Weighted order statistic and fuzzy rules CFAR ...

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Signal Processing 88 (2008) 558–570 www.elsevier.com/locate/sigpro

Weighted order statistic and fuzzy rules CFAR detector for Weibull clutter Amir Zaimbashia,1, Mohammad Reza Tabana,, Mohammad Mehdi Nayebib,2, Yaser Norouzib,3 a

Department of Electrical Engineering, Yazd University, Daneshgah Blvd., P.O. Box 89195-741, Yazd, Iran Deparment of Electrical Engineering, Sharif University of Technology, Azadi Ave., P.O. Box 11365, Tehran, Iran

b

Received 12 December 2006; received in revised form 8 July 2007; accepted 31 August 2007 Available online 8 September 2007

Abstract Order statistic (OS) CFAR processor is a powerful detector, but similar to many other CFAR detectors, suffers from clutter edge as well as interfering targets. To overcome these problems, two derivations of this detector have been developed. These are order statistic greatest of (OSGO) and order statistic smallest of (OSSO) CFAR processors. The cost, paid for this improvement, is a considerable decrease in detection probability for homogenous Weibull clutter. Furthermore, a substantial performance degradation of OSGO in interfering target (high target masking effect) and an excessive false alarm of OSSO in clutter edge, also, occur. This behavior is a result of non-soft rules used in these two detectors. This paper proposes a weighted order statistic and fuzzy rules-CFAR detector, which employs some soft rules based on fuzzy logic to cure the mentioned problems. Our simulation results demonstrate that considering realistic Weibull clutter, the proposed detector gives better detection probability than that of OSGO and OSSO detectors. Moreover, considering the problems of clutter edge and interfering target, the performance of the method is comparable to the best performance of the OSGO and OSSO detectors in different conditions. r 2007 Elsevier B.V. All rights reserved. Keywords: Radar detection; CFAR; Order statistic; Weibull clutter; Fuzzy rules

Abbreviations: CFAR, Constant false alarm rate; CUT, Cell under test; OS, Order statistic; ML, Maximum likelihood; GO, Greatest of; OSGO, Order statistic greatest of; OSSO, Order statistic smallest o; WOSF, Weighted order statistic and fuzzy; SCR, Signal to clutter power ratio; pdf, probability density function; CDF, Cumulative distribution function; CNR, Clutter to noise power ratio; ICR, Interfering target to clutter power ratio; C, Shape parameter of Weibull clutter; B, Scale parameter of Weibull clutter Corresponding author. Tel.: +98 913 151 9702; fax: +98 351 821 0699. E-mail addresses: [email protected] (A. Zaimbashi), [email protected] (M.R. Taban), [email protected] (M.M. Nayebi), [email protected] (Y. Norouzi). 1 Tel.:+98 342 422 1544. 2 Tel.: +98 912 307 5180. 3 Tel.: +98 21 66165957. 0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.08.017

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1. Introduction Constant false alarm rate (CFAR) detection involves the estimation of the parameters of the local clutter and the setting of a threshold for decision so that a constant false alarm probability (Pfa) is guaranteed for all values of unknown clutter parameters. The radar backscatter of different situations can be statistically modeled using various probability density functions (pdfs) [1,2]. A lot of works has been done on the Rayleigh clutter statistics and a large variety of CFAR detectors have been proposed. Most of these detectors are based on averaging [3] or ordering techniques [4,5] or a combination of them [6] and perform quite differently in various clutter situations [7]. For years, Rayleigh pdf was a good option for clutter modeling, but now a day with existence of high-resolution radars, models with longer tail are preferable [1,2]. Among these models, the Weibull pdf has been found to be applicable in a large variety of actual radar clutter situations [1], where deviation from Rayleigh pdf is encountered. This model describes satisfactorily many cases of land and sea clutter for low grazing angles and horizontal polarization at high frequencies (X-band). The pdf of Weibull depends on two parameters. The first parameter is related to the mean power of the clutter and is named scale parameter. The second is related to the skewness of the Weibull pdf and is called shape parameter. CFAR detectors based on the Weibull clutter assumption have been suggested in past [8–11]. The adaptive threshold is effectively obtained by estimating the scale parameter and shape parameter, using maximum likelihood (ML) estimation [8] or ordered statistics (OS) [9]. The OS–CFAR has a simpler structure compared with the ML–CFAR while its performance is slightly weaker. However, as many other CFAR detectors, it suffers from the problem of clutter edge as well as interfering targets. In order to reduce this problem, methods based on a combination of greatest of (GO) and smallest of (SO) concepts with OS–CFAR have been proposed in the literature, result of which an OSGO and OSSO–CFAR detectors [5]. The OSGO is considered as a solution to the problem of clutter edge while the OSSO is used to solve the interfering targets problem. The cost paid by OSGO and OSSO detectors to solve the problem of non-homogeneous situations, is a decrease in the detection ability of these detectors in homogeneous clutter [10]. More-

559

over, these two detectors can solve the mentioned problems only somewhat. In this paper, first we adapt the OSSO–CFAR detector, developed for a Rayleigh clutter to the case of Weibull clutter and derive its false alarm probability expression. We also obtain the integral expression of the detection probability for the OS, OSGO and OSSO–CFAR detectors. Next, for improving the performance of the OSGO and OSSO detectors in various situations, we generalize their structure and introduce some new CFAR schemes. In these detectors the non-soft rules, used in the OSGO and OSSO–CFAR, are replaced with some other soft rules inferred from fuzzy logic. We have named these new schemes weighted order statistic and fuzzy (WOSF) CFAR. The performance of the WOSF–CFAR detectors are evaluated and compared with that of OSGO and OSSO– CFAR detectors via computer simulations. The results represent that the WOSF–CFAR detector outperforms the OSGO and OSSO–CFAR detectors in the homogenous and non-homogenous situations of clutter. The paper is organized as follows. Section 2 summarizes the main assumptions used through the paper. In Section 3, structure of the OS–CFAR detector is reminded and the obtained closedform formulation for false alarm probability of the OSSO–CFAR detector is expressed. We also propose the integral expressions of detection probability of the OSGO and OSSO–CFAR detectors. In Section 4, several fuzzy-based OS–CFAR detectors are proposed as good candidates to replace the OSGO and OSSO detectors. In Section 5, the performances of these detectors are compared with each other and advantages and drawbacks of each scheme are highlighted. Finally, as a conclusion, Section 6 summarizes the results of our contribution. 2. General assumptions In any detection problem, there exist one or more observation samples and two or more hypotheses. In each hypothesis, it is assumed that the observed samples belong to a certain class. For example, in the problem of radar detection, there are only two hypotheses, H0 and H1. Under H0 hypothesis, it is assumed that the observed samples of a radar cell are only interference (i.e.; clutter and noise); while under another hypothesis H1, it is assumed that the observed samples are the returns of a target plus the interference. The mean power of clutter often

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Fig. 1. Configuration of CFAR detectors in the Weibull clutter.

dominates that of noise and generally, the interference is considered equal to clutter. In a conventional CFAR detector, the returned samples from every cell, at first, are converted to a non-negative variable by an envelope or square law detector. This single sample, which is used for target detection is named cell under test (CUT). Through this paper, the samples of every radar cell are assumed single which have been passed through an envelope detector. Also, the samples of every two different cells are assumed to be independent. In many detectors, the detection is performed simply by comparing the CUT with a threshold T: H1

4 CUT T. o

(1)

H0

In many practical problems, the false alarm probability (Pfa) of detector should remain constant. If the distribution of the interference (or clutter) is known then it is possible to select the threshold of Eq. (1), so that the Pfa never exceed the maximum tolerable value. In many situations, the parameters of clutter distribution change through the time. In these situations, a detector based on the simple structure of Eq. (1) cannot give rise to a constant Pfa and special detectors, which have CFAR property should be used. In these detectors, the threshold parameter is set adaptively. To do this, in the CFAR detectors in addition to CUT, some other clutter samples belonging to the adjacent cells are also collected (reference samples). The CFAR detector estimates the unknown parameters of the distribution of clutter from reference samples and adjusts the threshold so that the maximum tolerable Pfa is not violated in all situations. As was mentioned previously, the Weibull pdf is a good option for modeling the sea and ground clutter [1]. A Weibull random variable has a two para-

metric distribution, as follows:     C  x C1 x C exp  , f x ðxÞ ¼ B B B

(2)

where B and C are scale and shape parameters of the random variable x, respectively. In the Weibull clutter when B and C are unknown, in order to hold the CFAR property, the detection is performed by comparing the CUT with a threshold T as [8,9,12] 1=C^

^ T ¼ Bg 1

(3)

in which g1 is a complicated function of desired Pfa, the number of reference samples and the method used for parameter estimation. In many situations, the shape parameter changes slowly through the time. In this case, the shape parameter can be estimated from the geometry and physical condition of the environment [13] or from a large number of reference samples [14]. In both cases, the estimation error is enormous and it is almost realistic to assume C as a known parameter. By this assumption the detection strategy will be simplified as H1

CUT B^

4 o

1=C

g1

¼ g.

(4)

H0

As shown in Fig. 1, in CFAR detector the CUT is surrounded with N reference samples (xi i ¼ 1,y,N). Different CFAR detectors use these samples in different manners to estimate the parameter B. 3. OS–CFAR detector family In OS–CFAR, first the reference samples are sorted in ascending order and then the kth ordered sample (X k : 1pkpN) is selected as the estimation of the parameter B [9]. In OSGO–CFAR detector, the reference samples of the leading (le) and lagging (la) windows are

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sorted separately, and the kth sorted sample (X lak ; X lek : 1pkpN=2) of each window is selected. Then these samples are compared and the greater one is selected as the estimation of the parameter B [10]. Therefore, this detector has the following structure:

Since both target and clutter in the CUT are Rayleigh distributed, then the pdf of the CUT will also be Rayleigh distributed as ! 2x x2 exp  2 . (8) f CUT ðxÞ ¼ 2 Bc þ B2t Bc þ B2t

  CUT CUT CUT ¼ min ; maxfX lak ; X lek g X lak X lek

As it is shown in Appendix B, the detection probability (Pd) of the OS–CFAR can be calculated by the following integration: ! ! k1 M X k1 i PdOS ¼ k ð1Þ k i i¼0 Z þ1  expðS C u2=C  ðL þ iÞuÞ du ð9Þ

H1

4 o

g.

H0

(5) In OSSO–CFAR, after finding the kth sorted samples in the both windows; the smaller one is selected as the estimation of the unknown parameter B. The closed-form expression of Pfa for the OS and OSGO–CFAR detectors has been presented in [9,10]. We have obtained such an expression for the OSSO–CFAR detector as ! M GðgC þ M þ 1  kÞGðkÞ pfa ¼ 2k GðgC þ M þ 1Þ k ! ! M M Gðk þ jÞGðgC þ 2M  k  j þ 1Þ X ,  GðgC þ 2M þ 1Þ j j¼k ð6Þ which is figured out in Appendix A. In above equation, G(.) is the Gamma function and M ¼ N/2 is the number of lagging or leading samples. Here, Pfa value is independent of B, and the CFARness is guaranteed for the OSSO algorithm. It is also necessary to calculate the detection probability (Pd) of the different OS detectors to be able to compare their performances. It is common to model a fluctuating target with a Rayleigh random variable with a mean power Bt2. As was mentioned, a suitable model for clutter is Weibull distribution. Unfortunately, using this model, it is not possible to find a closed-form expression for signal plus clutter distribution. Therefore, similar to [8], we approximate the clutter in the CUT by a Rayleigh random variable with a mean power BC2, which is same as the mean power of Weibull clutter in the reference cells. This approximation results in a decreased version of calculated Pd for low signal to clutter power ratio (SCR), but for moderate and high values of SCR the difference between real and calculated value of Pd will be minor. The clutter mean power BC2 is related to the scale and shape parameters of Weibull pdf, B and C, as   2 B2C ¼ B2 G 1 þ . (7) C

0

in which g2 , Gð1 þ 2=CÞð1 þ SCRÞ L ¼ 2M  k þ 1

Sc ¼

ð10Þ

and the SCR value is equal to SCR ¼

B2t . B Gð1 þ 2=CÞ 2

(11)

Also for the OSGO–CFAR, Pd is obtained as ! ! M M M X PdOSGO ¼ 2k j k j¼k ! kþj1 X kþj1 i  ð1Þ i i¼0 Z þ1    exp S C u2=C  ðL þ i  jÞu du. 0

ð12Þ Pd of the OSSO–CFAR can be calculated from above equations as PdOSSO ¼ 2PdOS  PdOSGO .

(13)

Unfortunately, integral in Eqs. (9) and (12) generally does not have a closed-form solution. Nevertheless, for C equal to 1 and 2, it can be solved. The associated solution and results can be found in Appendix B. 4. Detection in fuzzy space As was mentioned previously, the performance of OSGO and OSSO–CFAR detectors degrade in homogeneous clutter comparing to the OS–CFAR. This degradation is basically the result of non-soft

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rules used in these two detectors for combining the statistic of leading and lagging windows. In fuzzy logic algorithms, conventional binary functions are replaced with soft functions. Recently, some authors have used fuzzy logic for CFAR detection [15–18] and distributed detection [16,17], in order to improve the system performance. Similarly in our problem, we can replace the nonsoft rules used in the OSGO and OSSO–CFAR, with some soft rules inferred from the fuzzy logic. In fact, the structure of the OSGO and OSSO detectors is as follows: 4 o

Z

þ1

f Z ðzi Þ ¼

jxjf CUT ðxzi Þf Xk ðxÞ dx.

g.

Under H0 hypothesis, the distribution of CUT is Weibull with parameters B and C, and the distribution of X lek (or X lak) is equal to 

M k

 Mk F k1 f x ðxÞ. x ðxÞð1  F x ðxÞÞ

(14)

(18)

H0

In this structure, z1 and z2 are the results of dividing the CUT by the kth samples of leading and lagging windows. The parameters a and b are the weighting coefficients. For the case of OSGO and OSSO–CFAR, these weights are selected as ( OSGO : ( OSSO :

Here, Fx(x) is the Weibull CDF with parameters B and C. The derivation of w(zi) based on Eq. (16) is given in Appendix C which results in wðzi Þ ¼

k 1  Y j¼0

a ¼ 1; b ¼ 0 if

z1 pz2 ;

a ¼ 0; b ¼ 1 if

z1 4z2 ;

a ¼ 1; b ¼ 0 if

z1 Xz2 ;

a ¼ 0; b ¼ 1 if

z1 oz2 :

(15)

Generally, such discrete weights cause significant loss of information. In order to reduce this problem, we employ the soft weighing function, wðzi Þ which can be implemented as a fuzzy membership function, assigning membership to the hypothesis H0. Furthermore, it is defined so that the membership values are distributed uniformly in the interval [0, 1] under H0 [17]: wðzi Þ ¼ Prðz4zi jH 0 Þ ¼ 1  F Z ðzi Þ;

i ¼ 1; 2,

(16)

where F Z ðzi Þ is the cumulative distribution function (CDF) of random variable z ¼ CUT  X 1 k under hypothesis H0 (which Xk is a random variable indicating both random variables X lek and X lak). Since w(zi) is the probability that an observation exceeds threshold zi under the hypothesis H0, it maps the observation set to a false alarm space corresponding to the probability of false alarm, Pfa. It is necessary first to find the pdf of random variables z. The quantity z is the result of dividing the CUT by the kth sorted sample of leading (or

(17)

0

f Xk ðxÞ ¼ k

H1

f ðz1 ; z2 Þ ¼ az1 þ bz2

lagging) window (X lek or X lak). If we name the pdf of CUT and X lek (or X lak) as f CUT ð:Þ and f Xk ð:Þ, respectively, then it is easy to show that the distribution of z can be calculated as [19]

1þ

zC i Mj

1 ;

i ¼ 1; 2.

(19)

It is important to remember that, if we substitute zi by the threshold value (g) in Eq. (19), the result will be the false alarm probability of the OS–CFAR detector with M reference cells [9]. After finding w(zi), we should specify the structure of the combining function to make the decision. Here, we will examine several different combining functions [20,21]. They are algebraic product (AP), algebraic sum (AS), Einstein product (EP), max and min, whose resultant detectors will be called OSAP, OSAS, OSEP, OSmin and OSmax, respectively. Referring to Eq. (16), it is evident that the weighting function in Eq. (19) is a monotonically decreasing function. This guarantees that stronger observations provide smaller values of weighting function. Such a property often inverts the inequality of detection test in the fuzzy CFAR. The structure of the above mentioned detectors are as follows. 4.1. OSAP detector In OSAP–CFAR, the product of two w(.) functions of leading and lagging windows is compared with a threshold H1

f ðz1 ; z2 Þ ¼ wðz1 Þ  wðz2 Þ

o 4 H0

g.

(20)

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4.2. OSAS detector

5. Performance study

The structure of OSAS–CFAR is H1

f ðz1 ; z2 Þ ¼ wðz1 Þ þ wðz2 Þ  wðz1 Þwðz2 Þ

o 4

g.

H0

(21) 4.3. OSEP detector OSEP–CFAR works as H1

o

wðz1 Þwðz2 Þ f ðz1 ; z2 Þ ¼ 2  wðz1 Þ  wðz2 Þ þ wðz1 Þwðz2 Þ

4

g.

H0

(22) 4.4. OSmax detector The structure of OSmax detector is as follows: H1

f ðz1 ; z2 Þ ¼ maxfwðz1 Þ; wðz2 Þg

o 4

g.

(23)

H0

As it was mentioned previously, w(.) is a monotonically decreasing function; Therefore, the above structure is the same as comparing the minimum between z1 and z2 with a threshold as follows: H1

maxfwðz1 Þ; wðz2 Þg

563

o 4 H0

H1

4 g ) minfz1 ; z2 g gw . o

5.1. Homogenous situation

H0

(24) Thus, it is exactly the same as the OSGO–CFAR detector (see Eq. (5)). As a result, the performance of the OSmax and OSGO is the same and it is sufficient to evaluate the performance of only one of them. 4.5. OSmin detector Similar to OSmax, the OSmin detector is H1

f ðz1 ; z2 Þ ¼ minfwðz1 Þ; wðz2 Þg

While the clutter distribution is Weibull, it is not easy to find closed-form expressions for the Pd and Pfa of many detectors. Even, if we could find such an expression, they are too complicated to produce any insight about the behavior of the detector. In such a situation, a better method is to use computer simulations to evaluate the performance of the detector numerically. To do such a simulation, as many other authors, we consider a Rayleigh model for the fluctuating targets. In single pulse detection, it covers both Swerling I and Swerling II models [22]. In OS type CFAR detectors, it is possible to select different values of k between 1 and the number of available reference samples. Among all these possible choices, we have chosen k in such a way to maximize the detection probability. Our simulation results show that while the number of available samples is equal to M, the optimum value of k will be approximately equal to 0.83M [23]. Accounting for this fact, in all the following simulations, the above-mentioned value of k is used. As it is shown in [4,9], under H0 assumption, the distribution of zi s used in all detectors is independent of the unknown scale parameter. As a result, the distribution of any function of these variables is also independent of the scale parameter, and Pfa of all these detectors is also independent of this parameter, which means that all these detectors preserve CFAR property.

o 4

g:

(25)

H0

Similarly, this detector is the same as the OSSO–CFAR.

The detection performances of the mentioned CFAR detectors, in the case of homogenous situation, are shown in Fig. 2. Here, the shape parameter of Weibull clutter is set equal to 0.4 and 0.8 and the Pfa value is selected 104. It is seen that all of the WOSF–CFAR detectors perform similarly and are somewhat worse than the OS–CFAR. However, the detection performance of proposed detectors with soft rules is better than that of the OSGO and considerably superior to the OSSO detector. Fig. 2 also demonstrates that as the shape parameter becomes smaller, the detection performance decreases. One reason is, as C gets smaller values, the tail of clutter pdf becomes longer, which forces an increase in the adaptive threshold. Another cause is the reduction of accuracy in the estimation of B [8].

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Fig. 2. Detection performance of the WOSF, OS, OSGO and OSSO in homogenous situation for C ¼ 0.8 and 0.4.

So far, we have investigated the performance of the WOSF, OSGO and OSSO–CFAR in the homogenous clutter situations, while in practice, presence of interfering target (e.g. one or more secondary target) and clutter edges (e.g. the trailing or leading edges of rain clutter) may affect the performance of a detector considerably. In what follows, the effect of these situations on the proposed and OSGO and OSSO detectors will be investigated. 5.2. Presence of interfering targets CFAR detectors operating in the presence of one or more interfering targets suffer from an additional detection loss with respect to the homogenous situations. Fig. 3 presents a comparison between the proposed detectors (WOSF–CFAR detectors) and the OSGO and OSSO–CFAR detectors in such a situation. Here the shape parameter of Weibull clutter is equal to 0.4 and the Pfa value has been selected equal to 104. Also, there are two interfering targets in the lagging window whose mean powers are equal to that of primary target (located in the CUT) i.e. the interfering target to clutter power ratio (ICR) is equal to SCR. Here, substantial performance degradation for the OSGO is seen while the performance of the OSAP and OSEP remain relatively unaffected. In this situation, the OSAP and OSEP are superior to OSSO detector. The detection performance is also examined when

there are two interfering targets in the lagging window and two interfering targets in the leading window with the ICR ¼ SCR. Simulation results are presented in Fig. 4. Here, substantial performance degradation for the OSSO is seen while the performance of the OSAP and OSEP remain relatively unaffected. In this situation, the OSAP and OSEP outperform the OSGO detector. Figs. 5 and 6 show the detection performance of the OSAP and OSGO in the presence of I interfering targets in one side of the CUT, respectively. We observe that in the OSGO, Pd approximately approaches to zero for I42, while the OSAP has delectability even for I42. We conclude from the above study, the detection performance of the OSAP and OSEP is considerably better than that of OSGO and OSSO detectors. 5.3. Clutter edge situations A clutter edge situation is an abrupt clutter transition, which may represent the boundary of a rain storm or a precipitation area or seashore. In simulations, it is assumed that the clutter samples prior to the clutter edge have power equal to P1, while the samples after the clutter edge are supposed to have power equal to P2 (P2oP1). Fig. 7 shows the false alarm probability of the WOSF, OSGO and OSSO–CFAR detectors versus the distance d between the CUT and the clutter edge (expressed as the range cell number). Here, do0 implies that CUT

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Fig. 3. Detection comparison of the WOSF, OSGO and OSSO–CFAR detectors in the presence of two interfering targets within lagging window versus SCR.

Fig. 4. Detection comparisons of the WOSF, OSGO and OSSO–CFAR detectors in the presence of four interfering targets within lagging and leading windows versus SCR.

is in the clear clutter (clutter with lower power). The ratio of the clutter powers in the two regions is given by the parameter CNR that is equal to P1/P2. Referring to this figure, we observe that, for do0, Pfa regulation of the OSAP (and OSEP) is better than that of OSGO (in this case, the Pfa regulation of the OSSO is superior to that of the other detectors). However, when d40, the Pfa regulation

of the OSGO and OSAS is superior to the other detectors, respectively. As we observed from Figs. 3–7, presence of nonhomogeneity in the reference cells of CFAR detectors results in an increase in the detection loss while decreasing the probability of false alarm (Pfa). The decreased value of Pfa is acceptable, since we want to have the Pfa less than or equal to

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Fig. 5. Detection performance of the OSAP–CFAR detector in the presence of I interfering targets.

Fig. 6. Detection performance of the OSGO–CFAR detector in the presence of I interfering targets.

the designed value but such a decrease, results in the target masking effect. Thus, in this situation, the detection system should have maximized Pd in addition to minimizing the target masking effect (Pfa is close to designed value). By referring to these figures, we observe that, OSAP (and OSEP) detector gives the highest Pd, in addition to suitable Pfa regulation (suitable performance in the target masking effect). 6. Summary and conclusion In this paper, first, we presented a theoretical analysis for the OS, OSGO and OSSO–CFAR

detectors. We also proposed a new WOSF–CFAR detector with various combining functions such as AP, AS, EP, min and max. The simulation results show that utilizing output of two OS processors on both lagging and leading windows around the test cell along with employing fuzzy weighting function instead of binary weighting function in OSGO and OSSO–CFAR and appropriate combining function improve the performance of the OSGO and OSSO–CFAR detectors in the homogenous and non-homogenous situations. By investigating the WOSF–CFAR detectors, we conclude that employing the AP and EP as the combining function result in the best performance.

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-10-1

567

N=24, C= 0.8, Design Pfa=10 -3, CNR =10dB OSAP OSGO(OSmax)

-10-2

OSSO(OSmin) OSAS

Pfa

OSEP

-10-3

CUT

-10-4

-17

-12

-9

-6

-3

0

3

6

9

12

Fig. 7. Pfa comparison of the WOSF, OSGO and OSSO detectors in clutter edge situation.

In this manner, as was shown, the max and min combining functions result in the same performance as the conventional OSGO and OSSO–CFAR detectors, respectively. In this study, we assumed that the samples are independent. This assumption is not always valid, especially in the case of sea clutter at high sea states. Therefore, an important issue as the further research work will be investigating the effect of correlated samples on the proposed detectors. Also, the optimization of the fuzzy detection rules is a matter of interest for this problem. The statement of slowly varying C can be true for ground clutter, not always for sea clutter. So, the generalization of this study when the shape parameter, C, is unknown can be another issue in the future works.

lagging and leading windows, we have f z min ðzÞ ¼ 2f Xk ðzÞð1  F Xk ðzÞÞ in which f X lak ðzÞ ¼ f X lek ðzÞ ¼ f Xk ðzÞ.

Z

þ1

PfCUT4gzjH 0 gf Xk ðzÞ dz  2

PfaOSSO ¼ 2 0

Z

þ1

PfCUT4gzjH 0 gf Xk ðzÞF Xk ðzÞ dz:

 0

ðA:4Þ Now regarding that Z

Acknowledgments The authors are grateful to the reviewers for their constructive comments.

and

þ1

PfCUT4gzjH 0 gf Xk ðzÞ dz

0

Here f zmin ðzÞ is the pdf of variable zmin ¼ min {X lak, X lek}. Considering identical distribution for the kth ranked sample ð1pkpMÞ in the both

(A.5a)

0

Z

þ1

PfaOSGO ¼ 2

The generic expression of Pfa for OSSO–CFAR detector is Z þ1 PfaOSSO ¼ PfCUT4gzjH 0 gf zmin ðzÞ dz. (A.1)

(A.3)

Substituting Eq. (A.2) into (A.1), we obtain

PfaOS ¼

Appendix A. Pfa of OSSO–CFAR

(A.2)

PfCUT4gzjH 0 gf Xk ðzÞF Xk ðzÞ dz. 0

(A.5b) It is easy to conclude PfaOSSO ¼ 2PfaOS  PfaOSGO .

(A.6)

Note that above PfaOS is considered according to the number of reference samples equal to M ð1pkpMÞ. For Weibull clutter, the expression of Pfa for the OS and OSGO–CFAR can be found

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in [9,10] as   M GðgC þ M  k þ 1ÞGðkÞ , PfaOS ¼ k GðgC þ M þ 1Þ k M pfaOSGO ¼ 2k 

M M X

j

!

ðA:7bÞ Consequently, we obtain Pfa for OSSO–CFAR detector as ! M GðgC þ M þ 1  kÞGðkÞ pfaOSSO ¼ 2k GðgC þ M þ 1Þ k ! M M X  j j¼k  Gðk þ jÞGðgC þ 2M  k  j þ 1Þ  , GðgC þ 2M þ 1Þ ðA:8Þ i.e. Eq. (6). Appendix B. Pd calculation In this section, we derive an integral equation of detection probability for the OS, OSGO and OSSO–CFAR detectors and solve it for the OSGO–CFAR when C is equal to 2 or 1. The generic expression of Pd for OSGO–CFAR is Z þ1 PfCUT4gzjH 1 gf z max ðzÞ dz: (B.1) PdOSGO ¼ 0

Here f z max ðzÞ is the pdf of variable zmax ¼ max {X lak, X lek}. By using Eqs. (7) and (8), the first term in the integral is obtained as ! g 2 z2 PfCUT4gzjH 1 g ¼ exp  2  . B G 1 þ 2=C ð1 þ SCRÞ (B.2) By using the assumption in Eq. (A.3), the second term in the integral of Eq. (B.1) is (B.3)

in which  f Xk ðzÞ ¼ k

M k

F jx ðzÞð1  F x ðzÞÞMj

(B.4b)

and

Gðk þ jÞGðgC þ 2M  k  j þ 1Þ . GðgC þ 2M þ 1Þ

f z max ðzÞ ¼ 2f Xk ðzÞF Xk ðzÞ

!

j

j¼k

(A.7a)

!

k

j¼k

F Xk ðzÞ ¼

M X M

 Mk F k1 f x ðzÞ, x ðzÞð1  F x ðzÞÞ (B.4a)

    C  z C1 z C f x ðzÞ ¼ exp  , B B B

(B.5a)

    z C F x ðzÞ ¼ 1  exp  . B

(B.5b)

Inserting Eqs. (B.2) and (B.3) into Eq. (B.1), and substituting u ¼ (z/B)C, we obtain ! ! M M Z þ1 M X expðS C u2=C Þ PdOSGO ¼ 2k j k 0 j¼k  ½1  expðuÞkþj1  expðð2M  k þ 1  jÞuÞ du,

ðB:6Þ

where Sc ¼

g2 . Gð1 þ 2=CÞð1 þ SCRÞ

(B.7)

Using the binomial expansion, it result that ! ! M M M X PdOSGO ¼ 2k j k j¼k ! kþj1 X kþj1 i  ð1Þ i i¼0 Z þ1  expðS C u2=C  ðL þ i  jÞuÞ du. 0

ðB:8Þ Here L is equal to 2Mk+1. For the special case of C ¼ 2 and C ¼ 1, this integral can be solved. For example, when C ¼ 2, we easily have M PdOSGO ðC ¼ 2Þ ¼ 2k



k

!

M M X

!

j

j¼k

kþj1 X

kþj1

i¼0

i

!

ð1Þi . ðL þ i  j þ S 2 Þ

ðB:9Þ For C ¼ 1, we can use the below integral equation: Z þ1 expððax2 þ bx þ cÞÞ dx 0 rffiffiffi  2    1 p b  4ac b ¼ exp erfc pffiffiffi . ðB:10Þ 2 a 4a 2 a

ARTICLE IN PRESS A. Zaimbashi et al. / Signal Processing 88 (2008) 558–570

Where erfc (.) is the error function complement, which is defined as Z þ1 2 erfcðyÞ ¼ pffiffiffi expðx2 Þ dx. (B.11) p y Applying Eq. (B.10) into Eq. (B.8), we obtain ! rffiffiffiffiffi M ! X M M p PdOSGO ðC ¼ 1Þ ¼ k S1 k j j¼k ! kþj1 X k þ j  1  ð1Þi i i¼0  2 ðL þ i  jÞ  exp 4S1   Lþij pffiffiffiffiffi . erfc ðB:12Þ 2 S1

For the special case of C ¼ 2 and 1, we have    X M k1 k  1 ð1Þi , PdOS ðC ¼ 2Þ ¼ k ðL þ i þ S 2 Þ i k i¼0 (B.17) !! rffiffiffiffiffi M ! X k1 k  1 p ð1Þi S1 k i i¼0 ! 2 ðL þ iÞ 2  exp  pffiffiffi p 4S1

1 PdOS ðC ¼ 1Þ ¼ k 2

þ1 X 2 2m ¼ pffiffiffi expðy2 Þ y2mþ1 . p 1  3      ð2m þ 1Þ m¼0

ðB:13Þ We can rewrite Eq. (B.12) as ! rffiffiffiffiffi M ! X M M p PdOSGO ðC ¼ 1Þ ¼ k S1 k j j¼k 

kþj1 X

ð1Þi

i¼0



exp

kþj1

ð1  expðsÞÞk1 ¼

(B.15)

PdOS ¼ k Z

k

 0

i¼0

ð1Þ

k1

k1 X j¼0

By similar discussion, the Pd of OS–CFAR (for M numbers of reference samples) can be obtained as i

ðC:1Þ

Using binomial expansion as

  ðL þ i  jÞ2 4S 1

PdOSSO ¼ 2PdOS  PdOSGO .

k1 X

The weighting function of the WOSF detector (Eq. (19)) is obtained as follows. By substituting Eqs. (2) and (18) into Eq. (17) and using s ¼ (x/B)C, we obtain ! Z þ1 M f Z ðzi Þ ¼ kC sð1  expðsÞÞk1 zC1 i k 0

i

We can also obtain the equation of Pd for the OSSO–CFAR detector by the relationship

!

Appendix C

 expððM  k þ 1 þ zC i ÞsÞ ds.

!

þ1 2 X 2m  pffiffiffi p m¼0 1  3      ð2m þ 1Þ !   L þ i  j 2mþ1 pffiffiffiffiffi . ðB:14Þ  2 S1

M

þ1 X

2m 1  3      ð2m þ 1Þ m¼0 !   L þ i 2mþ1  pffiffiffiffiffi . ðB:18Þ 2 S1 

Again, using erfðyÞ ¼ 1  erfcðyÞ

569

!

i

þ1

expðS C u2=C  ðL þ iÞuÞ du. ðB:16Þ

ð1Þj

k1 j

! expðjsÞ. (C.2)

We can solve the integral in Eq. (C.1) as ! M zC1 f Z ðzi Þ ¼ kC i k ! k1 k1 X j 2 ðM  k þ 1 þ j þ zC  ð1Þ i Þ . j j¼0 ðC:3Þ Using the definition of CDF, we find ! M F Z ðzi Þ ¼ 1  k k ! k1 k1 X 1 j  . ð1Þ  M  k þ 1 þ j þ zC j i j¼0 ðC:4Þ

ARTICLE IN PRESS A. Zaimbashi et al. / Signal Processing 88 (2008) 558–570

570

By using the below expression [24] k X j¼1

k Y ð1Þjþ1 1 , ¼ ðk  jÞ!ðj  1Þ!ðy þ 1Þ j¼1 ðy þ j  1Þ

(C.5)

we obtain F Z ðzi Þ ¼ 1 

k 1 Y

M j . M  j þ zC i j¼0

(C.6)

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