Exact and Approximate Detection Probability Formulas in Fundamentals of Radar Signal Processing

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1 Exact and Approximat tction robabiity Formuas in Fundamntas of Radar Signa rocssing Mark A. Richards Sptmbr 8 Introduction Tab 6. in th txt Fundamntas of Radar Signa rocssing, nd d. [], is rproducd bow. It givs formuas for th probabiity of fas aarm FA and th probabiity of dtction for th four standard Swring targt fuctuation mods and th nonfuctuating mod (dnotd Swring or 5, as is oftn don) in additiv Gaussian nois whn a squar aw dtctor and -fod noncohrnt intgration ar usd. χ is th signa-to-nois ratio. Th sam tab aso appars as Tab 6. of th first dition of th book. Tab 6. from [], robabiity of tction for Swring Mod Fuctuating Targts with a Squar-Law tctor. Cas Commnts r or 5 ( T+ χ ) T QM ( χ, T ) + Ir ( T χ ) r= χ onfuctuating cas T + xp χ + χ T I, ( + χ) 3 4 T ( ) T + xp χ + ( χ /) χ + + χ / k k ct! c ( ct ) c T > ( c) k= k!( k)! c k ct! c ( ct ) c T < ( c) k!( k)! c k= k T = I, in a cass Approximat for and χ > ; xact for = Approximat for and χ / > ; xact for = or + ( χ / ) I(, ) is arson s form of th incompt Gamma function; I k( ) is th modifid Bss function of th first kind and ordr k. ag

2 Th particuar forms of th dtction quations in this tab wr obtaind from th txt by Myr and Mayr ( M&M ) []. Howvr, th vrsions prsntd in this tab, whi usfu, ar a somwhat inconsistnt mix of xact and approximat rsuts. Th first goa of this mmo is to prsnt a mor consistnt st of xact rsuts, and to show brify how som approximat, simpifid rsuts ar thn drivd. A scond issu is that [] and thrfor th origina tab uss arson s form of th incompt gamma function to xprss som of th rsuts. arson s incompt gamma function is dfind in [] as a b+ a b+ b t b t I ( a, b) = t dt = t dt b! Γ + () ( b ) Myr and Mayr us ony th first vrsion bcaus th argumnt b is aways intgr; th scond vrsion is a gnraization for non-intgr b. spit th I (, ) uss just I (, ) for arson s form bcaus that is th way it was prsntd in []. notation adoptd hr, th tab abov It now is mor common to us an incompt gamma function or normaizd incompt gamma function dfind as [3][4] c c d t d t I ( c, d ) = t dt = t dt Γ! () ( d) ( d ) whr th scond vrsion is for an intgr argumnt d, which wi aways b our cas. Eqn. () is aso th dfinition usd by MATLAB for thir gammainc function. Th two vrsions ar ratd by (, ) (, ) I ab = I a b+ b+ (3) This convrsion is aso notd in [5]. Th scond goa of this mmo is to rstat th tabuar rsuts in trms of th mor modrn, common, and MATLAB -compatib normaizd incompt gamma function. Exact Equations for, in Origina arson s and Modrn MATLAB -Compatib Forms Th formuas for in th Swring and 3 cass of th origina Tab 6. abov ar approximations, as notd in th ast coumn. Tab rpats th rsuts of th tab without th approximations, and aso idntifis whr ach quation appars in []. Th quations in Tab sti us th arson s form of th incompt gamma function. Howvr, th notation has bn changd to dnot arson s form as I (, ), as in Eqn. (); th notation I (, ) wi now b rsrvd for th mor common and MATLAB - compatib form of Eqn. (). ag

3 Tab. Exact robabiity of tction for Swring Mod Fuctuating Targts with a Squar-Law tctor and arson s Form of th Incompt Gamma Function. Cas Commnts r or 5 ( T+ χ ) T QM ( χ, T ) + Ir ( T χ ) r= χ onfuctuating cas. Q, is Marcum s Q function. M M&M qn. (3-37). T I, + T T + xp I, χ χ + ( + ( χ )) T I, ( + χ) 3 4 = or : + ( χ /) T T + xp + ( χ /) χ / + ( χ /) T T c T T ct > : + +! ( c) ( ) + ct ( c) ct ( c T c) k k ct! c ( ct ) c, T > ( c) k= k!( k)! c k ct! c ( ct ) c, T < ( c) k!( k)! c k= k M&M qn. (3-56). M&M qn. (3-6). M&M qns. (3-69) and (A-85) + ( χ / ) + ( χ / ) M&M qns. (A-7) and (A-) T = I, in a cass M&M qn. (-7) I (, ) is arson s form of th incompt gamma function; I k( ) is th modifid Bss function of th first kind and ordr k. In ordr to mor asiy transat ths xprssions into computr cod, Tab rstats Tab in trms of th MATLAB -compatib normaizd incompt gamma function I (, ). ot that this affcts ony th Swring and xprssions, and th FA xprssion. In addition to rtaining which M&M quation is th sourc of ach cas, w aso not whr th sam xprssions can b found in Barton [5] for th Swring and cass. 3 ag

4 Tab. Exact robabiity of tction for Swring Mod Fuctuating Targts with a Squar-Law tctor and th MATLAB -Compatib Form of th Incompt Gamma Function. Cas Commnts r or 5 ( T+ χ ) T QM ( χ, T ) + Ir ( T χ ) I T, + T I, ( + χ) 3 4 r= χ T T + xp I, χ χ + ( + ( χ )) = or : + ( χ /) T T + xp + ( χ /) χ / + ( χ /) T T c T T ct > : + +! ( c) ( ) + ct ( c) ct ( c T c) k k ct! c ( ct ) c T > ( c) k= k!( k)! c k ct! c ( ct ) c T < ( c) k!( k)! c k= k = I T, in a cass onfuctuating cas. Q, is Marcum s Q function. M M&M qn. (3-37) Equivant to M&M qn. (3-56) Barton qn. (4.8) Equivant to M&M qn. (3-6). Barton qn. (4.35) M&M qns. (3-69) and (A-85) + ( χ / ) + ( χ / ) M&M qns. (A-7) and (A-) Equivant to M&M qn. (-7) Barton qn. (4.4) I(, ) is th normaizd incompt gamma function; I k( ) is th modifid Bss function of th first kind and ordr k. 3 Simpifid Equations for for Som Swring Cass roprtis of th normaizd incompt gamma function can b usd to simpify th xprssions for in a fw cass, producing what ar crtainy bttr-known and mor cacuator-frindy xprssions. Ths xprssions ar xact for som Swring cass and vaus of, and approximat for othrs, as notd in th discussion. First, not th foowing proprtis of I(, ) [4]: 4 ag

5 For any c, I( c,) = For any c, I c, = c For any c, (,) = ( + ) For d >, im I( cd, ) = Ic c c c 3. Swring Cas Whn =, appying th proprty I( c,) = immdiaty rducs th Swring xprssion in Tab to th much simpr but sti xact xprssion T = xp + χ [Swring, = ; M&M Eqn. (3-5) ] (4) Aso in th = cas, th proprty I( c,) xp( c) = can b appid to th xprssion for FA to gt T = [ = ] (5) Eiminating T from qns. (4) and (5) givs, for a sing samp of a Swring targt in nois, ( +χ = ) [Swring, = ] (6) FA ow assum that and χ >. (Th quantity χ can b thought of (vry) crudy as th intgratd signa-to-nois ratio at th input to th squar aw dtctor.) ifranco and Rubin argu on p. 39 of [6] that both of ths conditions must hod tru if w ar going to hav any chanc of dtcting a targt whi maintaining a sma FA. Thy furthr argu, apparnty rying on th proprty im I cd, =, that th two incompt gamma functions in th Swring xact rsut ar both c approximaty qua to on. Undr ths conditions, th xact xprssion in th tab thn rducs to xp T + χ + χ [Swring ; and χ > ; Barton qn. (4.3) ] (7) This rsut is xact (and matchs Eqn. (4) ) for =. 3. Swring 3 Cas Using th sam argumnts as in th Swring cas, ifranco and Rubin show on p. 4 of [6] that th xact xprssion givn in Tab for th Swring 3 cas with = or is a vaid approximation for argr as w, providd again that and χ > and using im I( cd, ) =. Thrfor, c 5 ag

6 T T + xp ( χ /) + + ( χ /) χ / + ( χ /) [Swring 3; and χ > ; M&M Eqn. (3-69), Barton qn. (4.38) ] (8) Again, this xprssion is xact for = and =. 4 Rationships Btwn Crtain Swring Cass 4. Fuctuation Mods Ar Moot Whn = Th ffct of noncohrnt intgration is moot whn =, i.. thr ar not mutip samps to intgrat. Consqunty, th Swring and cass, which shar th sam xponntia targt probabiity dnsity function (F), shoud produc idntica rsuts in th = cas. Simiary, th Swring 3 and 4 cass, which shar a chi-squar targt F, shoud b idntica whn =. Th quivanc of Swring and is asiy shown. It was sn in Eqn. (4) that xp T ( χ ) for = in th Swring cas. Using = + I c, = c shows that th Swring rsut is idntica, as xpctd. Th quivanc of th Swring and cass whn = aso mans that Eqn. (6) appis to th Swring cas as w. Concrning th quivanc of Swring 3 and 4 mods whn =, it is trivia to writ down th xprssion for th Swring 3 cas, but this cannot b radiy ratd to th Swring 4 xprssions in Tab with =. Howvr, it asy to s that th charactristic functions for both cass, which ar givn in M&M as Eqns. (3-63) (Swring 3) and (3-7) (Swring 4) ar idntica for =, so th rsuting Fs and thn dtction probabiitis must aso b qua. Th radr is rfrrd to [] for th dtaid xprssions. 4. Swring = Swring 3 Whn = In a prvious tchnica mmorandum [7] it was shown using charactristic functions that th rcivr oprating charactristic (ROC) curvs (i.., for a givn FA and signa-to-nois ratio χ ) ar idntica for th Swring and 3 cass whn =. This is aso confirmd by th xact quations in Tab. Appying th proprty (,) ( ) = ( + T ( + χ) ) xp T ( + χ) producs th idntica rsut. Ic = + c c to th Swring xprssion with = rducs it to. Stting = in th first xprssion for th Swring 3 cas 5 Rfrncs [] M. A. Richards, Fundamntas of Radar Signa rocssing, scond dition. McGraw-Hi, 4. [].. Myr and H. A. Mayr, Radar Targt tction: Handbook of Thory and ractic. Acadmic rss, ag

7 [3] Sction 6.5 in M. Abramowitz and I. A. Stgun, ds., Handbook of Mathmatica Functions. ationa Burau of Standards, Appid Mathmatics Sris, vo. 55. [4] Sction 8. in IST Handbook of Mathmatica Functions, Ovr t a, ds. Cambridg Univrsity rss,. [5]. K. Barton, Radar Equations for Modrn Radar. Artch Hous, 3. [6] J. V. ifranco and W. L. Rubin, Radar tction. SciTch ubishing, 4. [7] M. A. Richards, Swring = Swring 3 Whn =, tchnica mmorandum, Juy 4. Avaiab at 7 ag

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!