suppression mitigates the severe threshold penalties inherent in traditional coprime designs, yielding a consistent equivalent Signal-to-Clutter-and-Noise Ratio (SCNR) gain under stringent system false alarm requirements. Furthermore, the framework exhibits robust spectral generalizability and topological scalability. Extended evaluations confirm that these detection benefits are stably maintained across different spectral bands, such as the L-band, as well as under more complex higher-order $M/N$ decision criteria.

Future research will expand this architecture toward the joint optimization of fast- and slow-time domain parameters, such as varying sampling rates and PRIs, to achieve broader Pareto optimality in advanced radar regimes. In summary, this study provides a mathematically grounded methodology at the waveform level, offering theoretical guidance for the development of robust Multi-PRF radar systems.