Abstract— In this paper, we analyze the performance of themodified CA-CFAR detector, namely the GO-CFAR detectorin the presence of nonhomogeneous and non-Gaussian clutter.The nonhomogeneity is modelled as a step functiondiscontinuity in the reference window and the clutterbackground is assumed to contain independent non-Gaussiannoise modelled as a Pearson-distribution. We derive a closedform expression for the probability of false alarm innonhomogeneous Pearson distributed clutter. We compare thefalse alarm performance of the two detectors and show that thefalse alarm regulation of the GO-CFAR detector is better inpresence of clutter edge.

I. INTRODUCTION

n radar systems, detection procedures involve thecomparison of the received signal with a threshold under

the constraint of constant false alarm rate. This constraint isimpaired by the presence of clutter returns which arise fromreflections from the sea, land, etc. Since the clutter power isunknown, fixed threshold techniques cannot be applied. Onesolution to this problem is to set the detection thresholdadaptively.

The received signal is sampled in range by the rangeresolution cells. The clutter level in the test cell is estimatedby averaging the outputs of the nearby resolutions cells. Thedetection threshold is obtained by scaling the noise levelestimate with a constant T to achieve the design probabilityof false alarm. This is the conventional CA-CFAR (cell-averaging constant false alarm rate) detector proposed byFinn and Johnson [1].

This detector is optimal when the clutter powers in thereference window are independent, identically distributedand Gaussian modelled.

In much practical application, the clutter returns may notbe uniformly distributed. In the presence of clutterboundaries (clutter edge) the false alarm performance of theCA-CFAR detector degrades drastically. To alleviate thisproblem the ‘Greatest-of’ constant false alarm rate (GO-CFAR) detector was proposed [2]. In the GO-CFAR detectorthe leading and lagging reference samples are separately

H. A. MEZIANI is with Université de Skikda, Départementd’électrotechnique, BP 26, 21000 Skikda, Algeria (e-mail: hameziani@yahoo.fr).

F. SOLTANI., is with Université de Constantine, Départementd’électronique, Route de Ain Bey, 25000 Constantine, Algeria (e-mail:f.soltani@caramail.com).

summed and the larger of the two is used to set a threshold.Recent studies showed that the Gaussian distribution cannotappropriately model clutter returns whose amplitudedistribution is more impulsive and suggest that the cluttercan be modelled as a positive alpha-stable distribution witha shape parameter equal 0.5 which presents a heavy-taileddistribution [3-5].

II. PROBLEM FORMULATION

In a general CFAR processor, the square-law detectedsignal is sampled in range by the range resolution cells. Therange samples are sent serially into a shift register of lengthN+1=2M+1 as shown in Fig. 1. The leading M samples andthe lagging M samples form the reference window. The dataavailable in the reference window are processed to obtain thestatistic Z that is the estimate of the total noise power.

To maintain the probability of false alarm (Pfa) at adesired constant value when the total background noise ishomogeneous, the detection threshold is obtained by scalingthe statistic Z with a scale factor T to achieve the desiredfalse alarm probability.

Fig. 1. Block diagram of the GO-CFAR Detector

The test cell Q0 from the centre tap is compared with theadaptive threshold to make a decision about the presence(H1) or the absence (H0) of a target in the cell under test.

TZq

H

H

0

1

0 <>

(1 )

In the nonhomogeneous background, we assume that theclutter probability density function dominates the noisebackground. This paper deals with such a situation andassumes that in a homogeneous case, we assume that thereference window samples are independent but not

False Alarm Analysis of The GO-CFARDetector in Nonhomogeneous Pearson Distributed Clutter

H. A. MEZIANI, F. SOLTANI

I

identically distributed modeled as a step discontinuity whereR of the reference cells are occupied by the high level cluttermodeled as a Pearson distribution with dispersion parameter� (1+ C) and the remaining N-R cells are immersed in lowclutter level modeled as a Pearson distributed with dispersionparameter � as shown in Fig. 2. It is assumed that there isonly one clutter transition in the reference window.

Rieq

Cqp iq

C

iiiQ ,...,1

1

2

)1()( 2

)1(-

2/3

22

=+=+γ

πγ

(2a)

NRieq

qp iq

iiiQ ,...,1

1

2)( 2

-

2/3

2

+==γ

πγ

(2b)

Where � is the dispersion or scale parameter of the Pearsondistribution, R is the position of the step transition and C isthe ratio � 2 / � 1 of the dispersion parameters of the twoclutter patch.

Fig. 2. Nonhomogeneous clutter model.

To evaluate the performance of these CFAR detectors insuch environment, we derive the probability of false alarm(Pfa) which is given by

∫ ∫+∞ +∞

=0

000 )(])|([0

dzzpdqHqpPfa Z

TZ

Q (3)

Where pZ (z) is the probability density function of theestimate Z.

III. ANALYSIS OF THE GO-CFAR DETECTOR

In the GO-CFAR processor, the clutter level estimate Z isobtained from the larger of two separate sums computed forthe leading and lagging window

),( VUMAXZ = (4)

where

∑=

=M

iiq

MU

1

1 (5a)

and

∑+=

=M

Miiq

MV

2

1

1 (5b)

where M=N/2. The pdf of the estimate Z defined in (4) isgiven by

)()()()()( zFzvpzFzpzp UVUGOZ += (6)

Where FU(z), FV(z) and pU(z) , pV(z) are the cumulativedensity function (cdf) and probability density function (pdf)respectively of the leading window U and the laggingwindow V.

To analyse the performance of the GO-CFAR detector innonhomogeneous background, we assume that the leadingwindow U contains R cells coming from clutter withdispersion parameter � � 1 � C � and M-R cells from clutterwith dispersion parameter � . We distinguishes twosituations, when the test cell is in the low clutter region andthe case were the test cell is from the high clutter region.

A. Test cell in low clutter region:

the output of the leading window U is written as a sum oftwo independent random variables

2111

)(1

UUqqM

UM

Rii

R

ii +=+= ∑∑

+==

(7a)

and

∑+=

=M

Miiq

MV

2

1

1 (7b)

Since V are the average sum of M Pearson distributed

random variables, the dispersion of V is ��

M . The pdf andthe cdf of V are given by

z

M

V ez

Mzp 2

-

2/3

2

1

2)(

γ

πγ

= (8)

))(1(2)(Z

MzFV

γΦ−= (9)

Where x� denotes the cdf of the standard Gaussiandistribution

dyexyx

2

2

2

1)(

−

∞−∫=Φ

π (10)

while U1 and U2 are the average sum of statisticallyindependent R Pearson distributed random variables withdispersion � 1 C � , and M-R Pearson distributed random �respectively. Hence, the dispersion parameter of the randomvariable U representing the output the leading window is

MRMRC ))()1(( −++γ (11)

therefore, the pdf and the cdf of U are

MqRMRC

U

e

qM

RMRCqp

2/))()1((

2/3

22

.1

'2

))()1(()(

−++−

−++=

γ

πγ

(12)

and

)]))()1((

(1[2)(Mq

RMRCqFU

−++Φ−=

γ (13)

Substituting (8), (9), (12), and (13) into (6) and evaluating(3) we obtain the false alarm probability of the GO-CFAR innonhomogeneous clutter as

dyeM

RMRCyErfcM

eMy

Erfc

M

RMRC

T

yErfPfa

My

M

RMRCy

GO

}]2

))()1(([

]2

[

))()1(

({]2

[2

2

22

2

1

))()1((

2

1

0

−

−++−

∞

−++

+

−++= ∫π

(14)

Where Erf (x) is the error function and Erfc (x) is thecomplementary error function.

dtexErfx

t∫∞−

−=22

)(π

(15)

)(1)( xErfxErfc −= (16)

Equation (14) shows that the false alarm probability of theGO-CFAR detector does not depend on the parameter of thePearson distribution.

B. Test cell in high clutter region:

the output of the leading window V is written as a sum oftwo independent random variables

∑=

=M

iiq

MU

1

1 (17a)

21

2

12

2

1

)(1

VVqqM

VM

RMii

RM

Mii +=+= ∑∑

+−=

−

+=

(17b)

We consider that V1 represents the output of the cellsimmersed in the high clutter level and V2 represents theoutput of the cells contain low clutter level. In this case thepdf of U and V are given by

qCMU e

q

MCqp 2/)1(

2/3

221

2

)1()( γ

πγ +−+= (18)

and

MqRMMRC

V

e

qM

RMMRCqp

2/))2()()1((

2/3

22

1'

2

))2()()1(()(

−+−+−

−+−+=

γ

πγ

(19)

while the cdf of U and V are

)])1(

(1[2)(q

MCqFU

+Φ−= γ (20)

and

)))2()()1((

(1[2)(Mq

RMMRCqFV

−+−+Φ−= γ (21)

Replacing (18), (19), (20) and (21) into (6) ,andevaluating (3) we obtain the false alarm probability of theGO-CFAR in nonhomogeneous clutter as

dye

M

RMMRCyErfcMC

eMCy

Erfc

M

RMMRC

T

CyErfPfa

MCy

M

RMMRCy

GO

}

]2

))2()()1(([)1(

]2

)1([

))2()()1(

({]2

)1([

2

)1(2

1

))2()()1((

2

1

0

2

22

+−

−+−+−

∞

−+−++

++

−+−++= ∫π

(22)

IV. RESULTS AND DISCUSSION

In this section, the results obtained in the previous sectionare used to determine the threshold multiplier T whichachieves a constant false alarm rate and to illustrate theperformance of the GO-CFAR detector at clutter edge. Thethreshold multiplier T was determined under the hypothesisthat the reference window is in homogeneous clutterbackground [6], for fixed reference window size N and adesign probability of false alarm.

Figures 3-4 show the false alarm rate performance whenthe reference window is not homogeneous, as in the case ofclutter edges power transition of C= 5 dB, 10 dB and 15dB versus the number of clutter cells R present in thereference window, at design Pfa=10-4 and N=16. We observea dip in Pfa from cells 1 to 7 is due to the raise of thethreshold even through the test cell is in the low clutterregion. On the other hand, a sharp spike in the Pfa form 9 to16 due to the lower threshold which allows a large increasein Pfa. As expected the Pfa exhibits a sharp discontinuity atR equal the test cell. Figure 5 compares the performance ofthe GO-CFAR and CA-CFAR detectors in non homogeneousclutter environment The best false alarm rate regulation atclutter boundary is obtained for the CA-CFAR detector whenthe test cell is in lower clutter, while the GO-CFAR detectorexhibits better performance when the test cell comes fromhigher clutter as expected.

Fig. 3. Performance of the GO-CFAR detector in clutter power transitionfor R < N/2.

Fig. 4. Performance of the GO-CFAR detector in clutter power transitionfor R > N/2 .

Fig. 5. Performance comparison of the GO-CFAR and CA-CFARdetector in clutter power transition.

V. REFERENCES

[1] Finn H. M., Johnson R. S. “Adaptive detection mode with thresholdcontrol as a function of spatially sampled clutter level estimates”,RCA Review, 29, 414-463, Sept. 1968.

[2] Hansen H. M. “Constant false alarm rate processing in search radar”,In Proceedings of the IEE International Radar Conference, London,325-332, Oct. 1979.

[3] Pierce R. D. “RCS characterisation using the alpha-stableDistribution”, In Proceedings of the IEEE national RadarConference, 154-159, May 1996.

[4] Pierce R. D. ”Application of the positive alpha-stable distribution”. InIEEE signal processing workshop on higher- Order statistics, Banff,Alberta, Canada, 420-424, July 1997.

[5] Tsakalides P., Trinci F., Nikias C. L., “Performance assessment ofCFAR processors in Pearson-distributed clutter”, IEEE Transactionson Aerospace and Electronic Systems, Vol. AES-36, 4, 1377-1386,Oct. 2000.

[6] Meziani H. A., Soltani F. “Performance analysis of some CFARdetectors in Homogeneous Pearson-distributed clutter”, InProceedings of the third International conference : Sciences ofElectronic, Technologies of Information and Telecommunications,sousse, Tunisia, CD-Rom, Ref. 165, March 2005.

0 1 2 3 4 5 6 7 8-4.7

-4.6

-4.5

-4.4

-4.3

-4.2

-4.1

-4

-3.9

LOG

10 (

Pfa

)

Number of cells in high clutter (R)

C=5 dB C=10 dB C=15 dB design Pfa=10- 4

N=16

8 9 10 11 12 13 14 15 16

-4.05

-4

-3.95

-3.9

-3.85

-3.8

LOG

10 (

Pfa

)

Number of cells in high clutter (R)

C=5 dB C=10 dB C=15 dB design Pfa=10- 4

N=16

8 9 10 11 12 13 14 15 16

-4.05

-4

-3.95

-3.9

-3.85

-3.8

-3.75

-3.7

LOG

10 (

Pfa

)

Number of cells in high clutter (R)

CA-CFAR detector GO-CFAR detector design Pfa=10- 4

C=10dB N=16