Detection performance of the trimmed-mean CFAR processor with noncoherent integration

M.B. El Mashade

Indexing terms: Constant false alarm rate processors, Radar detection, Trimmed-mean processors

Abstract: The ordered-statistic constant false alarm rate (OS-CFAR) detector has some advant- ages over the cell-averaging (CA) CFAR scheme, especially in cases where more than one target is present within the reference window or where this reference window is crossing clutter edges. The trimmed-mean (TM) CFAR processor is a more generalised OS-CFAR scheme which combines ordering and arithmetic averaging. The author analyses the detection performance of the TM-CFAR scheme when noncoherent integration is used under the chi-square target fluctuation model. For specific values of the trimming param- eters, the performance of the TM-CFAR processor reduces to that of the CA-CFAR and OS-CFAR schemes. Results are depicted for the Swerling case I1 target model and for a homogeneous environ- ment. As the number of noncoherently integrated pulses increases, lower threshold values and better performances are obtained for the three schemes considered.

1 Introduction

Constant false alarm rate (CFAR) processors are the commonly used detectors in radar to maintain control of the false alarm rate in the face of local variations in the background noise level. A simple approach to CFAR is to set the detection threshold on the basis of the average noise power in a given number of reference cells, where each of these cells is assumed to contain no targets [l-51. There are two major problems that require careful inves- tigation in the CFAR detection scheme. The first one is that presented by regions of clutter transitions. This situ- ation occurs when the total noise power received within a single reference window changes abruptly. The presence of such a clutter edge may result in severe performance degradation in an adaptive threshold scheme leading to excessive false alarms or serious target masking depend- ing upon whether the cell under test is a sample from clutter background or from relatively clear background with target return, respectively. The second problem is encountered when there are two or more closely spaced targets in range. The interfering targets that appear in the reference window along with the target under investiga- tion may raise the threshold unnecessarily and a serious

6 IEE, 1995 Paper 1626F (EIS), first received 25th November 1993 and in revised form 28th September 1994 The author is with the Electrical Engmeering Department, Faculty of Engmeering, AI Azhar University, Cairo, Egypt

18

degradation in detection performance of the processor will result [ 6 ] .

The ordered-statistics (OS) CFAR scheme has been introduced to alleviate these problems to some degree [ 4 ] . The OS-CFAR processor estimates the noise power simply by selecting the Kth largest cell in the reference window of size N . This scheme of detection suffers only minor degradation in detection probability and resolves closely spaced targets effectively for K different from the maximum. However, the OS-CFAR processor is unable to prevent an excessive false alarm rate at clutter edges unless the threshold estimate incorporates the ordered sample near the maximum; but in this case the processor suffers greater loss of detection performance.

The trimmed-mean (TM) scheme is a more generalised OS-CFAR processor in which the ordered range cells of a particular reference window are trimmed or censored from both the upper and the lower ends. The threshold is estimated by forming the sum of the remaining range cells. The TM-CFAR scheme reduces to the CA-CFAR and OS-CFAR schemes for specific trimming values. Our goal in this paper is to study the detection performance of a TM-CFAR processor that employs M-pulse non- coherent integration when the primary target fluctuates in accordance with the Swerling case I1 model.

In Section 2 an analytic expression is derived for the probability of detection of the processor under consider- ation when the background noise samples are homogen- eous. The performances of CA-CFAR and OS-CFAR processors are obtained by setting the trimming param- eters equal to specified values. The numerical results for the three detectors are given in Section 3.

2

Radar detection of a known signal in additive white Gaussian noise of unknown variance is often accom- plished by comparing the test statistic for a single-range cell with an adaptive threshold equal to a scaled estimate of the unknown noise variance. For a system that quad- ratically rectifies the output of a matched filter, the problem can be modelled by the following hypothsis- testing formulation

Analysis of the trimmed-mean detector

(1) under H , under H , ~ [ ( x + a)' + (y + b)']

The variables x and y are the inphase and quadrature Gaussian noise samples with zero mean and unit variance. The target echo produces a fixed signal-to-noise ratio (SNR) when the target is stationary, while the amplitude of the echo produces a SNR which varies ran- domly in the case of target fluctuation, where

S N R = +(U' + b 2 ) (2) I E E Pro<.-Radar, Sonar Nat'rg., Vol. 142, No. I , February 1995

Implementing the generalised likelihood-ratio test, the radar decides whether Ha or H I according to

HI

HO v S Z T (3)

where Z is an estimate of the noise variance and T is a positive threshold parameter. Typically, the radar uses the N range cells surrounding the cell under test to compute Z [6, 71.

In the TM-CFAR processor, the N background samples are first ranked according to their magnitude in an increasing order

X(1) < x(2) < X ( , ) < ' ' ' < x ( k ) < . ' ' < X(N) (4)

where X ( , , , X , , , , . . . and XI,, are the order statistics of the sample ( z , , z , , . . . , z N ) and

( 5 ) zi are independent and identically distributed (i.i.d.) random variables with common probability density func- tion (p.d.f.) of the exponential form

zi = )(xZ + y z ) i = 1, 2, ..., N

where $ denotes the total clutter-plus-thermal noise power.

The TM-CFAR scheme then trims N I cells from the lower end and N , cells from the upper end before summing the rest to construct the statistic Z which in this case takes the form

N - N >

z = c - XIj, j = N i + 1

(7)

The order statistics XI,,, X ( , , , ..., X ( N ) are not i.i.d. random variables even when the original samples z,, z,, ..., z N are i.i.d. random variables. However, since the samples zl, z 2 , . . . , zN are exponentially distributed, the following transformation to new random variables Y,, Y, , . . . , Y,, results in independent quantities:

N s X I N i + 1) fo r j= 1 I ; . = { ( N s - j + t ) [ X ~ N ~ + j ) - X l ~ ~ + j - l l l for 2 < j < N s

(8) where N s = N - N , - N , .

The distribution of I;. can be easily calculated, taking into account that the distribution of z j is governed by the exponential form, and the following result is obtained:

j = 1

N - N , - j + 1 e x p [ - N - N l - j + l ] $ ( N s - j + 1 ) $ ( N s - j + 1)

2 < j C N s

(9) In terms of the independent random variables I;. the sta- tistic Z is given by

Pr,(Y) =

N S

z= XI;. j = I

Since the samples are statistically independent, the estimate Z has a characteristic function given by the product of the individual characteristic functions of the

I E E Proc.-Radar, Sonar Navig.. Vol. 142, No . I , February 1995

variables V , . Thus

where l-(N + l ) T ( N - NI + $ N s S )

'yl(S) = T ( N - N , ) T ( N + $ N s S + 1)

= ( N ~ N I ) N ~ + ~ / ( N - N I + $ N ~ S ) N , + , ( 1 2 ) ( N ) ; is the Pochhammer symbol which is defined as

( N ) ; = T ( N + i ) / T ( N )

1 for i = O N ( N + 1)(N + 2 ) . . . ( N + i - 1 )

for i = 1, 2, . . . and

with

c,= N - N , - j + 1 ( L ( N s - j + I )

The expansion of eqn. 12 into its factors gives a simplified form for 4,,(S) which becomes

with

The substitution of eqns. 14 and 16 into eqn. 11 gives the analytic expression of the characteristic fu1:ction of the statistic Z .

When noncoherently integrated M-pulses are considered, the following procedure is taken into account to derive the detection performance of the processor.

The reference cells of each sweep are first ordered in an increasing magnitude. Then N I cells from the lower end and N , cells from the upper end are rejected. This algorithm is repeated for all the M-sweeps and the rest of the ordered samples are summed to construct the statistic Z , (Fig. 1). Thus,

where X , , [ , ) represents the jth ranked cell in the ith sweep.

In terms of V , (eqn. l o ) , the above equation can be written in another form

M Ns M

z M = c 1 x . J = czZ (20) , = 1 ,=1 , = 1

where 2, is the statistic formed from the ordered cells of the ith sweep.

Integration of M pulses means that each column is added up. For the cell under test we have

M

vhl= cui ;= I

(21)

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The random variables ui are independent and identically distributed with a common p.d.f. of exponential form

2.7 CA-CFAR processor If NI = 0 and N, = 0 the performance of the TM-CFAR detector reduces to that of the CA-CFAR Drocessor

1 which has the same expression for P , with 4 z M ( S ) given P"14 = ~ $(1 + A) exp (- &) (22) by

where A represents the average SNR of the primary &,(S) = [&IM fi, [&r (28)

target.

given by Since U, are i.i.d. random variables, VM has a P.df. This result is the same as that obtained in Reference 10.

1 M M - I 2.2 OS-CFAR scheme When N , = K - I and N, = N -- K we obtain the per-

+ A ) r ( M ) $(I + A ) formance of the OS-CFAR detector which orders the p , , ( v ~ ~ ) = 1-1 !!-- [- A] (23)

' 1 Fig . 1 Block diagram of the TM-CFAR processor with noncoherent integration ofM pulses

Based on the previous discussion, the characteristic func- tion of the statistic Z , becomes

M

42,V) = n 4 Z h Y (24) , = 1

Since 2, are assumed to be independent we have

The performance of the TM-CFAR processor is com- pletely specified by calculating the probability of detec- tion which is given by

p , = F P Z M ( Z ) ~ P d U / A ) du dz (26)

The substitution of eqn. 23 in the definition of P , gives

(27) The above formula describes the performance of the TM-CFAR scheme which is characterised by the trim- ming values N , and N, .

20

range samples according to their magnitude and takes the Kth largest sample as a statistic Z. For M sweeps, the statistic Z , has a characteristic function given by

where Cj and hk are as previously defined (eqns. 15 and 17).

3 Results

In this Section we have dealt with the performance evalu- ation of the TM-CFAR detector processing M independ- ent sweeps (echoes) reflected from a Swerling I1 target. Results are shown in Figs. 2 and 3. The threshold multi- plier T is computed iteratively from eqn. 27 after setting A = 0 and P , = P,, . Numerical results are obtained for a nominal probability of false alarm of and with a sample set of size 32. In Fig. 2, the performance is evalu- ated for N, = 0 and N , = 0, which corresponds to the cell-averaging processor (CA-CFAR). When N , = 26 and

I E E Proc.-Radar, Sonar Naviy., Vol. 142, No . I , February 1995

N , = 5, which corresponds to the order-statistic scheme (OS-CFAR) with K = 27. the obtained results are also shown in Fig. 2. The optimum trimming values that give better performance are 3 , O (N, = 3 and N , = 0) and the results are plotted in Fig. 3. In each case, the performance

1 -

0 8 -

c 0 -

0 6 - D

0 - - 0 0 4 - n e 0

0 2 -

1-

0 8 - c a - U

D

0 2,

X 0 6 - - - E 0 4 - 0 n e

01 5 10 15 20 25 30

signal-to-noise ra110,dB

Performance ofCA and OS(27) detectorsfor M sweeps Fig. 2 N = 32. P,. = IO- ' . K = 27 A OS(li" a os (2) 0 OS (4)

/ P

10 15 20 25 30 signal-to-noise ratio,dB

Fig. 3 N = 32, P I . = IK6, M = 1,2,4 ForCA N , = O . N , = O For TM N . = 3. N , = O

Perjormance of CA and T M detectors for M sweeps

A TM(I ) ' . A TM(2) 0 TM (4)

is calculated for a single sweep, a double sweep and four sweeps (M = 1, 2, 4). These Figures are used to provide a specific comparison between CA, OS and TM processors under the Swerling I1 target fluctuation model when the background noise is homogeneous. The OS-CFAR scheme exhibits minor degradation. However, it has a unique feature in that the threshold is calculated from a single cell selected from the N ordered reference cells [SI. The very important property of the OS-CFAR processor is its immunity to interfering targets [SI. I E E Prur.-Radar, Sonar Naaig, Vol. 142, No . I , February 1995

In Figs. 4-8, we show the threshold parameter T required to achieve a prescribed probability of false alarm of for different values of the trimming param- eters. The threshold constant is given as a function of the

2 6 IO 14 I8 22 26 30 32 number of samples, N

Fig. 4 in M with N , = 0 and N , = U TM(O.0). Swerlmg case II

A M = l A M = 2 0 M = 4

M = b 0 M = X 0 M = l O

Threshold parameter 7 against N , sample set size, parametric

P,. = 1 0 - 6

ordered sample, K

Fig. 5 metric in M with N , = K - I and N , = N - K TM(K ~ 1. N ~- K ) , Swerling casc I 1 P,, = IO-'. N = 32 A M = l A M = 2 0 M = 4 8 M = h 0 M = X 0 M = l O

Threshold parameter T against K , rank-order sample, para-

size of the reference window N in Fig. 4, the rank-order parameter K, when N is held constant at 32, in Fig. 5, the trimming value NI when N , = 0 in Fig. 6, the trimming value N, when NI = 0 in Fig. 7 and the equal trimming values N I = N, in Fig. 8. It is shown that as M increases, lower values for the threshold constant are obtained in all cases, as expected.

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Finally, the corresponding CFAR loss values, relative to the optimum M-pulse fixed threshold detector, are

0 4 8 12 16 20 24 28 32 tr imming va1ue.K

Fig. 6 in M with N , = K and N , = 0 TM(K, 0). Swerlmg case I I Plr = 10 6. N = 32 A M = l A M = 2 0 M = 4 H M = 6 0 M = 8 0 M = l O

Threshold parameter T against K , trimming value, parametric

m

_o 3 - LL

2 2-

I -

0 4 8 12 16 20 24 28 32 trimming va1ue.H

Fig. 7 in M with N , = 0 and N , = K TMIO. K ) , Swerling case 11 P,. = IO-', N = 32 A M = l A M = 2 0 M = 4 H M = 6 0 M = 8 0 M = l O

Threshold parameter T against K , trimming value, parametric

shown in Figs. 9-13. The loss is computed as the addi- tional SNR required by the TM-CFAR detector to achieve a probability of detection of 90% with a false alarm probability of over the optimum Neyman- Pearson detector when the background noise is uniform and of known level. When P , = 0.9 and P,, = the optimum detector requires S N R = 21.15, 14.83 and 10.51 dB for M = 1, 2 and 4, respectively. As the number of noncoherently integrated pulses increases, lower losses

consequently better performances are obtained as shown from our numerical results.

l o /

0 2 4 6 8 IO 12 14 16 trimming va1ue.K

Fig. 8 in M with N I = K and N , = K TMIK. K). Swerling case I I P," = IO-'. N = 32 A M = l A M = 2 0 M = 4

M = 6 0 M = 8 0 M = l O

Threshold parameter T against K . trimming value, parametric

& 01 o 5 IO 15 20 25 30 I

number of samp1es.N

Fig. 9

TM(0, 0). Swerling case I I A M = l A M = 2

M = 4

C F A R loss against N, $ample set size, /or P,, = IO-', P, =

0.9, N I = 0 and N , = 0

4 Conclusions

In the present paper, we have analysed the performance of the TM-CFAR detector as a function of the number of noncoherently integrated pulses used by the detector for noise power estimation. A Swerling case 11, which is widely accepted for modelling radar returns, has been used. An analytic formula was derived for the detection probability when the environment surrounding the cell under test is homogeneous. The performances of CA and

I E E Proc.-Radar, Sonar Navig., Vol. 142, No. I , February 1995 ^^ LL

OS processors are obtained as special cases of the TM scheme by setting the trimming parameters equal to specified values for each case. These analytical results

6-

5-

m '- 2 3 -

111

4 L L U

2 -

1 -

O l 0 5 10 15 20 25 30

ordered samp1e.K

Fig. 10 P , = 0.9, N , = K - I and N , = N ~ K TM(K - I , N ~ K). Swerling case I 1 N = 32 A M = l A M = 2 H M = 4

CFAR loss against K , rank-order sample, /or P,, = 10 6.

01 i

0 5 10 15 20 25 30 trimming va1ue.K

Fig. 11 0 9 , N , = K a n d N , = V TM(K. 0). Swerling c u e I1 N = 32 A M = l A M = 2 H M = 4

CFAR loss agalnst K , trimming value,Jor P,, = P , =

have been used to develop a complete set of performance curves including required threshold versus N , a set of samples in the reference window, and the CFAR loss relative to the optimum M-pulse fixed threshold detector.

The presented technique for noncoherent integration is different from the conventional one, where the reference samples of each sweep are first processed and then each sample in a specified sweep is added to the corresponding samples in other sweeps. In the conventional case, the reference cells of each sweep are added to the corres- ponding ones in other sweeps and the final sums are then processed to form the noise level estimate. In the mean- level (ML) CFAR processors, the two techniques lead to the same result. In the CFAR detectors based in their operation on the OS procedure, the presented technique may be anticipated to give better results, especially in multiple target situations. The interfering target returns among the cells of each sweep occupy the top ranking reference samples. Therefore, there is no chance for any IEE Proc.-Radar, Sonar Navig., Vol. 142, No. I , Fehruary 1995

interfering target return to come before the thermal noise return in the reference cells of any sweep so that the OS-CFAR tends to estimate the noise power using only

O L 0 5 10 15 20 25 30

trimmlng value,K

Fig. 12 0.9, N , = 0 and N , = K TM(0, K). Swerling case II N = 32 A M = l A M = 2

M = 4

CFAR loss against K , trimming valwfor P,, = P , =

01 J 3 6 9 12 15

trimming value, K

Fig. 13 0.9, N , = K and N, = K TM(K. K ) . Swerling case I1 N = 32 A M = l A M = 2 H M = 4

CFAR loss againsr K , trimming value,/or P,, = P, =

thermal noise cells. As a result of this, the proposed tech- nique may lead to an adaptive threshold of smaller value relative to that obtained by using the normal integration technique where the noise level estimate may contain returns from extraneous targets, and this in turn gives better performance. In addition, the proposed technique simplifies the processor performance evaluation. Unfiortunately, the large processing time required by this technique in ordering the reference samples limits its use. However, it is possible to reduce this time by partitioning the reference window into smaller subwindows. The samples in the subwindows are processed and the statistic Z may be chosen by further processing the subwindow outputs.

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5 References

1 NITZBERG, R.: ‘Constant false alarm rate processors for locaily nonstationary clutter’, I€€€ Trans., 1973, AES9, (5). pp. 399-405

2 RICKARD, J.T., and DILLARD, G.M.: ‘Adaptive detection algo- rithms for multiple target situations’, IEEE Trans., 1977, AES13, (7), pp. 338-343

3 HANSEN, V.G., and SAWYERS, J.H.: ‘Detectability loss due to greatest-of selection in a cell averaging CFAR, I E E E Trans., 1980, AES-16, (I), pp. 115-118

4 ROHLING, H.: ‘Radar CF-AR thresholding in clutter and multiple target situations’, ! € E € Trans., 1983, AES-19, (7), pp. 608-621

5 RITCEY, 1.A.: ‘Performance analysis of the censored mean-level detector’, I€€€ Trans., 1986, AES-22, (7), pp. 443-453

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6 GANDI, P.P., and KASSAM, S.A.: ‘AndlySiS of CFAR processors in nonhomogeneous background: I€€€ Trans., 1988, AES-24, (7), pp. 427-445

7 AL HUSSAINI, E.K.: ‘Performance of the greater-of and censored greater-of detectors in multiple-target environments’, I€€ Proc. F,

8 LEVANON, N., and SHOR, M.: ’Order-statistics CFAR for Weibull background’, I € € Proc. F, 1990, 137, (3). pp. 157.~162

9 RITCEY, J.A., and HINES, 1.L.: ’Performance of MAX family of order-statistic CFAR detectors’, I € € € Trans., 1991, AES27, (1). pp. 48-57

10 EL MASHADE, M.B.: ‘M-sweeps detection analysis of cell averag- ing CFAR processors in multiple-target situations’, IEE Proc.- Radar, Sonar Nauig., 1994, 141, (2). pp. 103-108

198a ,ns , (3). pp. 193-195

I E E Proc.-Radar, Sonar Nauig., Vol. 142, No. I , February 1995