Sequential Decentralized Detection under Noisy Channels

Transcription

1 Squntial Dcntralizd Dtction undr Noisy Channls Yasin Yilmaz Elctrical Enginring Dpartmnt Columbia Univrsity Nw Yor, NY Gorg Moustaids Dpt. of Elctrical & Computr Enginring Univrsity of Patras 6500 Rion, Grc Xiaodong Wang Elctrical Enginring Dpartmnt Columbia Univrsity Nw Yor, NY Abstract W considr dcntralizd dtction through distributd snsors that prform lvl-triggrd sampling and communicat with a fusion cntr (FC via noisy channls. Each snsor computs its local log-lilihood ratio (LLR, sampls it using th lvl-triggrd sampling, and upon sampling transmits a singl bit to th FC. Upon rciving a bit from a snsor, th FC updats th global LLR and prforms a squntial probability ratio tst (SPRT stp. W driv th fusion ruls undr various typs of channls. W furthr provid nonasymptotic and asymptotic analyss on th avrag dtction dlay for th proposd channl-awar schm, and show that th asymptotic dtction dlay is charactrizd by a KL information numbr. Th dlay analysis facilitats th choic of appropriat signaling schms undr diffrnt channl typs for snding th 1-bit information from snsors to th FC. I. INTRODUCTION In this papr, w considr th dcntralizd dtction problm in which a numbr of snsors communicat to a fusion cntr (FC undr bandwith constraints. Th FC is rsponsibl for maing th final dcision basd on th limitd information it rcivs from th snsors. 1 and, which ar among th arly wors trating th problm, showd th optimality of lilihood ratio tst as a local dcision rul at snsors and a global dcision rul at th FC, rspctivly. Most of th wors on dcntralizd dtction, including th abov mntiond, followd th fixd-samplsiz approach whr th FC mas th global dcision at a fixd tim. On th othr hand, som wors,.g., 3 6, considrd th squntial approach whr th dcision tim of th FC is random. Among thos wors 5, 6 usd squntial probability ratio tst (SPRT both at snsors and th FC. It has bn obsrvd that SPRT rquirs, on avrag, approximatly four tims 7, Pag 109 lss sampls (for Gaussian signals to rach a dcision than th bst fixdsampl-siz tst, for th sam lvl of confidnc. Rlaxing th strict bandwith constraint, it is nown that data fusion (multi-bit mssaging is a much mor powrful tchniqu than dcision fusion (on-bit mssaging, albit it consums highr bandwith. Morovr, th lvl-triggrd samplingbasd squntial dtction schms, rcntly proposd in 5 and 6, ar as powrful as data fusion tchniqus, and at th sam tim thy ar as simpl and bandwith-fficint as dcision fusion tchniqus. In practic, thr ar two sourcs of uncrtainty in a dcntralizd systm, namly th nois in listning channls, corrupting th snsor obsrvations, and th nois in transmission channls, corrupting th mssags rcivd by th FC. Th convntional approach to dcntralizd dtction considrs th formr, but ignors th lattr,.g., 1 6. In othr words, th convntional approach assums idal transmission channls whil applying a fusion rul to th mssags rcivd by th FC. Following th convntional approach, bfor applying a fusion rul, a communication bloc can b indpndntly applid to rcovr th information bits transmittd from snsors. Howvr, this two-stp solution causs prformanc loss du to th data procssing inquality. For an optimal solution, th fusion rul should b channlawar, i.., should us th th nowldg on noisy channls 8. In this papr, w dsign and analyz channl-awar squntial dcntralizd dtction schms basd on lvltriggrd sampling, undr diffrnt typs of discrt and continuous noisy channls. In particular, w first driv channl-awar squntial dtction schms basd on lvltriggrd sampling. W thn prsnt an information thortic framwor to analyz th dcision dlay prformanc of th proposd schms basd on which w provid both nonasymptotic and asymptotic analyss on th dcision dlays undr various typs of channls. Basd on th xprssions of th asymptotic dcision dlays, w also considr appropriat signaling schms undr diffrnt continuous channls to minimiz th asymptotic dlays. Th rmaindr of th papr is organizd as follows. In Sction II, w dscrib th gnral structur of th dcntralizd dtction approach basd on lvl-triggrd sampling with noisy channls btwn snsors and th FC. In Sction III, w driv channl-awar fusion ruls at th FC for various typs of channls. Nxt, w provid analyss on th dcision dlay prformanc for idal channl and noisy channls in Sction IV and Sction V, rspctivly. Finally, th papr is concludd in Sction VI.

2 y 1 b S 1 n zn 1 t 1 ch1 Ii 1(t Î1 i (t Ĩ1 i (t y t y K t I i (t I K i (t S S K b n Îi (t Î K i b K n (t ch chk z n Ĩ i (t z K n Ĩ K i (t FC Fig. 1. A wirlss snsor ntwor with K snsors, and an FC. Snsors procss thir obsrvations {yt }, and transmits information bits {b n}. Thn, th FC, rciving {zn } through wirlss channls, mas a dtction dcision δ T. Ii (t, Î i (t, Ĩ i (t ar th obsrvd, transmittd and rcivd information ntitis rspctivly, which will b dfind in Sction IV. II. SYSTEM DESCRIPTIONS Considr a wirlss snsor ntwor consisting of K snsors ach of which obsrvs a Nyquist-rat sampld discrttim signal {yt : t N}, = 1,..., K. Each snsor computs th log-lilihood ratios (LLR {L t } t of th signal it obsrvs, sampls th LLR squnc using th lvl-triggrd sampling, and thn snds th LLR sampls to th fusion cntr (FC. Th FC thn combins th local LLR information from all snsors, and dcids btwn two hypothss, H 0 and H 1, in a squntial mannr. Obsrvations collctd at th sam snsor, {yt } t, ar assumd to b i.i.d., and in addition obsrvations collctd at diffrnt snsors, {yt }, ar assumd to b indpndnt. Hnc, th local LLR at th -th snsor, L t, and th global LLR, L t, ar computd as L t log f 1 (y 1,..., y t f 0 (y 1,..., y t = L t 1 + l t = δt t ln, (1 and L t = K =1 L t, rspctivly, whr lt log f 1 (y t f0 (y t is th LLR of th sampl yt rcivd at th -th snsor at tim t; fi, i = 0, 1, is th probability dnsity function (pdf of th rcivd signal by th -th snsor undr H i. Th -th snsor sampls L t via lvl-triggrd sampling at a squnc of random sampling tims {t n} n that ar dictatd by L t itslf. Spcifically, th n-th sampl is tan from L t whnvr th accumulatd LLR L t L sinc th last sampling tim tn 1, t n 1 xcds a constant in absolut valu, i.., { } t n min t > t n 1 : L t L t (,, ( n 1 whr t 0 = 0, L 0 = 0. Lt λ n dnot th accumulatd LLR during th n-th intr-sampling intrval, (t n 1, t n, i.., t n λ n lt = L t L n t. (3 n 1 t=t n 1 +1 Immdiatly aftr sampling at t n, as shown in Fig. 1, an information bit b n indicating th thrshold crossd by λ n is transmittd to th FC, i.., b n sign(λ n. (4 Not that ach snsor, in fact, prforms th Wald s wll nown squntial probability ratio tst (SPRT with thrsholds and. SPRT is th optimum squntial tst in th i.i.d. cas in trms of minimizing th avrag dcision dlay 9. At snsor th n-th local SPRT starts at tim t n 1 and nds at tim t n whn th local tst statistic λ n xcds ithr or. This local hypothsis tst producs a local dcision rprsntd by th information bit b n, and inducs local rror probabilitis α and β which ar givn by α P 0 (b n = 1, and β P 1 (b n = 1, (5 rspctivly, whr P i (, i = 0, 1, dnots th probability undr H i. Dnot th rcivd signal at th FC corrsponding to b n as z n cf. Fig. 1. Th FC thn computs th LLR λ n of ach rcivd signal and approximats th global LLR L t as L t N K t =1 λ n with λ n log p 1(z n p 0 (z n, (6 whr Nt is th total numbr of LLR mssags th -th snsor has transmittd up to tim t, and p i (, i = 0, 1, is th pdf of zn undr H i. In particular, suppos that th m-th LLR mssag λ m from any snsor is rcivd at tim t m. Thn at t m, th FC first updats th global LLR as L tm = L tm 1 + λ m. (7 It thn prforms an SPRT stp by comparing L tm with two thrsholds à and B, and applying th following dcision rul H 1, if L tm Ã, δ tm H 0, if L tm B, (8, othrwis. In othr words, it continus to rciv LLR mssags if L tm ( B, Ã. Th thrsholds (Ã, B > 0 ar slctd to satisfy th rror probability constraints P 0 (δ T = H 1 α and P 1 (δ T = H 0 β with qualitis, whr α, β ar targt rror probability bounds, and T min{t > 0 : L t ( B, Ã} (9 is th dtction dlay. With idal channls btwn snsors and th FC, w hav zn = b n, so from (5 w can writ th local LLR λ n = ˆλ n, whr ˆλ n log P1(b n =1 P 0(b n =1 = log 1 β α, if b n = 1, log P1(b n = 1 P 0(b = log β n = 1 1 α, if b n = 1. (10 In this papr, w will assum, as in 5, that th local rror probabilitis {α, β } ar availabl at th FC in ordr to comput th LLR λ n of th rcivd signals. For th cas of idal channls, w us th A and B to dnot th thrsholds in (8, i.., à = A, B = B, and us T to dnot th dcision dlay in (9, i.., T = T. In th cas of noisy channls, th rcivd signal z n is not always idntical to th transmittd bit b n, and thus th LLR λ n of z n can b diffrnt from ˆλ n of b n givn in (10. In th

3 nxt sction, w considr som popular channl modls and giv th corrsponding xprssions for λ n. III. CHANNEL-AWARE FUSION RULES In computing th LLR λ n of th rcivd signal z n, w will ma us of th local snsor rror probabilitis α, β, and th channl paramtrs that charactriz th statistical proprty of th channl. In this papr, w assum sampling (communication tims ar rliably dtctd at th FC undr continuous channls by using high nough signaling lvls at th snsors. Th unrliabl dtction of sampling tims undr continuous channls is analyzd in 10, Sction VI. A. Binary Erasur Channls (BEC Considr binary rasur channls btwn snsors and th FC with rasur probabilitis {ɛ }. Undr BEC, a transmittd bit b n is lost with probability ɛ, and corrctly rcivd at th FC, i.., zn = b n, with probability 1 ɛ. Thn th LLR of zn is givn by λ n = log P1(z n =1 P 0(z n =1 = log 1 β α, if z n = 1, log P1(z n = 1 P 0(z = log β n = 1 1 α, if z n = 1. (11 Not that undr BEC th channl paramtr ɛ is not ndd whn computing th LLR λ n. Not also that in this cas, a rcivd bit bars th sam amount of LLR information as in th idal channl cas, although a transmittd bit is not always rcivd. Hnc, th channl-awar approach coincids with th convntional approach which rlis solly on th rcivd signal. Although th LLR updats in (10 and (11 ar idntical, th fusion ruls undr BEC and idal channls ar not sinc th thrsholds à and B of BEC, du to th information loss, ar in gnral diffrnt from th thrsholds A and B of th idal channl cas. B. Binary Symmtric Channls (BSC Nxt, w considr binary symmtric channls with crossovr probabilitis {ɛ } btwn snsors and th FC. Undr BSC, th transmittd bit b n is flippd, i.., z n = b n, with probability ɛ, and it is corrctly rcivd, i.., z n = b n, with probability 1 ɛ. Th LLR of z n can b computd as λ n(zn = 1 = log P1 1(zn = 1P P 1 1 (z n = 1P 1 1 P 1 0 (z n = 1P P 1 0 (z n = 1P 1 0 = log (1 ɛ (1 β + ɛ β (1 ɛ α + ɛ (1 α ˆβ {}}{ = log 1 (1 ɛ β + ɛ (1 ɛ α + ɛ }{{} ˆα (1 whr P j i, i = 0, 1, j = 1, 1, is th probability undr H i givn b n = j,and {ˆα, ˆβ } ar th ffctiv local rror probabilitis at th FC undr BSC. Similarly w can writ λ n(z n = 1 = log ˆβ 1 ˆα. (13 Not that ˆα > α, ˆβ > β if α < 0.5, β < 0.5,, which w assum tru for > 0. Thus, w hav λ n,bsc < λ n,bec from which w xpct th prformanc loss undr BSC will b highr than that undr BEC. Th numrical rsults providd in Sction V-B will illustrat this. Finally, not also that, unli th BEC cas, undr BSC th FC nds to now th channl paramtrs {ɛ } to oprat in a channlawar mannr. C. Additiv Whit Gaussian Nois (AWGN Channls Now, assum that th channl btwn ach snsor and th FC is an AWGN channl. Th rcivd signal at th FC is givn by z n = h nx n + w n (14 whr h n = h,, n, is a nown constant complx channl gain; wn N c (0, ; x n is th transmittd signal at sampling tim t n givn by { x a, if λ n = n, b, if λ (15 n. whr th transmission lvls a and b ar complx in gnral. Th distribution of th rcivd signal is thn zn N c (h x n,. Th LLR of z n is givn by λ n = log pa (z np a 1 + p b (z np b 1 p a (z np a 0 + pb (z np b 0 = log (1 β xp( c n + β xp( d n α xp( c n + (1 α xp( d n, (16 whr th suprscript a (rsp. b dnots th conditioning on x n = a (rsp. x n = b, c n z n h a and d n z n h b. D. Rayligh Fading Channls If a Rayligh fading channl is assumd btwn ach snsor and th FC, th rcivd signal modl is also givn by (14-(15, but with h n N c (0, h,. W thn hav zn N c (0, x n h, + ; and accordingly, similar to (16, λ n is writtn as λ n = log 1 β a, xp( c n + β xp( d n b, α xp( c n + 1 α a, xp( d n b, (17 whr a, a h, +, b, b h, +, c n z n a, and d n z n. b, IV. PERFORMANCE ANALYSIS FOR IDEAL CHANNELS In this sction, w driv th xact non-asymptotic xprssion for th avrag dtction dlay E i T, and provid an asymptotic analysis on it as th rror probability bounds α, β 0. Bfor procding to th analysis, lt us dfin som information ntitis which will b usd throughout this and nxt sctions.

4 A. Information Entitis can dfin hypothtical KL information to srv analysis Not that th xpctation of an LLR corrsponds to a purposs. Kullbac-Liblr (KL information ntity. For instanc, Th KL information Ii (1 of a snsor whos information I1 (t E 1 log f 1 (y1,..., yt ratio, η i, is high and clos to 1 is wll projctd to th f0 (y 1,..., = E 1 L y t, FC. Not that thr ar two sourcs of information loss for t snsors, namly, th ovrshoot ffct du to having discrttim obsrvations and noisy transmission channls. Th lattr y t (18 and I0 (t E 0 log f 0 (y1,..., yt f1 (y 1,..., = E 0 L t appars only in η i, whras th formr appars in both ˆη i and η ar th KL divrgncs of th local LLR squnc {L i. In gnral with discrt-tim obsrvations at snsors t } t w hav undr H 1 and H 0, rspctivly. Similarly Îi(1 I i (1 and Ĩi(1 I i (1. Lastly, not that undr idal channls, sinc zn = b n,, n, w hav Î1 (t E 1 log p 1(b 1,..., b Ĩi(1 = Î i (1. Nt p 0 (b 1,..., = E 1 ˆL b t, Î 0 (t E 0 ˆL t B. Analysis on Dtction Dlay Nt Ĩ1 (t E 1 log p 1(z1,..., z Lt {τn : τn = t n t n 1} n dnot th intr-arrival tims Nt p 0 (z 1,..., = E 1 L z t, Ĩ 0 (t E 0 L t of th LLR mssags transmittd from th -th snsor. Not Nt that τn dpnds on th obsrvations y t (19 +1,..., y t and n 1 n, sinc {yt } t ar i.i.d., {τn} n ar also i.i.d. random variabls. ar th KL divrgncs of th local LLR squncs {ˆL t } t and { L t } t rspctivly. Dfin also I i (t Hnc, th counting procss {Nt } t is a rnwal procss. K =1 I i (t, Î i (t K =1 Î i (t, and Ĩi(t Similarly th LLRs {ˆλ n} n of th rcivd signals at th FC K ar also i.i.d. random variabls, and form a rnwal-rward =1 Ĩ i (t as th KL procss. Not from (9 that th SPRT can stop in btwn divrgncs of th global LLR squncs {L t }, {ˆL t }, and two arrival tims of snsor,.g., t { L t } rspctivly. n T < t n+1. Th vnt In particular, w hav I1 (1 = E 1 log f 1 (y 1 N f0 (y 1 = E 1 lt T = n occurs if and only if t n = τ τn T and t and I0 (1 = E 0 log f 0 (y 1 n+1 = τ τn+1 > T, so it dpnds on th first (n+1 f1 (y 1 = E 0 lt as th KL information LLR mssags. From th dfinition of stopping tim 11, pp. numbrs of th LLR squnc {lt 104 w conclud that N }; and I i (1 T is not a stopping tim for th K procsss {τ =1 I i (1, i = 0, 1 ar thos of th global LLR squnc {l t }. Morovr, I1 (t 1 = E 1 n} n and {ˆλ n} n sinc it dpnds on th (n + 1- log f 1 (y 1,...,y t th mssag. Howvr, NT + 1 is a stopping tim for {τ n} n 1 = and {ˆλ f0 (y 1,...,y t n} n sinc w hav NT + 1 = n N T = n 1 1 which dpnds only on th first n LLR mssags. Hnc, E 1 λ n, Î1 (t 1 = E 1 log p 1 (b 1 p 0 (b 1 = E 1 ˆλ n, and Ĩ 1 (t 1 = from Wald s idntity 11, pp. 105 w can dirctly writ th E 1 log p 1 (z 1 p 0 (z 1 = E 1 λ n, ar th KL information numbrs of following qualitis th local LLR squncs {λ n}, {ˆλ n}, and { λ N n}, rspctivly, T +1 undr H 1. Liwis, w hav I0 (t 1 = E 0 λ n, Î0 (t 1 = E i τ n = E i τ1 (E i NT + 1, (0 E 0 ˆλ n, and Ĩ 0 (t 1 = E 0 λ n undr H 0. To summariz, Ii (t, Î i (t, and Ĩ i (t ar rspctivly th obsrvd (at N T +1 snsor, transmittd (by snsor, and rcivd (by th and E i ˆλ n = E i ˆλ 1(E i NT + 1. (1 FC KL information ntitis as illustratd in Fig. 1. Nxt w dfin th following information ratios, ˆη i Î i (t 1 Ii (t 1 and η i Ĩ i (t 1 Ii (t 1, which rprsnt how fficintly information is transmittd from snsor and rcivd by th FC, rspctivly. Du to th data procssing inquality, w hav 0 ˆη i, η i 1, i,. W furthr dfin Î i (1 K =1 ˆη i I i (1 = K =1 Î i (1, and Ĩi(1 K =1 η i I i (1 = K =1 Ĩ i (1, as th ffctiv transmittd and rcivd valus corrsponding to th KL information I i (1, rspctivly. Not that Î i (1 and Ĩi(1 ar not ral KL information numbrs, but projctions of I i (1 onto th filtrations gnratd by th transmittd, (i.., {b n}, and rcivd, (i.., {zn}, signal squncs, rspctivly. This is bcaus snsors do not transmit and th FC dos not rciv th LLR of a singl obsrvation, but instad thy transmit and it rcivs th LLR mssags of svral obsrvations. Hnc, w cannot hav th KL information for singl obsrvations at th two nds of th communication channl, but w W hav th following thorm on th avrag dtction dlay undr idal channls. Thorm 1. Considr th dcntralizd dtction schm givn in Sction II, with idal channls btwn snsors and th FC. Its avrag dtction dlay undr H i is givn by E i T = Îi(T Î i (1 + K =1 Î i (t N T +1 E iy Î i (1 Î i (1 ( whr Y is a random variabl rprsnting th tim intrval btwn th stopping tim and th arrival of th first bit from th -th snsor aftr th stopping tim, i.., Y t N T +1 T. Proof: From (0 and (1 w obtain N N T +1 T E +1 i E i τn = E i τ1 ˆλ n E i ˆλ 1

5 whr th lft-hand sid quals to E i T + E i Y. Not that E i τ1 is th xpctd stopping tim of th local SPRT at th -th snsor and by Wald s idntity it is givn by E i τ1 = E iλ 1 E il1, providd that E il1 0. Hnc, w hav E i T = E iλ 1 E i ˆλ 1 E i N T +1 ˆλ n E i l1 E i Y = I i (t Î 1 i (T + Î i (t N +1 T Îi (t 1 Ii (1 E i Y N T whr w usd th fact that E +1 1 ˆλ n = E 1 ˆL T + Ẽ1ˆλ NT +1 = Î 1 (T + Î 1 (t NT +1 and similarly N T E +1 0 ˆλ n = Î 0 (T Î 0 (t NT +1. Not that Ẽi is th xpctation with rspct to ˆλ NT +1 and N T undr H i. By rarranging th trms and thn summing ovr on both sids, w obtain K E i T I (1Î i (t 1 i I =1 i (t 1 = }{{} Î i(1 Î i (T + K =1 Î i (t N T +1 E iy I i (1Î i (t 1 I i (t 1 }{{} Î i (1 which is quivalnt to (. Th rsult in ( is in fact vry intuitiv. Rcall that Î i (T is th KL information at th dtction tim at th FC. It naturally lacs som local information that has bn accumulatd at snsors, but has not bn transmittd to th FC, i.., th information gathrd at snsors aftr thir last sampling tims. Th numrator of th scond trm on th right hand sid of ( rplacs such missing information by using th hypothtical KL information. Not that in ( Îi (t NT +1 Î i (t 1, i.., Ẽiˆλ NT +1 E iˆλ 1, sinc NT and ˆλ ar not indpndnt. NT +1 Th nxt rsult givs th asymptotic dtction dlay prformanc undr idal channls. Thorm. As α, β 0, th avrag dtction dlay undr idal channls givn by ( satisfis E 1 T = log α Î 1 (1 + O(1, and E log β 0T = Î 0 (1 + O(1, whr O(1 rprsnts a constant trm. (3 Proof: W will prov th first quality in (3, and th proof of th scond on follows similarly. Lt us first prov th following lmma. Lmma 1. As α, β 0 w hav th following KL information at th FC Î 1 (T = log α + O(1, and Î0(T = log β + O(1. (4 Proof: W will show th first quality and th scond on follows similarly. W hav Î 1 (T = P 1 (ˆL T AE 1 ˆL T ˆL T A+ P 1 (ˆL T BE 1 ˆL T ˆL T B = (1 β(a + E 1 θ A β(b + E 1 θ B, (5 whr θ A, θ B ar ovrshoot and undrshoot rspctivly givn by θ A ˆL T A if ˆL T A and θ B ˆL T B if ˆLT B. From 5, Thorm, w hav A log α and B log β, so as α, β 0 (5 bcoms Î 1 (T = A + E 1 θ A + o(1. Assuming 0 < < and lt <,, t, w hav 0 < α, β < 1, hnc from (10 w hav ˆλ n <, and accordingly Î i (t 1 = E i ˆλ 1 <. Sinc th ovrshoot cannot xcd th last rcivd LLR valu, w hav θ A, θ B Θ = max,n ˆλ n <. Similar to Eq. (73 in 5 w can writ β B Θ and α A Θ whr Θ = O(1 by th abov argumnt, or quivalntly, B log β O(1 and A log α O(1. Hnc, w hav A = log α + O(1 and B = log β + O(1. From th assumption of lt <,, t, w also hav Î i (1 I i (1 <. Morovr, w hav E i Y E i τ1 < sinc E i l1 0. Not that all of th trms on th righthand sid of ( xcpt Îi(T do not dpnd on th global rror probabilitis α, β, so thy ar O(1 as α, β 0. Finally, substituting (4 into ( w gt (3. It is sn from (3 that th hypothtical KL information numbr, Î i (1, plays a y rol in th asymptotic dtction dlay xprssion. W can asymptotically minimiz E i T by maximizing Îi(1. Rcalling its dfinition Î i (1 = K =1 Î i (t 1 I i (t 1 I i (1 w s that thr information numbrs ar rquird to comput it. Not that I i (1 = E il 1 and I i (t 1 = E i λ 1, which is givn in (6 blow, ar computd basd on local obsrvations at snsors, thus do not dpnd on th channls btwn snsors and th FC. Spcifically, w hav I 1 (t 1 = (1 β ( + E 1 θ n β ( + E 1 θ n, and I 0 (t 1 = α ( + E 0 θ n (1 α ( + E 0 θ n (6 whr θ n and θ n ar local ovr(undrshoots givn by θ n λ n if λ n and θ n λ n if λ n. Du to having l t <,, t w hav θ n, θ n <,, n. On th othr hand, Î i (t 1 rprsnts th information rcivd in an LLR mssag by th FC, so it havily dpnds on th channl typ. In th idal channl cas, from (10 it is givn by and Î 1 (t 1 = (1 β log 1 β α Î 0 (t 1 = α log 1 β α β + β log, 1 α + (1 α log β 1 α. (7

6 Ĩ 1 (t 1 Sinc Î i (t 1 is th only channl-dpndnt trm in th asymptotic dtction dlay xprssion, in th nxt sction w will obtain its xprssion for ach noisy channl typ considrd in Sction III. V. PERFORMANCE ANALYSIS FOR NOISY CHANNELS In all noisy channl typs that w considr in this papr, w assum that channl paramtrs ar ithr constants or i.i.d. random variabls across tim. In othr words, ɛ, h ar constant for all (s Sction III-A, III-B, III-C, and {h n} n, {w n} n ar i.i.d. for all (s Sction III-C, III-D,??. Thus, in all noisy channl cass discussd in Sction III th intr-arrival tims of th LLR mssags { τ n} n, and th LLRs of th rcivd signals { λ n} n ar i.i.d. across tim as in th idal channl cas. Accordingly th avrag dtction dlay in ths noisy channls has th sam xprssion as in (, as givn by th following proposition. Th proof is similar to that of Thorm 1. Proposition 1. Undr ach typ of noisy channl discussd in Sction III, th avrag dtction dlay is givn by E i T = Ĩi( T K Ĩ i (1 + =1 Ĩ i (t NT +1 EiỸĨ i (1 (8 Ĩ i (1 whr Ỹ t N T +1 T. Th asymptotic prformancs undr noisy channls can also b analyzd analogously to th idal channl cas. Proposition. As α, β 0, th avrag dtction dlay undr noisy channls givn by (8 satisfis E 1 T = log α Ĩ 1 (1 + O(1, and E 0 T log β = Ĩ 0 (1 + O(1. (9 Proof: Not that in th noisy channl cass th FC, as discussd in Sction III, computs th LLR, λ n, of th signal it rcivs, and thn prforms SPRT using th LLR sum L t. Hnc, analogous to Lmma 1 w can show that Ĩ1( T = log α + O(1 and Ĩ0( T = log β + O(1 as α, β 0. Not also that du to channl uncrtaintis λ n ˆλ n, so w hav Ĩ i (t 1 Î i (t 1 < and Ĩi(1 Îi(1 <. W also hav E i Ỹ E i τ 1 < as in th idal channl cas. Substituting ths asymptotic valus in (8 w gt (9. Rcall that Ĩi(1 = K =1 Ĩ i (t 1 I i (t 1 I i (1 in (9 whr I i (1 and I i (t 1 ar indpndnt of th channl typ, i.., thy ar sam as in th idal channl cas. In th subsqunt subsctions, w will comput Ĩ i (t 1 for ach noisy channl typ. W will also considr th choics of th signaling lvls a and b in (15 that maximiz Ĩ i (t 1, i.., asymptotically minimiz E i T. A. BEC Undr BEC, from (11 w can writ th LLR of th rcivd bits at th FC as { ˆλ λ n = n, with probability 1 ɛ, (30 0, with probability ɛ. ǫ α = β Fig.. Th KL information, Ĩ 1 (t 1, undr BEC and BSC, as a function of th local rror probabilitis α = β and th channl rror probability ɛ. Hnc, w hav Ĩ i (t 1 = E i λ 1 = (1 ɛ Î i (t 1 (31 whr Î i (t 1 is givn in (7. As can b sn in (31 th prformanc dgradation undr BEC is only dtrmind by th channl paramtrs ɛ. In gnral, from (3, (9 and (31 this asymptotic prformanc loss can b quantifid as 1 1 min ɛ Ei T 1 E it 1 max ɛ. Spcifically, if ɛ = ɛ,, thn w hav Ei T E it = 1 1 ɛ as α, β 0. B. BSC Rcall from (1 and (13 that undr BSC local rror probabilitis α, β undrgo a linar transformation to yild th ffctiv local rror probabilitis ˆα, ˆβ at th FC. Thrfor, using (1 and (13, similar to (7, Ĩ i (t 1 is writtn as follows and Ĩ 1 (t 1 = (1 ˆβ log 1 ˆβ ˆα Ĩ 0 (t 1 = ˆα log 1 ˆβ ˆα + ˆβ ˆβ log, 1 ˆα (3 ˆβ + (1 ˆα log 1 ˆα whr ˆα = (1 ɛ α + ɛ and ˆβ = (1 ɛ β + ɛ. Notic that th prformanc loss in this cas also dpnds only on th channl paramtr ɛ. In Fig. w plot Ĩ 1 (t 1 as a function of α = β and ɛ, for both BEC and BSC. It is sn that th KL information of BEC is highr than that of BSC, implying that th asymptotic avrag dtction dlay is lowr for BEC, as anticipatd in Sction III-B. C. AWGN In this and th following sctions, w will drop th snsor indx of h, and for simplicity. In th AWGN cas, it follows from Sction III-C that if th transmittd signal is a, i.., x n = a, thn c n = u, d n = v a ; and if x n = b, thn c n = v b, d n = u whr u w n, v a w n +(a bh, v b w n +(b ah. Accordingly, from (16 w writ th KL information as in (33, whr E dnots th xpctation with rspct to th channl nois wn only, and Ē1 dnots th xpctation with rspct to both x n and wn undr H 1.

7 Ĩ1 (t 1 =Ē1 λ 1 = (1 β E log (1 β u + β va α u + β E log (1 β v b + β u + (1 α va α v b + (1 α u { }}{ = (1 β log 1 β β ( 1 β + β log +β E log 1 + β { }}{ 1 β u va α 1 α }{{ β } α + E log β β u v b α u va 1 + α 1 α u v b }{{} Î1 (t 1 C1 E 1 E (33 Sinc wn is indpndnt of x n undr both H 0 and H 1, w usd th idntity Ē1 = EE 1 in (33. Not from (33 that w hav Ĩ 1 (t 1 = Î 1 (t 1 + β C1 and Ĩ 0 (t 1 = Î 0 (t 1 + α C0. Similar to C1 w hav C0 E 1 1 α α E. Sinc w now Ĩ i (t 1 Î i (t 1, th xtra trms, C1, C0 0 ar pnalty trms that corrspond to th information loss du to th channl nois. Our focus will b on this trm as w want to optimiz th prformanc undr AWGN channls by choosing th transmission signal lvls a and b that maximiz Ci. Lt us first considr th random variabls ζ a u v a and ζ b u v b which ar th argumnts of th xponntial functions in E 1 and E in (33. From th dfinitions of u and v a, w writ ζ a = w n w n +(a bh = a b h γ whr γ R{(wn (a bh } and R{ } dnots th ral part of a complx numbr. Similarly w hav ζ b = a b h + γ. Not that γ N (0, a b h sinc wn N c (0,. If w dfin ν a b h γ, thn w hav ν N (0, 1. Upon dfining s a b h w can thn writ ζ a and ζ b as ζ a = s sν and ζ b = s + sν. If w dfin F 1 α α C0 =E C 1 =E log 1 + F 1 ζ b 1 + G ζ b log 1 + Gζ b 1 + F 1 ζ b and G 1 β β, thn w hav + F 1 E log 1 + F ζa + GE 1 + G 1 ζa log 1 + G 1 ζa. 1 + F ζa (34 Not from (14 that th rcivd signal, z n, will hav th sam varianc, but diffrnt mans, ah and bh, if x n = a and x n = b ar transmittd rspctivly. Hnc, w xpct that th dtction prformanc undr AWGN channls will improv if th diffrnc btwn th transmission lvls, a b, incrass. Toward that nd th following rsult givs a sufficint condition undr which th pnalty trm C i incrass with s, and hnc with a b. Th proof is givn in 10, Appndix. Lmma. Ci is an incrasing function of s, i = 0, 1, if F G and G F. Lmma indicats that for α, β valus insid th rgion shown in Fig. 3, Ci is incrasing in a b. Not that α, β ar local rror probabilitis which ar dirctly rlatd to th local thrshold. Thrfor, vn if th hypothss H 0 and H 1 ar non-symmtric, w can nsur that w will hav β α Fig. 3. Th rgion of (α, β spcifid by Lmma. α, β insid th rgion in Fig. 3 by mploying diffrnt local thrsholds, and, in (. In fact, vn for α, β valus outsid th rgion in Fig. 3 numrical rsults show that Ci is incrasing in s. Hnc, maximizing Ci is quivalnt to maximizing a b. If w considr a constraint on th maximum allowd transmission powr at snsors, i.., max( a, b P, thn th antipodal signaling is optimum, i.., a = b = P and a = b. D. Rayligh Fading It follows from Sction III-D that c n = u a, d n = a u b a whn x n = a; and c n = b u a b, d n = u b whn x n = b whr u a ah n +w n, u a b bh n +w n, and b a = a h +, b = b h + as dfind in Sction III-D. Dfin furthr ρ a. Hnc, using (17 w writ th KL information as in b (35, whr ζ a u a (1 ρ and ζ b u b (1 ρ 1. Not that whn a = b which corrsponds to th optimal signaling in th AWGN cas, w hav ρ = 1, ζ a = ζ b = 0 and thrfor Ĩ1 (t 1 = 0 in (35. This rsult is quit intuitiv sinc in th Rayligh fading cas th rcivd signals diffr only in thir variancs. Not that u a and u b ar chi-squard random variabls with dgrs of frdom, i.., u a, u b χ, thus w can writ th pnalty trm Ci as C 0 = 0 ( log 1 + F 1 ρ 1 u(1 ρ Gρ 1 u(1 ρ 1 F 1 log 1 + F ρu(1 ρ 1 + G 1 ρ u(1 ρ u du, (36

8 1 β a ua + β b ρua 1 β a ρ 1 u b + β u b b Ĩ1 (t 1 =(1 β E log α a ua + 1 α + β E log b ρua α a ρ 1 u b + 1 α u b b = (1 β log 1 β β + β log +β (E log 1 + Gρ 1 ζ b α 1 α }{{ 1 + F } 1 ρ 1 ζ + GE log 1 + G 1 ρ ζa b 1 + F ρ }{{ ζa } Î1 (t 1 C1 (35 Rayligh Fading btwn H 0 and H 1 around ρ = 1 will xist whnvr F = G, i.., α = β. C i Fig. 4. Th pnalty trm Ci for Rayligh fading channls as a function of ρ. and C 1 = 0 ( log 1 + Gρ 1 u(1 ρ F 1 ρ 1 u(1 ρ 1 + ρ G log 1 + G 1 ρ u(1 ρ 1 + F ρ u(1 ρ H1 H 0 u du. (37 Not that givn local rror probabilitis α, β th intgrals in (36, (37 is a function of ρ only. Howvr, maximizing Ci in (36, (37 with rspct to ρ sms analytically intractabl. As can b sn in Sction III-D, th rcivd signals at th FC will hav zro man and th variancs a and b whn x n = a and x n = b rspctivly. Thrfor, in this cas intuitivly w should incras th diffrnc btwn th two variancs, i.., a b. Considr th following constraints: max( a, b P and min( a, b Q, whr th first on is th pa powr constraint as bfor, and th scond is to nsur rliabl dtction of an incoming signal by th FC. W conjctur that th optimum signaling schm in this cas that maximizs Ci corrsponds to a = P, b = Q or a = Q, b = P. To numrically illustrat th bhavior of Ci as a function of ρ, w st α = β = 0.1, h = = 1, P = 10, Q = 1, and plot Ci in Fig. 4. It is sn that Ci has its global minimum whn ρ = 1, which corrsponds to th cas a = b as xpctd. Morovr, Ci, validating our conjctur, monotonically grows as ρ tnds to its minimum and maximum valus corrsponding to th cass a = Q, b = P and a = P, b = Q rspctivly. Not that in Fig. 4, th curvs for H 0 and H 1 ar mirrord vrsions of ach othr around ρ = 1 sinc w hav α = β in th xampl. From (36, (37 w can say that th symmtry VI. CONCLUSIONS W hav dvlopd and analyzd channl-awar distributd dtction schms basd on lvl-triggrd sampling. Th snsors form local log-lilihood ratios (LLRs basd on thir obsrvations and sampl thir LLRs using th lvl-triggrd sampling. Upon sampling ach snsor snds a singl bit to th fusion cntr (FC. Th FC is quippd with th local rror rats of all snsors and th statistics of th channls from all snsors. Upon rciving th bits from th snsors, th FC updats th global LLR and prforms an SPRT. Th fusion ruls undr diffrnt channl typs ar givn. W hav furthr providd non-asymptotic and asymptotic analyss on th avrag dtction dlay for th proposd channl-awar schm. W hav shown that th asymptotic dtction dlay is charactrizd by a KL information numbr, whos xprssions undr diffrnt channl typs hav bn drivd. Basd on th dlay analysis, w hav also idntifid appropriat signaling schms undr diffrnt channls for th snsors to transmit th 1-bit information. REFERENCES 1 R.R. Tnny, and N.R. Sandll, Dtction with distributd snsors, IEEE Trans. Aro. Elctron. Syst., vol. 17, no. 4, pp , July Z. Chair, and P.K. Varshny, Optimal data fusion in multipl snsor dtction systms, IEEE Trans. Aro. Elctron. Syst., vol., no. 1, pp , Jan V.V. Vravalli, T. Basar, and H.V. Poor, Dcntralizd squntial dtction with a fusion cntr prforming th squntial tst, IEEE Trans. Inform. Thory, vol. 39, no., pp , Mar Y. Mi, Asymptotic optimality thory for squntial hypothsis tsting in snsor ntwors, IEEE Trans. Inform. Thory, vol. 54, no. 5, pp , May G. Fllouris, and G.V. Moustaids, Dcntralizd squntial hypothsis tsting using asynchronous communication, IEEE Trans. Inform. Thory, vol. 57, no. 1, pp , Jan Y. Yilmaz, G.V. Moustaids, and X. Wang, Cooprativ squntial spctrum snsing basd on lvl-triggrd sampling, IEEE Trans. Sig. Proc., vol. 60, no. 9, pp , Sp V. Poor, An Introduction to Signal Dtction and Estimation, Springr, Nw Yor, NY, B. Chn, L. Tong and P.K. Varshny, Channl-Awar Distributd Dtction in Wirlss Snsor Ntwors, IEEE Signal Procssing Magazin, vol. 3, issu 4, pp. 16-6, July A. Wald and J. Wolfowitz, Optimum charactr of th squntial probability ratio tst, Ann. Math. Stat., vol. 19, pp , Y. Yilmaz, G.V. Moustaids, and X. Wang, Channlawar Dcntralizd Dtction via Lvl-triggrd Sampling, Sp S. Ross, Stochastic Procsss, Wily, Nw Yor, NY, 1996.