On Centralized Composite Detection with Distributed Sensors

Transcription

1 On Centralzed Composte Detecton wth Dstrbuted Sensors Cuchun Xu, and Steven Kay Abstract For composte hypothess testng, the generalzed lkelhood rato test (GLRT and the Bayesan approach are two wdely used methods. Ths paper nvestgates the two methods for sgnal detecton of a known waveform and unknown ampltude wth dstrbuted sensors. It s frst proved that the performance of GLRT can be poor and hence mproved for ths problem and then an approxmated Bayesan detector s proposed. Compared wth the exact Bayesan approach, the proposed method always has a closed form and hence s easy to mplement. Computer smulaton results show that the proposed method has comparable performance to the exact Bayesan approach. I. INTRODUCTION Detecton wth dstrbuted sensors has been studed for nearly three decades [] [] [3]. Wth respect to dfferent data assumptons, t can be categorzed to centralzed detecton and decentralzed detecton. In centralzed detecton, t s assumed that all data from all local sensors are avalable for processng. In decentralzed detecton, only compressed data or local decson from all local sensors are communcated to a central processor, where a central decson s made. Snce the centralzed detecton can largely resort to classcal detecton theory, many works focus on decentralzed detecton [] [] [3]. However, there are stll some problems left for centralzed detecton, especally when there are unknown parameters n hypotheses,.e. composte detecton. In [4], a mnmax constant false alarm rate (CFAR centralzed detector s proposed for composte detecton. However, t assumes that the unknown parameters take dscrete values and the detecton nvolves large computaton. In [3], many dstrbuted detecton technques are revewed, wth focus on the locally optmum dstrbuted detectors, whch are optmal only when sgnal s weak. Other common composte detecton methods nclude the generalzed lkelhood rato test (GLRT and the Bayesan approach. Because of ts ease of mplementaton, the GLRT s wdely used and usually has satsfactory performance. However, ts optmalty s hard to establsh, even though some attempts have been made, based

2 on dfferent crtera [8] [9]. On the other hand, the Bayesan approach s optmal f the assumed pror probabltes and pror probablty densty functons (PDFs of the unknown parameters are true. However, n most applcatons the true prors and true pror PDFs are unknown and hard to assume. Furthermore the ntegraton nvolved n calculaton of the margnal PDF may not be easy to evaluate []. In ths paper, we examne the mportant specal case of ndependent sensors (condtoned on hypothess. In partcular, we focus on stuatons that even when a sgnal s present only part of the sensors receve the sgnal. It s shown n Secton III that n ths case, the GLRT s not optmal. Some mprovements of the GLRT are also dscussed. In secton IV, an approxmated Bayesan detector s proposed based on vague pror PDFs, whch does not need a specfc form. Some computer smulaton results are shown n Secton V and fnally Secton VI offers concluson and further dscusson. II. STATEMENT OF THE PROBLEM To smplfy the mathematcal exposton, we assume the sensors are ordered wth respect to ther mportance. That s f j and sensor j receves sgnal then sensor must also receve sgnal. Ths order of mportance may be due to the physcal locatons of the sensors. When the sensors are not ordered, the dscusson s gven n Secton VI. We assume that there are M ndependent sensors or channels. The detecton problem s stated as H 0 : x [n] = w [n] n = 0,,,N H : x [n] = As[n] + w [n] n = 0,,,N ( where A s the M unknown ampltude vector, wth possbly dfferent elements, s[n] s a known waveform and w[n] s a Gaussan random vector. Each element of w[n] s whte both n tme doman and n space doman,.e. σ, n = m E [w [n]w [m]] = 0, n m for any M and σ, = j E [w [n]w j [n]] = 0, j for any 0 n N where w [n] denotes the th element of w[n] and E[ ] denotes the expectaton operator. Snce the sensors are ordered, under H only the frst L elements of A s nonzero. L s referred to as the true model order under H. Both L and the value of the frst L elements of A are unknown. If we assume there are nonzero elements n A, the correspondng model of ( s called model and s denoted as M. Under

3 3 model M, A s denoted as A and A = [A, A,,A, 0, 0,,0] M Clearly, under H 0 the data s from model M 0 and under H the data s from model M, M. Wth ths modelng, the detecton problem can equvalently be stated as to assgn the gven data to M 0 (H 0 or to M M M M (H. A. Performance of the GLRT III. GENERALIZED LIKELIHOOD RATIO TEST Frstly, suppose the varance σ s known. Snce L and A are unknown, the GLRT decdes H f p (X;Â, M L G (X = max > λ M p (X; M 0 where X = [x[0],x[],,x[n ]] T denotes the whole data and p (X;Â, M s the probablty densty functon (PDF under M parameterzed by the maxmum lkelhood estmator (MLE Â, whch s estmated assumng M s true. Snce f j, M can be thought of as a specal case of M j, t s readly shown that the GLRT wll always mplement the test statstc based on the maxmum model,.e. [5] or equvalently L G (X = p ( X;ÂM, M M p (X; M 0 > λ lnl G (X = ln p ( X;Â M, M M p (X; M 0 The test statstc of ( has the PDF [5] χ M lnl G (X, H 0 χ M (λ, H > λ ( where χ M denotes the ch-squared dstrbuton and χ M (λ denotes the noncentral ch-squared dstrbuton wth the noncentralty parameter λ. The noncentralty parameter s gven by where λ = ε = ε M A = σ N n=0 s [n] s the sgnal waveform energy. Snce only the frst L elements of A are nonzero, we have λ = ε L = A σ (3

4 4 Wth the PDF of the GLRT test statstc of (3, we now show the performance of the GLRT s generally not optmal and can be mproved. Suppose L < M and let K be an nteger such that L K < M. If there s a generalzed lkelhood rato test based on M K, whch decdes H f ( lnl GK (X = ln p X;Â K, M k p (X; M 0 the PDF of ths test statstc s gven by χ K lnl GK (X, H 0 χ K (λ, H We note that f the varance σ s unknown, ( and (4 should be changed to ( lnl G (X = ln p X;ÂM, ˆσ M, M M p ( X; ˆσ 0, M > λ 0 and ( lnl GK (X = ln p X;ÂK, ˆσ K, M K p ( X; ˆσ 0, M > λ 0 > λ (4 where ˆσ s the MLE of σ under M. However, asymptotcally (3 and (5 stll hold [5]. Snce K < M, from Theorem, whch s proved n Appendx I, the detector based on model M K has better performance than that of the GLRT, whch s based on M M. PDF Theorem 3.: Suppose there are two detecton statstcs T and T. The detecton statstc T has the χ ν T, H 0 χ ν (λ, H and the detector decdes a sgnal s present f T > γ. The detecton statstc T has the PDF χ ν T, H 0 χ ν (λ, H and the detector decdes a sgnal s present f T > γ. Then, f ν < ν, the performance of detector s better than that of detector n a Neyman-Pearson (NP sense,.e. for the same false alarm rate detector has a hgher probablty of detecton than detector. In other words, for ths dstrbuted detecton problem, f L < M the GLRT s not optmal. Ths s because the GLRT s based on M M and therefore the channels that contan only nose samples are ncluded n the test statstc. Includng these channels wll ncrease the degrees of freedom of the test statstc, however the noncentralty parameter remans the same. (5

5 5 B. Improved GLRT From Theorem, any detector based on M wth > L wll have a performance less than the detector based on M L. It s clear that the unknown order L s crtcal to the performance of detectors. In order to mprove the GLRT, L can frst be estmated and a generalzed lkelhood rato test based on model ˆL can then be conducted. Ths can be done by usng varous model order selecton crtera [6]. A wdely used crteron s the mnmum descrpton length (MDL crteron, whch selects the model that mnmzes MDL( = lnl G (X + lnn, M (6 The GLRT wth estmated model s referred to as rglrt n ths paper. Recently, a multfamly lkelhood rato test (MFLRT s proposed to modfy the GLRT to accommodate nested PDF famles [7]. The MFLRT decdes H f {[ ( ( ] ( } LG (X LG (X T MFLRT (X = max L G (X ln + u > γ (7 M where u( s the unt step functon. These two revsed GLRTs wll be compared wth the GLRT n Secton V. A. Approxmated Bayesan Approach IV. BAYESIAN APPROACH Frstly, suppose the varance σ s known. If we are wllng to assume Bayesan assumptons, that s model M, 0 M has a pror probablty π and under each model n H, A j, j M has a pror PDF gven as p(a j M j, the optmal NP detector decdes H f From the Bayesan assumptons Pluggng (9 nto (8, the detector decdes H equvalently f p (X H p (X H 0 > λ (8 M p (X H = π p (X M (9 = M π p (X M = p (X M 0 > λ (0 where the margnal PDF p(x M s gven by p (X M = p (X,A M da = p (X A, M p(a M da (

6 6 As mentoned prevously, the true pror PDFs are usually unknown and hard to assume. Furthermore the ntegraton n ( may not have a closed form. If the varance σ s unknown p (X A, M p(a M da should be replaced by p ( X A, σ, M p ( A, σ M da dσ and t s even more dffcult to assume the pror PDFs as well as calculate the ntegraton. It s shown n [0] that assumng vague pror PDFs, the margnal PDF p(x M can be asymptotcally approxmated as ( I p (X M const p X Â, M (Â (πe where denotes determnant and I (Â s the observed nformaton matrx, or I (Â = lnp(x A, M A A T For the model gven by (, t can be shown that asymptotcally and therefore I (Â N A=Â p (X M const p (X Â ( N, M πe Pluggng the above nto (0 we have M ( ( π const p X Â, M N πe = > λ p (X M 0 or M = When σ s unknown, smlarly we have ( ( N π L G (X > λ (3 πe p (X M const p (X Â (, ˆσ N, M πe + and (3 s stll vald. The detector gven by (3 s referred to as abayesan n the smulatons of Secton V. B. An Exact Bayesan Detector In order to make a comparson to the approxmated Bayesan detector n the smulatons n Secton V, n ths subsecton we provde a set of Bayesan pror PDFs that result n a detector wth a closed form.

7 7 Suppose σ s known, and under M, the elements of A are ndependent and dentcally dstrbuted, or p (A M = p (A k (4 Assume under any model M, M ( p (A k = exp πσ A σa (A k µ k, k k k (5 Denotng x k = [X k [0], X k [],,X k [N ]] T as the data vector from channel k, because each channel s ndependent to other channels we have Denotng p(xk M p (X M p (X M 0 = k= k= p (x k M p (x k M 0 p(x as R (x k M 0 k, wth the assumed pror PDFs, after some algebra we have σ R (x k = NσA k + σ exp N σa k ( x k + Nµ k x k σ Nµ k ( σ (7 σ NσA k + σ where x k denotes the mean of the data from sensor k, gven as x k = N N n=0 From (0 and (6 the Bayesan detector decdes H f X k [n] (6 M π R (x k > λ (8 = k= V. COMPUTER SIMULATIONS In ths secton the prevously dscussed detectors are compared va computer smulatons. We note, these detectors are derved from dfferent orgns and have quet dfferent propertes. The performance of each detector requres further nvestgaton. Ths secton only serves to gve some examples of these detectors and to show the GLRT can be mproved. A. Known Varance Defne the SNR of channel k as SNR k = 0 log 0 ( εa k σ Wth ths defnton, when SNR k 0dB all methods yeld pretty good results. Therefore the man concern s the stuaton when SNR k < 0dB,.e. when A k < 00σ ε. Hence we assume σa k =

8 8 00σ 9ε, k for p(a k gven by (5, so that 3σ Ak = 00σ ε and for most values that A k wll take the SNR k s less than 0dB. Snce the SNR k for each sensor s the same, we smply denote t as SNR. We also assume µ k = 0, k for p(a k and the prors π, M are all equal. Now, all the assumptons requred for the exact Bayesan detector gven by (7 and (8 are made. If all these assumptons are true, the exact Bayesan detector s optmal. However, snce n practce the assumptons are rarely completely true, the Bayesan approach s only to fnd a realzable detector and we hope the detector wll be robust. That s when the true prors and pror PDFs are not the same as the assumptons, the detector stll has satsfactory performance. In the smulaton the true prors under H are set to equal, whch s the same as the assumed prors. However, the true A k takes fxed value, n other words A k s determnstc. Snce the detecton performance only depends on the energy of the sgnal waveform, we smply let s[n] =, 0 n N and σ = N, so that ε σ =. The number of sensors M s set to be 5. Wth the parameters, t can be shown that ( ( L G (X = exp x k= We compare the recever operatng characterstcs (ROC curves of the GLRT gven by (, the rglrt wth model selected by the MDL, the MFLRT gven by (7, the approxmated Bayesan detector gven by (3 and the exact Bayesan detector gven by (7 and (8. When SNR= 6 or A k = the performances of detectors of 5000 realzatons are shown n Fg., and when SNR= or A k = 4 the performances of detectors of 5000 realzatons are shown n Fg.. For relatvely low SNR case, t can be seen n the Fg. that the Bayesan detector, abayesan detector and the MFLRT are better than the GLRT for all regon of the ROC curve. The rglrt are better than GLRT for the regon of the ROC curve where P FA > For relatvely hgh SNR case, t can be seen n the Fg. that the GLRT s the worst for all regon of the ROC curve. Comparng Fg. and Fg., t s seen that the abayesan detector has very smlar performance to the exact Bayesan detector. It s also seen from Fg. and Fg., when SNR s low, the rglrt s close to the GLRT, whch s the worst detector, and when SNR s hgh rglrt s the best detector. Compared wth the rglrt, the MFLRT s more robust. Actually, t can be shown the MFLRT has some mnmax propertes, whch wll be addressed n a future work. B. Unknown Varance When the varance σ s unknown, t can be shown (ˆσ L G (X = 0 ˆσ MN

9 9 0.9 ROC curves from 5000 trals; SNR = 6; M = P D MFLRT Glrt Bayesan abayesan rglrt P FA Fg.. ROC curves of the detectors n relatvely low SNR (known varance.. ROC curves from 5000 trals; SNR = ; M = P D MFLRT Glrt Bayesan abayesan rglrt P FA Fg.. ROC curves of the detectors n relatvely hgh SNR (known varance.. where ˆσ k = NM N k M (X [n] x + (X j [n] n=0 = j=k+

10 0 Because n ths case, t s even more dffcult to assume vald pror PDFs, we only compare the other 4 detectors. When SNR= 6 the performances of detectors of 5000 realzatons are shown n Fg. 3, and when SNR= the performances of detectors of 5000 realzatons are shown n Fg. 4. For relatvely 0.9 ROC curves from 5000 trals; SNR = 6; M = P D MFLRT Glrt abayesan rglrt P FA Fg. 3. ROC curves of the detectors n relatvely low SNR (unknown varance.. low SNR case, t can be seen n the Fg. 3 that the abayesan detector and the MFLRT are better than the GLRT for all regon of the ROC curve. The rglrt are better than the GLRT for the regon of the ROC curve where P FA > For relatvely hgh SNR case, t can be seen n the Fg. 4 that the GLRT s the worst for all regon of the ROC curve. The mnmax property of the MFLRT can also be observed comparng Fg. 3 and Fg. 4. VI. CONCLUSIONS AND DISCUSSION We have nvestgated the GLRT and the Bayesan approach for centralzed composte dstrbuted detecton. In partcular, In partcular, we have proved that the performance of GLRT s poor and can be mproved for ths problem. An approxmated Bayesan detector has also been proposed based on vague pror PDFs, and therefore the margnal PDF can be obtaned wthout ntegraton. As a result, the detector always has a closed form. In our exposton, we assume the sensors have already been ordered wth respect to ther mportance. Therefore when there are M sensors, only M + models need to be consdered. When the orgnal data

11 ROC curves from 5000 trals; SNR = ; M = P D MFLRT Glrt abayesan rglrt P FA Fg. 4. ROC curves of the detectors n relatvely hgh SNR (unknown varance.. from sensors are not ordered, for M sensors we have to consder M models as n [4]. Ths wll greatly ncrease the computatonal load. Alternatvely, some prelmnary processng can be conducted to order the sensors and smlar technques as n [] can be used. However, the performance degradaton as a result of orderng needs further nvestgaton. REFERENCES [] R. R. Tenney and N. R. Sandell Jr., Detecton wth dstrbuted snesors, IEEE Trans. Aerospace Elect. Syst.,, vol. AES-7, pp , July 98. [] R. Vswanathan and P. K. Varshney, Dstrbuted detecton wth multple sensors-part I: Fundamentals, Proceedngs of the IEEE,, vol. 85, pp , Jan [3] R. S. Blum, S. A. Kassam and H. V. Poor, Dstrbuted detecton wth multple sensors-part II: Advanced Topcs, Proceedngs of the IEEE,, vol. 85, pp , Jan [4] B. Baygun and A. O. Hero III, Optmal Smultaneous Detecton and Estmaton Under a False Alarm Constrant, IEEE Trans. Inform. Theory,, vol. 4, pp , may 995. [5] S. M. Kay, Fundamentals of Statstcal Sgnal Processng: Detecton Theory, Upper Saddle Rver, NJ: Prentce-Hall, 998. [6] P. Stoca and Y. Selen, Model-Order Selecton-A revew of nformaton crteron rules, IEEE Sgnal Processng Magzne,, vol., pp , July 004. [7] S. M. Kay, The Multfamly Lkelhood Rato Test for Multple Sgnal Model Detecton, IEEE Sgnal Process. Letters,, vol., no. 5, pp , may 005.

12 [8] L. L. Scharf and B. Fredlander, Matched subspace detectors, IEEE Trans. Sgnal Process.,, vol. 46, no. 8, pp , Aug [9] O. Zetoun, J. Zv and N. Merhav, When s the generalzed lkelhood rato test optmal, IEEE Trans. Inform. Theory,, vol. 38, pp , Sept. 99. [0] P. M. Djurc, Asymptotc MAP crtera for model selecton, IEEE Trans. Sgnal Process.,, vol. 46, no. 0, pp , Oct [] R. R. Hockng, The analyss and selecton of varables n lnear regresson, Bometrcs,, 3, pp. -49, Mar [] A. Gelman, J. Carln, H. Stern and D. Rubn Bayesan Data Analyss, London, U.K.: Chapman & Hall, 995. APPENDIX I PROOF OF THEOREM Snce ν < ν, consder a hypothess testng problem, n whch, H 0 : x[] χ ν, x[] χ ν ν H : x[] χ ν (λ, x[] χ ν ν and x[] s ndependent of x[] n each hypothess. The optmal NP detector decdes H f [5] Pluggng n the expresson of the PDF of χ ν p χ (λ (x[]p χ (x[] ν ν ν = p χ (λ (x[] ν > γ p χ ν (x[]p χ ν ν (x[] p χ ν (x[] and χ ν ( ν ] x[] 4 λ exp [ (x[] + λ k=0 ν Γ( ν The above expresson can be smplfed to x[] ν exp c k x[] k > γ k=0 (λ yelds [5] ( λx[] k+ ν k!γ( ν +k ( x[] > γ wth all postve c k s. Snce x[] 0, c k x[] k s a monotoncally non-decreasng functon of x[], the detector decdes H equvalently f k=0 T NP ([x[], x[]] = x[] > γ (9 Ths s the optmal detector n the NP sense. Now consder another detector whch decdes H f T G ([x[], x[]] = x[] + x[] > γ Ths detector s dfferent from the optmal detector (9 and therefore t has a poorer performance. Snce χ ν T NP, H 0 χ ν (λ, H

13 3 and χ ν T G, H 0 χ ν (λ, H The theorem s proved by lettng T = T NP and T = T G.