BEST AVAILABLE COPY

hvP L Ref at 67 96

STba. Detecbon of Nonfki4uabng Tarsefsnt Log-NOrina, Clutter

Odober 4, 196

BEST AVAILABLE COPY

a

a a

- _- -- IA

i~f01AiI IIlYt) )E'I.(A) ITN ItE~ 18 i

IKO\V N hi A~t ýNITT ll 18DI

ItI* 'E.It VN(EN 1

ADSTRACT

Mealsitlments of sel clutter using htigh-mrsolutiou radar indicatithat tho clutter-('ross-section returns follow at log-inormal probabilitydensity function more closely than the usually itisunllied Rayleigh law.This report develops the thlory for the detection of at steady stignal Inlog- norm'al cluttei" by first using it single 1)ulat1 and then by using the'uill Of N pulses integralted ionucOherenily. Plots of the probability deIn-sity of the evivelope of the signid plus clutter show the function to bebimodal, an unex)ected result. Curves are presented for the thresholdbias, normalized to the median clutter voltage, versus the probabilityof false alarm f(tr several values of the standard deviation , and forvarious values of N. Probability of detection curves are presented for,, -3, 6. and 9 dil, for N -. 1, 3, 10, and 30 pulses, :qud for false alarmprrobabilties from 10"- to 10s. 'rhe ratio of signal to median clutterrequired for detection increases markedly as , increases because ofthe hiighly skewed clutter density.

Mh n sic l ilpt s ubmitted August -11, 1968.

Ii

T1 VlETEWT2SION OF01 NOtNI l .'. 1' FIt

Si.,t Oill 114 111V tMll oelit 14 mailt, I NMt . klilsiv~ t% hgh l .ot 1 ,40411 i l r -li.w Owl lilh,V, I lilt Ii' I, v~ i 't11 1 )N tt i t m I vm ix I e I I g II, It t - ii1' MAt I 1v dtit M p! i ti lil, A itl tilt\, , it I ht I ., It t v I wI%I, 1titt , jpin.title , t h ict 'I e v , i'e mott ua pi opri' t l iiie titt 1 I h i l e orl ji v istvIhi lit, I i' Sthl N ilsu'll ;I studiy,

Viish a a di Iv . l i r ihulltw piv iiltneliii t~ ~'itl111v IIIn rl l 1u ivoit sl l kitosti Ivvt eN' lu tit'ut-t I) I 0 1lisd i famil ko i'u t V li I' ii ld a~~v (tiut'd 1,11m a~l arnI I~lt pr olk l the u vIllntl tr dottsl I0 sl . I hit 10

sl~motl lulle.i, as itul-ýt' el '. oItSUlt mug in g I'vtllv ilit'roaod Waus vd lktu'5 iis u.iltilia i U d it t hiItylv ligh noulell,

Nex , hi' diti I' hit loll Of I 11V O'twolope- tit higia I plIsum cjutte r tilt'' a sing Ii' pukev Is tit,, 1iv ( ' (, P1'(its (if pr ohthi lily dens iti I 41tssho a hii utoui ttl ulicioll, whitch Is an %ineilvelve d

vesti II I h Thi iiitlla IS hapi' ho~lds tor all v.%luis tit i gu ' i ouert ito t8 N I an tilto, alla luts tit , lill( tilti' dnt ,it peak. ki~ra w viloser l not- hi' r alid tilt (llt) h~koi' ons Wr's pitilollvi'd

.Is - illu'ii.uS's, 111,itii'iii ts lut' tile, peoilallilitv u ileitloul iiin air givetn, lintit Iuliv's vuutld littullit evalualt'd W,' cluiseuvtm ho tt) heiv wover progra ninlmedlilt, (he N1t I. CDC 31100) ciiiipo it

Thtcll thit i.'oblithi~tic' .I Ca Ist a tillii aund il~i'l ctlun ar~vt' dve Iiilied lilllit' t i'm teit \ijiulsk's iii egrauteil nlonvo''iht' i yei . gi' Inelra I mltilx hki se'd is that oft t i' u'hu racto'101t Ii'fulle ii~,wtion \01101i'tinlia liv rviuoi rvr Iwo appIi vat ions til t hi' Fouietom ini' r llva , live ai ga hil1111 4011ol~ u ll cl sdtit' 0 v I fr sei'vilivi l''adil l obtli Iaht', ski theo CDC( 30100 %aiN p etigra tit med'tf'il numenrical ev aloationl, 11sIng thlt faIst Fluori't'. (I ramtiloi'n ti'ehlllitime,

Suts of curl es u mu(' Iui'svlll i'd fll' the protihli lily of d'te't loll liii' vauluvt of 3' , and9 d113, for A 1, 3, 10, mnd 30 piulses , and for ta 1tiv a Iarem probatbilities from 10it)"t 10".The S N reijuirted filr iett'i-tiuun for - 3 (tit art' about thle samei ash t hose foll, lthe lay teighI~itidel, as Wiould be' ('qlYvctt,(l lintve thit log.l-novillit I tail for' this came v'ulnlpurv" to t litRay lvigh tallI. hlowever, as .. mlervases, thet roquired S N .list) i iss markedly its a resultof the' inret~asingly skewed (list ribult ions. It shoiuld he noted that the( S N defined in (thitsrepor't umses the niectiali clutte I' value fill' noist, aild no(01vth usual 1,'ti18 vautue.

INTRHODUCTION

Clutter -cross -sect ion measuremients of sea rviturn taken with high- resolution radarshow more of a tendency to follow it log-normal probaibility distribution than the Rjaylefirhdiatribution usually assumed. Data reported by NRL (1 ' using -,ni X-band radar wihth a0.02-,,see pulse are plotted in Fig. I on probability piaper' with the clutter cross section

,,in dB on anl arbitrary sc'ale. A Itog-nuulnilI chstx'ihution is indicated by these pointsfalling in a straight line. It is seen that the, cumulative probability clorwely fits a linebetween '~ and 80(,'( Accurate datat at the extremes are difficult to obtain, because ther -mnal noise limits the liiw ,,values -and tilt systvnm dynlamicv r'ange limits thle high , Values,

Studies made by Miner (2) and Blallard (3) also consider the log-normial description ofthe sea-cluttet' cr~oss section and relate thle standard doviation of' the distribution to the

IK All l~ i

NM N1II, A A

IN

v I Jk111 1ykI~ i a.t t, i b l l 1

IIH I a I kI W I v rt ~I tI mdo I , i~tt t 1vX1vvi eI Iv 111 t iw II k)ra1 1X t nhit11, v ith k I r itI-

,% 0 '0.~ -ptit,14 Irn.C

rail i lulmifilatted pat cli areal. Ria hard actuoally sot up it log-normal1,1 probability model but(11id iot da~lt'(lte'tiO(' lou t'i".(. IlThe presenOt report follows much of the sa ine theoreti-vat 1110ihod and etendIOucs tilet. miodel to Ilicludit, tilie 1flteginiioll of IN pulses.

l.t- NORNMAlc~r.,i CI1TIHmomDE

It wa' assumne thlat clutter cross section II, log- normally distributed, then the proba-bitity ovinsity Is givenl by

Where P, J the clutter cross setin is th meian voeo and Iis the standard

devvatlioi of In(natural logarithm). Since cross-section values are measures of power,Hie' standard deviation (s.d.) is expressed in -,natural units" (4) by the relationship

'' ( 1ilkir; Itillii~ I 0. 1 Ill 10) ahdll) 0,.230 3 ,(IIB).- (2)

Thev Value of inl Fig. 1 is inl dB3, and Imust be converted into natural units by Eq. (2) foruse in Eq. (1I)

In Fig. I thle log-normal probability distribution function is plotted for 3 dB3 and6 M1. The dashed line is a close fit to tile NRL experimental results 'on sea clutter

pr1eviously mentioned. For this line. 1,5.4 dB. From Ref. 3, a value of , -4.7 dBre1sulted from a c-lutter patch size of 3840 sq yd in a sea state q. From the NRL studies(1), 1,-5.4 dB corresponds roughly to a clutter patch size of 135 sq yd in sea state 1. *

To determine the probability of false alarm and subsequently the probability densityfunction of a signal plus clutter, Eq. (1) must first be transformed into a voltage- amplitude

It g en1 er atI inIc t I'a klsi.- 6ire ctLy with sea state andl inve r sely with k.lutter patch size,

NRI, ii'REPRT (171 t 3

density, If wV let v be the envelope of the radar echo from the clutter area , thenS 0.r' r Using this substitution In Eq. (1) yields

A-, +( Ia (1 V (3)

where V, is the median value of v and , remains the s.d. of In . Then, is still mea-sured according to Eq. (2).

The probability of false alarm for a voltage bias level /i is given by

PIA (v, •h' 1/_exp t- 2 1) , (4)

where ji %., is the normalized bias level. The integral in Eq. (4) is related to the errorfunction, which is well tabulated.

DETECTION OF A NONFLUCTUATING SIGNAL IN CLUTTER

Let us assume a radar pulse return from a target which is a steady signal of constantamplitude .I. The clutter return will be assumed to have the log-normal amplitude densityftution as �given in Eq. (3) and a phase ,. that is uniformly random over - . Theanalysis can be performed as though the radar were operating as a ew system. If thesignal and clutter voltages add linearly prior to envelope detection, then we can assumethat the signal plus clutter may be represented by the two components

X 1 ' cos ,o (5a)

and

Y sill (51))

where x and Y are the in-phase and quadrature components of the signal plus clutter.

We can define two new variables x' and Y" such that their joint probability is that ofthe clutter alone:

V' X 1'" 1, 'os,' (6a)

and

Y' Y I, sill (6b)

Then, the relationship between these joint distributions is

X,/x'ix',/IX ,JXp~ ,,/.)(h!,/.". p(X','Y') Ct,('f/Y" v p(X'.Y') dIX'd]Y'. (7)

~V

4 SF. GEORGE

But since v' and 6 are independent variables, p(v,,, )dv.do, p(v,)dv.p(,/v)d0P pvH)dv,(112,)d,. Hence, from Eqs. (3) and (7) we have

p(X', Y') dX'dY' _ I exP- L2 , (Ivf, d,. (8)

From Eqs, (6) this becomes

p(XY)= exp r-(2 (9)t o [ (X " vs) 2 + y 2 L 202 v\j

Finally, using Eqs. (5) in Eq. (9) yields

p(v.95) (v2-exp 1 (2 1n C " ŽbJJ (10)

where v 2 X 2 + y 2 .X v Cos ' y . v siin , and the transformation from Eq. (9) to Eq. (10)is analogous to the one used in Eq. (7) with o uniformly random from -n to 4 . Now vrepresents the envelope amplitude of signal plus clutter. The probability density functionof v alone is obtained from Eq. (10) by

p(v) : 2 f p(v,) do. (11)

Attempts to evaluate Eq. (11) in closed form were not successful, so computer numericalevaluation was employed. However, before illustrating the results, it is more meaningfulto transform Eq. (10) into an expression containing a "psuedo" S/N, actually a signal-to-clutter ratio, since v. is not easily obtained. If we let r v0 /'vm in Eq. (10) the probabilitydensity becomes

r, f') a (X2 - 2xr cos q5 , r2) ( o (12)

where x = v/yi is now the normalized envelope amplitude of signal plus clutter. Then theprobability density function of x is

p(x) 2 p(x,.) d0. (13)

0

The probability of detection for a single pulse of a steady signal in log-normal clutterfor a normalized bias level y is thus

PI) p(x) dx ý 2 p(x,o) d~dx - I - 2 $0 f: p(x,O) dodx. (14)

In Fig. 2 plots of Eq. (3) for various values of a are compared with the Rayleighdistribution for which the median value has been set equal to v. It should be noted thatthe tail of the Rayleigh distribution falls off about as rapidly as that of the log-normalfor , = 3 dB, but the tail falls off less rapidly as, becomes larger in the log-normaldistribution. For these largcr, values, then, the bias values for a fixed PF are expectedto be greater for the log-normal than for the lrtayleigh distribution. Plots of bias values,obtained from Eq. (4), illustrate in Fig. 3 how rapidly the bias does increase as , increases.

NRL., REPORT 6796

14 -- r.------- I7 __ -T

12 o'3d

010Cll

'4

o l o r'9,a -

06

1 0 20 3.0 4.0

RATIO OF CLUTTER CROSS SECTION TO THE MEDIANVALUE OF THE CLUTTER CROSS SECTION, vc/v

Fig. 2 - Log-normal probability density for (y 3, 6, and9 dB aind the Rayleigh density for the median value v,

100.000 C)( I-"'

10,000

a 12dB

E iO00

Fig. 3 - Bias levels as a function of false 1alarm probability with a - 3, 6, 9, and 12dB for N 1, a Single pulse of log-normial ED9dclutter

I 00

CT 6A~lS

41

IO.? 3 4 107 O !

logK 1F3

6 11. F. GEORGE

To plo t the probability density function in Eq. (13) the interral had to be evaluatednumerically, since no cl08ed form solution was found. The CDC 3800 was programmedusing Simpson's rule for the integration. Figure 4 shows a family of curves for thoprobability density of the envelope of signal plus clutter for , = 3 dB and for severalvalues of r . S/N which has been designated as a voltage S/N. The double liumpor bimodal curve is most unusual arid surely not predicted. Note that this curve is theenvelope distribution and hence assumes linear detection. The va)lues of S/N shown arevoltage ratios. Figure 5 shows curves for a = 6 dB. Again the double hump is presentbut less pronounced. In Fig. 6 by increasing o for a fixed value of r (i.e., S/N) the dis-tributions remain bimodal but tile humps tend to converge and the dip becomes lesspronounced. Figures 4 and 5 show that as r increases, the curve shapes do not changesubstantially.

A word of caution is in order regarding the definition of the signal-to-noise ratioS/N = v, v. it should be noted that this is not the usual definition of the signal-to-

noise ratio, where the noise is taken to be the rms value. Herein the noise (which isclutter) is taken to be the median value. This is a convenient definition to use and, also,one which would lave considerable engineering significance, since clutter is frequentlyexpressed in terms of its median value. However, in subsequent parts of this studywhere N pulses are integrated, care must be exercised in interpreting r. Since the log-normal model yields a two-pararneter distribution, its behavior is more difficult toanalyze and interpret than the Rayleigh model.

DETECTION USING TIlE SUM OF N PULSES

The straightforward method of determining the probability density for the sum of Nenvelope detected pulses, assuming they are independent, is by use of the characteristic

ýOH

M

12Ný S/N 6

02

Q • 4 5 ,6 1 8 '1

NORMALIED ENVELOPE AMPLITUDE OF '5GNAL PLUS CLUTIER, -/-

Fig. 4 - Probability densities of a single-pulse signalphis log-nor-nal clutter for a I 3 dB and voltage S/NS0, e_, 4, and 6. Note th0 dou bhiL pt-aks.

NRL REPORT 67967

ORF

~:06

= 0, 2, 4, and2

,04

-7 - - * - . ! - - -- -

12I dB.

8 S.F. Gi]ORGE

function. Starting with the probability density of the envelope of signal plus clutter givenby Eq. (13), the characteristic function is given by

6(w) J p(x) exp(iwx) dA, (15)which is the Fouxier transform (FT) of p(x), since x , 0. The characteristic function forthe sum of N independent pulses is

,JN(w) ,(w) ,N. (16)

It is noted that in general b,(w) is a complex function. Using the inverse FT of Eq. (16)gives the probability density for the sum of N pulses:

N(X) (W) exp(-iwx) dw. (17)

Plots of Eq. (17) :or N = 1, 2, and ',and for S/N 2 ore shown in Fig. 7. Note that thedouble hump disa,;pears as N increases. Also the distribution spreads out as N increasesand begins to lobk more like a normal distribution.

For the sun- of N pulses, the probability of false alarm is

PN(FA) pf1(X) dx I - pN(X) (Ix, (18)

where r = 0; and the probability of detection is

PN(Dt,. ) 1 - PN (X) (X, (19)

where p.(x) is evaluated for the parameter r A 0. All attempts to find a closed-formexpression for the integral in Eqs. (18) and (19) proved fruitless. Fortunately, the newdigital techLique known as the fast Fourier transform (FFT) made it possible to ev duatethese integrals numerically in a reasonable time.

PROBABILITY OF DETECTION RESULTS

Bias values for N z 3, 10, and 30 were determined from Eq. (18) by numerical hite-gration for lalse alarm probabilities from 10-2 to 10-11 and for , 3, 6, and 9 dB. Theseare plotted in Figs. 8 through 10. Curves for bias values are repeated in Figs. 11 through13 with N varied for a fixed , value.

Using these bias values, Eq. (19) is evaluated, giving the probability of detectionversus S/N per pulse, in values of PD from 0.1 to 0.9999. Figures 14 through 17 are for

3dB. A summary of P,) for PFA = 10- and N - 1, 3, 10, and 30 is presented in Fig. J8.In Figs. 19 through 23 detection curves are given for ,, 6 dB, and in Figs. 24 through 26for .... 9 dB. An inspection of these curves shows that the SI/N required for detectionincreases rapidly as , increases. In fact the S/N in dB appears to increase linearly as, increases, doubling as, goes from 3 to 6 dB and increasing by a factor of 3 as ,, goesfrom 3 to 9 dB.

It is interesting to compare the set of curves for 3 dB and N 1 (no integration)with those obtained using a Gaussian moxel. Skolnik (5) shows that the S/N required com-pares favorably with the results using a log-normal mxlel (FIg. 14) as should be expectedlor the 3 dB case.

NRL REPORT 6796 9

10 --- --

S0.8

>'.-

z

S0.

02

0 2 3 4 5 ' 8 9NORMALIZED ENVELOPE AMPC0TUDE OF OiGNAL PLUS CU TI, ER, Y/v,

Fig. 7 - Probability densities of a signal plus log-normalclutter for o = 3dB, voltageS/N =, and number of pasesN = I -, and 3. Note as N increases the curve loo,1s morelike a normal distribution.

IO(Y" . . . ... ---- 7 -- • - ~--- -- -- Ir-

IN~

1001 GI 9dH

Fig. 8 - Bia s levels for false alar m probabilitiesfrom 10-2 to 10-H for N 3 with 3, 6, and9 dB3

4 r

10 S. F. GEORGE

00'

I Fig. 9- Bias levels for false alarm prubabilitiesfrom I0-2 to I0"8 for N = 10 with , 3, 6, and

I0( 9) 7.. . . .- 1 - "'- r'T • • r

N Y )0,•

V" d I•

SI0

t'u t to, , ih, • ,

2 •, a 5 P I0I( ,J_)2

NRL REPORT 6796 I1

100 --- --____,_

70 cy '3dB

50

40

30

20-

Fig. 11 - Bias levels for false alarm proba- Tbilities with the number of pulses s.ntegraded, >- 0N = 1, 3, 10, ard 30 for ,y 3 dB

5.N:I

4

3

2

2 3 4 5 P •0

100 -PFA

7P d N'30 >

50

40N-10

30

20 N- 3

N-I

-10Fig. 12 - Bias levels for false alarm proba-bilities with the number of pulses integrated,

7N = 1, 3, 10, and 30 for a = 6 dB

5

4

3-

2

I 2 3 4 5 1 10

12, S. F. GEORGE

10I I I- "

1000 T-d8

500

400

300

200 N,30

N-10> 00 Fig. 13 - Bias levels for false alarm proba-

bilities with the number of pulses integrated,>N 1, 3, 10, and 30 for a 9 dB

N.0

50

40

20

101 ~ 11 2 3 4 5 7 1

log IOP-A10 5FA

09999

09995 - N-I0999 -0,3dB

0 9980995

z 099

0 98

S090

- 0,80 A "070 10 10-

060 IC

050

0.40030

0.20

0.10

2 4 6 6 10 2 14 6 I 20S/N (d8)

Fig. 14 - Probability of detection vs S/Nfor PFA 10.2 to 10-8 with a = 3 dB andN =

NKL RIEPORT 67V6 13

0ww

019

Fig. 15 Probability of dot ution vo 8w Nfor 1 1 10-2 to 10*8 witht ( I dB anti t1oN =3 •nlP0

0 4O

I'oil's

0400

0 10010 A

Olo• 0 a 4 0 0 0 It 14

U/N l4o1

09995 N-I00999 *10 998

C, 99Sl

099

098

095095 Fig. 16 - Probability of detection vs S/N

'. for PFA 10*2 to 10"a with , 3 dB and

:, oeo 1o0 5,N 10

0 0 %.10

0600 50 'Ic'

0400 30 • "•"•

020 "-t8

010-6 4 -2 0 2 4 6 a '0

r-1N (dB)

09999

0995

099 -

6090

Fig. 17 - Probability of detection vs S/N 1o

for PFA = 10"2 to 10.8 with a 3 dB and 090 10,40N 30 100180

t•-

D 080 10-1

060 •O050 i

040 0

0300

020

-8 -6 -4 -2 0 2 4 65/N (dB)

14

0 11t

i I

NV WO)

•t •

Fig. 18 - Probuhility of detv, 'timot vii S/Nfor N : 1, 3, 10, and 3') with d 3 d -3 and1.FA 10"t

Mil TT

0995•

099

0w -

~0a.1

a go Fig. 19 - Probability of detection vs S/N for0. A 10 -2 to 10-8 with, 6 dB and N

060050 o0,40 10-~

030 P

020

10 14 I .22 26 30 A4 38.9/N WEI)

Ni• tlt. REORT 67"•,16

0 99C-I NON.3

)998 Cr.6dOl

0995

o 99098

,095

Fi. 40 - Prubl0jility of dtetction \,w S/N for i ogoit FA - 10"J to 10"8 with 0 6 dil and N• 3

.5 080

070

060

0 0o05010in,

0 40 10

030 o0

020

0 to4 8 12 Its 20 24 28

S/N (dB)

0 9999 T~~

0999501999 N-10

0 9?8 o"6dB

0995°99

~.098

6 090

4j 080 1-So~o / / / + / ,o-,/o/ I -28070 10 1

N= 10

a0600 500 40

o030020-

0102 4 a 8 10 12 14 16 18 20

S/N (dB)

Fig . 21 -Probability of detection vs S/Nfor PFA =10-2 to 10,8 with Ila6 dB andN = 10)

16 S.F. GEORGE

009"

N-300""9 -a d

0*999099 -

0,39

0.f1

Fig. 22 - Probability of detection vs S/N0o,0 for PJ1A = 10"-2 to 10"- w ith a 6 d B3 and

0! N =30_0.?0 A 1,

S0.60 -C, 0.50 - 1 1 0 .8

0.40

0.30

0 20

01I0 -2 0 2 4 6 8 10 12 14S/N (d8)

099950,9995 -

0.995-z 0 990- 098

Fig. 23 - Probability of detection vs SIN ofor N = 1, 3, 10, and 30 with7 = 6 dB and

A = 10-6PFý0 80 N-30 N-10 N-3 N-1

ý0700.60

050

0 40

0.30

020

0100 8 12 16 20 24 28 32

S/N (d8)

0998 - N-(09995 - ,9dB

099

z0 10-s

2096

S10-4- 0 9 5 -K '090 -Fig. 24 - Probability of detection vs S/NO for 'A 10-2 to 10.8 with a= 9 dB and

_:0.80 'N

0.70 -O"O~bO

05004003C

0.20

01020 24 28 32 36 40 44 48 52

S/N (5B)

NRL REPORT 6796 17

Q9499 N-

0.9995 ,9d

0L999 - ý

0.995 - 0-3

2099

098

Fig. 25 - Probability of detection vs S/N for 100.8 0'si-PFA =10.-2 to 10 -8 with a 9 dB and N 3 00.9

0.80

S0.70CC 0.60

0- 0.50

0.40

0.30-

o 20

0,10LIs 20 24 28 32 S6 40 44

S/N (dB)

0.9999rIN -10

0.9990998

0.995

6098

0 Fig. 26 - Probability of detection vsS/N for0 90 PFA 10.2 to 10-8 with a = 9 dB and N 10

080 %,Ao a-30 70 100-6 C -6

8 060 1-(L 050

040

030-

0 20

010 -I -I - I -10 12 16 20 24 28 32 Se

S/N WO)

0,9999 T7

0,9995 -- d0.9990.998

099

S09

I- 095

Fig. 27 - Probability of detection ve S/N for 1ý 090 C" 0"PFA 10.2 to 10'g with a 9 dB and N 30 10-4

-0 80 10-Y

10

060

0.50

040

050

0.20-

0104 8 1? '6 20 24 28

SN/(40)

18 S.F. GEORGE

09999 7

09995 PFA* 10-6

0999 0"9d8

0998 -

0 995

099

S098

i 095

S090

S080 N.30 N-I0 N:3 NI0O70

o060

0.40

0.,300.20 ,

i2 1r ?0 24 28 32 36 40 44 48

S/N (dA)

Fig. ?8 - Probability of detection vs S/Nfor N = 1, 3, 10, and 30 with a = 9 dB andPFA = 10-6

ACKNOWLEDGMENT

The author expresses his appreciation to G. V. Trunk of the Radar Analysis Staff forhis many valuable discussions during the course of this work and Jon David Wilson of theRadar Analysis Staff for his helpfulness in the programming.

REFERENCES

1. Findlay, A.M., "Sea-Clutter Measurement by Radar-Return Sampling," NRL Report6661, Feb. 12, 1968

2. Miner, R.Y., "Review of ASW Radar Sea Clutter," TRW Systems Doc. 7234:1:001(Confidential Report, Unclassified Title), Feb. 8, 1966

3. Ballard, A.H., "Detection of Radar Signals in Log-Normal Sea-Clutter," TRW SystemsDoc. 7425-8509-TO-000, May 31, 1966

4. Fenton, L.F., "The Sum of Log-Normal Probability Distributions in Scatter Tranm-mission Systems," IRE Trans. on Comm. Systems CS-8, 57-67, Mar. 1960

5. Skolnik, M.I., "Introduction to Radar Systems," New York:McGraw-Hill, 1962

Securitv Classification

DOCUMENT CONTROL DATA- R & D(Se-rrity IagsitiCatioI o. titlo, body of oabstrurt aInld ndexing ainnotatiomn nost be entered wh.n the overall report is classitied)

I ORIGINATING ACTIVITY (Corporate author) 21. REPORT SECURITY CLASSIFICATION

Naval Research Laboratory Unclassified

Washington, D.C. 20390 2b. GROUP

3 REPORT TITLE

THE DETECTION OF NONFLUCTUATING TARGETS IN LOG-NORMAL CLUTTER

4A UESCRIPTIVE NOTES (Type of report and inclusive dales)

A final report on one phase of the problem5. AU THOR(SI (First neme. middle initial, last name)

Samuel F. George

6. REPORT DATE 7a. TOTAL NO, OF PAGES -T 37. NO. OF REF 5

October 4, 1968 22] 5B0. CONTRACT OR GRANT NO. 9.. ORIGINATOR'S REPORT NUMBERIS)

b NO. OJECTNO NRL Report 6796

c. 9b. OTHER REPORT NO(S) (Any other numbers that may be aesignedthis report)

d.

10. DISTRIBUTION STATEMENT

This document has been approved for public release and sale; its distributioni3 unlimited.

1I. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITYDepartment of the NavyNaval Research Laboratory

__ _ _ _Washington, D.C. 2039013. AVITR AC T

Measurements of sea clutter using high-resolution radar indicate that theclutter-cross-section returns follow a log-normal probability density function moreclosely than the usually assumed Rayleigh law. This report develops the theory forthe detection of a steady signal in log-normal clutter by first using a single pulseand then by using the sum of N pulses integrated noncoherently. Plots of the prob-ability denoity of the envelope of the signal plus clutter show the function to bebimodal, an unexpected result. Curves are presented for the threshold bias, normal-ized to the median clutter voltage, versus the probability of false alarm for severalvalues of the standard deviation a, and for various values of 4. Probablility of detec-tion curves are presented for a = 3, 6, and 9 dB, for N = 1, 3, 10, and -.0 pulses, andfor false alarm probabilities from 10-2 to 10-8. The ratio of signal to median clutterrequired for derection increases markedly as a increases because of the highlyskewed clutter density.

DD ... 1473 (PAGE 1) 19"S/N 010 •1.07.6- 01 Security Classification

Security Classification14 KEY WOROS LINK A LINK S LINK C

ROLE W' ROLE WT ROL E W:

High-resolution radarSea clutterSignal detectionLog-normal distributionsMathematical analysisProbability distributionsRadar targets

DDNORMl.,4 7 3 (BACK)(PAGF 2) S0