Probability of Detection

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MEMORANDUMRM-3095-PRAPRIL 1962

CUMULATIVE PROBABILITY OFDETECTION FOR TARGETS APPROACHINGA UNIFORMLY SCANNING SEARCH RADAR

J. D.Mallett and L.E. Brennan

PREPARED FOR:UNITED STATES AIR FORCE PROJECT RAND

---~--7kR_QnD~S...NT...MONC....CALFORNA----


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MEMORANDUMRM-3095-PRAPRIL 1962

CUMULATIVE PROBABILITY OFDETECTION FOR TARGETS APPROACHINGA UNIFORMLY SCANNING SEARCH RADAR

J. D.Mallett and L.E. Brennan

This research is sponsored by the United States Air Force under Project RAND - Con-tract No. AF 49(638)700 - monitored by the Directorate of Development Planning,Deputy Chief of Staff, Research and Technology, Hq USAF. Views or conclusions con-tained in this Memorandum should not be interpreted as representing the official opinionor policy of the United States Air Force. Permission to quote from or reproduce portionsof this Memorandum must be obtained from The RAND Corporation.

1100MANr, SANTAMONCACALfORNA-----


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P R E F A C E

This report presents the results or one study carried out as part orRAND's continuing work on the theory and advanced technology or radarsystems. The present study deals with the capability or search radars fordetecting approaching targets. The work is applicable to the currentproblem of the detection of ballistic missiles approaching a defenseradar. However, the results are in general form and will be useful indesigning and assessing the performance of search radars in detectingaircraft as well as ballistic missiles.


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S U M M A R Y

This Memorandum deals with the cumulative detection probability of asearch radar when it is scanning uniformly. This is the probability thata target approaching the radar at a constant radial ve10city is detectedat least once by the time it reaches a given range, as distinguished fromthe more common blip-scan ratio (a sing1e-scan detection probabi1ity). Itis shown that for constant-ve1ocity targets the range for a given cumula-tive detection probability varies as the cube root of the power-apertureproduct, rather than as the fourth root. curves of cumulative detectionprobability as a function of normalized r~ are given for three differenttarget scintillation models. A1so, curves of optimum (normalized) frametime are given as a function of the desired cumulative detection probabilityfor each of the three target scintillation characteristics.


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ACKNOWLEDGMENTS

Theauthors are indebted to Frank Katayamaf'or the calculations andcurves and to H. H. Bailey, F. TatumandJ. Hult for helpf'ul criticism.


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C O N T E N T S

PREFACE ............................................................S U M M A R Y ............................................................ACKNOWLEDGMENTS ....................................................L IS T O F FIG URE S ....................................................LIST OF SY?vfi30IS xiii

S ectionI. I N TRODUCT I ON ................................................II. E QU AT IO N FO R C UM U IA T IV E D E l'E C TIO N PRO BA BIL IT Y . . . . . . . . . . . . .

III. S E L E C T I O N O F O P T IMUM FRAM E T IM E TIl. CURVES O F C UM U IA T IVE D E l'E CT IO N PRO BA BIL IT Y VE RS US RA N GEv . N O N -C O H ER EN T S IG N A L IN T E G RA T IO N . .VI. C O N CW S I O N S . .

AppendixD ERIVA T IO N O F THE Q CORRECT I ON FO R NON -C OHER EN T :rn T E GRA T IO N . . 3 . 7

REFERENCES .........................................................

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126II133 . 6

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LIST OF FIGURES

1. R/Rl ve rsu s 5. No nfl uc tua tin g ta rg et mo del . 72. CUmulative probability of detection by range R. Nonfluctuating

target model ................................ 83. Cumulative probability of detection by range R. Targetf lu ct ua ti ng s ca n t o s ca n ( ca se I S, ve rl in g) 94. Cumulative probability of detection by a range R. Targetf luc tu ati ng sca n to sc an ( Ca se II I Swe rl ing ) 105. Q1 (N, n) 146. ~(N,n).......................................................... 15


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LIST OF SYMBOIB

A effective antenna areaB I F " ba nd wi dt hfr pu lse r ep eti ti on fre que nc yG ant en na gai nk Boltzmann I5 constantL los s f ac torN number of pulses (samples) integrated incoherently

number of pulses integrated coherentlyn average munber of noise samples bet,.,eenfalse alarms

a ver ag e tr ans mit te d p owe rc um ul at iv e d et ec ti on p ro ba bi li tybli p- sca n rat iopeak t ra ns mi tt ed p ow ercorrection factor for R1-- non-fluctuating target orscintillation model or Fig. 3correction factor for Rl-- target scintillation modelof Fig. 4

R rangea range defined in Eq. (4)range for unity signal-to-noise ratiodwell time, or time a target is within the radar beam

during a single scane ff ec ti ve r ec ei ve r t em pe ra tu reframe timeta rg et c los in g v el oci ty

v R/R1


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ra dia l d is tan ce t ra ver se d by target during a searchframe

target echo areasolid angle of search framesolid angle of radar beam


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Ie INTRODUCTION

In describing the capability of a search radar, one is often interestedin the cumulative detection probability, i.e., the probability that anapproaching target will have been detected at least once by the time itreaches a given range. This is to be distinguished from the probabilityof detecting a target at a given range during a single scan. The latter,which is sometimes called the blip-scan ratio, has been calculated and

presented in convenient graphs in Ref's. 1 and 2. This report discusses thecumulative detection probability (p ) and contains curves of P as ac cfunction of normalized radar range.


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II. E QUA T IO N FO R C UM UIA T IVE D E l'E CT IO N PRO BA BIL IT Y

In discussing blip-scan ratio, it was'convenient to normalize therange in terms of Ro' the range for unity signal-to-noise ratio. In termsof r ada r p ara met ers :

R 4 =oP G A (j Td

2(4,,) k Teff L (1)

assuming coherent integration for the time Td or equivalently that thesignal is passed through a matched ~ilter. This form of the range equationcan be derived from the more conventional form by substituting the fol-lowing appropriate relations: for a continuous wave (cw) radar

or for pulse radar

In Refs. 1 and 2, blip-scan ratio is given for several different targetscintillation models as a function of R/R. Here we will denote blip-scanoratio by p{R /R ).o

The cumulative detection probability is given in terms of this quantityby:

P (R,b.)c 1= -6R+~. J dR"'R

j,1 T (2)- m = 0

The quantity 6 is the distance a target travels radially during a searchframe. The quantity in large brackets in Eq. (2) is the probability that


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a target which is scanned by the radar beam at a range RI , is detected atleast once by this range. The integration, or averaging of this quantityover an interval~, accounts for the fact that the target could (with equalprobability) be anywhere in the interval ~ when last traversed by theradar beam. The value for j, is selected so that the blip-scan ratio isnegligible, i.e., comparable to false alarm probability, at ranges inexcess of (R J + 1~).

For simplicity it is assumed that the signals received from a targetduring each scan are integrated coherently. Noting that:

Eq. (1) can be rewritten in the following form:

4 3 R 3 P A 0"R = R ~.o 1 I 1. = 4" k Teff L n V c Q 3i (4)where Q.3 is a correction factor explained in Section V and the Appendix.~For the present, assume Qi = 1. (This is equivalent to the assumption ofcoherent signal integration and a false alarm number of 106 for the workthat follows.) In discussing ~tive detection probability it will beconvenient to normalize the range in terms of R1.' which is defined inEq. (4). T he i nt er val f : : , . is retained as a parameter and can be varied tooptimize the radar performance. All of the other independent radar andtarget parameters which influence the aumulative detection probability arecontained in R1 . in their proper relationship.

For a given Rl, i.e., for a fixed set of radar parameters (P,A,Teff,and target characteristics (o,V ), increasing the interval 6 (or thec,n)

frame time) increases the time for coherent signal integration and improves


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the signal-to-noise ratio. This results in an increased blip-scan ratio.On the other hand, increasing ~ decreases the number of looks at a targetand hence tends to decrease Pc. As might be expected, there is an optimumvalue of ~ and an optimum frame time for any assumed set of the other

*arameters.R ~Setting R: = v and R: = 8, Eq. (2) becomes:I I

v+5 [J dV' I- m=OAssume one is interested in obtaining a given cumulative detection

p robab ility , say 90 per cent. From Eq. (5), the normalized range for 90per cent P can be obtained as a function of 5 and the optimum value ofc5 determined. This optimum value of 5 is a function only of the targetscintillation characteristics and the value of P which is specified. Sincec5 = . f ' it follows that ~ for optimum performance is proportional to RI,Ii.e ., longer-range radars should use longer frame times (or, more exactly,larger ~).

For an optimum frame time (or 8 ) the range at which a given P is- cobtained varies with the cube root of the power-aperture product. This isapparent from the fact that for an optimized system, i.e., optimum 5 forthe desired Pc' or more generally for any fixed value of 5, cumulativedetection probability is a function only of v. Since v equals R/RI, the

*A similar problem was considered in Ref. 3 , w he r e p os t -d et e ct io nintegration rather than coherent integration was assu.med to vary with!scan interval. The assumption of varying coherent integration, which ismade here, leads to a simpler formulation of the problem. The factor Qallows for the introduction of a specified amount of post-detectioni n te gr a ti on .


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range for a given Pc is proportional to Rl and hence to the cube root ofthe power-aperture product. This is different from the well known fourth-root relationship which is applicable when one is interested in blip-scanratios.

It should also be noted, that a similar R3 relation holds in otherradar search problems, for instance, in the case of a fan-beam search radarsuch as ~iS, if coherent integration is assumed. In the case of an air-borne side-looking mapping radar the R3 relation also holds. In each ofthese cases the R3 relation holds because integration time increases withrange.


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III. SELECTION OF OPTlMJM FRAME TIME

Figure 1 gives the computed values of-v (i.e., R1R1) as a function of5 1'or several values of P. These curves were obtained by evaluatingcEq. (5) numerica1l.y and they apply to the case of a non-fluctuating target.For example, i1' one is interested in a cumulative detection probabilityof 75 per cent, the optimum 0 is 0.11. For this choice of 0, a 75 per centcumu.l.a.tiveetection probability is obtained at a range R = 2~6 R1

It can be seen 1'rom these curves that range (or R/R~) is a slowlyvarying function of 5 around the optimum and therefore that the choice ofo is not critical. The optimum value of 0 decreases with increasing P cFrame time is related to 5 by the simple expression:

(6)The optimum value 01' 5 is given in Fig. 2 as a function of P , againc

for a non-fluctuating target. Figures 3 and 4 show how the optimum 0 ,varies with P for two other target scinti~ation models as indicated inct he fi gu re s.


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coN_J-.::U> V ~ N C UI~ Z "'C


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IV . C U R Y I B . .O F. GUM1I4 ' . t 'mD~TION . l ? R O J 3 A B : t L I T Y v m S P S RANG E" "Figures 2, 3, ""and4aJ.so shoWhciwcumula-tlvedetection probability-

varies with librma.ii~~dre.:nge(R,!Ri)fbrthree target ~cintil1ation moclels.IIieach case the c u r v e labeled J i l a . x : : I J m i r iJ ~ /Rl applies to the case whereframe'time (or 8) has been selected to maximizethe range at which thegiveii Pcis obtailied. 1'hus8'1/8l"ies ~th Pc orR/Rl &longthese curves.For purposes ofillu~trat!on, jj_gs. 2 and 3 8l.so showPc as a function ofn6riD8.1ized~e for a fix~d'\'a.lue of '8 (8 =".0464).

To il1ustrtitethe wi~ot these curves, ~ s U m e one wishes to d.eteI'Illinethe r a n g e e.twhicna given rade.r-target combinationwouldYield an Boper~ent C'lllII1l.l.e.ti'J'e.etect:tonprobability.First, Rl can be ca1culatedusi~Eq.(4), the radar parameters (P~A,Teff,L, and n), and the target pa.r~metera . ( 1 and V"). A s s u m E ! the target:is non-fluctuating so that Fig. 2 iscapplidable. For 80 per centcumul.ative' detection probability, the optimumvalue ot6is.694. Theoptirrllllli " r a . m e t iJ r l e can be calculate( using thisv&lue ot 8 andEq. (6h TIlerl.ormalizedrange for 80 per cent Pc is R/Rl =0.21. Hence, the caJ.culated range in this case is 0.21 RI"

The formu.lation of the search problem is quite general in that onlythe f'undamentaJ.radar parameters are displ.a\Yed. The beamwidth andfrequency, for instance, do not appear in Eq. (4) for Rl Thus there isconsiderable latitude in radar design. For instance, the sameaperturecan be used for transmission and reception, or a. smaller one could bechosen for transmission, inwhich case the A applies to the receivingaperture and multiple receiving beamsmust be used. A narrow beamor a


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single large beam can be used to scan the search volume. In the lattercase the scan time is to be interpreted as an integration time. The lossfactor (L) in Rl should, of course, include scan loss, field degradation,a factor for transmitting antenna inefficiency (power radiated into thesidelobes), plumbing loss, and other losses appropriate to the particularsystem.

Curves of averaged single-scan detection probability, i.e., average ofblip-scan ratio over the interval 6, have also been computed but are notshovm here. The formulation of the problem in that case is very similarto the formulation for cumulative detection probability. Again the resultscan be presented conveniently in terms of v and 5. It is interesting thatthe averaged single- scan detection probability and the cumulative probabil-ity are almost identical for near-optimum values of 5 in the case of a non-fluctuating target. Because of the steepness of the blip-scan curve inthis case, the detection probability accumulated in earlier scans is ofl it tl e S ig ni fi ca nc e. In the case of fluctuating targets the gain due toc umu lat ion i s q uit e s ign ifi can t.


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v. N ON -C OH ER EN T SI GN AL I NT EG RA TI ONThe discussion thus far has assumed that the signal received from a

target during each search frame is integrated coherently. Also, the curvesassume n = 106, where n is the average number of noise samples between falsealarms. If' a different false alarm number is of interest (n " 106), orif the signal is integrated incoherently, a value of Qi d if fe re nt f ro munity should be used in ,Eq. (4) for computing Rl Aside from this difference,the procedures for determining optimum frame time and cumulative detectionprobability outlined above are still applicable. As explained in theAppendix, N = 1 in the Ql factor corresponds to coherent integration. If'a value Nl is assigned to N, this means that only l/Nl of the originalradar returns are integrated coherently, and that Nl samples are integratedaf ter en ve lop e de tec ti on.

T he q uan ti ty Ql should be used in the cases of a non-fluctuatingtarget or a target with a Rayleigh amplitude distribution, i.e., Ql shouldbe used in conjunction with Figs. 2 and 3 . The quantity ~ should be usedfor the target scintillation model of Fig. 4 . T he se q ua nt it ie s, Ql and ~,are given in Figs. 5 and 6 for three different false alarm numbers, as afunction of the number of samples integrated incoherently (N). The deri-vation of the Qi curves is discussed in the Appendix.


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VI. CONCIIJSIONS

The range at which a uniformly scanning radar achieves a given cumula-tive detection ~robabil1ty against an a~proaching target varies as the cuberoot of the radar's ~ower.a~rture product, assuming either optimum radardesign (i.e., o~timum scan interval and coherent signal integration) ora fixed deviation from o~timum (e.g., scan interval equal to a constanttimes its optimum value in all cases).

CUrves are given in Figs. 2 through 6 for computing radar range as afunction of cumulative detection probability and for optimizing frame time.Three different target scintillation models were included, as indicated inFigs. 2 to 4. Since the work was done graphically the accuracy is notbetter than a few ~r cent. However, in most cases target echo area andfield degradation are not k n o w n with sufficient accuracy to warrant moreaccurate range predictions. The accuracy is better for n = 106 a n d N = 1than for other values, since an ap~roxima.tion is made to arrive at theo th er v al ue s.

The results are, of course, ap~licable to either mechanically scannedor p hased -arr ay rad ars. An interesting area for further analysis 1s the'use of sequential detection techniques with phased-array radars in order too pt im iz e c um ul at iv e d et ec ti on p ro ba bi li ty .


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AppendixD ERIVA TIO N O F THE Q C O RR EC T IO N FO R N O N -C O HE RE N T IN T E G -R A TIO N

In Ref. 2 it was shown that blip-scan ratio (p) can be expressed as afunction of a single variable, u, to a good approximation in the case of atarget fluctuating scan-to-scan with a Rayleigh amplitude distribution.In this case:

Ru= - g (n,N)Rowhere g (n,N) is a correction factor. It is defined and plotted in Ref. 2.The b1.ip-scan ratio for any arbitrary (n,N) can be obtained from theblip-scan curve for (n = 1.06, N = 1) by multiplying the value of R ocomputed from Eq. (1.)by:

( 1. g (1.06, 1) )~4 g (n,N) (8)

For a given dwell time in Eq. (l), when the number of samples integratedincoherently increases from 1o N, the amount of coherent integrationdecreases by the factor N. This accounts for the ~4 term above.

Since:

(9)multiplying the value of R by the factor of Eq. (8) is equivalent toomu 1. ti pl yi ng R l by :

~6 ] 4/3Q = g (10 j 1)1 g (n,N 1. :-mN (10)

This is the quantity plotted in Fig. 5 and it is applicable to the targetsc in ti 1. la ti on c ha ra ct eri st ic of F ig. 3 . T he q ua nt it y Ql can also be used


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to a good approximation for non-fluctuating targets (Fig. 2) to extend theresults to other values of N. The extension to different values of n doesnot hold to as good an approximation in this case but it is not far offfor small changes in n, For example, one calculation for the case of an on -f lu ct ua ti ng t ar ge t i nd ic at ed "" 5 per cent range error in using Ql to6 10change n from 10 to 10 An identical argument, based on the scale factorf (n,N) of Ref. 2, leads to a similar curve for ~ (Fig. 6) which isapplicable for the target scintillation model of Fig. 4.


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REF 'EEENC:ES

1. Marcum, J. I., A Statistical Theory of Target Detection by PulsedRadar, The RANDCorporation, MemorandumRM-754, December 1, 1947;Marcum, J. I., A Statistical Theory of Target Detection by PulsedRadar: Mathematical Appendix, The RANDCorporation, ~morandumRM-75" Juy 1, 1948.2. Swerling, Peter, Probability of Detection for Fluctuating Targets,The RANDCorporation, MemorandumRM-1217, March 17, 1954.,. Mallett, J. D. and P. Swerling, Detection Range of an Active Radar

Seeker, The RANDCorporation, MemorandumRM-1238 , April 20, 1954.


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