Performance analysis of the OS family of CFAR schemes with incoherent integration of Ad-pulses in the presenlce of interferers

M.B.EI Mashade

Indexing terms: Ordered statisti,:s, CFAR, Multiple targets, Background noise, Moving-target indicator

~~~ ~

Abstract: The author analyses the performance of the ordered statistics (OS) CFAR schemes for M- correlated sweeps when these processors operate in multiple-target environments. These schemes include the conventional OS scheme as well as its modified versions. Members of the OS family use local estimators based on ordered statistics and generate both a near and far range noise-level estimate. Local estimates are then combined through a mean-level (ML), a maximum (MX) or a minimum (MN) operation to estimate the final noise power level. Exact expressions are derived for the detection probabilities of these techniques under a Rayleigh fluctuating target model. The processor perform,mce for that case is characterised by an integral equation, which is solved numerically 1 o assess the radar system’s behaviour against the background conditions. It is shown that for a fixed M , the relative improvement, over the single sweep case, increases as the correlation between consecutive sweeps decreases. For the same parameter values, the ML-OS detector gives the best homogeneous performance. In multiple-target situations, the ML-OS scheme still has the best performance, given that the number of interferers in each half of the reference window is within its allowable values (R I N - K) . On the other hand, the MN- OS scheme is preferable when a cluster of radar targets appears among the candidates of the reference window.

1 Introduction

Radar systems are used to detect targets of interest, on the basis of the energy returned to the radar antenna resulting from a transmitted signal that illuminates and returns from the target. The basic problem in radar detection is the discerninent of signals in a background of noise of the thermal type and clutter, which origi- nates in both the receiving system and the external 0 TEE, 1998

Paper first received 2nd September 1996 and in revised form 29th May 1997 The author is with the Electiical Engineering Department, Faculty of Engineering, Al Azhar University, Nasr City, Cairo, Egypt

TEE Proceedings online no. 19W1607

IEE P m - R a d a r , Sonar. Navig., IJol. 145, No. 3, June 1998

environment as the result of natural and artificial phen- emena. Since fixed-threshold detectors require knowl- edge of the disturbance distribution, they are not useful in this situation.

A common approach, in white Gaussian back- ground, is to employ a detector that sets the threshold based on local estimates of the total noise power. A rel- atively simple algorithm uses the averaged received energy in the nearby range cells to obtain the detection threshold. This processor yields a CFAR when the clut- ter in the estimation cells is identically independent and Rayleigh envelope distributed [I]. When the size of the reference set approaches infinity, the variance of the clutter power estimate vanishes, which means the con- sistency of the estimator and the threshold is set opti- mally, in the sense that its probability of detection approaches that of the Neyman-Pearson scheme.

Recently, we have been interested in detection in clutter fields and in multiple target environments. In this situation, the CA-CFAR scheme can be shown to exhibit an intolerable rise in false alarm probability during abrupt transitions of clutter power levels, and significant degradation in detection probability in the presence of extraneous targets [2, 31. Since the presence of clutter returns among the reference cells degrades the processor performance, it is natural to attempt to remove these samples from the reference sequence. Sev- eral nonlinear estimation techniques have been pro- posed to carry out this task. Recent interest has focused on ordered statistics (OS) based algorithms for detecting targets in nonhomogeneous backgrounds [4]. The OS-CFAR is a detection algorithm in which the threshold is just a scalar times one of the ranked refer- ence cells. This concept provides inherent protection against drastic drop in performance owing to the pres- ence of outlying targets. However, the large processing time required by this technique in ordering the refer- ence samples limits its practical applications. To solve this problem, modified versions of this processor have been suggested [5]. These versions rely on dividing the estimation cells of the reference set equally into two groups on either side of the target cell being examined. The samples of each group are separately processed to estimate the group noise power level. The mean-level (ML), the maximum (MX) or the minimum (MN) of the two noise level estimates is taken to construct the final noise level estimate in the proposed detector. Sev- eral authors [6, 71 have analysed the three mentioned CFAR processors under homogeneous and nonhomo- geneous backgrounds. An extensive amount of research has been performed on OS schemes using a single pulse

181

only. However, the noncoherent integration of M pulses is a promising technique for improving the CFAR processor performance [3, 81. On the other hand, most radar systems need some form of Doppler processing to filter out clutter and thereby reveal faster targets. The use of a moving target indicator (MTI) as part of the detection process attenuates the input signal power at frequencies where the clutter return is domi- nant. However, the presence of an MTI complicates the analysis of the detection system performance, since its output sequence is correlated even though the input sequence may be uncorrelated. This is one situation in which the consecutive sweeps are correlated. Further- more, when the radar target detection takes into account the simultaneous effects of scanning antenna beam shape, input clutter interference, delay line can- cellers and weighted post detection integrators, the suc- cessive pulses become correlated. Therefore the perfomance evaluation of the OS based algorithms for M-correlated sweeps is of considerable practical impor- tance. When M of these pulses were summed, their probability density function was determined in 191 for arbitrary correlation at the input to the square-law envelope detector. This result was used in [lo] to evalu- ate the homogeneous performance of the CA scheme for the case where the correlation matrix has a tridiag- onal form. In this paper, we use this form of correla- tion matrix to evaluate the performance of the OS family of CFAR procedures in homogeneous as well as multiple target environments.

2 Performance evaluation model

select k,th

sample

2. I System model The system to be analysed is depicted in Fig. 1. In each channel, in-phase and quadrature, the input pulses con- sist of a signal from a slowly fluctuating target, which

select kith

sample

has been amplitude modulated by the two way gain pattern of a scanning antenna and contaminated by additive stationary Gaussian noise-plus-clutter. In the presence of this type of clutter, the moving target indi- cator (MTI) is sometimes used to detect moving targets by performing a high pass filter process. The weights of the filter taps are chosen such that the resulting trans- fer function of the MTI attenuates the input signal power at frequencies where the clutter return is domi- nant. This is achieved by coherently subtracting each range gate return from a delayed version of the previ- ous echo from that range gate. If nothing has changed, cancellation of clutter signals occurs which would be complete in the absence of noise. On the other hand, if the echo has slightly phase changed, because of its motion, the cancellation will be less complete.

The presence of an MTI complicates the analysis of the detection system performance, since its output sequence is correlated even if its input sequence is uncorrelated. The MTI outputs are quadratically detected, and then M successive detected outputs are incoherently integrated to form the input to the CFAR detection algorithm. Since the integrated variables are correlated, this poses two problems: firstly, the determi- nation of the detection threshold must take into account the noise correlation and secondly, the pulse- to-pulse correlation degrades the system detection per- formance, as indicated in the literature [9, 1 I]. The type of MTI considered in this paper is a nonrecursive dig- ital filter that linearly combines the inputs hk from L successive pulse repetition intervals (PRIs). Thus the MTI output xk is given by

L

xk = Wthk-t+l (1) t=1

The fixed weights Wes applied to the filter taps are

cell under test -

Fig. 1

182

Block diugrum of OS-CFAR detectors processing datu from M-correlated sweeps

IEE ProcRadar, Sonar Navig., Vol. 145, No. 3, June 1998

chosen in such a way that the dominant clutter return signal components are attenuated. The input samples h k s are assumed to be mutually independent white Gaussian random sequences with variance a2 and mean uk. Thus the sequence xk is Gaussian with mean f, where

L

5 k E { X k 1 = WeUk-L+l (2) !=1

and with covariance

E{xkxk+n-[k</c+n} =I { L g n ~ f ~ t + n In1 < L

1121 L L ( 3 )

Similarly, if the input sequence g k to the quadratic MTI has the same variance as the inphase channel and with mean vk, its output sequence yk has a Gaussian distri- bution with mean Ek and with the same covariance as that of xk. In addition,

Referring to Fig. 1, the box labelled 'cell under test' represents the radar range cell that is currently being examined for the presence of the target. To make this determination, the output of this cell Vo is compared with a threshold.

HI

E { X k Y k + % - S k & + n } = 0 (4)

L; ZT (5) HO

As in many algorithms designed for clutter scenarios, the OS family bases its tlireshold estimate on collecting an even number N of range cells surrounding the test cell. Buffer cells, not shown in Fig. 1, adjacent to the cell under test can be used to avoid contamination with the edge of the output coming from the target return. These reference samples are equally partitioned into leading and lagging windows on both sides of the tar- get cell. The samples of leach set are ranked in increas- ing magnitude and the K,th ordered cell is chosen to represent the noise level estimate of the leading win- dow. Similarly, the K2th ranked cell is chosen from the reference cells of the lagging window to represent its noise level estimate. The two noise level estimates are combined to create the final strength of noise level, 2, which is then multiplied by a predetermined constant 'T to construct the system detection threshold.

2.2 Mathematical model The performance of a lhreshold setting algorithm in various environments may be evaluated with either the Monte-Carlo approach or the closed-form approach using probability distrilsution functions. The latter, when mathematically tractable, is generally preferred since it yields more precise results in less computer time. The closed-form approach is used here to evalu- ate the radar detector performance. The key input to the closed-form approach is the distribution function used to model the clutter. A variety of probability den- sity functions may be used to model this clutter, with the Rayleigh, log-normal, Weibull and K-distribution being the most common in the literature. Here, the Rayleigh distribution is used to model the background clutter.

The processor performance is completely determined by calculating its false alarm and detection probabili- ties. The values of the detection threshold and the Pfa

IEE Proc -Radar Sonar Nuvcg VOI 145 No 3 June 1998

- 1 p12 p13 p14 . ' . P l M - P2l 1 p23 P24 . ' . P2M

p31 p32 1 p34 " * P3M

= p41 p42 p43 1 * ' * P4M

. . . .

- 1 - p M l . . . . . . . . . . . .

are related through the relation

Pfa = p 4 ./7uiPvo(wlHo)dudl (6)

where pvo denotes the pdf of the cell under test and the hypothesis Ho represents the absence of the radar tar- get signal. This relation gives a descriptive explanation of the way the pdf pz(z) affects the Pya. The output pdf pz(z) depends on the input pdf and the clutter mean power estimator. On the other hand, in the alternative hypothesis H I , Pd can be determined as

p d = i m P Z ( i ) ~ ~ ~ V o ( u l ~ l ) d u d ~ (7)

The data to be processed can be presented in a matrix form. Let the number of rows of this matrix be A4 and the number of its columns be N + 1; N columns for the reference noise samples and one column for the target cell being examined. This means that the mean noise power is estimated by collecting A4 sweeps of N refer- ence samples each. In other words, the total number of noise samples used in estimating the detection thresh- old is A4 x N . The noise cells are assumed to be inde- pendent within the same sweep, but they are correlated from one sweep to another. When considering the ith sweep, let dY and qL represent the content of the noise cell and the cell under test, respectively. Integration of A4 pulses means that each column in the reference matrix is added up. The random variable describing the sum of thejth column is denoted by XI while that rep- resenting the sum of the column under test is denoted by V,. Thus, the two created random variables are

M M

i=l i= 1 and

M M

where M M

G, b L ; ) L ~ ) (12) n=l m r l

and 0 for clear background

w = { $I for interfering target (13) $A for primary target

A, and L(zl denote the ith eigenvalue and the corre- sponding eigenvector of the covariance matrix, I) repre- sents the total clutter-plus-thermal noise power, I denotes the interference-to-total noise ratio (INR) and A indicates the SNR of the primary target. This means that the Swerling I model is assumed for the radar returns from the primary and the secondary interfering targets.

It is to be noted that eqn. 11 is derived for the case where the random variables qis are correlated and iden- tically distributed, each with a pdf given by

where U(q) represents the unit step function and for clear background

p = $(1 + I ) for interfering target (15) r $(l + A) for primary target The Laplace inverse of eqn. 11 gives the pdf of the inte- grator output V,. This pdf can be easily obtained by rewriting its CF in the following form

(16) 4 ~ 0 ( s ) = a o s M + a l s M - l + a 2 s M - Z + . . . + a ~ - l s + a ~ 1

The coefficients of the above polynomial are functions of A,, L(", I) and w. If the roots of this Mth order poly- nomial are determined, either analytically or numeri- cally, we can compute the corresponding pdf by evaluating its Laplace inverse, which in this case becomes a simple task. Let cJ, j = 1, ..., M , denote the roots of this polynomial in the case of the cell under test, then eqn. 16 can be put in another simpler form

M

The Laplace inverse of the above equation gives the pdf of the cell under test variate which becomes

M

PV, (U) = D, exP(-c,U) (18) 3=1

where

2 5 3

In the above expression, we assume that all the poles of the CF are simple. However, if there are repetitive poles, the partial fraction technique can also be applied to calculate the pdf.

Finally, the substitution of eqn. 18 in the definition of Pd gives

M -

j=1 "3

This is the basic formula of our analysis that is used in the OS processor performance evaluation. From the above relation, we note that the CF of the noise level estimate, &,(s), is the key quantity in our analysis.

184

3

A block diagram of the generic OS-CFAR processor is shown in Fig. 1. These detection schemes are based, in their operation, on estimating the strength of the noise level in each reference window and then combining them through mean-level, maximum or minimum processing to establish the final noise power estimate 2. The common theme of these techniques is the evalu- ation of 2 according to the OS algorithm. In this Sec- tion, we analyse three OS-CFAR processors, namely the ML-, MX- and MN-OS schemes, and we evaluate their performance in homogeneous as well as in multi- ple target situations, under the assumption that these detectors are processing data from M correlated sweeps.

3. I One-window OS-CFAR detector Let us denote Xlli, i = 1, 2, ..., N, the ith smallest sam- ple of the set { X I , X2, ..., X,}. Each sample in this set is, as previously defined in eqn. 8, obtained by sum- ming M dependent and identically distributed cells. In the OS-CFAR detection scheme, the samples of the ref- erence set are sorted in an increasing order, i.e.

Performance analysis of OS detectors

X(1) I X(2) I X(3) I . . . I X ( K ) I . . . I X ( N ) (21)

The detection threshold is obtained by selecting the Kth ranked cell to represent the noise and clutter level, and then multiplying the input to that cell by a constant scale factor T. Thus, the statistic 2 for this algorithm takes the form

Z = X ( K ) 1 < K < N (22) In the following, we will denote by OS(K) the OS detector with parameter K. The value of K is generally chosen so that the detection probability (in a homoge- neous background) is maximised. In a reference set of size N = 24, the optimum value of K, in the single sweep case, is 21 [2]. On the other hand, the OS(K) processor can resolve up to N - K outlying targets from the primary one. In order to analyse the detection per- formance of this procedure in a multiple-target situa- tion, suppose that the reference set contains R returns coming from extraneous targets and Q (Q = N ~ R) ones from the clutter-plus-thermal noise region. Hence, the estimated total noise power (2) from this set has a cumulative distribution function (cdf) given by [7]

N J 2

F z ( z ) = ( Q ) (. .) [1 - Ft(2)IQ-J z=K 3 = J l 3 a - 9

x [F, ( 4 1 9 [ 1 - F, (41 R-z+g [F, (z)]"-' U ( 2)

( 2 3 )

(24)

where J1 = max(0,i - R)

and

5 2 = min(i, Q) (25) F,(z) represents the cdf of the cell that contains only thermal noise and F,(z) denotes that for the interfering target cell. If the reference samples are all of the first type (R = 0), then the above equation becomes

IEE Proc.-Radar, Sonar Navig.. Vol. 145, No. 3, June 1998

In the case of M-correla ted sweeps, the Laplace trans- formation of F,(z) can be obtained from eqn. 11 after replacing U by VI. Thus

(27) Similarly to eqn. 17, this expression can be written in a more simple form as a function of the roots of its asso- ciated polynomial. By denoting these roots by b,, i = 1, ..., M , eqn. 27 takes the form

1

1 oi

s . z = 1 s + b , 4 F , . ( S ) = - rI -

The Laplace inverse of the above equation gives Fr(z), which becomes

k4

FT(z ) = 1 - :< BI, exp(-bkz)U(z) (29)

with k = l

M , 02

Bk == rI i = l a # S

On the other hand, the Laplace transformation of F,(z) has the same form as that given by eqn. 27 except that Zmust be zero. Thus

. M 1 (31)

1 ix, 4 F t ( s ) == - n 7

J=1 s+?Lx, The Laplace inversion of eqn. 31 gives the cdf of the content of the thermal noise cell. Therefore

with

(33 ) Z # S

By substituting eqns. 29 and 32 into eqn. 23, we can evaluate the cdf of the test statistic Z of the OS(K) detector. The final step is to compute its probability of detection. To derive it, we need to calculate &+), see eqn. 20. In terms of @,&), we can obtain &(s) from the following mathematical relation:

4z(4 = S 4 F Z (s) (34) Since &-(s) has no closed form, we evaluate it numeri- cally from the well known formula

3.2 Two windows OS-CFAR detectors Modifications of the OS-CFAR schemes have been proposed to overcome the problem associated with the large processing time required by this technique in ordering the elements of the reference set. These schemes constrwt their threshold by processing the contents of the two equal size subsets surrounding the cell under test. The samples of each subset are inde- pendently ordered, and subsequently compared under the mean, the maximum or the minimum criterion. The

IEE Proc -Radur, Sonar Navig , Vi11 145, No 3, June 1998

K,th ranked cell is taken from the leading subset and the K,th one from the lagging subset to represent their noise level estimates Z1 and Z,, respectively. Thus

(37) N

2 2 = X ( K * ) 1 L K2 I - 2 In terms of Z1 and Z,, we can analyse the modified OS detectors.

3.2. I Mean-level (ML) OS detector: In the ML- OS processor, the estimation of the noise level in the test cell is the sum of the leading and lagging noise level estimates.

Z M L = 2 1 + 2, Using eqn. 23 for the cdf of a Kth representative cell from a set of size N , the cdfs of Z1 and 2, can be eval- uated by replacing the parameters N, K, Q and R by NI 2, K,, Ql and RI for Z1, and by Nl2, K,, Q2 and R, in the case of 2,. In the obtained mathematical expres- sions, RI and R, represent the number of interfering target returns that exist among the reference samples of the leading and lagging subsets. On the other hand, Ql = NI2 - R I and Q2 = NI2 ~ R, denote the samples that contain thermal noise and clutter in each subset.

Since Z1 and 2, are assumed to be statistically inde- pendent, the CF of the estimated noise power is obtained by multiplying their CFs. Thus

(38 )

2

4 Z M L (SI = rI 42, (SI

4 Z 7 (SI = S 4 F 2 , (SI

(39)

(40)

3=1

where

As mentioned before, the Laplace transform of the Z,s cdf has no closed form and it has to be computed numerically from its basic definition. Finally, the detec- tion performance of the ML-OS processor can be eval- uated by replacing $z(s) in eqn. 20 by that given in eqn. 39.

3.2.2 Maximum (MX) OS detector: The reduc- tion of the number of excessive false alarms at clutter edges is the prime motivation for exploring the MX-OS procedure [5, 71. In addition, this processor can be used for detecting radar targets in the presence of spurious targets. In this type of detection, the total noise power is estimated from the larger of the two separate OS sta- tistics computed from the two reference subsets. There- fore the noise power is estimated as

Since Z,, is given by the maximum of 2, and Z,, its cdf may be written in terms of their cdfs as [7]

Z M X = max(Z1,22) (41)

2

F Z M X = n Fz, (.I (42) 3=1

The Laplace transformation of the above equation will give the s-domain representation of the cdf of ZMx By evaluating this transformation, the CF of the statistic under consideration can be obtained rrom

4 Z M X (s) = S 4 F Z M X (3) (43) Once the CF of ZMx is calculated, the detector per- formance can be easily evaluated.

185

3.2.3 Minimum (MN) OS detector: This proces- sor has been introduced to alleviate problems associ- ated with closely spaced targets leading to two or more targets appearing in the reference window. While test- ing for target presence at a particular range, the proces- sor must not be influenced by the extraneous target echo. This scheme of detection estimates the noise power from the smaller of the leading and lagging noise power estimates. Thus the statistic of this proces- sor takes the form

Consequently, this noise level estimate has a cdf given Z M ~ J = min(Z1,Zz) (44)

(45) by 131

FZMN (4 = Fz1(.) + Fz2 (.I - F Z M X (.I where the last term in the above equation is that given by eqn. 42. The Laplace transformation of eqn. 45 yields

4 F z M N = 4Fzl + ~ F z , - 4 F z n n X (46) It is to be noted that all the right terms of the above expression are previously calculated. Hence, the CF of the test statistic ZMN can be evaluated using the stand- ard mathematical relation

4ZMN (s) = (SI (47) and the evaluation of the processor performance becomes straightforward. We end this Section by stat- ing that the false alarm probability Pfa of the four men- tioned processors can be easily calculated by setting the SNR ( A ) equal to zero in the expression of their detec- tion performance.

4 Numerical results and performance comparisons

The performance of the OS family of detection schemes is now numerically evaluated in terms of the variation of Pfa with the threshold parameter T when the back- ground noise is homogeneous, the detection perform- ance in homogeneous as well as nonhomogeneous environments as a function of the primary target SNR and the SNR required to achieve a prescribed operat- ing point (Pfa, Pd). Our numerical results are obtained for a two pulse MTI (L = 2 in eqn. 1) canceller. The correlation matrix /3 for that type of MTI is a tridiago- nal symmetric Toeplitz matrix. Each element in its main diagonal is unity, while each one in the upper and lower diagonals equals p. Thus

0 0 ' . '

. . ... P ] 0 P 1

(48) This means that the correlation coefficient (Y) between the reference samples xY and x(,+k), can be described by the following relation:

p for lkl = 1 i 0 for lkl > 1 rz,%+k = 1 for k = 0 (49)

The behaviour of Pfa for the OS family of CFAR schemes is presented here as a function of T, M and p. In addition, we make a comparison between the per-

formance of these schemes in stationary clutter as well as in multiple-target situations. Fig. 2 depicts the varia- tion of Pfa with the thresholding parameter T for the ML-OSD. The curves of this Figure are parametric in M and p. In the curves of this plot, the notation 2,0.2 means that the indicated curve is drawn for M = 2 and for p = 0.2. Clearly, Pfa decreases as either M increases or p decreases. This result is expected because an increasing M results, on average, in a larger value for the statistic 2. Since the threshold parameter T is held constant, the estimated value for the detection thresh- old (ZT) increases and consequently Pfa decreases. The same thing occurs when the sweep-to-sweep correlation coefficient p decreases. We note also that, for fixed A4 and p values, Pya decreases as T increases. Since A4 and p are fixed, the estimated noise power (2) is constant and increasing T leads to increasing threshold (273.

1 1

0.5 2.5 4.5 6.5 8.5 10.5

threshold constant T Fig.2 -False alarm probability variation versus constant scale factor T, parametric in number of sweeps M and sweep-to-sweep correlation coej3- cient p, for ML-OSD with N = 24 and K, = K2 = 10 M and p values: A 2,0.2; A 2,0.5; 0 3,0.2; W 3,0.5; 0 4,0.2; 0 4,0.5

0.5 2.5 4.5 6.5 8.5 10.5

threshold constant T Fig. 3 ,False alarm probability variation versus constant scale factor T, parametric in number of sweeps M and sweep-to-sweep correlation coef3- cient p, for MX-OSD with N = 24 and KI = Kz = 10 M and p values: A 2,0.2; A 2,0.5; 0 3,0.2; W 3,0.5; 0 4,0.2; 0 4,0.5

A correspondingly smaller value for Pfa is therefore obtained. The same results for the MX-OSD are shown in Fig. 3. The curves behave like the previous ones, except that their rate of decrease is less than in the case

IEE Proc -Radar, Sonar Navig , Val 145, No 3, June 1998 186

of ML-OSD. In Fig. 4, we show the same Pfa plot for the MN-OSD but with a lesser rate of decrease, relative to the ML-OSD results. As a comparison, we evaluate the same characteristics for the conventional OSD with the optimum value fix its ranking parameter K, OS(21), see Fig. 5. Similar behaviour is also observed with a rate of decrease greater than that of MX- and MN-OSD and smaller tjhan that of ML-OSD. In all the cases, the numerical results are computed for N = 24 and for symmetrical parameter values ( K , = K2 = 10).

0.8-

0.6-

0.4 -

0.2-

m d E - ;ir

- P m

c c

0.5 2.5 4.5 6.5 8.5 10.5

tlhreshold constant T Fig. 4 False alarm probability variation versus constant scale factor T, purumetric in number of sweeps M and sweep-to-sweep correlation coefj- cient p, for MN-OSD with N = 24 and K, = K, = IO A4 and p values: A 2,0.2; A 2,OS; 0 3,0.2; II 3,0.5; 0 4,0.2; 0 4,OS

' x

0.5 2.5 4.5 6.5 8.5 10.5

theshold constant T Fig. 5 False alarm probubilit:~ variation versus constant scale factor T, parametric in number of sweeps M and sweep-to-sweep correlation coejfi- cient p, for the conventional OSD with N = 24 anti K = 21 M and p values: A 2,0.2; A 2,0.5; 0 3,0.2; W 3,0.5; 0 4,0.2; 0 4,0.5

We now turn to the processor performance evalua- tion. Using computer iterations, the threshold multi- plier T was obtained irnder the assumption that no interfering target returns are present among the con- tents of the reference windows. Numerical integration yields the relation between the desired Pfa and T (sev- eral iterations are needed if Pya is given and T is unknown). In Figs. 6-13, we compare the detection performance of the different OS schemes in terms of P, against the primary target SNR, for several values of p, when the number of integrated pulses equals 2 and 4.

U a t .- c 0)

U 4-

c

'1

I

I I I I I 0 5 10 15 20 25 30

0 1 signal to noise ratio,dB

Fig.6 Detection pevJormance of ML-OSD as a f i c t i o n of SNR (dB) in the absence as well as in the presence of four spurious targets for N = 24, M = 2, P, = IQ6 and K, = K, = IO A 0.2,0,4; A d5,0,4; 0 0.2,1,3; W 0.5,1,3; 0 0.2,2,2; 0 0.5,2,2

1

0.8

2 K ._ p 0.6 c

-0 c

!$ 0.5 m R

Q e

0.2

0 0 5 10 15 20 25 30

signal to noise ratio,dB Fig.7 Detection performance of ML-OS0 as a function of SNR (dB) in the absence as well us in the presence offour spurious targets .for N = 24, M = 4, P, = I@andK, = K, = IO A 0.2,0,4; A d5,0,4; 0 0.2,1,3; W 0.5,1,3; 0 0.2,2,2; 0 0.5,2,2

0.8 1

"I I I I I I 0 5 10 15 20 25 30

signal to noise ratio,dB Fig.8 Detection performance of MX-OSD as a function of SNR (dB) in the absence as well us in the presence of four spurious targets for N = 24, M = 2, P = IO6 and Ki = Kz = 10 A 0.2,0,4; A &5,0,4; 0 0.2,1,3; 0.5,1,3; 0 0.2,2,2; 0 0.5,2,2

IEE Proc-Radar, Sonar Navig., Vol. 145, No. 3, June 1998 187

1

0.8 U a

._ s e

Q 0.6 U r

._ F n m 0.4 n e a

- ._

0.2

0

I

0.8 - U a

._ s e

Q 0.6 - U r

._ F n m 0.4 n e a

- ._ -

0.2 -

0 5 10 15 20 25 30

signal to noise ratio,dB Fig.9 Detection performance of MX-OSD as a junction of SNR (dB) in the absence us well as in the presence of four spurious targets for N = 24 M = 4 P =1P6andK - K -10 Ab.2,0,4; A 0?5,0,4; 0 0 . 2 , 1 , $ : i 0%>,3; 0 0.2,2,2; 0 0.5,2,2

1

a"

0.1 5 10 15 20 25 30

signal to noise ratio,dB Fig. 10 Detection evformance of MN-OSD as a function of SNR (dB) in the absence as welf as in the presence of four spurious targets for N = 24 M = 2 P =1P6andK,=K,=10 Ab.2,0,4; b 0?2,1,3; 0 0.2,2,2; A 0.5,0,4; 0 0.5,1,3; 0 0.5,2,2

a" C ._ c a,

U c

is

1

0.1 5 10 15 20 25 30

signal to noise ratio,dB

Fig. 11 Detection erfonnunce of MN-OSD as a function of SNR (dB) in the absence as weias in the presence of four spurious targets for N = 24 M = 4 P = 1 0 - 6 a n d K , = K , = I 0 A b.2,0,4; b 0?2,1,3; 0 0.2,2,2; A 0.5,0,4; 0 0.5,1,3; 0 0.5,2,2

For each processor, the performance is evaluated for homogeneous as well as for multiple target environ- ments. In the presence of interferers, the detector per- formance is obtained for a possible practical application where the signal and the interference strengths are of the same order (SNR = INR). The nonhomogeneous performance is evaluated in the pres- ence of four spurious targets with their possible differ- ent locations: (R1 = 0, R2 = 4), (RI = 1, R2 = 3 ) and (R, = 2, R2 = 2). In the conventional OSD, the per- formance is obtained for different values of the number of extraneous targets: R = 1, R = 2, R = 3 and R = 4 when K = 21. In these Figures there are two families of curves: the homogeneous curves represent the processor performance when the background noise is homogene- ous, while the other family of curves represent the mul- tiple target performance. In each Figure, the notation 0.2,1,3 on a curve means that it is derived by setting p = 0.2, RI = 1 and R2 3. It is known that the OS proc- essor performance is highly dependent upon the value of the ranked parameter K. In other words, if a single interfering target return appears among the reference cells, it occupies the highest ranked cell with high prob- ability. If K is chosen to be N, the detector will almost set the threshold based on the value of the extraneous target. This results in an increase of the threshold and may lead to a target miss. If, on the other hand, K is chosen to be less than the maximum value, the OS processor will be influenced only slightly for up to N - K spurious targets. For a two-windows OS processor, if Kl is chosen to be 10, then the processor is able to dis- criminate the primary target from, at most, two inter- fering ones in each half of the reference window. In Figs. 6-7, the detection probability is plotted as a func- tion of the primary target SNR for the ML-OSD with two and four correlated sweeps. It is shown that there is a performance degradation when the above state- ment is not fulfilled, (R, = 0, R2 = 4), (RI = 1, R2 = 3) cases and the best nonhomogeneous performance is obtained in the case of (R, = 2, R, = 2), where the con- dition RI = NI2 - Kl and R2 = NI2 - K2 is verified. As expected, these Figures show a drastic decrease in the processor performance as p increases. The curve labelled optimum describes the ideal detector perform- ance. Figs. 8-9 depict the same results for an MX-OSD under the same conditions. The behaviour of this proc- essor in multiple-target environments is similar to that of the ML-OSD. Intolerable masking of the primary target occurs in the ML-OSD and MX-OSD when the four interfering target returns are not equally distrib- uted between the leading and $railing windows. The masking effect is greater in the MX-OSD than in the ML-OSD. On the other hand, thtt MN-OSD is the only scheme that is capable of resolvjng multiple targets as long as all the interferers appear in only one of the two reference windows, as shown in Figs. 10-11. It is important to note that the perfotmance of this scheme is plotted on a log scale for thLe results to be clear. Although The MN-OSD gives ithe best performance when the interferers are located in either reference win- dow, its performance degrades in the case where these interferers exist in both halves of the reference window. Figs. 10-1 1 clearly demonstrate this statement. For comparison, we evaluated the performance of OS(21) under the previous conditions and the results are drawn in Figs. 12-13. Here, the notation 0.2,4 means that the indicated curve is plotted for p = 0.2 and R = 4. In all

IEE Proc -Radar, Sonar Nuvig , Vol I45 No 3, June 1998 188

the cases, increasing the number of sweeps or decreas- ing the sweep-to-sweep correlation will improve the processor performance in any situation.

26 -

m 2- 22- z v)

U ._ P $ 18- E

1

0.8

a" c ._ c f, 0.6 c U c

._ b

e 0.4

m n a

0.2

0 0 5 10 15 20 25 30

signal to noise ratio,dB Fig. 12 Detection performance of the conventional OSD as a function of SNR (dB) in the absence as well as in the presence offour spur~ous targets f o r N = 2 4 M = 2 P = l Q 6 a n d K = 2 1 a 0.2,4: A 0.5,4; 0 O.$i; W 0.5,3; 0 0.2,2; 0 0.5,2

1

0.8 U a c ._ c

0.6 U r

._ b - 9 0.4 R

g 0.2

0 5 10 15 20 25 30

signal to noise ratio,dB Fig. 13 Detection performunte of the conventional OSD as a function of SNR (dB) in the absence as ?Jell as in the presence offour spurious tar- getAfor N = 24 M = 4 P = I O 6 and K = 21 A 0.2,4: A 0.5,4: 0 0.2,31 d0.5,3; 0 0.2,2; 0 0.5,2

There is an inherent loss of detection probability in a CFAR processor compared with the optimum proces- sor performance. This is owing to the fact that the CFAR scheme sets the detection threshold by estimat- ing the total noise power within a reference set of finite size. The optimum detector, on the other hand, sets a fixed threshold under the assumption that the total noise power is known a priori. Therefore it is obviously of value to have some idea about the loss of detection power for a proposed CFAR scheme relative to the optimum detector. In Figs. 14-17, we show the SNR (dB) required to achieve an operating point 0.9) versus the ranking order parameter (Kl = K2 = K ) for M = 2 and parametric in p . The required SNRs are computed in the absence as well as in the presence of interfering targets of the same cross-section as that of the primary target. In i;hese Figures, it is shown that the required SNR decreases monotonically as K

IEE Proc -Rudur. Sonur Navig., Vol. 145, No 3, June 1998

30 1 \\

p =0.2

I p =o

1 4 1 10 1 I I I I I 1

10 12 0 2 4 6 a ranking parameter K

Fig. 14 Required SNR against ranking parameter K ( K ~ K ~ K ) parametric in p for ML-OSD with N = 24, M = 2, Pfa = IO< id =";, RI = R , = I a p = 0.0; 0 p = 0.2; 0 p = 1.0

30

26

m U- 22 E z v) U P g 18 ??

._

14

10

p=0.2

~

p=o

I I I I I I 2 4 6 8 10 12

ranking parameter K Fig. 15 Required SNR againsi ranking parumeter K (Kj = K, = K) parametric in pjor MX-OSD with N = 24, M = 2, P, = IO', Pd = 0.9, R, = R , = I a p = 0.0; 0 p = 0.2; 0 p = 1.0

30 1

I p=o

"i 10 I I I I I I I

0 2 4 6 8 10 12

ranking parameter K Fig. 16 Required SNR against ranking parameter K (K - K - K ) parametric in p j o r MN-OSD with N = 24, M = 2, Pfu = IO-( jd ='Oj, R I = R , = l a p = 0.0; 0 p = 0.2: 0 p = 1.0

189

increases until it reaches a minimum value and then starts to increase. The value of K corresponding to this minimum value of SNR, in a homogeneous back- ground, is the optimum value that gives the maximum processor performance. It is also noted that the required SNR increases as the sweep-to-sweep correla- tion increases, since the correlation decreases the strength of the target signal power. In multiple target situations, we finish our curves before the number of outlying targets exceeds the difference between the set (or subset) size and K, since at that point the detection probability goes down.

1 4 ~ 10 0 4 8 12 16 20 24

ranking parameter K Required SNR against ranking parameter Kparametric in p for Fig. 17

conventional OSD with N = 24, M = 2, Pfa = a p = 0.0; 0 p = 0.2; 0 p = 1.0

Pd = 0.9, R = 2

5 Conclusions

The detection problem of a target, embedded in clutter, when the radar signal processor contains a nonrecur- sive MTI followed by a square-law integrator and a CFAR circuit, has been described. Since the output sequence of the MTI is correlated, the technique of whitening approach is applied in order for the consecu- tive samples to become independent. The density func- tion of the integrator output is given as a function of the number M of pulses integrated, the eigenvalues and eigenvectors of the correlation matrix of the MTI out- put. The CFAR processors considered in this paper are those based on the ordered statistic technique, where they are natural candidates for the outlier resistant CFAR design. Their performance has been evaluated in the absence of, as well as in the presence of, spurious targets. A two-pulse MTI system, the covariance matrix of which can be easily diagonalised, is chosen to

assess the behaviour of these schemes against the back- ground conditions. More precisely, a tridiagonal form is taken for the correlation matrix and the processor performance is evaluated for a different number of pulses and several values of the sweep-to-sweep correla- tion coefficient p. A Rayleigh fluctuating signal model, which is widely used for modelling both radar returns and fading communication channels, has been assumed in our analysis. It is noted that the considered detection schemes are of type CFAR (i.e. their probability of a false alarm is independent of the correlation matrix of the additive noise and clutter) only if the correlation coefficient p is assumed to be known. Since the correla- tion between the MTI outputs is a function of the number L of pulses processed by the MTI and its applied weights Wk, knowing p becomes a reasonable assumption.

From the results obtained, it is shown that for a fixed A4, the relative improvement, over the single sweep case, increases as p decreases. Furthermore, as p tends to 1, no gain is obtained for A4 > 1 over the sin- gle sweep case. In the presence of interferers, the ML- OS detector gives the best performance, and the MN- OS scheme is preferable when a cluster of radar targets appears in the reference window.

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References

RICKARD, J.T., and DILLARD, G.M.: ‘Adaptive detection algorithms for multiple target situations’, IEEE Trans. Aerosp. Electron. Syst., 1977, 13, (4), pp. 338-343 GANDHI, P.P., and KASSAM, S.A.: ‘Analysis of CFAR proces- sors in nonhomogeneous background’, IEEE Trans. Aerosp. Elec- tron. Syst., 1988, 24, (4), pp. 427445 EL MASHADE, M.B.: ‘M-sweeps detection analysis of cell-aver- aging CFAR processors in multiple targets situations’, IEE Proc. Radar, Sonar Navig., 1994, 141, (2), pp. 103-108 BLAKE, S.: ‘OS-CFAR theory for multiple target and nonuni- form clutter’, IEEE Trans. Aerosp. Electron. Syst., 1988, 24, (6),

ELIAS-FUSTE, A.R., DE MERCADO, M.G., and DAV- 0, E.R.: ‘Analysis of some modified order statistic CFAR: OSGO and OSSO’, IEEE Trans. Aerosp. Electron. Syst., 1990, 26, (I), pp. 197-202 YOU HE: ‘Performance of some generalised modified order-sta- tistics CFAR detectors with automatic censoring technique in multiple target situations’, IEE Proc. Radar, Sonar Navig., 1994, 141, (4), pp. 205-212 EL MASHADE, M.B.: ‘Performance analysis of modified ordered-statistics CFAR processors in nonhomogeneous environ- ments’, Signal Process., 1995, 41, (3), pp. 379-389 LIM, C.H., and LEE, H.S.: ‘Performance of order-statistics CFAR detector with noncoherent integration in homogeneous sit- uations’, IEE Proc. F, Radar Signal Process., 1993, 140, (5), pp.

KANTER, I.: ‘A generalization of the detection theory of Swer- ling’,Proceedings of EASCON74, 1974, pp. 198-205 AL HUSSAINI, E.K., and AL HUSSAINI, E.K.: ‘Performance

of mean-level detector processing M-correlated sweeps’, IEEE Trans. Aerosp. Electron. Syst., 1981, 17, (3), pp. 329-334 DILLARD, G.M., and RICKARD, J.T.: ‘Performance of an MTI followed by incoherent integration for nonfluctuating sig- nals’. IEEE International Radar conference, 1980, pp. 194-199

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