TELK OMNIKA T elecommunication, Computing, Electr onics and Contr ol V ol. 24, No. 5, October 2026, pp. 1699 ∼ 1709 ISSN: 1693-6930, DOI: 10.12928/TELK OMNIKA.v24i5.27784 ❒ 1699 Rob ustness analysis of distrib uted CF AR detection with K -out-of- L fusion in heter ogeneous en vir onments Pham V an Hung, Nguy en Dinh T inh Department of Radar , F aculty of Radio-Electronic Engineering, Le Quy Don T echnical Uni v ersity , Hanoi, V ietnam Article Inf o Article history: Recei v ed Jan 7, 2026 Re vised Apr 27, 2026 Accepted May 25, 2026 K eyw ords: Constant f alse alarm rate processors Distrib uted detection Non-homogeneous en vironments Order statistics constant f alse alarm rate Poisson binomial distrib ution ABSTRA CT T o operate ef fecti v ely in comple x en vironments, distrib uted radar system s re- quire a rob ust constant f alse alarm rate (CF AR) processor . Con v entional distrib uted cell-a v eraging (D-CA) CF AR detectors are optimal in homoge- neous noise. Ho we v e r , the y suf fer se v ere performance de gradation in non- homogeneous scenarios, specically under multiple interfering tar gets and abrupt clutter edges. This study presents a comprehensi v e performance analysis of distrib uted order statistics (D-OS) CF AR systems emplo ying the K -out-of- L fusion rule. W e deri v e the e xact global detection and f alse alarm probabilities using the Poisson binomial distrib ution. These models are implemented via v ari- able precision arithmetic to ensure numerical stability under e xtreme masking conditions. Detection thresholds are rigorously calibrated to maintain a x ed global f alse alarm rate of 10 − 4 in homogeneous backgrounds. Qua ntitati v e re- sults demonstra te that D-OS-CF AR signicantly outperforms the D-CA-CF AR baseline. It maintains stable detection performance ( P D > 0 . 9 ) under clutter po wer v ariation and suppresses f a lse alarm sur ges at clutter edges to approxi- mately 4 . 5 × 10 − 4 . The analysis concludes tha t lo wer - K rules (e.g., K = 1 , 2 ) of fer the optimal trade-of f, limiting the homogeneous CF AR loss to belo w 0.5 dB while ensuring stable operation in highly heterogeneous en vironments. This is an open access article under the CC BY -SA license . Corresponding A uthor: Nguyen Dinh T inh Department of Radar , F aculty of Radio-Electronic Engineering, Le Quy Don T echnical Uni v ersity 236 Hoang Quoc V iet, Hanoi, V ietnam Email: tinhnd k31@lqdtu.edu.vn 1. INTR ODUCTION The capability to automatically detect tar gets in non-stationary noise and clutter en vironments is a fundamental requirement for modern surv eillance systems, ranging from radar netw orks to dis trib uted wireless sensor netw orks (WSN) and internet of things (IoT) applications [1]-[5]. In these decentralized architectures, maintaining a predictable performance le v el is critical. Constant f alse alarm rate (CF AR) processors [6], [7] are widely emplo yed to adapti v ely set the detection threshold based on local estimates of the background interference. The con v entional cell-a v eraging CF AR (CA-CF AR) detector , rst proposed by Finn and Johnson [8], is the optimum processor under the assumption of a homogeneous background with e xponential distrib ution. Ho we v er , in realistic operational scenarios, the background en vironment is rarely homogeneous. The per - formance of CA-CF AR se v erely de grades in non-homogeneous en vironments, specically in the presence of multiple interfering tar gets or abrupt clutter edges [9], [10]. In multiple-tar get situations, the a v eraging logic J ournal homepage: http://telk omnika.uad.ac.id Evaluation Warning : The document was created with Spire.PDF for Python.
1700 ❒ ISSN: 1693-6930 causes the threshold to rise unnecessarily , leading to detection losses that can e xceed 10–20 dB (the “masking ef fect”), causing the primary tar get to be missed. Con v ersely , at clutter edges, the inclusion of high-po wer clutter samples in the estim ation windo w can cause the f alse alarm probability to sur ge uncontrollably , often increasing by orders of magnitude (e.g., from 10 − 4 to 10 − 1 ), thereby saturating the fusion center (FC) with f alse tracks [6], [7]. T o mitig ate these performance losses, rob ust CF AR techniques based on order statistics (OS) ha v e been de v eloped [11]. By selecting the k -th rank-ordered sample as the noise est imate, the OS CF AR can ef fecti v ely reject up to N − k interfering pulses, e xhibiting superior rob ustness compared to mean-le v el detectors [12]-[14]. While v arious comple x local v ariants ha v e recently emer ged to enhance single-sensor adaptability [15]-[22], the challenge shifts to maintaining this rob ustness across decentralized netw orks. P arallel to the adv ancements in local processing, distrib uted detection has g arnered signicant atten- tion. In a decentralized architecture with decision fusion, local sensors transmit compressed decisions (typi- cally 1-bit) to a FC, which combines them according to a specic rule [23]. Barka t and V arshne y [24] analyzed the performance of distrib uted CA-CF AR (D-CA-CF AR) using “ AND” and “OR” rules. Ho we v er , similar to the single-sensor case, D-CA-CF AR remains vulnerable to non-homogeneous clutter . Recognizing this limi- tation, Uner and V arshne y [25] e xtended the analysis to distrib uted OS-CF AR (D-OS-CF AR), demonstrating its superiority in multi-tar get scenarios. Subsequent studies ha v e focused on optimizing parameters [26]-[28], considering statistical dependence [29], [30], or modeling non-Gaussian clutter [31]-[33]. Despite this solid foundation, a critical research g ap remains: the e xisting literature predominantly restricts analytical models to simplied congurations (e.g., tw o-sensor systems or simple “ AND/OR” logic) and often o v erlooks numerical instability in e xtreme interference. T o bridge this g ap, the no v elty of this study lies in analyzing the e xact rob ustness of the generalized K -out-of- L fusion rule under non-homogeneous clutter conditions and precisely quantifying its performance de gradation. This study is among the rst to deri v e e xact closed-form global detection and f alse alar m probabilities for D-OS-CF AR under the generalized K -out-of- L fusion rule. Accordingly , the major contrib utions of this w ork are as follo ws: (i) we formulate e xact closed-form e xpressions for the global probability of detection ( P D ) and f alse alarm ( P F ) in non-homogeneous en viron- ments using the Poisson binomial distrib ution [34], generalizing the analysis for an y netw ork s ize L and v oting rule K ; (ii) we implement a v ariable precision arithmetic (VP A) approach [35] to eliminate catastrophic can- celation errors [36], establishing a precise numerical benchmark for e xtreme masking scenarios often missed by standard approximations; and (iii) we pro vide a comprehensi v e quantitati v e analysis, assessing the system’ s immunity to high-po wer interfering tar gets and its capability to suppress f alse alarm sur ges at clutter edges. These ndings pro vide critical design guidelines for implementing reliable distrib uted detection frame w orks in heterogeneous en vironments, which are applicable to both radar systems and cogniti v e sensor netw orks. The rest of the outline for the paper is belo w . Section 2 describes the proposed system model. Section 3 presents the methodology of performance analysis, including closed-form e xpressions. Section 4 presents the numerical results. Finally , section 5 concludes the paper with k e y ndings, limitations, and future w ork. 2. SYSTEM MODEL Consider a decentralized detection system consisting of L distrib uted radar sensors and a FC, as sho wn in Figure 1. Each radar sens or recei v es echoes from the tar get–en vironment scene and performs a processing chain comprising the radio frequenc y (RF) / antenna front-end, local signal processing, and a CF AR detector that produces a local decision u i ( i = 1 , . . . , L ). These local decisions are transmitted to a FC, where the global decision U is obtained by applying logical fusion rules (AND, OR, K -out-of- L ). This distrib uted structure e x- ploits the spatial di v ersity of multiple sensors to enhance the probability of detection at a gi v en f alse-alarm rate while preserving the o v erall system’ s modularity and scalability . W e assume a symmetric system conguration in which all local detectors share identical parameters (reference windo w size N , order number k , and threshold multiplier T ). 2.1. Local CF AR detector model Figure 2 depicts the CF AR detection mechanism emplo yed at each sensor . A sliding CF AR windo w e xtracts the reference cells around the cell under test (CUT), and the CF AR logic (e.g., CA or OS) forms the clutter es timate. The output samples in the reference windo w , denoted by X 1 , X 2 , . . . , X N , and the sample in CUT , denoted by Y i , follo w an e xponential distrib ution with the follo wing probability density function: TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 5, October 2026: 1699–1709 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 1701 f ( x ) = 1 λ exp − x λ , x ≥ 0 (1) where λ represents the sample’ s a v erage po wer . Figure 1. The distrib uted detection system Figure 2. A local CF AR processing The follo wing binary h ypothesis testing problem is considered: - Hypothesis H 0 : the CUT contains only thermal noise (or a clutter background). The a v erage po wer is λ = µ . - Hypothesis H 1 : the CUT contains a tar get plus noise. The a v erage po wer is λ = µ (1 + S ) , where S is the a v erage signal-to-noise ratio (SNR) of the primary tar get. W e assume a slo wly uctuating Swerling I tar get model. Each local sensor i ( i = 1 , . . . , L ) estimates the noise po wer Z i from the reference windo w using a CF AR processor . The local decision u i is made as (2): u i = ( 1 , if Y i > T Z i 0 , otherwise (2) where T is the scaling f actor used to set the desired f alse alarm rate. In this study , we analyze tw o CF AR schemes: - CA-CF AR: the noise estimate is the sum of the reference samples Z = P N j =1 X j . - OS-CF AR: the reference samples are rank ed X (1) ≤ X (2) ≤ · · · ≤ X ( N ) , and the k -th sample is selected as the estimate, Z = X ( k ) . 2.2. Fusion rule Local binary decisions u i are transmitted o v er error -free channels to the FC. The FC emplo ys the generalized “ K -out-of- L ” v oting rule to mak e the global decision U 0 . Specically , the global decision declares tar get presence if at least K out of L local detectors report detection: U 0 = ( 1 , if P L i =1 u i ≥ K 0 , otherwise (3) where K is an inte ger ( 1 ≤ K ≤ L ). Note that the logical “OR” rule corresponds to K = 1 , and the “ AND” rule corresponds to K = L . 2.3. Non-homogeneous en vir onment models T o e v aluate the system’ s rob ustness, we dene tw o distinct non-homogeneous scenarios af fecting the local sensors. Throughout this analysis, the background interference is assumed to follo w an e xponential distrib ution ( corresponding to a Rayleigh en v elope), and the heterogeneity is modeled as an abrupt v ariation in the a v erage po wer le v el: Rob ustness analysis of distrib uted CF AR detection with K -out-of- L fusion in ... (Pham V an Hung) Evaluation Warning : The document was created with Spire.PDF for Python.
1702 ❒ ISSN: 1693-6930 - Multiple interfering tar gets: in this scenario, m cells within the reference windo w of a “interfered” sensor contain high-po wer interfering tar gets with a v erage po wer µ (1 + I ) , where I is the interference-to-noise ratio (INR). The remaining N − m cells contain thermal noise µ . - Clutter edge: this scenario models the radar beam’ s transition across a background po wer discontinuity . The reference windo w contains q cells from a clutter re gion with po wer µ (1 + C ) , where C is the clutter -to-noise ratio (CNR), and N − q cells from the clear re gion with po wer µ . The CUT may reside in either the clear re gion (CUT in noise) or the clutter re gion (CUT in clutter). 3. PERFORMANCE AN AL YSIS In this section, the e xact e xpressions for the global probabili ty of f alse alarm ( P F ) and detection ( P D ) are deri v ed. In this study , we use uppercase and lo wercase subscripts to dif ferentiate between global and local probabilities, respecti v ely . 3.1. Local detection pr obabilities The probability that a local sensor i declares a detection, denoted as P i = P r ( u i = 1) , depends on the specic CF AR algorithm and the en vironment. 3.1.1. Homogeneous backgr ound When the en vironment is homogeneous ( m = 0 or q = 0 ), the local f alse alarm ( P f ) and detection ( P d ) probabilities for OS-CF AR are gi v en by [11]: P f = k − 1 Y j =0 N − j N − j + T , P d = k − 1 Y j =0 N − j N − j + T 1+ S (4) F or CA-CF AR, the e xpressions are [8]: P f = (1 + T ) − N , P d = 1 + T 1 + S − N (5) 3.1.2. Non-homogeneous backgr ound The reference windo w samples ar e no longer identically distrib uted for the non-homogeneous scenar - ios dened in section 2.3. W e adopt the general analytical form deri v ed by Gandhi and Kassam [10]. F or an OS-CF AR detector where the reference windo w contains tw o groups of samples (group 1 with size N 1 , po wer µ 1 and group 2 with size N 2 , po wer µ 2 ), the probability that the CUT (with po wer µ Y ) e xceeds the threshold T X ( k ) is as (6): P ( Y > T X ( k ) ) = N X i = k min( i,N 1 ) X j =max(0 ,i − N 2 ) N 1 j  N 2 i − j × A ( i, j ) (6) where the term A ( i, j ) is dened as (7): A ( i, j ) = j X a =0 i − j X b =0 j a  i − j b ( − 1) a + b 1 + µ Y T a + N 1 − j µ 1 + b + N 2 − ( i − j ) µ 2  − 1 (7) The term A ( i, j ) encapsulates the comple x summation of probability densities resulting from the OS operation on non-identically distrib uted samples. This accounts for all possible permutations where j interfering samples and i − j noise samples f all belo w the rank k , ensuring the e xact e v aluation of the local detection probability without approximation. This general formula is applied as: - Interfering tar gets: N 1 = N − m, µ 1 = µ ; N 2 = m, µ 2 = µ (1 + I ) - Clutter edge: N 1 = N − q , µ 1 = µ ; N 2 = q , µ 2 = µ (1 + C ) F or CA-CF AR in non-homogeneous backgrounds, we use the moment generating function approach as detailed in [25]. TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 5, October 2026: 1699–1709 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 1703 3.1.3. Numerical stability In strong masking scenarios (e.g., I ≫ 1 or C ≫ 1 ), the local probabilities can become e xtremely small (e.g., < 10 − 15 ). Standard double-precision arithmetic often suf fers fr o m catastrophic cancelation in the alternating sums of A ( i, j ) , leading to erroneous ne g ati v e probabilities [36]. T o address this is sue, we imple- mented VP A usi ng the MA TLAB Symbolic Math T oolbox [35] to ensure e xact e v aluation. This information is particularly useful and reliable for the observ ations in section 4.4 of this paper . It is important to ackno wledge that the implementation of VP A incurs a signicantly higher computa- tional cost compared to standard double-precision arithmetic. In this study , VP A is strictly utilized as an of ine numerical benchmark to ensure the rigorous v alidation of the theoretical deri v ations at these e xtreme tails. From a practical implementation perspecti v e, real-time distrib uted radar systems w ould not perform VP A cal- culations online. Instead, the e xact e xpressions deri v ed here should be used to generate accurate pre-computed look-up tables or threshold maps. This approach combines the precision of the e xact analytical model with the lo w-latenc y requirements of the operational hardw are. 3.2. Global perf ormance using poisson binomial distrib ution T o e v aluate the o v erall system performance, it is necessary to establish ho w local detection probabili- ties mathematically propag ate to the global decision le v el. Consider a netw ork where p sensors are located in a non-homogeneous en vironment (e.g., af fected by interference) and L − p sensors remain in a homogeneous background. Local detection probabilities are no longer identical across the netw ork. In such cases, the total number of loca l detections follo ws the Poisson Binomial distrib ution, which models the number of successes in independent b ut non-identically distrib uted Bernoulli trials [36]. Let P inter and P cl ean denote the detection probabilities of the interfered and unaf fected sensors, respecti v ely . Since the sensors within each respecti v e group share identical local probabilities, the discrete random v ariables j and i follo w independent Binomial distrib utions: - F or the interfered group: P ( j ) = p j P j inter (1 − P inter ) p − j , where 0 ≤ j ≤ p . - F or the clean group: P ( i ) = L − p i P i cl ean (1 − P cl ean ) L − p − i , where 0 ≤ i ≤ L − p . Because the tw o groups operate independently , the joint probability of obtaining e xactly j interfered detections and i clean detections is simply the product of their indi vidual probabilities. At the FC, the gener alized K -out-of- L rule dictates that a global detection is declared if the total number o f local detections ( i + j ) is greater than or equal to K . T o ele g antly capture all v alid combinatorial pairs of ( i, j ) that satisfy this condition without utilizing comple x summation boundaries, we introduce the indicator function I ( i + j ≥ K ) , which equals 1 if the condition is met and 0 otherwise. Consequently , the global probability ( P F or P D for either H 0 or H 1 ) can be deri v ed by summing the joint probabilities o v er all possible v alues of i and j , ltered by the indicator function. This yields the follo wing closed-form e xpression: P F ,D = L − p X i =0 p X j =0 I ( i + j ≥ K ) L − p i P i cl ean (1 − P cl ean ) L − p − i p j P j inter (1 − P inter ) p − j (8) This formulation allo ws for an e xact system performance analysis under an y arbitrary mix of sensor conditions. 4. NUMERICAL RESUL TS AND DISCUSSION In this section, we present the numerical e v aluation of the proposed symmetric D-OS-CF AR s ystem and compare its performance with that of the con v entional D-CA-CF AR. 4.1. Simulation setup T o ensure reproducibility , the simulation parameters are dened as: - Netw ork size: L = 4 sensors. - CF AR windo w: size N = 24 , guard cells G = 0 . - Order statistic: rank k = 20 (pro viding rejection capacity N − k = 4 , (approximately 4 N / 5 )). - Design probability: global P F = 10 − 4 in homogeneous background. T o ensure a f air comparison, the threshold parameters ( T ) for both systems are rigorously calibrated in a homogeneous en vironment ( p = 0 ) to achie v e a unied global f alse alarm probability for e v ery specic fusion rule K ( K = 1 , . . . , L ). Rob ustness analysis of distrib uted CF AR detection with K -out-of- L fusion in ... (Pham V an Hung) Evaluation Warning : The document was created with Spire.PDF for Python.
1704 ❒ ISSN: 1693-6930 Note that while a netw ork size of L = 4 is selected for simulation to clearly visualize the performance dif ferences between specic v oting rules (e.g., AND, OR, and majority), the proposed analytical frame w ork is scalable and applicable to an y netw ork size L . The computational comple xity of the e xact deri v ation scales linearly with L , making it feasible for lar ger sensor netw orks. Furthermore, the performance curv es presented in the subsequent s ections are generated using the e xact analytical e xpressions deri v ed in section 3. As these are theoretical v alues calculated via VP A, the y are free from statistical v ariance. 4.2. Baseline perf ormance in homogeneous backgr ound Figures 3(a) and 3(b) illustrate the global probability of detection ( P D ) v ersus SNR in a homogeneous en vironment for D-CA-CF AR (Figure 3(a)) and D-OS-CF AR (Figure 3(b)). Both systems e xhibit the standard S-curv e beha vior , with lo wer - K rules (e.g., OR rule) yielding higher detection probabilities at the cost of lo wer thresholds. Furthermore, the 2 out of 4 rule presents the best performance at lo wer SNR v alues. As observ ed in Figure 3, the D-OS-CF AR system requires a slightly higher SNR to achie v e the same detection probabili ty as the D-CA-CF AR sys tem. This performance de gradation represents the inherent CF AR loss e xchanged for the rob ustness g ained in non-homogeneous en vironments. A quantitati v e e v aluation of the required SNR re v eals that the CF AR loss increases mar ginally at higher detection requirements and v aries depending on the specic fusion rule. F or instance, to achie v e a standard detection probability of P D = 0 . 9 , the D-OS-CF AR system imposes an acceptable CF AR loss ranging from approximately 0.1 dB (for stricter rules lik e K = 3 and K = 4 ) to 0.4 dB (for the OR rule, K = 1 ). Ev en at a more stringent requirement of P D = 0 . 95 , the maximum observ ed loss remains bounded at around 0.5 dB for the K = 1 rule. This beha vior is consistent with the theoretical e xpectation that the sample mean in CA-CF AR is the maximum lik elihood estimator for e xponential noise. Ho we v er , as demonstrated in the subsequent sections, this minor homogeneous loss is a ne gligible price to pay for the signicant immunity achie v ed ag ainst masking ef fects and f alse alarm sur ges in heterogeneous scenarios. 5 10 15 20 25 SNR (dB) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P D K=1 (OR) K=2 K=3 K=4 (AND) (a) 5 10 15 20 25 SNR (dB) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P D K=1 (OR) K=2 K=3 K=4 (AND) (b) Figure 3. The P D v ersus SNR in a homogeneous en vironment: (a) CA-CF AR system and (b) OS-CF AR system 4.3. Rob ustness against multiple interfering tar gets 4.3.1. Impact of the number of interfer ed sensors ( p ) W e e v aluate the system’ s resilience when m = 4 interfering tar gets interfere with p sensors. Fig- ures 4(a) and 4(b) depicts P D as a function of p (with SNR = INR = 15 dB). D-CA-CF AR (Figure 4(a)): the performance suf fers catastrophic de gradation. As p increases, the thresholds at interfered sensors drastically increase due to the a v eraging of high-po wer interference, ef fecti v ely blinding those sensors (“masking ef fect”). Ev en with the rob ust “OR” rule ( K = 1 ), when all sensors are af fected, P D drops belo w 0.5. D-OS-CF AR (Figure 4(b)): in stark contrast, re g ardless of p , the system maintains stable high performance ( P D > 0 . 9 ). This stability is ph ysically e xplained by the sorting mechanism of OS-CF AR. Because the number of interfering tar - gets ( m = 4 ) is within the rejection capacity ( N − k = 4 ), the k -th ordered sample remains representati v e of the noise oor . Consequently , the interference po wer does not inated the local threshold, allo wing the clean sensors to successfully dri v e the global K -out-of- L decision. TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 5, October 2026: 1699–1709 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 1705 These results of fer critical guidelines for operat ional radar netw orks. The analysis demonstrates that lo wer - K rules (e.g., K = 1 , 2 ) act as a “survi v ability mode” in hostile en vironments. These rules require fe wer sensors to declare a detection, allo wing the global decision to be dri v en by the subset of sensors that remain unaf fected by interference, ef fecti v ely isolating the compromised nodes. This trend is fundamental to v oting logic and generalizes to lar ger netw ork sizes ( L > 4 ), suggesting that distrib uted systems should prioritize “OR” or “majority” logic o v er “ AND” logic when interference is detected. 0 0.5 1 1.5 2 2.5 3 3.5 4 The number of interfered sensors (p) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P D K=1 (OR) K=2 K=3 K=4 (AND) (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 The number of interfered sensors (p) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P D K=1 (OR) K=2 K=3 K=4 (AND) (b) Figure 4. The P D v ersus the number of interfered sensors: (a) CA-CF AR system and (b) OS-CF AR system 4.3.2. Impact of interfer ence po wer (INR) Figures 5(a) and 5(b) in v estig ates the sensiti vity of the system to the interference po wer le v el, as- suming p = 2 interfered sensors. D-CA-CF AR (Figure 5(a)): performance declines monotonically as INR increases. Stronger interference leads to deeper masking. D-OS-CF AR (Figure 5(b)): the performance curv es are remarkably at horizontal lines. This conrms the “interference immunity” property of the OS-CF AR. As long as the number of interferers is bounded, their po wer magnitude is irrele v ant to the sorting-based estimation logic. Furthermore, the analysis conrms that lo wer majority rules of fer superior rob ustness. This is because in a distrib uted netw ork, a lo wer K requires fe wer sensors to declare a detection. In a heterogeneous en viron- ment where some sensors are blinded by interference, a lo w- K rule ef fecti v ely “ignores” the non-functioning sensors, relying on the remaining operational sensors to maintain global detection. 0 5 10 15 20 25 30 INR (dB) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P D K=1 (OR) K=2 K=3 K=4 (AND) (a) 0 5 10 15 20 25 30 INR (dB) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P D K=1 (OR) K=2 K=3 K=4 (AND) (b) Figure 5. The P D v ersus INR of interfering tar gets: (a) CA-CF AR system and (b) OS-CF AR system Rob ustness analysis of distrib uted CF AR detection with K -out-of- L fusion in ... (Pham V an Hung) Evaluation Warning : The document was created with Spire.PDF for Python.
1706 ❒ ISSN: 1693-6930 4.4. F alse alarm contr ol at clutter edges This section analyses the system’ s beha vior during a transition from a clear re gion to a clutter re gion (CNR = 15 dB), assuming p = 1 sensor crosses the edge while L − p = 3 sensors remain in the clear re gion. 4.4.1. The “Sur ge” phenomenon Figure 6 sho ws the global f alse alarm rate ( P F ). Noise oor ( q ≤ N / 2 ): when the CUT is in noise b ut clutter enters the windo w , both systems reduce P F belo w the tar get 10 − 4 (masking). Interestingly , P F does not drop to zero b ut con v er ges to a “system noise oor” ( ≈ 5 × 10 − 5 for K = 2 ). This beha vior is in contrast to con v entional monostatic radars, where the f alse alarm probability decreases signicantly as the clutter–edge position approaches ( N / 2 ) within the reference windo w . This property w as v eried by performing a much more accurate numerical analysis based on VP A, as described in section 3. This oor is m aintained by the L − p clean sensors, demonstrating the distrib uted architecture’ s i nherent f ault tolerance. F alse alarm sur ge ( q > N / 2 ): as sho wn in Figure 6(a), when the clutter edge crosses the CUT ( q > N / 2 ), the D-CA-CF AR e xperiences a se v ere f alse alarm sur ge, peaking at 6 . 2 × 10 − 4 . In contrast, D-OS-CF AR ( K = 2 ) ef fecti v ely suppresses this sur ge, limiting the peak to approximately 4 . 5 × 10 − 4 , which corresponds to a reduction of roughly 1.4 dB in equi v alent po wer thresholding terms. 4.4.2. Sensiti vity to clutter po wer (CNR) Figure 6(b) sho ws the magnitude of the w orst-case sur ge peak as a function of CNR. D-CA-CF AR: the peak height increases continuously with CNR, posing a se v ere risk of saturating the FC with f alse tracks in strong clutter en vironments. D-OS-CF AR: the sur ge peak e xhibited a sat u r ation beha vior . This suggests that D-OS-CF AR pro vides a bounded “safety limit” for f alse alarms, making it a safer choice for dynamic en vironments with v ariable clutter po wer . 0 5 10 15 20 25 The position of edge clutter (q) 10 -4 10 -3 10 -2 P F D-OS-CFAR (K=2) D-CA-CFAR (K=2) The desired Pfa (a) 5 10 15 20 25 CNR (dB) 10 -4 10 -3 The peak of P F D-OS-CFAR (K=2) D-CA-CFAR (K=2) The desired Pfa (b) Figure 6. The global f alse alarm rate ( P F ) v ersus: (a) clutter edge position q and (b) CNR 5. CONCLUSION This study established a rigorous performance benchmark for symmetric D-OS-CF AR systems based on the e xact poisson binomial distrib ution and the VP A. The quantitati v e results demonstrate that the K -out- of- L scheme demonstrates impro v ed rob ustness when K is optimally selected under moderate heterogeneity . While incurring a minor SNR loss of approximately 0.5 dB in homogeneous backgrounds, the D-OS-CF AR system maintains stable detection performance ag ainst masking ef fects and safely caps the f alse alarm sur ge at clutter edges to a manageable le v el. Furthermore, the analysis conrms that emplo ying lo wer majority rules ( K ≤ L/ 2 ) is a critical survi v ability strate gy . This highlights the practical im po r tance of adapti v e fusion threshold selection for maintaining stable operations in distrib uted radar and cogniti v e sensor netw orks. This analysis focused on CA and OS detectors, error -free channels, and identical sensors to ensure an e xact closed form. Consequently , future research will e xtend this frame w ork to other rob ust detectors, non-Gaussian clutter models, and dependent sensors while addressing practical constraints such as channel f ading. Future w ork TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 5, October 2026: 1699–1709 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 1707 will focus on hardw are v alidation with real radar data, adapti v e K selection, and the inte gration of machine- learning-based threshold tuning to dynamically optimize performance in cogniti v e netw orks. A CKNO WLEDGMENTS The authors w ould lik e to thank the researchers and colleagues at the Department of Radar , Le Quy Don T echnical Uni v ersity , for their v aluable technical support and discussions. This research did not recei v e an y specic research grant or contract from funding agencies in the public, commercial, or not-for -prot sectors. FUNDING INFORMA TION Authors state no funding in v olv ed. A UTHOR CONTRIB UTIONS ST A TEMENT This journal uses the Contrib utor Roles T axonomy (CRediT) to recognize indi vidual author contrib u- tions, reduce authorship disputes, and f acilitate collaboration. Name of A uthor C M So V a F o I R D O E V i Su P Fu Pham V an Hung ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Nguyen Dinh T inh ✓ ✓ ✓ ✓ ✓ ✓ ✓ C : C onceptualization I : I n v estig ation V i : V i sualization M : M ethodology R : R esources Su : Su pervision So : So ftw are D : D ata Curation P : P roject Administration V a : V a lidation O : Writing - O riginal Draft Fu : Fu nding Acquisition F o : F o rmal Analysis E : Writing - Re vie w & E diting CONFLICT OF INTEREST ST A TEMENT Authors state no conict of interest. INFORMED CONSENT W e ha v e obtained informed consent from all indi viduals included in this study . D A T A A V AILABILITY Data a v ailability is not applicable to this paper as no ne w data were created or analyzed in this study . REFERENCES [1] J. C. Garc ´ ıa, L. A. Ri v era, and J. Perez, “ A Literature Re vie w on Outlier Detection in W ireless Sensor Netw orks, ” J ournal of Advances in Information T ec hnolo gy , v ol. 15, no. 3, pp. 372–388, 2024, doi: 10.12720/jait.15.3.372-388. [2] X. Zhou, X. Xu, W . Liang, Z. Zeng, and Z. Y an, “Deep-Learning-Enhance d Multitar get Detection for End-Edge-Cloud Surv eillance in Smart IoT , ” IEEE Internet of Things J ournal , v ol. 8, no. 16, pp. 12588–12596, 2021, doi: 10.1109/JIO T .2021.3077449. [3] L. Y an, S. Han, C. Hao, D. Orlando, and G. Ricci, “Inno v ati v e Cogniti v e Approaches for Joint Radar Clutter Classication and Multiple T ar get Detection in Heterogeneous En vironments, ” IEEE T r ansactions on Signal Pr ocessi ng , v ol. 71, pp. 1010–1022, 2023, doi: 10.1109/TSP .2023.3250084. [4] B. K. Chalise, D. M. W ong, M. G. Amin, A. F . Martone, and B. H. Kirk, “Detection, Mode Selection, and P arameter Estimation in Distrib uted Radar Netw orks: Algorithms and Implementation Challenges, ” IEEE Aer ospace and Electr onic Systems Ma gazine , v ol. 37, no. 11, pp. 4–22, 2022, doi: 10.1109/MAES.2022.3205568. [5] W . Jiang, Y . Ren, Y . Liu, and J. Leng, “ Articial Neural Netw orks and Deep Learning T echniques Applied to R adar T ar get Detection: A Re vie w , ” Electr onics , v ol. 11, no. 1, p. 156, 2022, doi: 10.3390/electronics11010156. [6] M. A. Ri chards, Fundamentals of Radar Signal Processing, 3rd ed., v ol. 58, no. 12. McGra w-Hill Education, 2014. [7] M. A. Richards, J. A. Scheer , and W . A. Holm, Principles of modern radar: Basic principles, 2nd ed., v ol. 1. SciT ech Publishing / The Institution of Engineering and T echnology , 2010, doi: 10.1049/sbra021e. [8] H. M. Finn and R. S. Johnson, “ Adapti v e detection mode with threshold control as a function of spatially sampled clutter le v el estimates, ” RCA Re vie w , v ol. 29, no. 3, pp. 414–464, 1968. Rob ustness analysis of distrib uted CF AR detection with K -out-of- L fusion in ... (Pham V an Hung) Evaluation Warning : The document was created with Spire.PDF for Python.
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