Effect of the Output of the System in Signal Detection

We analyze the consequences that the choice of the output of the system has in the efficiency of signal detection. It is shown that the signal and the signal-to-noise ratio (SNR), used to characterize the phenomenon of stochastic resonance, strongly depend on the form of the output. In particular, the SNR may be enhanced for an adequate output.

T he phenom enon ofstochasti c resonance (SR ) [ 1{11] hasem erged i n the l astyearsasone ofthe m ostexci ti ng i n the el d of nonl i near stochasti c system s. Its i m portance as a m echani sm for si gnaldetecti on has gi ven ri se to a greatnum berofappl i cati onsi n di erent el ds,asfor exam pl e el ectroni c devi ces [ 12] ,l asers [ 2] ,neurons [ 13,14] and m agneti c parti cl es [ 15,16] .
T he m ost com m on characteri zati on of SR consi sts of the appearance of a m axi m um i n the output si gnal -tonoi se rati o (SN R ) at non-zero noi se l evel ,al though di fferent de ni ti ons have been used i n the l i terature. T he de ni ti on through the SN R accounts for practi calappl icati ons,because the SN R i s the quanti ty that gi ves the am ount ofi nform ati on that can be transfered through a m edi um aswel lasm easuri ng the qual i ty ofa si gnal .A ddi ti onal l y,the SN R quanti es the possi bi l i ty to detect a si gnalem bedded i n a noi sy envi ronm ent. A nother deni ti on ofSR ,apparentl y si m i l ar to the one ofthe SN R , hasbeen proposed i n term sofa m axi m um i n the output si gnal . A l though both the SN R and the output si gnal have been anal yzed i n term s of the param eters of the system ,e. g. the frequency or the am pl i tude ofthe i nput si gnal ,there i s an i m portant aspect w hi ch has not been consi dered i n depth up to now . A n adequate el ecti on of the output of the system m ay have i m pl i cati ons i n the behavi or ofthe quanti ti es used to m ani fest the presence of SR .T hi s i s preci sel y the probl em we address i n thi s paper.
It i s i nteresti ng to real i ze that norm al l y the output of thesystem i sthesam easthedynam i cvari abl ex(t)enteri ng the stochasti c di erenti alequati on, al though som eti m es the si gn functi on ofx(t) has al so been consi dered. N o m atterthe system ,the outputm ay i n generalbe any functi on ofx(t) w hi ch i s usual l y xed through the characteri sti cs of the probl em . H owever,i n order to detect a si gnalem bedded i n a noi sy envi ronm ent any functi on m ay be used. T hus,i nstead ofFouri ertransform i ng x(t) we can transform the functi on v(x(t)).
Letusdi scussone ofthe m ostsi m pl estcases,i n w hi ch the dynam i csi sdescri bed by an O rnstei n-U hl enbeck process,w herethei nputsi gnalm odul atesthestrength ofthe potenti ali n the fol l ow i ng way: H ere h(t) = k(1 + si n(! 0 t)), w i th k and constants and (t)i sG aussi an w hi te noi se w i th zero m ean and second m om ent h (t) (t+ )i = D ( ), de ni ng the noi se l evelD . T he e ect ofthi s force m ay be anal yzed by the averaged power spectrum To thi s end we w i l l assum e that i t consi sts of a del ta functi on centered at the frequency ! 0 pl us a functi on Q (!) w hi ch i s sm ooth i n the nei ghborhood of! 0 and i s gi ven by Let us now assum e the expl i ci t form for the output of the system ,v(x)= j xj ,w here i sa constant.A l though thi sm odeldoesnotexhi bi tSR ,i ti sadequateto i l l ustrate the form i n w hi ch si gnaland SN R vary as a functi on of the output. C onsi derati onsaboutourm odelbased upon di m ensi onal anal ysi s enabl e us to rew ri te the averaged power spectrum as w here q(!=! 0 ;k=! 0 ; ) and s(k=! 0 ; ) are di m ensi onl ess functi ons. In spi te of the si m pl i ci ty of thi s resul t, a num ber of i nteresti ng consequences can be deri ved. From Eq. (4) we can obtai n the expressi on for the output si gnal T hreequal i tati ve di erentsi tuati onsarepresentdependi ng on the exponent . For > 0 the si gnal di verges w hen the noi se l evelD goesto i n ni ty,w hereasfor < 0 the si gnaldi verges w hen D goes to zero. Even m ore i nteresti ng i s the case = 0,i n w hi ch the si gnaldoes not depend on thenoi sel evel .T heprevi ousresul tshavebeen veri ed num eri cal l y for som e parti cul arval ues of (Fi g. 1),by i ntegrati ng the correspondi ng Langevi n equati on fol l ow i ng a standard second-order R unge-K utta m ethod for stochasti c di erenti alequati ons [ 17,18] . It i s i nteresti ng to poi nt out that the si gnali ncreases forl ow orhi gh noi sei ntensi ti es,dependi ng on theval ueof the exponent . From the previ ous consi derati ons i t becom es cl ear that the output si gnali tsel fdoes not al ways consti tute a useful quanti ty to el uci date the opti m um noi se l evelto detect a si gnal . In contrast,the SN R overcom es thi s am bi gui ty. Its expressi on strai ghtforwardl y fol l ow s from Eq. (4) SN R = k s(k=! 0 ; ; ) q(!=! 0 ;k=! 0 ; ; ) ;  A l l the functi ons we are consi deri ng as outputs are scal e i nvari ant,and di m ensi onalanal ysi s can be readi l y perform ed. H owever,w hen thi s requi rem ent about v(x) does not hol ds,the previ ous resul ts do not appl y. T hi s coul d be the case ofthe H eavi si de step functi on v(x) = (x ), w here represents a threshol d. In fact, thi s si tuati on i s qui te si m i l ar to standard threshol d devi ces [ 9]consi dered previ ousl y.In thi scase,both the SN R and the output si gnalexhi bi t a m axi m um at non-zero noi se l evel(see Fi g.4).A l though the evol uti on equati on ofthe vari abl e x(t) i s l i near,SR appears due to the fact that the output i s a nonl i near functi on. H avi ng di scussed the rol e pl ayed by the output i n a si m pl e m onostabl e system ,l etusnow anal yze the case of the bi stabl e quarti c potenti al ,w hi ch hasbeen frequentl y proposed i n orderto descri bethephenom enon ofSR .T he dynam i cs of the system i s then gi ven by the fol l ow i ng equati on: w here a,b and A are constantsand (t)i sthe sam e noi se as the one i ntroduced through Eq. (1).
To study thi s system ,one usual l y takes as output the vari abl e x(t) and som eti m es the si gn functi on sgn(x(t)). In the l i m i t w hen the am pl i tude ofthe i nput si gnalgoes to zero, these two form s of the output gi ve the sam e resul ts (see ref. [ 3] for m ore detai l s). H owever, w hen the i nput si gnalhasa ni te am pl i tude,the SN R for x(t) di verges, w hereas for sgn(x(t)) goes to zero w hen the noi se l eveldecreases.D espi te the di vergence ofthe SN R for x(t),i fthe am pl i tude ofthe i nput si gnali s not l arge enough,the SN R has a m axi m um at non-zero D .
A s output,we coul d take i n generalx(t) . T he choi ce of hasi m portantconsequencesasthe SN R m ay depend on thi s param eter. T hus to better detect a si gnal , the noi se l eveli snotnecessari l y the onl y tunabl e param eter. In Fi g.5(a)we have pl otted the SN R fordi erentval ues of , observi ng i ts strong dependence on thi s param eter. In parti cul ar,for l og 10 (D ) 1: 25,upon varyi ng from 1 to 7 the SN R i ncreasesi n about12 dB .M oreover, w hen i ncreasi ng the m axi m um i n the SN R becom es l esspronounced and di sappearsfora su ci entl y l arge , as occurs for the case = 25 . In regards to the si gnal , vari ati onsof changei tsbehavi ordrasti cal l y.T hi spoi nt i s i l l ustrated i n Fi g. 5(b) w here we can see that,w hen i ncreasi ng the noi se l evel , for = 1 the si gnalgoes to zero,w hereas for the rem ai ni ng cases the si gnalal ways i ncreases at su ci entl y hi gh noi se l evel . In Fi g. 5(c) we have al so di spl ayed the output noi se. From Fi g. 5(a) i t fol l ow s that a si m pl e vari ati on on the output changes the qual i tati ve form ofthe SN R ,i n such a way that the m axi m um at non-zero noi se l evelm ay di sappear. T hus, the SN R i sa m onotoni c decreasi ng functi on ofD and apparentl y the i nput si gnalcan al ways be better detected by decreasi ng the noi se l evel . H owever,w hen the SN R i s a decreasi ng functi on ofD ,there exi sts a regi on around the m axi m um ,correspondi ng to the curves = 1;3;5;7 i n w hi ch the SN R for = 1;3;5;7 i s greater than that for = 25. W e then concl ude that w hen i ncreasi ng the noi se l evel ,the si gnalcan be betterdetected i fone si m ultaneousl y changes the val ue of . To end our anal ysi s,i n Fi g. 6 we have di spl ayed two tem poralseri es for two di erent val ues of at the noi se l evelfor w hi ch the e ect of the vari ati on on i s m ore pronounced. W e can see that i ntrawel losci l l ati ons,for = 7 are better observed than for = 1. T hi s fact expl ai ns the i ncrease ofthe SN R . In sum m ary,we have show n thatthe quanti ti es(si gnal and SN R ) used to characteri ze the phenom enon of SR strongl y depend on theform oftheoutput.In thi sregard, the behavi or ofthe SN R has reveal ed to be m ore robust than the one correspondi ng to the si gnal . O ur ndi ngs have i m portantappl i ed aspects si nce an adequate choi ce of the output of the system m ay be cruci ali n order to better detect a si gnal .
T hi s work was supported by D G IC Y T of the Spani sh G overnm ent under G rant N o. PB 96-0881. J. M . G . V . w i shes to thank G eneral i tat de C atal unya for nanci al support.