Simultaneous Detection and Estimation, False Alarm Prediction for a Continuous Family of Signals in Gaussian Noise


 Thomasina Scott
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1 Sultaneous Detecton and Estaton, False Ala Pedcton fo a Contnuous Faly of Sgnals n Gaussan Nose D Mchael Mlde, Robet G Lndgen, and Mos M Bean Abstact New pobles ase when the standad theoy of jont detecton and estaton s appled to a set of sgnals dawn fo a contnuous faly; decson thesholds ust be detened as a functon of the contnuous paaete x chaactezng the sgnals, and false alas occu, not wth a dscete pobablty, but wth a densty n x A Bayes decson stuctue ove the doan of sgnal paaetes yelds a state estate of the sgnal paaete x as an ntegal pat of a sgnal declaaton The decson cteon s conveted to a fo n whch detecton and false ala denstes appea and fo whch s deved a elaton between the fo all x The ltng case of addtve Gaussan nose and a hgh detecton theshold allows a splfed decson cteon and a state estate of sgnal locaton n x that appoaches the CaeRao bound Also n ths lt, an analytc fo fo the false ala pobablty densty ove x, a quantty not eadly obtaned n geneal, s evaluated hee though ts elaton to the detecton pobablty The false ala densty expesson and state accuacy pedcton ae tested though Monte Calo sulatons, and the copason deonstates excellent ageeent Index Tes Contnuous paaete estaton, decson theoy, falseala pobablty, jont detecton and estaton I INTRODUCTION Detecton of known sgnals n nose s seldo unaccopaned by estaton The unknown vecto of paaetes x chaactezng the faly of sgnals coonly occus n ts splest fo as a te of aval o a locaton n an age, and the opeaton of sgnal detecton eques, at least plctly, a seach ove the paaete space Mddleton and Esposto [] poneeed the unfcaton of detecton and estaton as a poble n Bayesan decson theoy, exanng seveal scenaos of sequental decsons wth both ndependent and coupled cost functons fo the two opeatons Subsequent papes have elaboated on ths fundaental appoach, fo Gaussan nose [], fo bounds on detecton and estaton pefoance [3], and fo the advantage of jont, athe than sequental optzaton [4] The paaete estaton decsons n these papes, howeve, ae dscete Moustakdes et al extend the analyss to contnuous sgnal paaetes and deve decson stuctues to nze soe vaant of the Bayes sk (sestaton cost plus ssedsgnal cost n [5], sestaton cost condtonal on a sgnalpesent decson n [6] subject to a false ala constant (constant on Bayes cost unde a nosgnal hypothess n [7] In ths pape we also appoach the optzaton of sultaneous detecton and estaton though the nzaton of the Bayes sk, but we avod the coplcaton caused by a false ala constant by assgnng Bayesan costs to all decsons A decson stuctue s then eadly obtaned that nzes the Bayes sk ove all easueent possbltes Ths setup epesents the statng pont fo ou analyss, whch focuses on the detenaton of sgnal detecton pobabltes, state estaton eos, and false ala ates as functons of the sgnal paaete x Vaaton of the false ala cost, whch we shall use to deve an potant elaton between sgnal detecton and false ala pobabltes as functons of the sgnal paaete x, s the effectve equvalent to vaaton of a falseala constant We begn by specfyng an optzaton cteon based on the Bayes sk, and fo t we constuct a decson statstc T n whch the lkelhood ato fo sgnal vesus nose, foed fo a possble sgnal at any x, has a cental ole When ths ethodology s appled to the poble of a contnuous faly of sgnals n addtve Gaussan nose, and the estate of x s allowed to be contnuous, soe novel and useful popetes eege The decson test at hgh thesholds (hgh falseala costs can be cast n a fo that s geatly splfed fo the geneal stuctue that nzes the Bayes sk And, pehaps not supsngly, t s found that the detecto/estato esultng fo the nzaton of the Bayes sk conssts of an nfnte bank of lnea coelatos (atched fltes whose output, as a contnuous functon of the coponents of the hypotheszed sgnal state vecto x, s an essental coponent of the lkelhood ato We show how the geoetc popetes of ths lkelhood ato feld ove x allow one to deve a sple expesson fo T n the asyptotc lt of hgh sgnaltonose ato (SNR Ths asyptotc expesson, vald n any applcatons, elnates uch of the buden of wokng n the contnuous state paaete doan, and the splfed pocessng yelds a concse expesson fo the detecton cteon and a sultaneous estate fo the sgnal state In addton to splfyng the decson pocess, ths pape poduces pefoance pedctons n the fo of detecton and false ala pobabltes as functons ove the sgnal paaete doan x These quanttes ae denstes n x fo the pobablstc occuence of axa at x n the decson statstc T The detecton pobablty, as t elates to the detecton cteon, s teated as a functon of two state vectos the estated sgnal locaton whee the detecton statstc assues ts axu (ove theshold and the actual locaton
2 of the sgnal The pedcted statstcal dscepancy between the two povdes the state estaton accuacy Evaluaton of the false ala densty ove the doan x s, n geneal, a poble of fundaental dffculty one that affods no easonably usable expesson that s dectly obtanable Howeve, n ou applcaton to addtve Gaussan nose and hgh declaaton theshold, we ae able to evaluate detecton pobabltes dectly fo the geoetc popetes we establsh ove the sgnal paaete doan We then use statonaty elatons obtaned between sgnal and false ala pobabltes to solve the dffcult poble of obtanng a concse analytcal expesson fo the false ala densty ove the doan x The expessons we obtan fo the detecton and false ala pobabltes ove the sgnal paaete doan x at hgh theshold n Gaussan nose eveal the followng behavo Fo a fxed decson theshold the false ala densty and detecton pobablty ae stong functons of SNR; the detecton pobablty can ange fo nea zeo to nea unty The false ala densty s sall when the detecton pobablty s close to one o zeo, and geate by odes of agntude when the detecton pobablty s onehalf When the SNR vaes ove the sgnal state doan x by a facto well n excess of unty, egons of nealy cetan and nealy possble detecton ae sepaated by egons of agnal detecton that contan ost of the false alas The detecton and false ala descptos we obtan addess the pay concens of a pactcal applcaton fo what sgnal chaactestcs (what state s a sgnal detectable and how sevee s the false ala poble The pape s oganzed as follows Secton ntoduces the notaton and deves the Bayes decson ule fo jont detecton and estaton A decson statstc T follows fo an optzaton cteon based on nzng the Bayes sk The decson statstc T s used n Secton 3 to establsh an potant elaton between detecton and false ala pobabltes by eans of a geneal statonaty popety Secton 4 then specalzes to Gaussan addtve nose, and the decson statstc s used agan n a fo that explctly ntoduces the lkelhood ato as a feld ove the sgnal paaete doan x Though an asyptotc appoxaton to the decson statstc at hgh theshold, the geoety of the decson statstc ove the contnuu of the doan x s developed Ths geoety s key to the analyss n Secton 5 of the detecto/estato pefoance as a functon of x, patculaly when the SNR vaes wth x Secton 6 apples Monte Calo sulaton to a sple sgnal and easueent odel to test the theoetcal expessons of Secton 5 State accuacy and false ala occuence ae seen to agee vey well wth the pedctons Secton 7 sus the pefoance ove the doan x to fo a global opeatng chaactestc that s used to evaluate a global false ala pobablty a quantty that would be vey tedous to obtan by othe eans Sgnfcant new esults ae labeled as theoes FORMULATION The easueent vecto conssts of nose n and possbly an addtve sgnal s(x descbed by an unknown paaete x, s( x = f x = x = s( x + n ; ( s( x f x χ χ, whee x actually epesents the absence of sgnal and the set χ s a contnuous doan of denson d, whch n typcal applcatons ay nclude such densons as sgnal appeaance te, poston, velocty, apltude, shape paaetes, etc The easueent o obsevaton vecto space s of fnte denson and s assued to be contnuous n ts eleents, adttng a pobablty densty n fo all values of x In a typcal applcaton the eleents would nclude saples ove space, te, and possbly fequency, etc The poble s to decde whethe sgnal s pesent, and f so, wth what value of x The easueent odel povdes the condtonal pobablty fo wth sgnal absent, The paaete space s the unon { x } p ( f x = x, ( and the condtonal pobablty fo gven sgnal pesent at x, p ( x f x χ (3 These two cases ae pesued to occu espectvely wth a po pobablty a and densty a( x, whee a + a( x dx = (4 x χ It should be noted that the geneal popetes developed n ths secton and the next ae good fo a oe geneal easueent odel, n whch the nose andoly altes the sgnal; the only assupton s that a useful easueent yelds values of the lkelhood ato L( x = p ( x/ p ( (5 sgnfcantly exceedng unty fo pobable values of when sgnal s pesent The addtve fo ( s ntended fo the analyss of Gaussan nose n late sectons The uncondtonal pobablty densty fo s p( = a p ( + a( x p ( x dx (6 x χ It follows that the Bayes pobabltes fo no sgnal ( x = x o fo sgnal at x χ, condtoned on a patcula easueent, ae
3 3 b( = ap( / p(, bx ( = axp ( ( x/ p( ; that s, the ando vaable x occus wth pobablty b (, and the ando vaable x χ occus wth condtonal pobablty densty b( x These ae the consttuents of an optu detecton/estaton ule, to be defned and assebled fo a cost atx as follows [8] The standad Bayesan appoach to assgnng costs to decsons s appopate wth the notaton extended to accoodate dffeent costs at dffeent sgnal locatons x and declaaton locatons x We let the fst aguent of the cost functon Cx (, x epesent the declaaton locaton and the second the actual sgnal locaton Futheoe, we epesent the sgnalabsent paaete x explctly and assue the locaton paaetes x, x ae othewse fo the set χ Accodngly we have: Cx ( ', x: Estaton eo: cost fo a detecton at x' of a sgnal at x Cx (, x : Mssed detecton: cost fo declang sgnal absent when t s pesent wth paaete x Cx ( ', x : False ala: cost fo a sgnal declaaton at x ' when sgnal s absent Cx (, x : Cost fo declang sgnal absent when t s absent The cost functon s assued to be contnuous and bounded n x and x, and Cx (, x s assued to attan a nu ove x at x = x, the coect state estate The notaton becoes splfed by settng C( x, x to zeo (wthout loss of C x, x as genealty, by edefnng the falseala cost ( C( x, and by defnng a ewad functon R( x, x = C( x, x C( x, x as the cost dffeence between declang a sgnal at x to be absent and declang t to be pesent at x We assue a easonable cost assgnent that would leave R( x, x nonnegatve but at o nea zeo fo x fa fo x The Bayes sk s defned as the cost functon aveaged ove the Bayesan pos and the easueent dstbuton: (, = (, ( ( Bsk x MC C x x a x p x dxd { M, x } C + { M, x } C (, ( ( ( ( + C x ap d { M, x } C C x x a x p x dxd (8 (7 The ntegaton doan { M x } epesents the ctcal egon of easueent space that s assgned a sgnalpesent decson at x Measueent values that le n the copleentay egon { MC, x } eceve a nosgnal decson Ou optzaton cteon fo sultaneous detecton and estaton can now be stated explctly: selecton of x and the coespondng ctcal egon { M C } to nze the Bayes sk Ths optzaton balances the cost of a ssed detecton, the cost of sestatng the sgnal state fo a vald detecton, and the cost of a false detecton The decson stuctue that satsfes ths cteon s eadly obtaned, and n antcpaton we defne the decson statstc T ( x ' = R( x ', x p( x a( x dx C( x ' p( a p ( (9 The decson statstc T( x s eadly seen to be the dffeence between the posteo loss fo a sgnalabsent decson and that fo a sgnalat x decson condtonal on : C, ( ( ( ( ( T( x' = C x, x C x, x b x d x + C x, x C x, x b( ( Wth the ecognton that the easueent pobabltes ae noalzed to ntegate to unty ove the cobned egon, { MC, x } { MC, x }, the Bayesan sk can be epesented as sk (, = (, ( B x M C x x a x dx C T( x p( d { M, x} C The equaton above shows clealy that choosng x and ( M C to axze the ntegaton of T( x' ove nzes the Bayes sk, and t s vey obvous how the choce should be ade: values of that yeld a negatve T( x' fo all x should be excluded fo the ctcal egon M C, and x should be chosen to axze the postve values that ae ncluded n M C The defnton of T( x' ensues that t s negatve when a sgnalabsent decson nzes the expected cost fo the easueent and postve fo soe x when a sgnalpesent decson nzes the cost Thus the cobned detectonestaton decson fnds the sgnal paaete estate x = x that axzes T( x' (nzes the net expected cost,
4 4 { T x } T( x ax ( ' = x', ( and then, as the decson test, chooses between sgnal pesent and sgnal absent: >, choose x T( x <, choose x, (3 esultng n the ctcal egon { MC, x } = { T( x >, x = x} (4 The developent to ths pont has poduced a statstc that povdes the pope choce between sgnal and nosgnal and sultaneously specfes the sgnal paaete estate x that best explans a sgnal decson Howeve, analyss of the decson statstc can povde uch oe nfoaton It has the potental to descbe the dstbuton of sgnal locaton about the obseved axu at x = x, and t can be used n a pedctve sense (wth as a ando vaable to detene sgnal detecton and false ala pobabltes ove the sgnal paaete doan In the next secton we anpulate the fo of the decson statstc to poduce geneal expessons fo these pobabltes In the succeedng sectons on Gaussan nose, we develop explct analytc expessons fo these pobabltes (at hgh SNR, hgh theshold 3 GENERAL PROPERTIES Ou objectve s to old the decson statstc nto a fo that eveals pefoance ove the doan x of sgnal paaetes The followng basc opeatons ae equed We adopt as ou geneal easue of pefoance, ( ( T = T x p d (5 T( x > It s obvous fo ( that T epesents the axu obtanable educton n the Bayes sk fo C ( x, x a ( x d x, the expected cost of a ssed detecton as calculated fo the Bayes po alone The declaaton theshold, whch depends on ewad and cost functons, appeas as a lt of ntegaton, and to exane dependence caed by ths theshold, we ntoduce a scale facto nto the cost functon The pefoance easue T, now dependent on a scaled falseala cost, s conveted to the ntegal of a densty ove the paaete space x, and the expesson eveals quanttes that can be dentfed as pobablty denstes fo sgnal detecton and false ala occuence Fnally the senstvty of the pefoance easue densty to the scaled cost functon povdes a elaton between the detecton and false ala pobabltes ove all ponts x of the paaete doan Whle the expessons fo these pobabltes ae too coplex fo dect evaluaton, we shall fnd we can detene explct analytc fos fo the unde the assuptons and appoxatons of the Gaussan analyss We begn the developent sketched above by eplacng Cx ( wth a scaled cost functon µ Cx (, whee the scala facto µ wll povde a echans fo the constucton of an opeatng chaactestc We denote the decson statstc of (9 coespondng to the scale facto µ as T( x ', µ Consde now, as the pefoance easue, the decson statstc weghted wth espect to the agnal pobablty densty p( ove all easueent ealzatons that poduce a sgnal decson ( T( x, µ > : ( T µ = T( x, µ p( d (6 T( x, µ > Senstvty of the pefoance odel to the cost facto µ can be edately obtaned fo the expesson n (9 (wth Cx ( eplaced by µ Cx ( as T( µ = C( x p( ad µ (7 T( x, µ > The ntegaton egon depends on µ, but ts vaaton does not contbute to the expesson, because the ntegand s zeo at T x, µ = the theshold ( We wsh to dstbute the aveaged decson statstc ove the paaete space of potental declaaton locatons x as the fst step towad ntoducng sgnaldetecton and falseala denstes ove that space The Dac delta functon δ ( x, wth the popety that δ ( x f ( x dx = f ( (8 fo any sutably contnuous functon f ( x [9], seves as a convenent vehcle fo ths pattonng The delta functon can be nseted nto (6 as (, { } µ > ( = ( T µ δ x x T ( x, µ dx p( d, (9 T x and an ntechange of the ode of ntegaton yelds the desed densty fo: whee ( µ τ ( µ T = x dx ( ( x = ( τ µ T( x, µ > δ x x T( x, µ p( d (
5 5 Substtuton of the expesson fo T( x, µ fo (9 enables the ntoducton of the sgnaldetecton and falseala denstes we seek: whee τ ( x µ = R ( x, x pd ( x x, µ a ( x dx ( µ C x ν x µ a, ( ( F (, µ = δ ( pd x x x x p ( xd F T( x, µ > ( x = δ ( ν µ T( x, µ > x x p ( d (3 Hee p ( x' x, µ can be ecognzed as the pobablty densty, D n x, fo detecton at x of a sgnal whch actually occued at x, and ν ( x ' µ as the densty of false detectons at x The F delta functons povde the densonalty fo denstes n x, and the ntegaton ove sus the pobabltes fo whch x = x The senstvty of the densty τ ( x µ to the cost facto µ can now be used to geneate the elaton between sgnal detecton densty and falseala densty that we seek Dffeentaton of the expesson n ( yelds τ µ ( x = ( µ T( x, µ > µ = T( x, µ > δ x x T( x, µ p( d ( ν F ( µ = C x x a ( ( x δ x x C p( ad (4 As n the devaton of (7, the vaaton n the theshold does not contbute because the ntegand s zeo thee Dffeentaton of the sae quantty fo the expesson n ( yelds x R x x p x x a x dx µ µ C( x ν F ( x µ a C( x ν ( x µ a µ τ µ µ D µ ( = (, (, ( F (5 Copason wth (4 shows that the fst two tes on the ght su to zeo The cobnaton of these two tes can be dentfed as the vaaton of τ ( x µ n ( wth espect to the change n ntegaton doan, and ths change occus at the theshold whee the ntegand tself s zeo and esults n the cancellaton of the tes Now the vaatonal elaton we seek follows fo ths equalty of the fst two tes and takes the fo of the followng dffeental equaton: and The detecton and falseala denstes pd ( x x, µ ν F ( x µ ae denstes n the sgnal paaete space Dffeentaton wth espect to µ endes the devatves as denstes wth espect to apltude as well We shall see when we analyze the hghsnr, Gaussannose case n Secton 5 that the detecton pobablty can be appoxated n a easonably dect anne Howeve, the false ala pobablty nvolves the pobablstc occuence of local axa ove the feld x of possble sgnal paaetes, and dect evaluaton s not so eadly attaned Fo false alas we shall use the dffeental elaton (6 to evaluate the false ala pobablty fo the detecton pobablty, as shown n geneal n Theoe Theoe : The statonaty elaton of (6 can be ntegated fo an uppe lt µ =, whee the nfnte false ala cost elnates all taget declaatons and whee both detectons and false alas vansh, to µ =, whee the assgned cost functon Cx ( apples, to gve ( R( x, x ax ( D( µ µ (7 C( x a ν x = p x x, µ dx dµ F If the lowe ntegaton lt s set at µ nstead of unty, the elaton above seves as a genealzed veson of the local opeatng chaactestc of the detecto as a functon of abtay theshold (povded µ Cx ( s taken as the cost functon detenng the theshold These elatons bea a eseblance to those fo the opeatng chaactestc fo the bnay decson poble addessed n, fo exaple, Van Tees [] and Schee and Schaf [], wth µ coespondng to a lkelhood theshold Howeve, n the bnay poble the opeatng chaactestc s obtaned vey dectly fo the lkelhood ato theshold that esults fo both the NeyanPeason cteon and a twopossblty Bayes cteon In ou analyss, on the othe hand, the elaton between false ala and detecton pobabltes s obtaned only ndectly fo the basc detecton cteon of Equatons (9, (, and (3 Ou object of pay nteest s the detecton and false ala behavo ove the sgnal paaete space, and the basc detecton cteon only plctly copehends the dependence on x ' The conveson of the detecton cteon to an expesson that ncludes pobablty denstes ove x ' ncopoates the theshold dependence equed to balance ewads and costs ove the ente space Scalng the falseala cost functon by µ povdes the vaaton n the pefoance easue that yelds a statonaty elaton connectng sgnaldetecton and falseala pobabltes R ( x, x pd( x x, µ a( x dx µ C ( x ν F ( x = µ a (6 µ µ
6 6 4 ADDITIVE GAUSSIAN NOISE The specfc case of Gaussan nose addtve on a faly of known sgnals s both wdely useful and eadly analyzable The edate a of the followng analyss s to constuct an appoxaton, good n the asyptotc lt of hgh SNR, to the decson statstc T( x' n (9 In the couse of ths analyss a nube of nteestng popetes of T( x' wll eege, ncludng the ntnsc estaton accuacy and estates of both local and global detecton pefoance ove x ' n the fo of detecton and false ala pobabltes Unde the condtons we pose fo the analyss, the paaete estate whch axzes the decson statstc s essentally the sae as that whch axzes the lkelhood ato ove the sae local egon In fact, the lkelhood ato of (5 appeas explctly n the decson statstc f we ove p ( outsde the backet n (9: p ( T ( x ' = R( x ', x L( x a( x dx C( x ' a p( (8 We begn by explong the popetes of the loglkelhood ato (LLR The easueent odel of Equaton ( apples wth a nose vecto n of ean zeo and covaance C In the Gaussan case the lkelhood ato of (5 has as ts denonato, the Gaussan nose pobablty of n, and as ts nueato, the Gaussan nose pobablty of s The esultng expesson fo the LLR s ( λ( x = logl x = s*( x C s*( x C s( x, (9 whee the astesk denotes vecto tanspose We note that the fst te, epesentng the output of a lnea atched flte fo the hypotheszed sgnal appled to the easueent, has a Gaussan pobablty dstbuton wth ean zeo n the absence of sgnal and wth ean s*( x C s ( x when the patcula hypotheszed sgnal s pesent The vaance of ths te s ( x = s*( x C s ( x, whee (x can be ecognzed as the apltude SNR fo s(x Defnng the atchedflte output noalzed to unt vaance (wth sgnal ethe pesent o absent, we can wte y( x = s*( x C / ( x, (3 x (, sgnal pesent at x y( x = (3, sgnal absent The expectaton opeato, hee and subsequently, epesents the statstcal expectaton wth espect to the dstbuton of the nose n In soe eleentay teatents the quantty y s tself used as the decson statstc [] Ths splfcaton s pessble fo one sgnal but not fo seveal copetng sgnal hypotheses of dffeng SNR, whee the ente fo above s essental The quantty y( x nevetheless plays an potant nteedate pat n the analyss of the LLR The esponse of the wong atched flte s ( x' to a sgnal at x can be coputed n tes of the scaled nne poduct wth σ ( x', x = s*( x' C s ( x / x ( ' x (, (33 σ ( xx, =, σ ( x', x <, x x by the Schwaz nequalty It follows fo ths that (34 y( s ( x x' = x ( σ ( x, x (35 so that the wong flte peseves the apltude of the sgnal wth a decoelaton loss fo the sgnal dsslaty Fo notatonal convenence we now dop the explct easueent dependence of y and epesent y( x as yx ( (and wll late do the sae fo λ ( x In the absence of an actual sgnal n the easueent, the gadent of yx (, epesented hee as devatves wth espect to the d paaetes of the hypotheszed sgnal, yx ( / x, =,,, d, conssts of Gaussandstbuted vaables wth the popety that they ae statstcally ndependent of the apltude at the hypotheszed sgnal paaete x, as seen fo y ( x / N yx = = ( yx ( N, x x (36 whee the condton N (nose ndcates no sgnal pesent The coponents of the gadent ae n geneal utually coelated, λ ( x = xy ( ( x ( x (3 y y x x j N j j (,,,,, = M = M j = d (37 and, when the hypotheszed sgnal atches the actual sgnal, Slaly, the gadent of (36 gves
7 7 = Mj + yx ( yx ( N, x x j (38 so that the atx of second devatves s negatvely coelated wth apltude Moe patculaly, f we ntoduce the atx U as the dffeence between the second gadent of yx ( and what happens to be ts condtonal ean (condtoned by apltude yx (, y x x j = ym + U j j, (39 we can use the basc chaactestcs we nfe about U to povde soe geneal geoetc popetes of the feld of noalzed atchedflte outputs Snce yx ( and all ts devatves ae ean zeo n the absence of sgnal, takng the expectaton of (39 as the equaton stands and afte ultplcaton by yx ( (and usng (38 and the noalzaton condton y ( x N = yelds U N =, yu = (4 Ths sgnfes that, fo an obseved value yx (, the second gadent has a ean pat popotonal to yx ( tes what we ay call the cuvatue atx M, plus a ando Gaussan pat U ndependent of apltude As yx ( nceases, the ando pat becoes asyptotcally unpotant and the cuvatue becoes oe accuately descbed by M Thus as we suvey the feld of noalzed atchedflte outputs ove the sgnal paaete doan, an abent peak (sgnal absent that ght tgge a detecton declaaton n the lt of hgh theshold (hgh false ala cost Cx ( would have ts shape chaactezed by the cuvatue atx M Note also that (3 and (38 ply and ( N cov yx (, yx ( = σ ( x, x (4 σ ( x, x x = x = yx ( x = x yx ( N j x x j xx = M j ( x (4 As one ght expect, the shape of a local peak n the atchedflte output povdes no addtonal nfoaton about the sgnal s pesence; fo (35 a sgnal at x wth apltude ( x poduces a ean cuvatue x x j ( ( = ( ( y x x = x s x x Mj x (43 ndstngushable fo a spontaneous nose peak havng a copaable apltude y The foegong enables us to descbe the statstcal geoety of axa n the LLR Rensetng the scalng n (3, we fnd that λ x x x ( x = ( x y( x + ( y( x ( x ( x and at a axu, λ / =, x y y =, x x (44 (45 so that vaable SNR ples that axa n yx ( and λ ( x do not qute concde Wth ths constant on y/ x, the atx of second gadents of λ becoes λ y = x x x x x x j j j y + x x j x x j (46 We now ague that the last te of (46 can be neglected because y/ wll be sall fo a declaable peak n a hgh SNR egon of the sgnal paaete doan, e, that yx ( wll be lage and nea x ( n agntude at a local axu n loglkelhood whethe o not a sgnal s pesent A detaled explanaton follows Fo sgnal pesent t has been establshed n (3 that y = fo y evaluated at a hypotheszed locaton x that atches the locaton x of an actual sgnal Ou analyss s estcted to hgh thesholds whee only sgnals esdng n egons of hgh SNR, >>, can be expected to be detected The quantty y s a noalzed Gaussan vaable easued n unts of standad devatons A dffeence of any standad devatons at the sgnal locaton s vey unlkely, and so, wth hgh pobablty, y/ wll be sall When no sgnal s pesent ( y =, t λ = y / y / (Equaton (3 wll be lage at any patcula locaton x Howeve, a lage nube of effectvely ndependent egons wll be seached ove the doan of sgnal paaetes, and the pobablty of encounteng a false ala wth a hgh log s also vey unlkely that (
8 8 lkelhood value ay be substantal If such a false ala occus, y ust be lage, snce λ y / The second te of λ, ( y /, s subtacted, and f t s not sall (f y s not nea n value, achevng a gven level of λ would eque an even lage and less pobable value of y Thus n a paaete doan n whch vaes sgnfcantly ove x, the ost pobable confguaton fo achevng a hgh specfc (theshold loglkelhood value s wth y λ and theefoe ( x y Ths bef analyss, by the way, povdes an ntutve glpse of one of the fndngs of the pape that s deved oe foally late: that false alas tend to concentate n egons of the doan x of agnal detectablty Wth the last te of (46 otted and y eplaced by as the coeffcent of the cuvatue atx (39, the second gadent of LLR at a local axu n the sgnal paaete feld becoes whee x x λ j ( x ( x Q ( x j j, (47 Q j M j ( log = + ( log (48 x x Note that ths equaton s exact fo the expected value of λ / x x j when the sgnal s pesent at x = x The atx descbng the LLR axu shows an nceased cuvatue along the axs of the local SNR gadent; the easued SNR povdes addtonal localzaton nfoaton n that decton Now we etun to evaluatng the decson statstc Appoxatng the ntegal n T( x' (8 eques us to specfy the functon R( x ', x, and patculaly the toleance fo estaton eo To do ths eanngfully we note that a lkelhood peak has ntnsc densons, as seen n the asyptotc appoxaton obtaned by etanng the fst two tes n the Taylo expanson of λ aound the locaton x of the axu λ = λ( x (whch, unde the condtons of ou analyss, s essentally the sae as the locaton of the coespondng local axu n T( x' : ( x L( x ex λ p ( x x* Q( x ( x x (49 Ths appoxaton, good fo lage, ndcates that ost of the lkelhood occus wthn a egon X( x defned by ( * ( ( / x x Q x x x c (5 wth c beng a odest constant; eg, fo 95 pecent lkelhood contanent, c = 3, 38, 47 fo denson d =, 3, 4, espectvely One othe assupton we shall adopt thoughout s that the sgnal po ( R x, x, the a x, the ewad functon ( cost functon C( x, and the cuvatues ae slowly vayng on the scale of the peak densons Ths chaactezaton shows that, fo ou hghsnr Gaussan applcaton, the ewad functon R( x ', x, whch n geneal enfoces the equeent that the estated locaton be nea the tue sgnal locaton, has lttle quanttatve pact povded that t extends at sgnfcant apltude ove the egon of a peak In that case t becoes convenent to defne ts egon of sgnfcance to atch the extent of the peak: (, x X x R( x, x =, othewse In choosng unty above nstead of a vaable R( x, x (5, we ecognze that no genealty s lost, because the esults to follow depend only on the ato R( x, x / C( x Then fo ntegaton ove the doan x X( x we have, fo the ntegal n (8, evaluated at x = x, ( ( ( λ d Rx (, xaxl ( ( xdx = e x a x V x (5 whee ( / V x = Qx ( /π (53 defnes a ddensonal volue scale assocated wth the local lkelhoodato peak It povdes a easue of the local egon n sgnal state space ove whch the atchedflte output would be affected by the pesence of a sgnal, and t also epesents the egon acoss whch sgnfcant nose coelatons would exst Thus a lage volue scale ndcates few ndependent nose ealzatons (sepaate false ala possbltes n the vcnty of x and coespondngly boosts the decson statstc though the ncease n the te (5; a sall volue scale adjusts the decson statstc downwads as copensaton fo a geate densty of effectvely ndependent nose daws
9 9 5 PERFORMANCE We ae now equpped to evaluate the fo the decson cteon assues as a theshold on LLR, to calculate the estaton eo n sgnal paaete space fo a detecton declaaton, and to calculate detecton and falseala pobabltes, all n the hghsnr lt of ou Gaussan applcaton The decson test at hgh thesholds n addtve Gaussan nose can be geatly splfed fo the full pocessng ndcated by Equatons (9, (, and (3 The dect use of the Bayesan decson ule nvolves a ultdensonal ntegaton ove the doan x fo evey hypotheszed sgnal locaton x a fodable undetakng followed by seekng the axu But n the Gaussan case n the hghsnr lt, the ntegatons (on the decson statstc n the fo (8 can be pefoed analytcally, as shown n (5 The seach fo x ust stll be caed out, but t s accoplshed by fong the LLR ove the doan of x, fndng the local axa, and copang the to the local thesholds If t happens that oe than one local axu s ove theshold (and f that attes n an applcaton, t s a sple atte to check the factos of the theshold to detene whch local axu coesponds to the global axu x of the ognal Bayes ule The decson ule can be conveted to a test of the LLR aganst a theshold a theshold that vaes ove the sgnal paaete doan though the dependence of po pobabltes, the cost functon, and the fundaental geoety of the lkelhood peaks The use of (5 n (8 poduces the quanttatve pescpton of Theoe Theoe : The applcaton of the decson test (3 n the >> takes hghsnr lt of ou Gaussan applcaton ( the fo, > λ ( x choose sgnal at x λ, < λ ( x choose nosgnal (54 d ( ( ( ( λ ( x = log ac x x / a x V x The theshold λ ( x does not depend on the easueent and can be coputed anywhee ove the sgnal paaete doan, but t need be evaluated only whee the LLR has a local axu: x = x Ths equaton consttutes a ajo esult of ths pape, funshng an asyptotcally coect, sple pleentaton of the Bayes soluton fo detecton and estaton The vaable theshold λ ( x contans the ente po nfoaton and geoetc chaactestcs ( V( x of the sgnal paaete doan as well as a odest dependence on SNR Note patculaly that the scale volue V( x, whch depends on the choce of the coodnate syste x, occus n the cobnaton axvx ( (, whch s nvaant to the coodnate choce and epesents the po sgnal pobablty encopassed by the volue The accuacy of the state estate x can be eadly analyzed though a Taylo expanson about the tue sgnal locaton x s Use of quanttes defned n Secton 4 n the expanson povdes a statstcal epesentaton of both the sgnal state estate and the peak LLR apltude Equatons (33 to (35 show dectly that y( x s( xs has a local axu at x = x s It then follows edately by takng expectatons of the equaton fo the LLR gadent (44 that λ ( x s( xs also has a zeo gadent and hence a local axu at x = x s Although the expected value of the LLR has a local axu at x = x s, the peak LLR wll be dsplaced by nonzeo nosenduced slope (44, whose covaance s [see (36 and (37] λ λ x x x x ( x ( x ( = ( x M ( x + ( x ( x s s s x j s s s s s j j = ( xs Qj ( xs (55 Denotng the LLR gadent vecto at the sgnal locaton as λ λ λs = ( xs,, ( xs (56 x xd and asyptotcally eplacng the cuvatue by ts aveage n the hgh SNR lt [see (47], the expanson about the sgnal λ x s s denoted λ s becoes locaton (whee ( ( λ x = λ + ( x x * λ s s s ( xs x xs Q( xs x xs ( * ( T (57 Takng the gadent of (57 and settng t equal to zeo, we see that the LLR s a axu at the locaton x (unbased unde ou appoxatons and assuptons, wth the value x xs = Q λs, (58 λ = λs + λ * s Q λs = λs + ( x xs * Q( x xs (59 Aveagng the estaton eo of (58 and the LLR peak apltude of (59 leads to the followng fundaental esult
10 Theoe 3: The estaton eo n the hghsnr lt appoaches the CaeRao bound [3]: ( ( s s = s s (6 ( x x ( x x * x Q x Note that the s estaton eo n each denson s nvese to the SNR The expected apltude of the LLR peak s based upwad fo the expected apltude at the sgnal by the second te of (59, whch nvolves a quadatc fo wth the eanzeo ando vecto λ s and ts nvese covaance Q / Ths cobnaton poduces a chsquae vaable wth ean d and vaance d [4] and yelds the expected LLR peak apltude, ( λ = λs + d/ = xs / + d/ (6 p ( x = pob[ z λ ( + d / ] D π zd z / ( λ dz, z = d /4 / D = e (63 As shown n Fg, ths pobablty s sall fo d < / < λ, becoes appecable fo / λ and closely appoaches unty fo />> λ Ths has the effect of dvdng the sgnal paaete space nto egons of hgh and low pobablty sepaated by agnal egons of odeate pobablty, whch we wll soon show to contan ost of the false alas At ths pont we ae equpped to pedct pefoance of the Bayes test though pobabltes of detecton and false ala occuence The detecton pobablty s eadly obtaned unde ou Gaussan, hghsnr assuptons though the expanson about the sgnal locaton x s pesented above The false ala pobablty, on the othe hand, depends fundaentally on the occuence of local axa ove a ando feld and s not so eadly obtaned Equaton (3 povdes a theoetcal epesentaton of the pobablty but does not povde a feasble ethod fo evaluaton Howeve, the elaton developed n Theoe wll povde a eans of detenng the falseala pobablty ove the sgnal paaete doan fo the detecton pobablty The detecton pobablty can be detened fo the dstbuton of the LLR peak epesented by (59 and the theshold n the fo (54 A sgnal at x wth SNR = x ( poduces a peak log lkelhood λ = λ + z = ( + d/ + z, = + d /, (6 whee we have ntoduced a untvaance vaable z: the exta quadatc vaance n the second te of (59, (whch, as noted n Theoe 3, s half a chsquae vaable of d degees of feedo, has been absobed nto the ando te contanng z Ths appoxaton s legtate n the asyptotc SNR doan unde consdeaton ( >> d /, and we shall gnoe the slght alteaton fo Gaussan n the copound dstbuton of z, snce the added vaance s sall and the focus fo the detecton analyss s on the an body of the dstbuton We now nfe fo (6 that the pobablty of exceedng the local theshold λ ( x n the pesence of sgnal s Fgue Pobablty of detecton as a functon of apltude SNR (x; LLR theshold λ (d ange value of 5 coesponds to D sulaton odel to be ntoduced n Secton 6 A false ala would appea as a spontaneous local axu ove theshold n λ ( x Fo pedcton, the appng of the easueent, as a ando quantty n ts vecto space, to the sgnal paaete doan though the geneaton of the LLR at each value of x ceates a ando feld λ ( x Pedctng the occuence dstbuton of local axa ove a ultdensonal ando feld s a dffcult poble that yelds anageable analytc expessons only n the lt of asyptotcally hgh values of a Gaussan feld Hasofe [5] pesents a devaton of the asyptotc foula fo the ean nube of axa above hgh levels n a Gaussan feld, and Adle n hs onogaph, The Geoety of Rando Felds [6], coves the analyss and povdes any efeences to vaous appoaches to the poble But both of those authos teat only hoogeneous felds ( x ( = constant n ou settng, and ou pay nteest s n a stuaton n whch the SNR vaes appecably Fotunately detecton and false ala quanttes ae connected though the dffeental elaton of Theoe n ou optzed Bayes appoach, and we can use that elaton to obtan an analytc asyptotc expesson fo the false ala densty ove the sgnal paaete doan even n the nonhoogeneous case
11 We begn by anpulatng the ntegals on the ght sde of (7 to solate the detecton pobablty n a fo sutable fo ou use Unde ou assupton that po pobabltes and the geoetc quanttes (cuvatue change only neglgbly ove the egon x X( x, we can evaluate the at x = x, ecognzng that pd ( x x, µ s neglgbly sall outsde that egon The ewad functon R( x ', x s unty thee [see (5], and Equaton (7 fo the false ala densty can be wtten as ν ax ( F ( x = pd ( x x, d ac ( x µ µ A change of vaables fo µ to λ, µ µ dx (64 ( ( d ( ( a x V x λ µ = e = e ac x x λ λ connects to the apltude loglkelhood theshold of (54: ν, (65 d ( x λ F ( x = e pd ( x x, λ dλdx V( x (66 λ λ ( x The detecton quantty n the ntegal above s not the detecton pobablty of (63, but t can be constucted fo elatons we have avalable As dentfed afte (3, t s the densty ove the sgnal paaete doan fo detecton at x ' of a sgnal at x (ove a loglkelhood theshold λ, and dffeentaton by λ ceates a densty n loglkelhood apltude as well We epesent ths densty as pd( x x, λ = pd( λ, x x (67 λ and decopose t nto factos that we can evaluate: ( λ ( λ ( p, x x = p x, x p x x (68 D D x The pe on λ ndcates that t s the peak loglkelhood value at ' λ = λ x as ndcated by (59: whee the bas b, x, obtaned fo ( λ = λ + b, (69 b = ( x x * Q( x x, (7 s detenstc when x ' and x ae both condtons Both pd ( λ x, x, the pobablty densty fo obsevng a loglkelhood peak value of λ condtonal on ts occung at a locaton ' px x x, the pobablty densty that a loglkelhood peak value fo a sgnal at x occus at x ', ae goously Gaussan dstbutons (unde the condtons of ou analyss and can be evaluated fo the dstbuton paaetes The basc quanttes needed ae the covaances cov ( λ, x x and cov( x x, avalable fo the ando eleents of (3, (44, and (58 (wth the use of (36 and dectly fo (6: x fo a sgnal at x, and ( * cov ( λ, x x = Q cov ( x x = Q (7 The gadent opeato s epesented as n (56 but wth the dffeentaton occung at x The Gaussan dstbuton paaetes can now be detened n the standad way: ( ( ( λ x, x= λ x+ cov λ, x x cov x x x x * = ( x + ( x ( x x ( x = ( λ x x = ( λ x cov ( λ, x x cov ( x, x cov ( x, λ x va, va whee the paaetes = ( x( ζ =, + γ γ = M ζ (7 (73 (74 γ = Q = + eflect the nhoogenety of the feld Note that when the loglkelhood peak occus at x ', the condtonally expected value of λ at x s ts uncondtonally expected value at x ' fo a sgnal at x ' The condtonal dstbuton paaetes fo λ ae sply detened by ncopoatng the bas b, and we can constuct the condtonal densty as ( + γ + γ pd ( λ x, x = exp λ b π γ (75 The ntegaton ndcated n the nne ntegal of (66 can now be pefoed (afte the algebac execse of copletng the squae n λ to yeld the expesson,
12 d F = ζ G x b ( ( ( ν ( x exp z x e p x x dx, (76 V whee and ( γ / γ + z ( x = λ + b γ + (77 z ( / G z = e dz π (78 z s the cuulatve Gaussan pobablty ove the theshold z Note that the x dependence of z ases fo the bas te b of (7 In evaluatng the ntegal, we have substtuted = ( x fo ( x, an appoxaton that sees acceptable snce the pay plcaton of a vaable SNR has been ncopoated though the paaetes ζ and γ (nonzeo The dstbuton of the vecto x ' s edately avalable fo ts covaance (7 (contnung the substtuton of fo, ( d ( π / Q b p x x = e, (79 x d / and a change of vaables fo x to the vecto z of ndependent, standad Gaussan coponents, / ( ( z = Q x x, (8 splfes the expesson fo the false ala densty: d ζ / b b ν F( x e G ( z ( b e e dz d / V ( π = (8 The ntegand depends on x, and now z, only though the bas te b, whch s evaluated as * b= zz= z, (8 and suggests a conveson to pola coodnates n the d densons Wth u as the adal coponent, u = z, we can ntegate ove the d angula coponents to poduce S d, the total angula easue (o the suface aea of the unt ddensonal sphee, whee Γ ( n s the gaa (factoal functon ( ( n Γ + = n! (Note that S = π and S3 = 4π fo fala densons Theoe 4: Ou fnal expesson fo the false ala densty now assues the fo, d/ d ( d/ V ( x ζ / u ( ( d ν F( x = e G z u e u du, Γ (84 whee z fo (77 s now gven by ( γ / γ + z ( u = λ + u γ + (85 If the ntegaton vaable u s eplaced by the chsquae vaable u, the gaa densty fo the chsquae dstbuton appeas explctly n (84 [4], and the ntegal (ncludng the nuecal coeffcent becoes ecognzable as the expected Gexp u / value of ( We now have a fo fo the false ala densty that could hadly be obtaned dectly fo the abent dstbuton fo the LLR Howeve, the fo does not eadly eveal qualtatve chaactestcs of the densty Theefoe, we ake one addtonal appoxaton to obtan an expesson fo the falseala densty that eveals the basc chaactestcs of the densty ove the sgnal paaete doan wth oe tanspaency but wth less accuacy In the next secton we shall exploe the dstbuton futhe wth a D exaple and copae t wth sulaton esults We etun to Equaton (66 but now pefo the ntegaton ove x fst We assue that the SNR ( x as well as the po pobablty and geoetc popetes ae constant ove the egon x X( x and neglect the coelatons between λ and x ' that esult fo a nonzeo [see (7] Then the elaton between sgnal locaton and detecton locaton exhbted n (58 and (6 yelds an ntechange syety, pd( x x, λ = pd( x x, λ, and the ntegaton ove x n (66 becoes pd( x x, λ dx pd( x x =, λ dx (86 But ntegaton ove the densty vaable leaves a quantty that s ecognzable as the detecton pobablty of (63 A change of apltude vaables (based on (63 n the eanng ntegal, S d d / π =, (83 Γ ( d / leads to the fo, ( λ /4 /, (87 z = d
13 3 d d/4 e z / z / ν F ( x = e e dz dz V( x z π z z d d/4 D e = V( x π d d/4 e = V( x π whee fo (63 zd zf ( z+ e e z / / dz, dz (88 The theoetcal pojectons on false ala behavo and detecton estaton accuacy developed above ae suppoted wth sulatons n the followng secton F D ( λ z = z + = d /4 + / (89 Ths expesson, except fo the facto exp ( d / 4 and a soewhat copensatoy ddependence n the theshold and n [see (6], s the exact expesson gven n Refeences [5] and [6] fo the paaetespace densty of axa above a theshold y = λ / + / n a hoogeneous standad T Gaussan feld of d densons (the abent feld y( x of Equatons (3 and (3 The qualtatve behavo of the false ala densty wth SNR n (88 s eadly detened though an exanaton of the ntegaton lt The lt z F s nu, and the densty axu, nea /= λ, and the densty declnes apdly fo hghe and lowe SNR Note that fo both the expesson above and the oe geneal expesson (84 the densty at constant LLR theshold λ ( x and constant SNR s nvese to the lkelhood volue scale, showng that V ( x s a easue of the ntnsc ndependent exposue to false alas as dscussed afte Equaton (53 But n fact the optzed theshold λ also depends on the volue scale [see (54] n a way that effectvely cancels the explct dependence on V( x to poduce an appoxate constant false ala ate (CFAR detecto wth egad to the volue scale dependence Ths behavo, calculated fo (88 and based on the D sgnal odel to be ntoduced n Secton 6, s llustated n Fg Ths fgue also llustates the dffeence between ou theshold effect, based on nzaton of a Bayes sk cteon, and that of the genealzed lkelhood ato test (GLRT Moustakdes [7] pesents condtons unde whch the GLRT s optal (dscete estaton choces, constant Bayes costs and pos; n contast, we show that t s dstnctly nonoptal n a stuaton n whch the sgnal paaete V x vaaton acoss the sgnal dependence confes a stong ( paaete doan The GLRT appoach lacks the ntegaton ove the contnuous Bayes sgnal densty that poduces the volue dependence n the theshold, and t would esult n a false ala dependence followng the dashed cuve of Fg Fgue Relatve false ala densty (88 as a functon of the scale volue (53 Sold cuve coesponds to optal theshold λ (54; dashed cuve to theshold λ wth V held constant (V= 6 SIMULATIONS We apply a sple sgnal odel and easueent settng to assess the accuacy of the theoetcal expessons fo false ala denstes ((84 and (88 and state estaton accuacy (6 though Monte Calo sulatons The odel has two state densons, scala poston ξ s and sgnal wdth σ, ξs x =, σ and coponents of the sgnal vecto s take the fo, ( ( ( (9 sξ x = Aexp ξ ξ s / σ, (9 whee A s a known constant and the coponent ndex ξ uns ove ntege values In the sulatons the nose content n the coponents ξ of easueent [see (] conssts of ndependent daws of standad Gaussan vaables, so the easueent covaance atx C s the dentty atx The easueent conssts only of nose ( = n n ou sulatons to test the theoetcal false ala expessons, and t ncludes a sgnal wth specfed state paaetes n the sulatons to test state estaton accuacy Futheoe, fo the false ala assessent the odel above wll be educed to two sepaate odels of a sngle unknown paaete by specfyng ethe ξ s o σ as known a educton necesstated by coputatonal constants
14 4 The LLR foulaton of (9, the fundaental eleent of the analyss, apples a sgnal at a hypotheszed state x to the easueent as λ ( x = s( x ( x, (9 whee the fst te n (9, the atched flte output, s evaluated as and the SNR as MFO ( x = s( x = sξ ( x ξ (93 ( ( ( ( ξ x = s x s x = sξ x (94 Although coputatons fo the theoysulaton copason ae pefoed as ndcated n (93 and (94, the nne poduct fo the SNR can be appoxated though ntegaton n a way that eveals the basc paaete dependence: ξ ( ( x A exp ( ξ ξs / σ dξ = A πσ (95 Ths expesson s actually vey accuate except when the sgnal wdth paaete σ s sall: the factonal eo s less than 3 when σ > Cetan othe quanttes that depend on the fo of the sgnal odel (9 ust be detened to pefo the theoysulaton copason These nclude the LLR theshold and paaetes n the theoetcal expessons The Q atx leads dectly to seveal of these quanttes, and fo ou sgnal and easueent odels, t can be calculated fo the elaton n (55 as s s s s cov ( ξs ξs ξs σ Q = λ = s s s s σ ξs σ σ (96 The atx expesson on the ght ases though the ecognton fo (9 that the ando pat of λ s ( s n A shot analytcal execse utlzng contnuous ntegaton as n (95 povdes the Q atx n the appoxate fo, /σ Q 3/4σ (97 Hee, as thoughout the sulaton, the actual calculatons wee pefoed dectly as vecto nne poducts The elatve SNR vaaton wth the state paaetes s slaly appoxated by / ξ = s / σ /σ (98 and the nhoogenety paaetes becoe, usng the second equaton of (74, ζ /3 γ / (99 Fo false ala sulaton an exteely lage nube of Monte Calo tals s needed to poduce a statstcally sgnfcant nube of false alas above a hgh theshold Theefoe, fo coputatonal feasblty we educed the D state odel to D: fst wth σ as the state vaable and ξ s set to zeo; then wth ξ s as the state vaable and σ set to σ = 4 Note that the D odel s sply adopted as a convenent souce fo the D odels Results fo the D odels have no dect beang on false ala expectatons fo the scenao wth two unknown paaetes The full D odel s used to assess the estaton pedcton of Theoe 3 The theshold that s appled to the LLR values poduced dung the Monte Calo tals of the false ala sulaton depends on what the state vaable s and what ts value s Wth σ as the sole state vaable, the volue scale facto of (53, needed fo the evaluaton of λ, becoes V ( σ π π = σ ( 3 Q σσ Whle we have gven appoxate expessons n Equatons (97 though (, these quanttes wee goously calculated though the nne poducts of (96 and wee used n (84 fo copason of the pedcted densty wth the Monte Calo esults Wth the paaete dependence of ( x and V ( x, the theshold λ of (54 can be wtten as ( log ( ( / ( log ( ( / ( λ σ = λ + σ σ V σ V σ ( and, usng (95 and (, n appoxate fo as λ ( σ λ log ( σ / σ, ( whee the dependence on constants (ncludng A, costs, and the a po pobabltes ae luped togethe n the constant λ It s to λ that we assgn the vaous theshold values dentfed n the fgues and table Wth σ as the state vaable, ou Monte Calo tals copsed ando daws of standad Gaussan vaates fo the
15 5 9 coponents of Ntal = 5 easueent vectos wth the nube of coponents suffcent to contan all coputatonally sgnfcant values of the sgnal fo (9, and to each we appled the hypotheszed sgnal (flte as n (9 ove a wde ange of σ values spaced by The σ values wee seached, afte ntepolaton to a fne gd ( spacng, to fnd the one that axzed the LLR (9 fo each Monte Calo tal These axzng values, togethe wth the assocated axa λ( σ, wee placed n an aay ove the fnely spaced gd fo copason to a ange of LLR thesholds λ ( σ ( The nube of values above theshold n the fnely spaced sga bns wee sued ove a wdth σ =, centeed about each pont of the fne gd wth 5 bns on ethe sde, as a soothng pocedue The nube of values ove ths wdth ( N σ was scaled to poduce a sulaton false ala densty as A = λ = 7 λ = 37 Fgue 3 Copason of false ala denstes n σ fo theshold λ = 7 Sulaton fo (3; theoy fo (84 νf σ ( σ s N = (3 σ N tal The coespondng theoetcal densty was coputed fo (84 wth the theshold taken fo ( and the paaetes ζ and γ calculated fo (74 fo the chaactestcs of ou odel Results ae pesented n Fgs 3 to 5 fo vaous LLR thesholds λ Whle the denstes wee coputed as functons of the state vaable σ, they ae plotted aganst the coespondng SNR apltude (94 to substantate ou cla that false ala poducton s axzed whee the detecton pobablty s agnal (whee λ To povde a clea confaton of ths behavo, we epeated the sulaton fo a dffeent known sgnal apltude A (9 to poduce Fg 6 Copae Fgs 4 and 6 wth dffeent sgnal apltude factos A but dentcal LLR thesholds λ False ala poducton peaks at essentally the sae SNR; t s clea fo (95 that σ adjusts to a dffeent A value to poduce the sae SNR peak locaton fo the densty; the densty values change soewhat, but the peak occus at essentally the sae SNR The pedcted densty agees vey well wth the obseved densty n all cases; the axu value appeas less by just a sall facto whle the absolute densty changes by about fou odes of agntude ove the LLR theshold ange of ou calculatons ( λ = 5 to 4 A = λ = λ = 45 Fgue 4 Copason of false ala denstes n σ fo theshold λ = Sulaton fo (3; theoy fo (84 A = λ = 4 λ = 53 Fgue 5 Copason of false ala denstes n σ fo theshold λ = 4 Sulaton fo (3; theoy fo (84
16 6 Fgue 6 Copason of false ala denstes n σ fo theshold λ = Sulaton fo (3; theoy fo (84 Sgnal/flte apltude facto changed fo to 3 Integaton of the denstes ove σ povdes a pedcton of the total false ala pobablty, and copason of sulaton and theoetcal values at the vaous thesholds s pesented n Table The sulaton values ae actually obtaned dectly as the total count ove the σ doan of axzng sga values above theshold dvded by the nube of tals N tal The theoetcal values ae obtaned though nuecal ntegaton of the densty n (84 ove σ The ageeent s clealy excellent despte the slght dstoton noted above n the fo of the denstes Table Total false ala pobablty false ala denstes n ( F ν σ ntegated ove σ doan Paaete values: A = ; σ = 4 Theshold Integated FA Rate λ Expected (78 Sulaton 5 75E4 76E4 6 8E4 83E4 7 8E4 9E E5 4E5 9 5E5 5E5 563E6 56E6 8E6 9E6 767E7 777E E7 8E7 4 4E7 3E7 A = 3 λ = λ = 45 The second D false ala copason, wth ξ s as the state vaable, nvolves a hoogeneous doan snce the SNR s ndependent of ξ s Futheoe, f we lt ou seach ove the doan to ntege values of ξ s, we can pefo the seach vey effcently wth the use of the Fast Foue Tansfo (FFT The sgnal of (9 can be epesented as s ξ ξs, syetc n ts subscpt, and the atched flteng opeaton of (93 takes the fo of a convoluton: The FFT of ths expesson yelds the FFT of MFO as the poduct of the FFTs of s and, and the nvese tansfo then poduces the atched flte output of (4 fo all hypotheszed ntege state locatons ξ s Fo ths hoogeneous case we geneated easueent vectos of hgh densonalty (, ealzatons, each wth,, ξ saples and pocessed each easueent vecto as ndcated by (4 usng Foue technques to poduce the atched flte output s as a functon of ξ s Local axa of λ (9 n ξ s wee dentfed as potental false alas, and a false ala densty was obtaned by dvdng the nube of local axa ove theshold by the length of the doan (nube of ξ saples Results wee ensebled ove the nube of easueent vecto ealzatons to povde addtonal statstcal stablty Ths obseved densty, ν F ( ξ s s, s, of couse, ndependent of ξ s ; because of hoogenety t apples to the ente ξ s doan To ntoduce an SNR dependence nto the false ala obsevaton, we scale the sgnal odel (9 ove a ange of apltude factos A Note that ou ognal ntepetaton of A eans: A s not a state vaable we apply no opeaton to axze λ ove A The vaaton n A sply epesents a vaaton n the known sgnal apltude The sae Monte Calo tal ealzatons obtaned fo one A value can be eadly adapted to othe values The local axa n the atched flte output occu at the sae ξ s locatons; the apltudes sply scale wth A And the SNR te that s subtacted to fo the LLR (9 scales as A These scalng wee used to obtan the plots of Fgs 7 and 8 fo two dffeent thesholds Note that the abscssa on these plots s not A but the SNR apltude as detened fo A though (95 The appled theshold s unchanged as A vaes: λ = λ The densty of (88 s the appopate theoetcal copason standad, and t povdes the theoetcal pedcton n the plots Agan the ageeent between theoy and sulaton s vey good Whle the geneal level of false ala densty changes by nealy two odes of agntude between the two λ theshold values of and 4, at each theshold theoy and sulaton agee wthn about 5 to pecent The theoyvesussulaton copason could have been caed out equally well usng the geneal foula (84 wth the nhoogenety paaetes set to zeo ( γ = ζ = ; ou calculatons show that, although they nvolve slghtly dffeent appoxatons, the two theoetcal expessons agee wthn one pecent fo these cases The advantage of the expesson n (88 fo a hoogeneous doan s that the algebac fo eveals the paaetc dependence oe clealy MFO = s, (4 ξs ξs ξ ξ ξ
17 7 λ = λ = 45 σ = 4 Fgue 7 Hoogeneous settng Copason of false ala denstes n ξ s ove hypotheszed SNR apltudes Theoy fo (88, theshold λ = Fgue 9 Copason of covaance of ξ s estate to CaeRao bound as SNR apltude vaes λ = 4 λ = 53 5 ξ s 5 Fgue 8 Hoogeneous settng Copason of false ala denstes n ξ s ove hypotheszed SNR apltudes Theoy fo (88, theshold λ = 4 Ou fnal sulaton tests the accuacy pedcton of Theoe 3 (6 The sgnal (9 was njected nto a lage nube of easueents ( = s+ n wth σ =4 and wth ξ s selected unfoly between 5 and 5 ove Monte Calo tals (to avod any unwanted bas n the dgtal pocessng ove ξ saples The apltude A of the njected sgnal was vaed to povde an SNR dependence n the accuacy test The sgnal (9 was appled as a flte to the easueent n a D seach ove σ and ξ s (ove appoxately ±6 standad devatons n each paaete about the sgnal njecton locaton to axze λ (9 Dscepances between the state estate and the njected state wee tabulated and suazed n a saple covaance fo copason wth the CaeRao bound of Theoe 3 Copasons of the dagonal entes, the vaances fo σ and ξ s, appea n Fgs 9 and The appoach of the sulaton vaances to the CaeRao bound wth nceasng SNR s clea The offdagonal eleents, whch ae zeo theoetcally, scatteed about zeo ove the vaous SNR values of the sulaton They typcally showed a saple coelaton n the tenths of a pecent, and the coelatons wee less than 5% fo all SNR values Fgue Copason of covaance of σ estate to CaeRao bound as SNR apltude vaes 7 GLOBAL OPERATING CHARACTERISTIC In a epesentatve applcaton, knowng the detecton pobablty pd ( x as a functon ove the doan locatons x would typcally be of sgnfcant value, but on the false ala sde, the pobablty of a false ala anywhee n the paaete doan ght be the quantty of ost nteest It s possble, of couse, to ntegate the densty ν F ( x ove the paaete doan, but the ntegaton expesson eveals lttle about the basc dependence of false ala pobablty on theshold, and the ntegaton tself s soewhat nuecally awkwad wth the nvolveent of nubes extendng ove a vey lage dynac ange Howeve, the statonaty elaton between detecton and false ala denstes pesented n Theoe can be ntegated ove the seach doan n x to povde a global opeatng chaactestc In an envonent n whch the SNR vaes sgnfcantly ove the paaete doan x, ths opeatng chaactestc can be used to obtan a sple analytc expesson fo the global false ala pobablty
18 8 Wth ou focus on global easues fo detecton and false alas, we ncopoate the xdependence of Bayesan costs and ewads togethe wth that of the sgnal po pobablty a( x nto a pseudo po a ( x Constant factos can be absobed nto the theshold µ (whch s vaed to fo the opeatng chaactestc so that a ( x s noalzed to unty fo the sgnalpesent condton: a ( x dx = Usng the appoxatons appled to obtan the expesson (64 fo Theoe (7, the false ala densty coespondng to a theshold µ becoes ν F( x, µ = a x p ( x x, µ dx dµ µ µ ( D (5 µ We fst pefo the ntegaton ove x to obtan the global false ala pobablty P F ( µ, as P ( µ = ν ( x, µ dx, (6 F PF ( µ = a ( x pd ( x, µ dx dµ µ µ µ F, (7 ecognzng n the pocess that a ( x a ( x whee pd ( x x, µ s of sgnfcant value [see the dscusson pecedng (64] The net a posteo pobablty of a detecton ove the sgnal paaete doan s eadly dentfed as P ( µ = a ( x p ( x, µ dx (8 D D to eveal the basc elaton between false ala and detecton pobabltes Theoe 5: The net pobablty of a false ala can be detened fo the net a posteo pobablty of detecton though the elaton PF( µ = dpd( µ ' (9 µ ' µ Whle ths elaton s obtaned vey sply fo Theoe, t povdes eady access to a pedcted false ala pobablty that would be vey dffcult to calculate dectly The elaton s oe eadly appecated wth a change of ndependent vaables fo µ to λ = log µ n P F and P D A change of ntegaton vaables λ log ( µ / µ = wth thesholds elated by λ = log µ then leads dectly to the expesson T λ dp T F T T dλ λ ' D ( ( P ( λ = e e λ + λ' dλ' Restct now to the case of stongly vayng SNR, n whch the paaete space s pattoned nto slands of nealy cetan detectablty suounded by egons of neglgble detectablty In ths case, the net detecton pobablty, beng cudely equal to the facton of the doan ove whch detecton s nealy cetan (wth the weghtng facto a ( x ncopoated, s actually a slowly vayng functon of the theshold Then the devatve dpd / dλ n the ntegand can be expanded to fst ode about λ T and the ntegaton pefoed to yeld the analytc expesson λ dp T D PF( λt e ( λt + ( dλ Ths expesson shows that the pay behavo of the net false ala pobablty s a sple exponental dependence on the effectve loglkelhood theshold λ T 8 CONCLUSIONS When the geneal popetes of the optu Bayes detecto/estato n a paaetc contnuu of sgnals ae appled to the specfc case of addtve Gaussan nose, seveal nteestng new esults eege Fst, n the asyptotc lt of hgh decson theshold, the sgnal state estate, obtaned fo the locaton of a local axu n the loglkelhood ato, has an accuacy that appoaches the CaeRao bound The decson theshold, dependng on the sgnal paaetc fo, can vay acoss the paaete space, and n ts applcaton to the loglkelhood ato, can dvde that space nto egons of nealy cetan and nealy possble detecton, accodng to whethe the squaed SNR s geate than o less than twce the theshold value Inteestngly, the egons of agnal detectablty sepaatng the egons of hgh and low detectablty contan the pepondeance of false alas Ths last fndng coesponds to a geneal ntegal elaton between detecton pobablty and false ala densty, a elaton that can be ntepeted as the opeatng chaactestc when expessed as a functon of abtay theshold Ths ntegal elaton s of especal value fo detenaton of the false ala densty ove the sgnal paaete doan The dect detenaton of false ala pobablty s a fodably dffcult poble, but detecton pobabltes ae oe eadly accessble and povde a easonable path to false ala pedcton though the ntegal elaton Sulaton esults deonstate a eakably accuate pedcton of false alas The fo of the theoetcal false ala densty ove sgnal paaete space s well atched n the sulaton esults, and the ageeent n total false ala pobablty s alost exact
19 9 ove a ange of thesholds that causes the pobabltes to vay ove odes of agntude Fnally, the opeatng chaactestc, extended globally ove the ente sgnal paaete doan, leads to a sple expesson fo the net false ala pobablty as a functon of the net detecton pobablty In a sgnal paaete doan of stongly vayng SNR, whee dect estaton of the false ala pobablty would be vey dffcult, the expesson can povde a easonable false ala estate fo a cude estate of whee a sgnal s detectable The optu Bayes detecto/estato, as developed hee fo a hghsnr, paaetcally detened sgnal n addtve Gaussan nose, exhbts featues that dstngush t fo othe coon detecton/estaton appoaches Its elaton to two of these the CFAR detecto and the GLRT ae entoned befly n the dscusson centeed on Fg The vaaton of the Bayes theshold wth egad to the paaetespace volue of a lkelhood ato peak poduces a false ala dependence sla to that of a CFAR detecto Such a dependence s notably ssng n a GLRT appled to a contnuu sgnal paaete doan [6] R J Adle, The Geoety of Rando Felds, John Wley and Sons, Inc, 98 D Mchael Mlde eaned hs BS degee n physcs at the Calfona Insttute of Technology, Pasadena CA USA, 959, and hs PhD n physcs at Havad Unvesty, Cabdge, MA USA, 968 He woked at Teta Tech fo 967 to 97, R&D Assocates fo , and joned Aete Assocates, Nothdge, CA USA as a foundng patne n 976 Hs eseach papes have appeaed n the Astophyscal Jounal, the Jounal of the Acoustcal Socety of Aeca, the Jounal of Flud Mechancs, Rado Scence, and Waves n Rando and Coplex Meda D Mlde passed away Decebe, 9 Robet G Lndgen eaned hs BS degee n checal engneeng at the Unvesty of Mnnesota, Mnneapols, MN USA n 965 and hs PhD n checal engneeng wth a no n appled atheatcs at the Calfona Insttute of Technology, Pasadena CA USA n 97 He woked at R&D Assocates fo and joned Aete Assocates, Nothdge, CA USA n 98 Mos M Bean eaned hs BS degee n Matheatcs at the Calfona Insttute of Technology, Pasadena CA USA, 975, and hs MS n Engneeng and Appled Scence at UCLA, Los Angeles, CA USA, 978 He woked befly at R&D Assocates fo , and has been wth Aete Assocates, Nothdge, CA USA snce ts foundng n 976 ACKNOWLEDGMENT The authos ae gateful fo the suppot of Aete Assocates and fo the geneous help of Phlp Peale of Halton College REFERENCES [] D Mddleton and R Esposto, Sultaneous optu detecton and estaton of sgnals n nose, IEEE Tans Inf Theoy, vol IT4, no 3, pp , May 968 [] A Fedksen, D Mddleton, and D VandeLnde, Sultaneous detecton and estaton unde ultple hypotheses, IEEE Tans Inf Theoy, vol IT8, no 5, pp 6764, Sep 97 [3] B Baygun and A Heo, Optal sultaneous detecton and estaton unde a false ala constant, IEEE Tans Inf Theoy, vol IT4, no 3, pp , May 995 [4] X R L, Optal Bayes jont detecton and estaton, Poc th Int Conf Infoaton Fuson, pp8, Jul 7 [5] G V Moustakdes, Optu jont detecton and estaton, n Poc Int Syp Inf Theoy, St Petesbug, Russa,, pp [6] G V Moustakdes, Jont detecton and estaton: optu tests and applcatons, IEEE Tans Inf Theoy, vol 58, no 7, Jul [7] G V Moustakdes, Fnte Saple Sze Optalty of GLR Tests, Nov 9 [Onlne] Avalable axv: [8] R N McDonough and A D Whalen, Detecton of Sgnals n Nose, nd ed, New Yok: Acadec Pess, 995, pp [9] M J Lghthll, Intoducton to Foue Analyss and Genealzed Functons, Cabdge Unvesty Pess, 964 [] H L Van Tees, Detecton, Estaton, and Modulaton Theoy, Pat I, John Wley & Sons,, pp [] P J Schee and L L Schaf, Statstcal Sgnal Pocessng of Coplex Valued Data, Cabdge Unvesty Pess,, pp 88 [] C W Helsto, Statstcal Theoy of Sgnal Detecton, Oxfod: Pegaon Pess, 96, pp 956 [3] H L Van Tees, Detecton, Estaton, and Modulaton Theoy, Pat I, John Wley & Sons,, pp [4] B W Lndgen, Statstcal Theoy, Macllan Copany, New Yok, 96, pp883 [5] A M Hasofe, The ean nube of axa above hgh levels n Gaussan ando felds, J Appl Pob 3, (976
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