Southern Illinois University CarbondaleOpenSIUC

Conference Proceedings Department of Electrical and ComputerEngineering

5-2008

Switched Order Statistics CFAR Test for TargetDetectionAnas TomSouthern Illinois University Carbondale

R. ViswanathanSouthern Illinois University Carbondale, viswa@engr.siu.edu

Follow this and additional works at: http://opensiuc.lib.siu.edu/ece_confsPublished in Tom, A, & Viswanathan, R. (2008). Switched order statistics CFAR test for targetdetection. IEEE Radar Conference, 2008. RADAR '08, 1-5. doi: 10.1109/RADAR.2008.4720923©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republishthis material for advertising or promotional purposes or for creating new collective works for resaleor redistribution to servers or lists, or to reuse any copyrighted component of this work in otherworks must be obtained from the IEEE. This material is presented to ensure timely dissemination ofscholarly and technical work. Copyright and all rights therein are retained by authors or by othercopyright holders. All persons copying this information are expected to adhere to the terms andconstraints invoked by each author's copyright. In most cases, these works may not be repostedwithout the explicit permission of the copyright holder.

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Recommended CitationTom, Anas and Viswanathan, R., "Switched Order Statistics CFAR Test for Target Detection" (2008). Conference Proceedings. Paper 89.http://opensiuc.lib.siu.edu/ece_confs/89

Switched Order Statistics CFAR Test for TargetDetection

Anas Tom and R. ViswanathanDepartment of Electrical & Computer Engineering

Southern Illinois University CarbondaleCarbondale, IL 62901-6603

Phone: (618) 453-7052, 453-4321, Fax: (618) 453-7972, anas@siu.edu: viswa@engr.siu.edu

Abstract- In this paper we introduce a new switched orderstatistics CFAR test (SW-OS) for detecting a radar target in thepresence ofnonhomogeneous clutter and/or multiple interferingtargets situation. Whereas a switching CFAR test (S-CFAR)was recently proposed in the literature for addressing a similarbackground scenario, unlike the S-CFAR test, the test proposedhere does not utilize the test cell statistic in classifying the cellssurrounding the test cell as homogeneous or not. The SW-OStest has some similarity to the selection and estimation (SE) test,which was co-authored by the second author ofthis paper, but issimpler to design. Probability of detection performance resultsobtained for Rayleigh clutter and Rayleigh target indicate thatthe SW-OS performs nearly as good as an order statistic test(OS) in the homogeneous background condition and performsmuch better than OS under the presence of many interferingtargets. When compared to S-CFAR, SW-OS performs slightlyworse in the homogeneous background, but performs betterunder the condition of many interfering targets.

Index terms: CFAR detection, Order statistic, switchedCFAR test.

I. INTRODUCTION

Procedures for constant false alarm rate (CFAR) detection ofradar targets in clutter have been investigated extensivelysince latel960's (see [1], [2] for a review). Variations ofcellaveraging CFAR tests do not perform well in multipleinterfering targets or in step-clutter transitions [1 ]-[3].Almost all the tests that perform reasonably well in a varietyofbackground clutters are based on order statistics of cluttercells surrounding the test cell [2]. Some of them haveappeared in the literature as censored mean level detector,variably trimmed mean, LI CFAR, MAX family of OS­CFAR, selection and estimation (SE) test, an intelligentCFAR processor not based on order statistic, distributedCFAR test, and OS-CFAR detector for Weibull clutter [4]­[12]. Even though ordering of samples with currentlyavailable economical processors should not be acomputational burden, the switching CFAR test (S-CFAR)was proposed recently as a new and competitive test that doesnot require ordering ofsamples [13]. However, the S-CFARtest utilizes the test cell sample along with the samples fromthe reference (surrounding) cells in order to classify thebackground, i.e, the reference cells, as from an homogeneous

1-4244-1539-X/08/$25.00 ©2008 IEEE

clutter or not. In the second stage ofthe decision process, thetest cell sample needs to be used again in order to make adetermination ofthe presence or the absence ofa target in thetest cell [13]. While the use of test cells in both stagesintroduces statistical dependency, the author ofthis paper wasable to derive analytical expression for detection probabilityof Rayleigh-target in Rayleigh-clutter without too muchdifficulty. However, philosophically speaking, it must bepossible to classify the background clutter as homogeneous ornot, without having to take into account the test cell sample.Hence, in this paper, we revisit the approach taken in [8],wherein the SE test was proposed. Whereas both SE and theswitched order statistic (SW-OS) method proposed heredepend on the ordered values ofthe samples of the referencecells, the SW-OS is simpler to design.

In section II we propose the SW-OS test and deriveexpression for the probability of detection of a Rayleightarget in homogeneous Rayleigh clutter. In section III, theperformances of a simple OS test, S-CFAR and SW-OS testare compared for homogeneous and interfering targetssituations. These results indicate superior performance ofSW-OS test in interferers involving many targets.Conclusions from this study are presented in section IV.

II SWITCHED ORDER STATISTICS TEST

Let Xl, X 2, .. , X N denote the samples of the referencecells coming out of a radar signal processor, where Nindicates the total number of reference cells. Let Yi denote

the ith order statistic of the samples, XI,X2, ..,XN' viz.,

Yi is the lh largest value when Xl, X 2, .. , X N are ordered

in an ascending order, i = 1,2,.. ,N. By appropriately

choosing two order statistics, Y I and Y k , 1> k, the proposed

switched order statistic (SW-OS) test classifies the referencecell as homogeneous (hypothesis H 2 ) or nonhomogeneous

(hypothesis HI) according to the following rule:

HI

Y/ ~ {3Yk' (1).<

H2

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where f3 2 1is a constant. In designing the test, f3 needs to

be determined along with 1and k. When all the referencesamples are from homogeneous background, with high

probability y / will be less than f3 y k , whereas the converse

will be true when there are significant number of interferersin the reference cells. Since a simple OS test based on y/,

3with I ~ - N will provide a reasonable probability of

4detection in homogeneous background, while tolerating up toN/4 interfering targets, this choice of 1seems reasonable forthe SW-OS test also. The choice ofk depends on whether thesituation of a step clutter transition in the middle of thereference window is important. Assuming that this is the

Ncase, a choice of k ~ - is reasonable. Having fixed 1and k

2through qualitative requirements, an approximate value of f3can be obtained by requiring that the probability of falselydeclaring a nonhomogeneous background, given that thereference samples had actually come from the homogeneousclutter, be below a number. This number could be chosen tobe in the range, say (0.0001, 0.001). Of course, the final

choice for f3 is arrived by looking at the detection

performance in various environments. Finally, if we denotethe test cell sample as Z, the decision rule for declaring thepresence of a target in the test cell is given byDecide target present iff

{z ~ Cl Y / when H 2 is decided in (1)

, (2).Z ~ C 2 Y k when HI is decided in (1)

where CI and C2 are two positive constants. Using the

customary definition of probability of false alarm (pF) as

the probability ofthe test declaring the presence ofa target inthe test cell, when it is in fact absent, an expression for PF

for the test (2), as a function of I, k, f3, and CI and C2 can

be obtained. Since other parameters, except CI and C2 ,

could be fixed as stated earlier, the variations of PF , as

CI and C2 are varied, could be studied. However, to make

the design of the test simple, in the sequel we assume thatCI = C2· In the remainder of the paper, we assume that the

clutter cells as well as the target returns have Rayleighamplitudes. Moreover, the reference samples are statisticallyindependent and are all independent of the test sample.

2.1 Probability of Detection in Homogeneous Rayleighclutter.

First, we determine the probability, Ph, that the test (1)

decides hypothesis HI, given that all the reference

samples are from homogeneous Rayleigh clutter, viz.,

Xl' X 2,··, X N are all i.i.d exponential with identical

mean, which can be assumed to be unity, without any lossof generality. Using the general theory of order statistic,

the joint density of order statistics yk, y/ can be written as

[14]:

f k /Lv k ' Y/)=(N) (I ) (I - k + 1) (I - k ), 1 k-l

.Fk-1(Yk) (F(y/)- F(Yk)Y-k-l

.(I-F(y/))N-/ f(Yk)f(v/) , O<Yk < Y/

(3).

where f(x) = e- x and F(x) = 1- e- x, x 2 o. Use of

binomial expansion and simplification of (3) yields, with

0< Yk < Y/,

fk /(vk,y/)=(N)(I )(/-k+lXZ -k), 1 k-l

.k~l/-i-l(k -1)(/- k-1)(_I)P+mp=O m=O p m

.{e- (r\Yk+ Y2yJ}, O<Yk < Y/

(4).

where r1 =P + I - k - m, r2 =(N + m -I + 1).Hence,

Pb=P(Y/>!JYkI H 2)= 77 fk /(x,y)dydxop x '

(5).Using (4) in (5), and doing the integration, Ph can be

obtained as the right hand side of (4), with the expression

within the curly brackets replaced by { (jI 1 )} .Y2 1+ Y2/3

Numerical evaluation yields Ph = 2.908E-4, when 1= 19, k

=10,f3= 11,.In homogeneous clutter, the probability of

detection for the test (2) can be obtained from

PD =P~ ~ Cl Y/ Idecide H 2' true H2' target presen~

.p(decideH21 H 2)

+P(Z~CIYkldecide HI' true H 2 ,target present)

. p(decideHII H 2)

(6).

Upon denoting the two summands on the right hand side of

(6) as TI and T2'

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30

15

2520

--SW-OS, p=20, c=8.243

--SW-OS, p=15, c=8.345- - - SW-OS, p=11, c=9.602- - - OS, r=18

10 11 12 13 14

Homogeneous

Pf= 1E-4

o~_"__--'-____'_____.L--_--'-____'____-----'

o 10 15SNR(dB)

10-6'--------'---_--'---------L._----'--_'"-------'---_--'---------L._----'---------'

5

0.8

0.9

0.2

0.1

c 0.7.2

~ 0.6~'0 0.5>.

~ 0.4en.. 0.3

Fig 1. SW-as probability of false alann.

signal Z is generated to be an independent exponential samplewith mean equal to SNR. Then, appropriate tests areimplemented using the generated samples to determine ifthese tests detect the presence of target correctly or not. Acount ofthe number oftimes a test detects the target, dividedby the total number of times the samples are generated,provides an estimate of the probability of detection for thattest. The number of reference cells, N, equals 24. We take kto be 10 so that SW-OS would tolerate a little more than N/2interfering targets. For 1, two choices, 18 and 19 wereconsidered. I is fixed at 19, because this choice provides aslightly larger detection probability when seven interferingtargets are present in the reference cells (see Fig. 6).

PZ/C2Y/P 00 Z/C2

P2= I Ilk l(x,y)dxdy+ I Ilk l(x,y)dxdyo 0 ' PZ/C2 0 '

(10).Using (4) and (6) through (10) and some simplification, afinal expression for the probability ofdetection is obtained asthe right hand side of(4) with the expression within the curlybrackets replaced by

p-l CI

(11 + Y2 P)(11 + Y2t YI (11 + Y2)(e(11 + Y2)+CI)p2 Cl ()

YI (11 + Y2 P)(e(YI + Y2P)+ PCIt Y2 (e(YI + Y2P)+C2)The expression for the probability of false alarm, PF , is

simply the expression for PD' with () replaced by 1

(corresponding to SNR of zero).

III PROBABILITY OF DETECTION PERFORMANCE OFSW-OS, OS, AND S-CFAR TESTS

OOP(Y I < min (3.-, f3 YkJIZ = Z, true H 2)TI = f CI

O. f z (zl target present ) dz

(8),

where f Az Itarget present) = ..!- e-~, z > 0 ,and ()()

equals one plus target signal power to noise clutter powerratio (SNR). Upon denoting the probabilities inside theintegral in (7) and (8), respectively, as PI and P2 ,

Z/(PCl)pX z/Cl z/ClP}= f ffkl(x,y)dydx + f ffkl(x,y)dydx

Ox' Z /(pCl) x '

(9).

(7).

OOP(Y k < min (~, LLJIZ = Z, true H 2)T2 = f C2 f3

O. f z (zl target present ) dz

In this section we analyze the detection performances ofSW­OS, OS, and S-CFAR tests. The design parameters of alltests that guarantee a specified false alarm probability inhomogeneous clutter can be obtained through appropriateanalytical expressions, viz., (41) in [1] for OS test, (20) in[12] for S-CFAR test, and by the procedure stated belowequation (10), for SW-OS test. Once appropriate designparameters are obtained, the detection performances ofvarious tests under nonhomogeneous background areobtained by generating independent exponential samples of

Xl ,X2 '00' X N' with Nl of them having a mean value of

one plus interfering target power to noise (low clutter) powerratio (INR) and the rest having a mean value of 1. The target

Fig 2. Probability of detection in homogeneousbackground, SW-OS, as.

Fig. 1 shows the variation of false alarm probability as a

function of c, for 1= 19, k = 10, and three values of p. As

explained earlier, p is chosen by targeting a reasonable value

of Ph. In Fig. 2 we show the probability of detection of

SW-OS at PF = 10- 4 , for homogeneous clutter and for

three values ofp. As mentioned earlier, for the sake of

simplicity of design, we consider only the case,

CI = C2 = C . It is observed that

p = 11, C= 90602 provides a slightly lower detection

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30

30

25

25

20

20

--SW-OS, p=20, c=8.243

--SW-OS, p=15, c=8.345

- - - SW-OS, p=11, c=9.602

- - - OS, r=18

--SW-OS, p=20, c=8.243

--SW-OS, p=15, c=8.345

- - - SW-OS, p=11, c=9.602- - - OS, r=18

--SW-OS, p=20, c=8.243

--SW-OS, p=15, c=8.345

- - - SW-OS, p=11, c=9.602

- - - OS, r=18

/

I/

/

/

/I

II

II

I

II

I

II

/

//

/

10

10

/

/

I

II

II

I

II

I

II

II

I

I/

/

//

,/ ;/

,/ ,/

,/ v/ /

/ // /'

/ // /

/ /

/ // /

/ // /

/ /

/ // /

/ /

/ /

/ // I

/ I/ /

/ // /

/. /

... :-/

11 targets

INR =SNR

5 targets

INR = SNR

7 targets

INR= SNR

15SNR(dB)

Fig 4. Probability of detection for SW-OS andOS, twelve interfering targets.

15SNR(dB)

Fig 3. Probability of detection for SW-OS and OS,five interfering targets.

O-=~--=-----t_------'---_-L-_-----"------'--_----l

o

0.8

0.2

0.8

0.9

0.9

0.2

0.1

0.1

c 0.7o

~ 0.6CD'0

'5 0.5

~~ 0.4.0e0.. 0.3

0.2

0.9

0.8

0.1

c 0.7o

~ 0.6CD'0

'5 0.5>-

~ 0.4.0e0.. 0.3

c 0.7oUQ) 0.6

~'0 0.5>-

~ 0.4.0e0.. 0.3

IV CONCLUSIONS

probability when compared to /3 =20,c =8.243. The

detection performance ofSW-OS, for /3 =II,c =9.602 ,although below that of as, for r=18, is reasonably close tothe performance of the as test.Fig. 3 shows the detection performance for the sameparametric values in Fig. 2, but with five interfering targets,with interfering target power to noise ratio (INR) equal to

SNR of the target. As before, /3 = 11,c = 9.602 provides

a slightly smaller detection probability when compared to theother combinations and the as test. Fig. 4 shows theprobabilities ofdetection ofSW-as and as tests under seveninterferers situation. The as test was designed with a rankorder of r = 18, N = 24, which implies that this test couldtolerate only up to six interfering targets. We can observethat the as test provides a detection probability ofonly 0.43,as the SNR asymptotically approaches infinity. Superiority

of the SW-as test with /3 = 11, c = 9.602 over other two

combinations of (,8, c) values and the as test can be

observed. For 11 interfering targets situation (Fig. 5), SW­as completely outperforms as. Since the SW-OS wasdesigned to reasonably differentiate between thehomogeneous and nonhomogeneous situation, correspondingto a large number of interfering targets approximating halfthe reference window size, the switched order statistic test(SW-OS) is able to handle the case of a large number ofinterfering targets much more efficiently than a simple astest. Fig. 6 shows that I = 19 is a better choice than I = 18,when seven interfering targets are present ( of course, eachcase is designed to provide false alarm probability of 10-4

).

Figs. 7, 8, 9 and 10 show detection performance comparisonof SW-as and S-CFAR. S-CFAR was designed with

(a = 0.5,/3 = 12.293) in order to achieve the same false

alarm probability of 10-4 [13]. Although S-CFAR provides alarger detection probability than SW-as in five interferingtargets situation (and only a slightly larger detectionprobability in homogeneous background), SW-as performsbetter than S-CFAR under 9 and 11 interfering targets.

In this paper we proposed a switched order statistic CFARdetector for detecting a radar target in the presence of asignificant number of interfering targets or varying clutterbackground. For detecting a Rayleigh target in Rayleighclutter, the analysis performed shows that SW-OS performsmuch better than a simple as test and the S-CFAR formultiple interfering targets.

REFERENCES30252010 15

SNR(dB)

Fig 5. Probability of detection for SW-OS and S-CFAR, fiveinterfering targets.

[1] P.P. Gandhi and S.A. Kassam, "Analysis of CFAR processors innonhomogeneous background," IEEE Trans. AES, July 1988, pp. 427-445.[2] R. Viswanathan, "Order Statistics Applications to CFAR Radar TargetDetection," in Handbook of Statistics, Vol. 17- Order Statistics and theirApplications, N. Balakrishnan and C.R. Rao Eds., North-Holland Publishers,1998, pp. 643-671.

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oL....-_==-__----I... ....L.....-__--l- L---__---.J

o 10 15 20 25 30SNR(dB)

Fig 6. Probability of detection for SW-OS, seven interfering targets.

30252010

//

/

//

//

//

//

/

//

//

//

//

/ / --SW-oS, p= 11, c = 9.602

- - - S-CFAR, p= 12.293, a = 0.5

5 targets

INR=SNR

oL...-~==~__------l ...l...-__~ L--__---.J

o

0.8

0.1

0.2

0.9

c: 0.7o

~ 0.6CD1j

15 0.5

:S~ 0.4

~c.. 0.3

15SNR(dB)

Fig 8. Probability of detection SW-OS and S-CFAR, five interferingtargets.

-- SW-OS, p=11, c=9.602, 1=19

- - - SW-OS, p=10, c=10.1051, 1=18

/'/'

//

I/

/

//

/

//

/

/

/

//

//

//

/V r------------..., I

INR= SNR

7 targets

0.8

0.2

0.9

0.1

6 0.7

~ 0.6CD1j

15 0.5

~~ 0.4ec.. 0.3

30

/'/

//

//

/

//

/

//

//

//

//

'I

/,

/

11 targets

INR= SNR

9 targets

INR= SNR

0.1

0.2

0.8

0.9

0.9

.§ 0.7

~ 0.6CD"0

15 0.5

:S~ 0.4

~c.. 0.3

--sw-OS, /3= 11, C =9.602

S-CFAR, p= 12.293, a = 0.5oL_.!!!!=::::E::::::::::=-----l...--_--=t:::====::::L===::::::L::=====::Jo 10 15 20 25

SNR(dB)

Fig 9. Probability of detection SW-OS and S-CFAR, nine interferingtargets

[3] H. Rohling, "Radar CFAR thresholding in clutter and multiple-targetsituations," IEEE Trans. AES, July 1983, pp. 608-621.[4] lA. Ritcey, "Performance analysis of censored mean-level detector,"IEEE Trans. AES, Jan. 1986, pp. 213-223.[5] I. Ozugunes, P.P. Gandhi, and S.A. Kassam, "A variably trimmed mean

CFAR radar detector," IEEE Trans. AES, Oct. 1992, pp. 1002-1014.[6] J.A. Ritcey and J.L. Hines, "Performance of MAX family of order­statistics CFAR detectors," IEEE Trans. AES, Jan. 1991, pp. 48-57.[7] M. Lops and P. Willett, "LI-CFAR: A flexible and robust alternative,"IEEE Trans. AES, Jan. 1994, pp. 41-54.[8] R.Viswanathan and A. Eftekhari, "A selection and estimation test formultiple target detection," IEEE Trans. AES, April 1992, pp. 505-519.{9] M.E. Smith and P.K. Varshney, "Intelligent CFAR processor based ondata variability," IEEE Trans. AES, July 2000, pp. 837-847.[10] M.K. Uner and P.K. Varshney, "Distributed CFAR detection inhomogeneous and nonhomogeneous backgrounds," IEEE Trans. AES, Jan.1996, pp. 84-96.[11] N. Levanon and M. Shor, "Order statistics CFAR for Weibullbackground," lEE Proc. F. Radar & Signal Process., June 1990, pp. 157­162.[12] F. Gini, F. Lombardini, and F. Verrazzani, "Decentralized CFARdetection with binary integration in Weibull clutter," IEEE Trans. AES, April1997, pp. 396-407.[13] T.-T. Cao, "A CFAR thresholding approach based on test cell statistics,"IEEE Radar 2004, pp. 349-354.[14] B.C. Arnold, N. Balakrishnan, and H.N. Nagaraja, A first course inorder statistics, Wiley, New York, 1992.

0.8

302520

--SW-OS, P=11, C =9.602

- - - S-CFAR, P= 12.293, a = 0.5

10Ol....-_-=~=--_---.l.. ....L-__--...L L-__......J

o

0.2

0.1

c: 0.7.2~ 0.6CD1j

15 0.5

:S~ 0.4.0ec.. 0.3

15SNR(dB)

Fig 10. Probability of detection SW-OS and S-CFAR, eleven interferingtargets.30252010

/'/'

/

//

//

//

//

//

//

//

//

//

/' / --sw-os, fJ =11, C =9.602

- - - S-CFAR, fJ = 12.293, a = 0.5

Homogeneous0.9

0.8

0.76

i 0.6"0

'0 0.5~:c 0.4111

~a.. 0.3

0.2

0.1

00 15

SNR(dB)

Fig 7. Probability of detection for SW-OS and S-CFAR, homogeneousbackground.

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