M-Sweeps detection analysis of cell-averaging CFAR processors in multiple-target situations

M.B. El Mashade

Indexing terms: CFAR detector, Signal processing

Abstract: The performances of the conventional cell-averaging (CA), greatest-of (GO) and smallest- of (SO) constant false alarm rate (CFAR) detectors are analysed for the case where one or more inter- fering target returns are present in the set of cells used in estimating the clutter-plus-noise level. Exact expressions are derived for the detection probability of these detectors when noncoherent integration is used under the chi-square target fluctuation model. Results are depicted for a Swerling I target model. As M increases, better performances are obtained for the CA, GO and SO detectors under both homogeneous and non- homogeneous background models. For the same parameters, the SO scheme gives the best per- formance in the case of a multiple-target environ- ment.

1 Introduction

The signal received from a radar target is usually buried in thermal noise and clutter. As the clutter-plus-noise power is not known U priuri, a fixed threshold detection scheme cannot be applied to the radar return if the false alarm is to be controlled. To overcome this problem, constant false alarm rate (CFAR) detectors are com- monly used. This type of processing scheme sets the threshold adaptively based on local information of total noise power [l-51. The radar uses the range cells sur- rounding the cell under investigation as a reference window from which an estimate of the unknown back- ground noise level is derived. When the reference window consists of independent and identically distributed (IID) noise samples from the same population as that of the cell under test under the null hypothesis, the probability of false alarm is invariant to the noise level. As the number of reference cells increases, the detection prob- ability approaches that of the optimum detector.

To solve the problems associated with nonhomo- geneous noise background, the CFAR processors split the reference window into leading and lagging parts, sym- metrical about the cell under test. The simplest CFAR detector is the cell-averaging (CA) one, which adaptively sets the threshold by estimating the mean level of the sums in the leading and lagging windows. The CA detec- tor is the optimum CFAR processor in an homogeneous

Q IEE, 1994 Paper 9887F (E5, E15), first received 10th June 1992 and in revised form 2nd June 1993 The author is with the Electrical Engineering Deparlmenl, Faculty of Engineering, AI Azhar University, Cairo, Egypt

I E E Proc.-Radar, Sonar Navig., Vol. 141, No. 2, April IYY4

background when the reference cells contain IID obser- vations governed by an exponential distribution. However, its detection performance is significantly affected when the assumption of an homogeneous refer- ence window is not verified. In the case of multiple-target detection, the noise estimate includes the interfering signal power leading to an unnecessary excess in overall threshold. Interfering targets that fall within the sur- rounding cells raise the threshold and lead to serious deg- radation in detection performance of the detector. On the other hand, if a clutter edge is present in the reference window with a target return in the test cell, severe masking of targets results owing to the increase in thresh- old [ 6 ] .

To minimise the etrect of clutter edge on the false alarm probability, the greatest-of (GO) CFAR processor is proposed. This detector takes the maximum of the sample means drawn from the leading and lagging windows. Although the false alarm rate performance of the GO detector in regions of clutter transition is better than the CA scheme, its detection performance in multiple-target, environments is quite poor [8]. On the other hand, the smallest-of (SO) CFAR processor is sug- gested to alleviate the problems associated with closely spaced targets. However, this processor is unable to prevent excessive false alarms in the presence of clutter edges. Although the SO-CFAR scheme is preferred in the multiple-target situation, it has undesirable effects when interfering targets are h a t e d in both halves of the refer- ence window [9] .

In this paper, we study the detection performances of CA, GO and SO detectors that use M-pulse noncoherent integration, under the assumption that arbitrary numbers of interfering targets are present in the leading and lagging cells of the reference background noise.

2 Basic assumptions and problem formulation

In the CFAR detection scheme, the square-law detected range samples are sent serially into a shift register of length N + 1 = 2n + 1. The target decision is commonly performed using the sliding-window technique. The data available in the reference window enter into an algorithm for the calculation of the decision threshold. The statistic Z , which is proportional to the estimate of the total noise power, is formed by processing the contents of N refer- ence cells surrounding the cell under test, whose content is Y,. The estimate Z is then multiplied by a constant scale factor T, depending, first, on the estimation method applied and, secondly, on the false alarm rate required. A target is desclared to be present if Y, exceeds the thresh- old T Z . The procedure for estimating Z varies with different CFAR schemes. For the mean level CFAR

103

detectors, the processor consists of two summers forming sums X and Y for the leading and lagging windows. In the CA scheme, 2 is simply the sum of X and Y and, in the GO and SO processors, it is the larger or smaller of the outputs X and Y , respectively.

To analyse the detection performance of the CFAR processors in the nonhomogeneous background noise, we assume that the target under test and the interfering targets in the reference noise cells follow the chi-squared target fluctuation model with a probability density func- tion (PDF) of the following form

1 p ( y ) = -j e-YIA (1) .

Under the null hypothesis of no target in a range cell and homogeneous background, i is the total clutter-plus- thermal noise power which is denoted $. If an interfering target is present in the reference noise cell, 1 is $(I + I ) , where I is the average interference-to-total noise ratio (INR). On the other hand, under the alternative hypothe- sis of presence of a target, /z is $(l + A), where A is the average signal-to-total noise ratio (SNR) of the target. This means that a Swerling I model is assumed for the radar returns from the primary and the secondary inter- fering targets. We also assume that the background noise is stationary Gaussian, and the observations in the N + 1 reference cells are statistically independent. Therefore, for the cell under test, the value of 1 is

for clear background noise 1 = G(1 + I ) for the interfering target (2) i" $(l + A ) for the primary target

The statistic Z is formed by processing the contents of the leading and lagging windows. Suppose that the leading window Z , contains r l cells coming from inter- fering targets and q1 = n - r l cells from clutter-plus- thermal noise. Hence, the estimated total noise power of this window is obtained from

V I 4 1

r = 1 1 = 1 2, =pi+ c y j p x , + Y1 (3)

Similarly, if the lagging window Z , has r2 samples of the first type and q2 = n - r z samples of the second one, the statistic Z , is given by

I 2 (I2

I = l , = 1 z, = ,I xi + y j 42 x2 + Y, (4)

As XI and Yl are statistically independent and as the samples of each type are identically distributed, the char- acteristic function [I 11 of 2, becomes

4&) = 4XI(S)4Y,(S)

= (']'("->'I

s + a s + h

where a = 1/$(1 + I ) and h = l/$.

resulting characteristic function has the following form When noncoherent integrated M-pulses are taken, the

Similarly,

(7)

Let p(u/A) denote the probability density function of the cell under test when target is present. In the case of M - sweeps, this PDF is given by

where T ( M ) is the gamma function. Eqns. 6-8 are the basic formulas of our analysis.

3 Analysis of detectors

The analytical results for the probability of detection of the CA, GO and SO processors are derived as follows.

3.1 Cell-averaging (CA) detector In the CA detector, the total noise power is estimated by summing the range cells of the leading and lagging refer- ence windows. Thus the statistic Z is given by

(9) As 2, and 2, are statistically independent, the chracteris- tic function of Z in the case of M-sweeps is simply the product of eqns. 6 and 7, as follows

z = z1 + 2,

The Laplace transform of the cumulative distribution function (CDF) of 2 is obtained by dividing eqn. 10 by s, as follows

M(ri + r d M((I1+42)

4PA4 = + (&) (A) (1 1)

The probability of detection P , of this processor becomes

3.2 Greatesr-of (GO) detector In this situation, the noise power is estimated by selecting the greater of the leading and lagging sets of the reference noise cells [7]. Therefore the statistic Z is given by

2 = max (Zi, Z , ) (13)

FA4 = F,,(Wz*(4 (14)

The CDF of 2 in this case has the following formula

F,,(u) is obtained by taking the Laplace inverse of 4&)/s, which gives

where

D,, i = - a r l { ~ + ( - l y 1

104 I E E Pror.-Radar, Somr Nauig., Vol. 141, No. 2, April 1994

and

sAx) = ( X ) m + 1) (18)

(x), is the Pochhammer symbol, which is defined as ( X ) i = r(x + i)/r(x)

= 1 for i = 0

= X(X + 1Xx + 2), . . . , (x + i - 1) for i > 0 (19)

The evaluation of the probability of detection P , of this scheme gives

where r l = Mr,, r , = Mr,, q1 = Mq,, q2 = Mq,.

3.3 Sma//est-of (SO) defector In the SO-CFAR scheme, the noise power estimate is the smaller of the sums 2, and 2,. That is,

2 = min ( Z , , Z , )

FA4 = F,,(U) + F&) - F,,(~)FZ,(4

(21)

(22)

Therefore, upon substituting eqn. 15 into eqn. 22, we obtain for the SO-CFAR processor

In this case, the CDF of 2 is given by

The substitution of eqn. 23 into the definition of the detection probability gives

d M + r ~ - i - 2

dSMfrl--i-2 [ '$F~a(~)js=a+(TI@(l + A ) )

I E E Proc.-Radar, Sonar Navig., Vol . 141, No. 2, April 1994

4 Results

In this Section, the exact detection probabilities deter- mined in the preceding Section are applied to the per- formance evaluation of CA, GO and SO processors. Numerical results are obtained for both homogeneous and nonhomogeneous backgrounds, taking into account that there are M noncoherent integrated pulses in each case. In the multiple-target environment, the per- formances are evaluated for a possible practical situation of equal INR and SNR when the false alarm probability equals The threshold multiplier T has been deter- mined on the assumption that no interfering targets are present in the cells of the reference windows. The primary and secondary interfering targets are assumed to fluctu- ate in accordance with the Swerling I target model [lo]. Results are shown in Figs. 1-5. In Figs. 1-3, we show the detection probability, in the absence as well as in the presence of extraneous targets, as a function of the primary target SNR for the proposed detectors. These results are obtained for N = 24, r l = 0, r , = 2, P,, =

and for a number of sweeps M = 1, 2, 3, 4. As expected, when the number of noncoherently integrated pulses increases, the detection performance of the CFAR processor is improved. From these Figures, we note that intolerable masking of the primary target occurs in the CA and GO schemes, and the SO scheme has the best performance in a multiple-target situation. Therefore the SO processor is capable of resolving multiple targets in the reference window when all the interferers appear in either side of the cell under test. However, the SO detec- tor exhibits severe degradation in its performance when

0.8 -

C 0

U I

0.6 - d r 0 I -

0.4 - (3 .n e

0.2 -

d 4 Ib 1: 2; 2; 3b signal-to-noise ratlo. dB

Fig. 1 interfering targets when r I = 0 and r z = 2 __ without interfering targels

Performance of C A detector for M-sweeps without and with

A M = I A M = 2 0 M = 3 I M = 4

N = 24. P,. =

105

.- ~ - /--

M = 4 3 2 1,'

C 0 c

$ 0 6 /

0 8 -

signal-to-noise ratio, dB

- M = 4 3

/

Fig. 2 interfering targets when r , = iJ and r2 = 2

PerJormance of GO detertor for M-sweeps without and with

1.0

0 8 C ... " 2 0.6 U

__ without interfering targets A M = l

r

-

A M = 2 0 M = 3 1 M = 4

N = 24, P , = IO-'

+ 0 >. c

.: 0 4 - n 0 0 a

0.2

0-

-

& , I I I I 10 15 20 25 30

signal-to-noise ratio, dB Fig. 3 interfering targets when r , = 0 and r2 = 2 ~ without interfering targets

Performance of SO detector for M-sweeps without und with

A M = l A M = 2 0 M = 3 1 M = 4

N = 24, P, = 10.'

the interfering targets are located in both leading and lagging windows. In Figs. 4 and 5, the detection per- formances of GO and SO processors are plotted for the same values of the parameters as before, except that we take rl = 1 and r2 = 1. The performance of the CA scheme in this case is unchanged as the actual position of the interferers is irrelevant for this processor. The per- formance of the SO detector is still superior to that of the other two detectors for large SNR.

The actual probability of false alarm (PA) for the three processors is shown in Figs. 6-8. The displayed results show that PA decreases monotonically with the increase of M in the case of CA and GO schemes. For the SO processor, PA is approximately constant at a value near the desired false alarm probability and this value approaches P , as the number of sweeps M increases.

106

Fig. 4 interjenny targets when r , = I and rl = 1

Performunce of CO detector for M-sweeps without und with

~~ without interfering targets a M = l A M = 2 0 M - 3 I M = 4

N = 24, P , = I O

0.8 l ' O [

E

c

2 0.6 - 0

0 x

t

- < 0.4 - 0 n 0 a

0.2 -

I I I 1 5 10 15 20, 2 5 30

Performance of SO detector for M-sweeps without and with signal-to-noise ratio. dB

Fig. 5 interfering targets when r , = I and r2 = I __ without interfenng targets a M = l A M = 2 0 M = 3

M = 4 N = 24, P,o = IO-"

The curves for the CFAR loss, relative to the fixed threshold detector, for the processors under consider- ation, are shown in Figs. 9-11. From the displayed results we note that, as M increases, the number of refer- ence cells required to achieve a prescribed loss decrcases.

5 Conclusions

The performances of CA, GO and SO detectors pro- cessing M independent sweeps are derived. Analytical expressions for the detection probability of these schemes are obtained under the assumption that arbitrary numbers of interfering targets are present in the leading and lagging windows of the reference background noise. The primary and secondary interfering targets are

I E E Proc.-Radar, Somr Navig., Vol. I41, No. 2, April 1994

I

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Fig. 6 ( C A ) proressor for M-sweeps

I N R = O A M = I A M = 2 0 M = 3 I M = 4

Actual probability of false alarm (PA) of the cell-averaging

N = 24. ,, = 0, r 2 = 2, P I. - - 10-6

- 1 8 1

'1 2 -4 ; lb 1; 2b 'iz; %3; average interference-to-noise ratio, dB

Fig. 7 processor for M-sweeps 0 I N R = O A M = l A M = 2 0 M = 3 1 M = 4

N = 24, r , = 0, r 2 = 2, P , = LO-"

Actual probability of false alarm (PA) of the greatest of (GO)

assumed to fluctuate in accordance with the Swerling I model.

The detection performance of CFAR detectors men- tioned in this paper is superior in the case of homoge- neous background noise. However, a serious degradation of detection probability of the CA-CFAR processor is noted owing to the presence of interfering returns in the reference cells. The GO-CFAR scheme is proposed to maintain a constant false alarm rate at clutter edges, when the clutter samples are located in either the leading or the lagging side of the test cell. For this reason, the performance of the GO detector when interfering returns are located in both sides of the test cell is better than in the case where the interferers are present in either side.

I E E Prw-Rudar, Sonar Nauig. Vol. 141, No . 2, April 1994

average interference-to-noise ratio. dB

Fig. 8 processorfor M-sweeps A M = l A M = 2 0 M = 3 1 M = 4

N = 24, r , = 0. r z = 2, P,. =

Actual probubility of false alarm ( P A ) of the smallerr-of (SO)

1 i ,

'

-03

V

2 -

1 -

OO k l b 15 210 215 3b number of samples , N

Fig. 9 ~ without interfering targets

SNR(IB) loss of cell-averaging (CA) processorfor M-sweeps

A M = 2 0 M = 3

PO = 0.5, P,, = 1 M = 4

I , = 0, rz = 2

However, the GO processor is incapable of resolving closely spaced targets. On the other hand, the SO-CFAR detector resolves the primary target in multiple-target environments when all the interferers are located on either side of the test cell. However, the SO processor degrades considerably if the interfering targets are located in both the leading and lagging windows. This is simply due to the fact that at least one of the interfering targets will influence the threshold by raising its value, leading to masking of the primary target. Therefore the performance of the SO scheme when the interfering returns are present in one side of the test cell is better than in the case where the interferers are located in both sides.

Our results can be extended to the chi-square target fluctuation model to include all the Swerling cases.

107

7 5 -

m 4 -

VI n - 0 3 - a d

2 -

number of samples N

Fig. 10 SNR(dB) loss ofereatest-of(G0) processorfor M-sweeps without interking targets ~~

U M = 3 I M = 4

PD = = 05. Pfn = r , = I , r z = 1

number of samples, N

Fig. 11 __ without interking targets

SNR(dB) loss ofsmallest-of(S0) processorfor M-sweeps

A M = l A M = 2 I M = 4

PD = 0.5, P, = r , = 0. r l = 2

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108 IEE Proc.-Radar, Sonar Nauig., Vol. 141, No. 2, April 1994