On Power Allocation for Distributed Detection with Correlated Observations and Linear Fusion


 Byron Newman
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1 1 On Power Alloation for Distributed Detetion with Correlated Observations and Linear Fusion Hamid R Ahmadi, Member, IEEE, Nahal Maleki, Student Member, IEEE, Azadeh Vosoughi, Senior Member, IEEE arxiv:1794v1 [eesssp] 6 Ot 17 Abstrat We onsider a binary hypothesis testing problem in an inhomogeneous wireless sensor network, where a fusion enter (FC) makes a global deision on the underlying hypothesis We assume sensors observations are orrelated Gaussian and sensors are unaware of this orrelation when making deisions Sensors send their modulated deisions over fading hannels, subjet to individual and/or total transmit power onstraints For parallelaess hannel (PAC) and multipleaess hannel (MAC) models, we derive modified defletion oeffiient (MDC) of the test statisti at the FC with oherent reeption We propose ransmit power alloation sheme, whih maximizes MDC of the test statisti, under three different sets of transmit power onstraints: total power onstraint, individual and total power onstraints, individual power onstraints only When analytial solutions to our onstrained optimization problems are elusive, we disuss how these problems an be onverted to onvex ones We study how orrelation among sensors observations, reliability of loal deisions, ommuniation hannel model and hannel qualities and transmit power onstraints affet the reliability of the global deision and power alloation of inhomogeneous sensors Index Terms Distributed detetion, oherent reeption, modified defletion oeffiient, power alloation, orrelated observations, linear fusion, parallelaess hannel, multipleaess hannel I INTRODUCTION The lassial problem of binary distributed detetion in a network onsisting of multiple distributed sensors and a fusion enter (FC), has a long and rih history Eah sensor (loal detetor) proesses its single observation loally and passes its binary deision to the FC, that is tasked with fusing the binary deisions reeived from the individual sensors and deiding whih of the two underlying hypotheses is true [1] [3] Motivated by the potential appliation of wireless sensor networks (WSNs) for event monitoring, researhers have further studied this problem and extended its setup, taking into aount that bandwidthonstrained ommuniation hannels between sensors and the FC are errorprone, due to limited transmit power to ombat noise and fading (soalled hannel aware binary distributed detetion [4] [6]) Given eah sensor makes its binary deision based on one loal observation, they have investigated how the reliability of the final deision at the FC is affeted by performane indies of loal detetors (sensors) as well as wireless hannel properties Following these works, we onsider hannel aware binary distributed detetion in a WSN with oherent reeption at the FC [7] [9] In this paper, our goal is to study transmit power alloation, when eah sensor has an individual transmit power onstraint and/or all sensors have a joint transmit power onstraint, suh that the reliability of the final deision at the FC is maximized This work is supported by the National Siene Foundation under grants CCF and CCF Power alloation for hannel aware binary distributed detetion in WSNs has been studied in [], [11] More speifially, [] studied the power alloation that maximizes the Jdivergene between the distributions of the reeived signals at the FC under two different hypotheses, subjet to individual and total transmit power onstraints on the sensors, with parallel aess hannel (PAC) 1 and oherent reeption at the FC (ie, hannel phases are known and ompensated at the sensors) Leveraging on [], [11] studied detetion outage and detetion diversity, as the number of sensors goes to infinity, and sensors have idential performane indies Note that [], [11] assume the sensors have unorrelated observations under eah hypothesis Power alloation in WSNs has also been studied for distributed estimation [13] [], where some works minimized the mean square error (MSE) of an estimator subjet to ertain transmit power onstraints [14] [1], while others minimized total transmit power subjet to a onstraint on the MSE of an estimator [13], [] These works, exept [13], [19], [], mainly fous on PAC with oherent reeption at the FC In [13], [19], [], sensors and the FC are onneted differently via a multipleaess hannel (MAC), where the individual sensors send their signals simultaneously, albeit after hannel phases are ompensated at the sensors, and the FC reeives the oherent sum of these transmitted signals Most of these works assume the sensors observations are unorrelated, with the exeption of [14], [16], [] In [19], [] sensors ollaborate with eah other by linearly ombining their independent observations before sending to the FC For binary distributed detetion in WSNs, [1] ompared the detetion performane using both PAC and MAC, with linear fusion rule and nonoherent reeption at the FC (ie, no hannel phase ompensation at the sensors), albeit without imposing any transmit power onstraint Assuming the sensors observations are unorrelated under eah hypothesis and the FC utilizes a linear fusion rule when using PAC, [1] showed that oherent MAC outperforms oherent PAC, whereas nonoherent PAC (MAC) outperforms nonoherent MAC (PAC) when sensors deisions are (un)reliable Distributed detetion with orrelated observations has been studied assuming errorfree [3], [3], [4] and erroneous ommuniation hannels [] The fous of these works though is on how to design optimal loal and global deisions rules to improve the detetion reliability at the FC, assuming sensors know the orrelation among their observations Different from [3], [3] [] we fous on how to optimally transmit the sensors deisions to 1 In PAC, hannels between the sensors and the FC are orthogonal (noninterfering) This an be realized by either time, frequeny, or ode division multiple aess [1], [13]
2 the FC within ertain transmit power onstraints, with a linear fusion rule at the FC and assuming sensors are unaware of the orrelation among their observations Our Contributions: We onsider a binary hypothesis testing problem using M sensors and a FC, where under H, sensors observations are unorrelated Gaussian with ovariane matrix σi and under H 1 they are orrelated Gaussian with a nondiagonal ovariane matrix Σ We relax the assumption in [3], [3] [] that sensors know the orrelation among their observations and onsider a more pratial senario, where the sensors are unaware of suh orrelation Sensors send their modulated binary deisions over nonideal fading hannels, subjet to individual and/or total transmit power onstraints We onsider PAC and MAC with oherent reeption at the FC, assuming that hannel phases are ompensated at the sensors similar to [13], [19], [] To urb the hardware and omputational omplexity and also have a fair omparison between PAC and MAC, we assume that, when the sensors and the FC are onneted via PAC, the FC utilizes a linear fusion rule to obtain the global test statisti T We propose a transmit power alloation sheme, whih maximizes modified defletion oeffiient (MDC) of T We hoose MDC as the performane metri, sine unlike detetion probability and J divergene that require the probability distribution funtion of T, obtaining MDC only needs the first and seond order statistis of T, and often renders a losedform expression [6] [8] Also, an MDCbased optimization problem an lead into nearoptimal solutions for its orresponding detetion probabilitybased optimization problem with muh less omputational omplexity [7], [6], [9] We obtain the MDC of T for oherent PAC and MAC in losedforms that depends on the orrelation among sensors observations Considering three different sets of transmit power onstraints, we investigate transmit power alloation shemes that maximize the MDC Under the onditions that analytial solutions to our onstrained optimization problems are elusive, we disuss how these problems an be onverted to onvex ones and thus an be solved numerially Paper Organization: Setion II details our system model and three different sets of transmit power onstraints Setion III derives the MDC of T for oherent PAC and MAC in losedform expressions Setion IV formulates three different sets of onstrained optimization problems and desribes our approah to solve these problems Setion V presents our numerial results for different orrelation values, sensors observations and ommuniation hannel qualities Setion VI onludes the paper Notations: Salars, vetors and matries are denoted by nonboldfae lower, boldfae lower, and boldfae upper ase letters, respetively A Gaussian random vetor x with mean vetor µ and ovariane matrix Σ is shown as x N (µ, Σ) Transpose and omplex onjugate transpose (Hermitian) of vetor a are denoted as a T and a H, respetively DIAG{a} represents a diagonal matrix whose diagonal elements are the omponents of olumn vetor a A (A ) indiates that A is a positive (semi)definite matrix a b (a b) indiates that eah entry of a is greater than (or equal to) the orresponding entry of b Re{x} is the real part of x = [,, ] T and 1 = [1,, 1] T are two M 1 vetors The (i, j) entry of matrix A is indiated with [A] ij For vetor a we have a = a T a and a = a T a II SYSTEM MODEL AND PROBLEM STATEMENT Our system model onsists of an FC and M distributed sensors with observation vetor x = [x 1, x,, x M ] T The FC is tasked with solving the binary hypothesis testing problem H : x N (, σ I), H 1 : x N (, Σ), where σ is the variane under H and Σ is a nondiagonal ovariane matrix under H 1 with diagonal entries different from σ, ie, under H 1 (H ) sensors observations are orrelated (unorrelated) Gaussian variables with different energy levels Suppose sensor k, only based on its own observation x k, makes a binary deision [], [], [1] and maps it to u k = 1 (u k = ) when it deides H 1 (H ), ie, we assume that sensor k is unaware of the orrelation among sensors observations, Σ We denote p fk = P(u k = 1 H ) and p dk = P(u k = 1 H 1 ) as the false alarm and detetion probabilities of sensor k and assume p dk > p fk The deision u k is ommuniated to the FC over a fading hannel with transmit power P tk Let h k = h k e jϕ k denote the omplex fading oeffiient orresponding to sensor k, with h k and ϕ k being the hannel amplitude and phase, respetively Let y k and y denote the hannel output orresponding to the hannel input u k and (u 1, u,, u M ), when the sensors and the FC are onneted via PAC and MAC, respetively Sine hannel phases are ompensated at the sensors, we have [1] PAC : y k = P k h k u k + n k, k = 1,, M and M MAC : y = Pk h k u k + n (1) k=1 where ommuniation hannel noises are n k CN (, σn) and n CN (, σn) Fading oeffiients h k s and noises n k s and n are all mutually unorrelated and h k is assumed to be onstant during a detetion interval Also P k = P tk θ k, where θ k = Gd ɛ F S k, d F Sk is the distane between sensor k and the FC, ɛ is the pathloss exponent, and G is a onstant We assume that the FC obtains est statisti T from the hannel output(s) and makes a global deision u {, 1} where u = 1 and u = orrespond to H 1 and u =1 H, respetively In partiular, the FC applies T τ u = where the threshold τ is hosen to maximize the total detetion probability P D = P(u = 1 H 1 ) at the FC, subjet to the onstraint that the total false alarm probability satisfies P F = P(u = 1 H ) β F at the FC, where β F (, 1) In a PAC, we assume that the FC is restrited to utilize a linear fusion rule to obtain the test statisti T [], [1] Implementing the linear fusion rule has low omplexity and allows a fair omparison between PAC and MAC Furthermore, the authors in [] have shown that, when idential sensors and the FC are onneted via PAC, the linear fusion rule is a good approximation to the optimal Likelihood Ratio Test (LRT) rule at low signaltonoiseratio (SNR) regime We let T be
3 3 PAC : T = M Re(y k ), MAC : T = Re(y) () k=1 We onsider oherent PAC and MAC with hannel phase ompensation at the sensors [], [] Our goal is to find the transmit powers at sensors suh that the MDC of T is maximized, subjet to different sets of power onstraints We refer to these as the MDCbased transmit power alloation We onsider three different sets of transmit power onstraints: (A) there is otal power onstraint (TPC) suh that M k=1 P t k P tot, where P tot is the total transmit power budget among sensors, we refer to this set as TPC; (B) there is an individual power onstraint (IPC) for eah sensor suh that P tk P k as well as a TPC M k=1 P t k P tot, where P tot < M k=1 P k, we refer to this set as TIPC; (C) there are only IPCs for sensors suh that P tk P k, we refer to this set as IPC Setion III drives the MDC of T for oherent PAC and MAC The MDCbased transmit power alloations under these three different sets of power onstraints are disussed in Setion IV III DERIVING MODIFIED DEFLECTION COEFFICIENT Before delving into the derivations, we introdue the following definitions and notations Consider the signal model in (1) and () We let a k = P k, w k =Re(n k ), w =Re(n) We define the olumn vetors h = [h 1,, h M ] T, h = [ h 1,, h M ] T, y = [y 1,, y M ] T, a = [a 1,, a M ] T, w = [w 1,, w M ] T, n = [n 1,, n M ] T, p d = [p d1,, p dm ] T, p f = [p f1,, p fm ] T, u = [u 1, u,, u M ] T, P = [P 1,, P M ] T, ψ = [ψ 1,, ψ M ], φ=[φ 1,, φ M ], and the square matrix H =DIAG{ h } We define the MDC of T as [6] ( E{T H1, h} E{T H, h} ) MDC =, (3) Var{T H 1, h} where E{} and Var{} are performed with respet to the hannel inputs u k s and the hannel noises To alulate E{T H i, h} for i =, 1 and Var{T H 1, h} in (3), we use the Bayes rule and the fat that H i u k y k (y) u in PAC(MAC) form Markov hains for i=, 1 Hene E{T H i, h} = E{T u, h}p(u H i ), i =, 1 (4) u Var{T H 1, h} = P(u H 1 ), () u where { } = E (T E{T H 1, h}) u, h, and the sums are taken over all values of vetor u To simplify in (6), we add and subtrat E{T u, h} to the terms inside the parenthesis in (6) and expand the produts We have = (6) E { (T E{T u, h}) u } + (E{T u, h} E{T H 1, h}) }{{}}{{} + E {(T E{T u, h})(e{t u, h} E{T H 1, h}) u} We observe that the last term in (6) is zero Thus in (6) is simplified to = + Using (4), (6) and (6), we derive the MDC in the following A PAC Considering the signal model in (1) and (), we have Re(y k ) = a k h k u k + w k where w k N (, σ n ) We write T =a T H u+1 T w Therefore E{T u, h}=a T H u Substituting E{T u, h} into (4) and using the fats p d =E{u H 1 }= u up(u H 1) and p f =E{u H }= u up(u H ) we find E{T H 1, h}=a T H p d, and E{T H, h}=a T H p f (7) Next, we derive, for Var{T H 1, h} Sine T E{T u, h} = 1 T w, we find = E{1 T ww T 1} = M σ n Also, beause E{T u, h} E{T H 1, h} = a T H (u p d ), we have = a T H (u p d )(u p d ) T H a Substituting = + into (6) and using the fats u P(u H 1) = 1, u (u p d)(u p d ) T P(u H 1 ) = E{uu T H 1 } p d p T d, we reah to Var{T H 1, h} = M σ n + at H ( P d p d p T d ) H a, (8) where P d = E{uu T H 1 } is a square matrix with diagonal entries [ P d ] ii = p di and offdiagonal entries [ P d ] ij = P(u i = 1, u j = 1 H 1 ) for i, j = 1,, M, i j Note that the orrelation among sensors observations affets the offdiagonal entries of Pd, ie, for independent observations [ P d ] ij =p di p dj for all i j and equivalently P d =DIAG{p d }(I DIAG{p d }) + p d p T d (9) Substituting (7), (8) into (3) we have where MDC(a) = at bb T a a T Ka + () b = H (p d p f ), = M σ n, K = H ( P d p d p T d ) H B MAC Considering the signal model in (1) and (), we have Re(y) = M k=1 a k h k u k + w where w N (, σ n ) We write T = a T H u + w Therefore E{T u, h} = a T H u Substituting E{T u, h} into (4) and applying similar fats as stated above, we find E{T H 1, h}=a T H p d, and E{T H, h}=a T H p f (11) Next, we find and Sine T E{T u, h} = w, we find = E{w } = σ n Also, sine E{T u, h} E{T H 1, h} = a T H (u p d ), we have =a T H (u p d )(u p d ) T H a Substituting = + into (6) and using similar fats as stated above we reah Var{T H 1, h} = σ n + at H ( P d p d p T d ) H a (1) Substituting (11), (1) into (3) we have MDC(a) = at bb T a a T Ka + (13)
4 4 where b = H (p d p f ), = σ n, K = H ( P d p d p T d ) H Regarding the results in () and (13), a remark follows Remark: For both PAC and MAC, the MDC takes the following form MDC(a) = at bb T a a T Ka + (14) Vetor b and matrix K are idential for PAC and MAC, whereas salar, whih aptures the effet of the hannel noises, is M times larger in PAC Note that b and K depend on the hannel amplitudes H and the loal performane indies Furthermore, K depends on the spatial orrelation among sensors observations IV MDCBASED TRANSMIT POWER ALLOCATION Reall P k = P tk θ k where P tk is transmit power of sensor k and θ k aptures the pathloss effet Sine a k = P k, we define k = P tk = a k θk Let = [1,, M ] T, P t = [P t1,, P tm ] T, and Θ be the omponentwise square root of Θ = DIAG{[θ 1,, θ M ] T } We an rewrite (14) expliitly in terms of vetor as MDC( ) = at t b t b T t a T t K t +, (1) where b t = Θb and K t = ΘK Θ In this setion, we maximize the MDC in (1), with respet to, subjet to different sets of power onstraints speified in Setion II: (A) TPC, where a T t P tot ; (B) TIPC, where a T t P tot and P We define vetor P = [P 1,, P M ] T and P is the omponentwise square root of P ; (C) IPC, where P Setions IVA, IVB, IVC disuss the analytial solutions for MDCbased power alloations under these different sets of power onstraints A Maximizing MDC in (1) under TPC The MDCbased transmit power alloation under TPC is the solution to the following problem max st a T t btbt t at a T t Ktat+ (O 1 ) a T t P tot We start with Lemma 1 whih states that the solution to (O 1 ) satisfies TPC at equality Lemma 1 The maximum values of MDC in (1) are ahieved when the inequality onstraint a T t P tot turns into equality onstraint Proof: Suppose 1 maximizes MDC and a T t11 <P tot Define = at1 Ptot, whih satisfies a T t =P tot We have MDC( ) = 1 a T t1 btbt t at1 a T t1 Ktat1+( at t1 1 P tot ) > at t1 btbt t at1 a T t1 Ktat1+ = MDC(1 ), whih ontradits the optimality assumption of 1 ie, the that maximizes MDC, must satisfy a T t = P tot When the inequality onstraint in TPC is turned into equality onstraint, we an rewrite MDC in (1) as MDC( )= at t b t b T t a T t Q a, where Q a =K t + P tot I (16) Hene, (O 1 ) redues to max st a T t btbt t at (O a 1) T t Qaat a T t = P tot To analytially solve (O 1), we use the result of Lemma given below Lemma For Q the funtion f(x) = xt b tb T t x x T Qx maximized at x =Q 1 b t and its nonzero sales Proof: See Appendix A To be able to use Lemma to solve (O 1), we need to examine whether symmetri matrix Q a is positive definite Note P d p d p T d sine it is a ovariane matrix Thus K, K t Also P tot I Therefore Q a To solve (O 1), we find ˆq = q q where q = Q 1 a b t If ˆq we let a t = ˆq P tot and if ˆq we let a t = ˆq P tot But if all the entries of ˆq do not have the same sign, we resort to numerial solutions In partiular, we turn the problem (O 1) into a onvex problem and solve it numerially We disuss these numerial solutions in Setion IVD Analytial Solution to (O 1) with Independent Observations: Pd is given in (9) and K simplifies to K = H DIAG{p d }(I DIAG{p d }) H Let g k = θ k h k It is easy to verify Q a is a diagonal matrix with diagonal entries [Q a ] kk = p dk (1 p dk )gk + P tot Let q k be the kth entry of q = Q 1 (p a b t Then q k = dk p fk )g k p dk (1 p dk )gk +, k = 1,, M, P tot whih is positive for p dk >p fk We observe q k (p d k p fk ) p dk (1 p dk )g k for large Ptot, whereas q k Ptot (p d k p fk )g k for small Ptot For homogeneous sensors where p fk = p f and p dk = p d, we find the MDCbased power alloation strategy as q k 1 g k for large Ptot (inverse water filling) and q k g k for small Ptot (water filling) B Maximizing MDC in (1) under TIPC The MDCbased transmit power alloation is the solution to the following problem max st a T t btbt t at a T t Ktat+ (O ) a T t P tot P While analytial solution to (O ) remains elusive, we find suboptimal power alloation via solving the following optimization problem max st a T t btbt t at (O a ) T t Qaat a T t = P tot P where Q a is given in (16) Note that (O ) is idential to (O ), exept that the inequality in TPC is turned into equality, ie, is
5 the feasible set of (O ) is a subset of the feasible set of (O ) and the objetive funtion of (O ) is rewritten aordingly Indeed, this suboptimal solution is an aurate solution when κ = PtotgT g 1 for (O ), as we show in the following Examining K and K t when κ 1, we an establish the following inequalities a T (a) t K t a T t Θ H 11 T H Θ = a T t gg T (b) (a T t )(g T g) () P tot g T g (d) where (a) is obtained noting that all entries of Pd p d p T d are less that 1, (b) is found using CauhyShwarz inequality, () omes from the inequality onstraint in (O ), and (d) is due to κ 1 This implies that when κ 1, (O ) an be approximated as (O l ) in (17) min (O a ) l T t btbt t at st a T t P tot (17) P In Appendix B, we show that the solution to (O) l satisfies the equality a T t = P tot This onfirms that the solution to (O ) (suboptimal solution) is an aurate substitute for the solution to (O ) under the ondition κ 1 To solve (O ), we first ignore the box onstraints of IPC and onsider only TPC at equality The problem solving strategy is similar to solving (O 1) in Setion IVA In partiular, to solve (O ), we find ˆq = q q where q =Q 1 a b t If ˆq we let a t1 = ˆq P tot and if ˆq we let a t1 = ˆq P tot If a t1 satisfies the box onstraint P, it is the solution to (O ) However, if a t1 does not satisfy its orresponding box onstraint, following Appendix A, we an easily show that f(x) = xt b tb T t x does x T Qx not have loal maximum or minimum in the set {x : x } This means that, in this ase, the losest feasible point to a t1 is the solution to (O ) That is, the solution to (O ) when a t1 P is the solution to (O ) given below min a t1 (O ) st a T t = P tot P Our analytial solution to (O ) is presented in the appendix C Note that (O ) is not onvex In Appendix C we show that, despite this fat, the solution to KarushKuhnTuker (KKT) onditions for (O ) is unique C Maximizing MDC in (1) under IPC The MDCbased transmit power alloation is the solution to the following optimization problem max a T t btbt t at a T t Ktat+ (O 3 ) st P Similar to Setion IVB, we show below that, when ξ = 1 T P g T g 1, (O 3 ) an be approximated as (O3) l in (18) Examining K and K t when ξ 1, we an establish the following inequalities a T (a) t K t a T t Θ H 11 T H Θ = a T t gg T (b) (a T t a)(g T g) () 1 T P g T g (d), where (a) is beause all entries of P d p d p T d are less that 1, (b) is found using CauhyShwarz inequality, () omes from the inequality in IPC, and (d) is due to ξ 1 This implies that when ξ 1, (O 3 ) an be approximated with (O3) l in (18) min a T t btbt t at (O l 3) st P (18) In Appendix D we show that the solution to (O3) l is = P Analytial Solution to (O 3 ) with Independent Observations: Suppose P k = P, k = 1,, M We showed in Setion IVA that K, K t With independent observations, these matries beome diagonal To solve (O 3 ), we minimize 1 under IPC Assume ψ, φ are the Lagrange multi MDC() pliers of the onstraints P and, respetively Then KKT onditions are (b T t ) ([K t] kk k b tk η) + ψ k φ k =, k = 1,, M, where η = at t K t + b T t ψ k (k P )=, ψ k, k P, and φ k k =, ψ k, k, (19) Sine a T t K t, b t and >, we find η > Solving the KKT onditions yields k = { btk [K t] kk η, for η [Kt] kk b tk P, otherwise P () In Appendix E, we show that at least one of k s in () obtains its maximum P Suppose we sort the sensors suh b ti1 that [K t] i1 bt i M i 1 [K t] im, ie, i1 im Let i M = [i1,, im, im+1,, im ] T and i1 = = im = P, 1 m M Solving a T t K t + ηb T m t = for η, ombined with (), we find η = P j=1 [Kt]i j i j + m P j=1 bt i j If [Kt]imim b P tim η [Kt]i m+1 i m+1 b P tim+1, then the above assumption is valid, and we substitute η in () and alulate im+1,, im and MDC in (O 3 ) Note that η depends on m Otherwise, we inrease m by one and repeat the proedure, until we reah η that lies within the proper interval In Appendix E we also show that, although (O 3 ) is not onvex, the KKT solution in () is unique D Disussion on Maximization of MDC Using Convex Optimization Program Reall that in Setion IVA we ould not find a losed form solution for (O 1 ) when all the entries of ˆq do not have the same sign Also, the analytial solution to (O ),
6 6 formulated in Setion IVB, remains elusive Hene, we have provided a suboptimal solution, via solving (O ) that is aurate solution when κ 1 Similarly, we have derived a suboptimal solution to (O 3 ), formulated in Setion IVC, that is aurate solution when ξ 1 In this setion, we turn (O 1 ),(O ),(O 3 ) into onvex optimization problems, in order to solve them numerially using CVX program We start with (O ), in whih we wish to minimize 1 = at t Ktat+ (b T t at), under TIPC Let x a = at b T t at MDC() t a = 1 b Therefore T bt t at t x a =1 and = xa and t a Employing these definitions, (O ) an be rewritten in the following equivalent form min x T a K t x a + t a (O 1 ) x a,t a st x T a x a P tot t a x a t a P x a b T t x a = 1 We an reformulate (O 1 ) as where min za T D a z a (O ) z a [ ] I st za T T z P a < 1 tot [I, P ]z a [b T t, ]z a = 1, z a z a = [x T a, t a ] T, D a = [ Kt T ] (1) Examining (O ), we realize that it is a quadrati programming (QP) onvex problem sine D a and hene it an be solved using CVX program One an take similar steps to turn (O 1 ) and (O 3 ) into a problem whose optimal solution an be found using CVX program In partiular, we formulate (O 11 ),(O 1 ) via deleting the seond inequality onstraints orresponding to IPC and (O 31 ),(O 3 ) by removing the first inequality onstraints orresponding to TPC from (O 1 ),(O ), respetively Sine D a, (O 1 ) and (O 3 ) are also QP onvex problems V NUMERICAL RESULTS In this setion, through simulations, we orroborate our analytial results We study the effet of orrelation between sensors observations on the MDC, the performane improvements ahieved by the MDCbased transmit power alloations (we refer to as DPA ), and the impat of different sensing and ommuniation hannels on DPA For our simulations, we onsider the signal model H : x k = z k, H 1 : x k = s k + z k, for k = 1,, M, where z k N (, σ ) and s k N (, σ s k ) is a sample of an external Gaussian signal soure s N (, σs) We assume σs k = σ s, where d d ɛs Sk is the distane between sensor k and s and ɛ s is the pathloss exponent S k We assume z k and s k are mutually unorrelated, however, s k s are orrelated Let s = [s 1, s,, s M ] T have ovariane matrix K s =E{ss T } We assume [K s ] ij =ρ ij σs i σs j where ρ ij = ρ dij, ρ 1 is the orrelation at unit distane and depends on the environment and d ij is the distane between sensors i and j [14] Eah sensor employs an energy detetor that maximizes p dk, under the onstraint p fk < 1 Sensors are deployed at equal distanes from eah other, on the irumferene of a irle with diameter m on the xy plane, where the oordinate of its enter is (,, ) For sensing part, we assume M = 8, ɛ s =, σs = dbm, σ = 7 dbm, and for ommuniation part we let σn = 7 dbm, G = db [], ɛ =, and P 1 ==P M = P Performane of DPA when p dk s and pathloss are idential: Suppose the oordinates of signal soure s and the FC, respetively, are (,, 3m), and (,, m) With this onfiguration, p dk = 661, k and pathloss are idential We assume h k CN (, 1), k and we average over, number of hannel realizations to obtain the results We explore the MDC enhanements ahieved by DPA and ompare the MDC values with those of obtained by uniform power alloation (we refer to as UPA ), in whih sensors transmit at equal powers Fig 1 ompares optimal power alloation (OPA), whih finds the sensors powers that maximize P D under the onstraint P F < β F, DPA and UPA, for linear fusion rule and the optimal LRT rule To find OPA with both linear and the LRT rules and DPA with the LRT rule, we use brute fore searh to find the power values, and we simplify the network and only onsider s 1 and s We assume ρ = 1, β F = 1 and plot P D versus P tot under TPC for PAC Fig 1(a) ompares OPA, DPA and UPA, given linear fusion rule We observe that at low P tot, they are lose to eah other but as P tot inreases, they diverge and DPA outperforms UPA but performs worse than OPA Fig 1(b) ompares OPA, DPA and UPA, given the LRT rule Similarly, we see that DPA performs between UPA and OPA We note that at low P tot, DPA approahes OPA Also we plot P D for the LRT rule, where the transmit power values are obtained from maximizing the MDC of the linear fusion rule (in Fig 1(b) we refer to it as Power allo by DPA with linear rule ) We observe that, exept at low P tot, this urve is very lose to P D orresponding to DPA for the LRT rule, implying that the MDCbased power alloation with linear fusion rule is very lose to the MDCbased power alloation with LRT rule Figs and 3 show P D and maximized MDC versus P tot, respetively, under TPC for PAC and MAC, ρ = 1, 9, and β F = in Fig Comparing Figs and 3, we observe that the MDC and P D follow similar trends Hene, to make our omputations faster and less omplex, in the rest of this setion we only alulate the MDC From Fig 3, we note that the MDC inreases by inreasing P tot or by dereasing ρ Comparing PAC and MAC, we note that MAC outperforms PAC at low P tot, whereas PAC onverges to MAC at high P tot These are due to the fats that, at low P tot the effet of ommuniation hannel noise haraterized by in the MDC expression of PAC is M times larger than that of MAC (see equations () and (13)) and thus MAC outperforms PAC However, at high P tot this differene in values is negligible and hene PAC onverges to MAC We also observe that, at low P tot the performane gaps orresponding to DPA and UPA are negligible, despite the fat that sensors experiene different ommuniation hannel fading This is beause at low P tot the dominant effet of ommuniation hannel noise
7 7 renders the deisions of sensors equally important to the FC, regardless of the hannel realizations and the atual (different) deisions On the other hand, at high P tot the performane gaps orresponding to DPA and UPA are signifiant Note that this performane gap in MAC is wider than that of PAC This is expeted, sine the larger value in PAC undermines the differenes between sensors and narrows the performane gap between DPA and UPA As ρ inreases, the hanes that sensors make similar deisions inrease and therefore the performane gaps between DPA and UPA shrink Fig 4 shows the MDC maximized under TIPC versus P tot for PAC and MAC, P = 3mW, and ρ = 1, 9 Similar to Fig 3, the MDC inreases by inreasing P tot or dereasing ρ and MAC outperforms PAC We also ompare the MDC obtained from solving (O ) and (O ), in whih we have the inequality onstraint (I) P tot and the equality onstraint (E) = P tot, respetively We observe that at low P tot, there is no performane gap orresponding to DPA with E and DPA with I, whereas at high P tot, the performane of DPA with E degrades from that of DPA with I This performane degradation in MAC is due to the inreasing interferene of sensors deisions at the FC when sensors are assigned higher transmit power At very high P tot the performane of DPA with E redues to that of UPA This is beause the maximum value that P tot an assume is M P Hene, at very high P tot we have =M P, implying that k = P, k Fig shows the MDC maximized under IPC versus P for PAC and MAC and ρ=1, 9 We note that at low P, the performanes of DPA and UPA are similar, sine k = P, k This observation is in agreement with our analytial results in Setion IVC, where we showed for ξ 1 we have = P Examining the effet of inreasing ρ from 1 to 9 in Figs 3, 4 and, we observe that the performane gap (ie, the differene between the two maximized MDC values) inreases as P tot or P inreases Trends of DPA when p dk s are different and pathloss are idential: We hange the oordinate of signal soure s to (m,, 3m), right above sensor S 1 With this onfiguration, p dk s hange to p T d = [739, 688, 683,, 37,, 683, 688], while pathloss are still idential (note that p d1, p d, p d8 are the three largest) Assuming h k = 1, k, we investigate the impat of different p dk s on DPA via plotting P tk, k Consider Fig 6 whih plots P tk for MAC Figs 6(a), 6(b) orrespond to the ase when the MDC is maximized under TPC, ρ = 1, 9, and P tot = 3 mw, 1 mw, 4 mw Figs 6(), 6(d) orrespond to the ase when the MDC is maximized under TIPC, ρ = 1, 9, P = 3 mw, and P tot = 3 mw, 1 mw, 4 mw Figs 6(e), 6(f) orrespond to the ase when the MDC is maximized under IPC, ρ = 1, 9, and P = 4 mw, 1 mw, 3 mw These figures show that for the ases when the MDC is maximized under TPC or under TIPC, sensors with higher p dk values (ie, more reliable loal deisions) are assigned higher P tk for all P tot values For the ase when the MDC is maximized under IPC, at low P, P tk = P, k (we have UPA) However, as P inreases, for those sensors with smaller p dk values (ie, less reliable loal deisions) we have smaller P tk We also note that as ρ inreases from 1 to 9 the variations of P tk aross sensors inrease: for the ase when the MDC is maximized under TPC, sensors with larger and smaller p dk s, respetively, are assigned further more and lesser P tk ; for the ase when the MDC is maximized under TIPC or IPC, sensors with smaller p dk s are assigned less P tk, suh that for p di < p dj we have P ti < P tj P Fig 7 plots P tk for PAC Comparing Figs 6 and 7, we note that similar trends hold true, while the variations of P tk s aross sensors in MAC, espeially in TIPC and IPC, are wider than those of PAC (ie, P tk s aross sensors in MAC are more different than UPA), due to the fat that the value in MAC is smaller Trends of DPA when p dk s are idential and pathloss are different: Suppose the oordinates of s and the FC, respetively, are (,, 3m), and (m,, 3m), where the FC is right below sensor S 1 With this onfiguration, p dk = 661, k, whereas the pathloss are different (note that the pathloss orresponding to S 1, S and S 8 are the three smallest) We observed that P tk s for different ρ values remain the same Hene, in this part we fous on ρ = 1 DPA is shown in figures 8 and 9 for MAC and PAC, respetively We observe that, for both PAC and MAC under TPC or TIPC, sensors with larger pathloss are assigned higher P tk (we refer to as inverse water filling) Examining the ase when the MDC is maximized under IPC and P = 4mW, we have P tk = P, k (UPA) in PAC, whereas sensors with larger pathloss are assigned higher P tk (inverse water filling) in MAC This is due to the fat that the value in MAC is smaller and therefore, the effetive reeived signaltonoise ratio in MAC is larger, leading to variations of P tk s aross sensors To investigate more the effet of different pathloss on DPA, we move the FC further from the sensors and hange its oordinate to (m,, m), to effetively inrease the pathloss between all the sensors and the FC (and derease reeived power at the FC), while still S 1, S and S 8 have the three smallest pathloss We observe that in TPC and TIPC sensors with smaller pathloss are assigned higher P tk (we refer to as water filling), whereas in IPC, P tk = P, k (we have UPA) VI CONCLUSION We onsidered a hannel aware binary distributed detetion problem in a WSN with oherent reeption and linear fusion rule at the FC, where observations are orrelated Gaussian and sensors are unaware of suh orrelation when making deisions Assuming that the sensors and the FC are onneted via PAC or MAC, we studied power alloation shemes that maximize the MDC at the FC Our numerial results suggest that when MDCbased power alloation and optimal transmit power alloation are employed at low P tot, the resulting P D is very lose for both linear fusion rule and the LRT rule For homogeneous sensors with idential pathloss, MAC outperforms PAC at low P tot under TPC and TIPC (low P under IPC), whereas PAC onverges to MAC at high P tot Compared with equal power alloation, performane enhanement offered by the MDCbased power alloation is more signifiant in MAC and this improvement redues as orrelation inreases For inhomogeneous sensors with idential pathloss, sensors with more reliable deisions are assigned higher powers As
8 8 orrelation inreases, the variations of power aross sensors inrease: sensors with more (less) reliable deisions, are assigned further more (lesser) powers For homogeneous sensors with different pathloss, power alloations are invariant as orrelation hanges At low (high) reeived power at the FC, sensors with smaller (larger) pathloss are assigned higher powers under TPC and TIPC APPENDIX A Proof of Lemma Consider Q = DΛD T, where Λ = DIAG{[λ 1,, λ M ] T } and λ k s are the positive eigenvalues of Q and olumns of D are the eigenvetors of Q We an rewrite f(x) as f(x) = (x T (D Λ)(D Λ) 1 b t) x T D Λ = xt bt btt x ΛD T x x T x, where x = ΛD T x and b t =(D Λ) 1 b t Using the Rayleigh Ritz inequality [3], we find f(x) λ max ( b T t bt ) and the equality is ahieved when x is the orresponding eigenvetor of λ max ( b T t bt ) Sine T b t bt is rankone with the eigenvalue bt and the eigenvetor b t, we have f(x) b tt bt = b T t Q 1 b t and the equality is ahieved at x = b t or x = Q 1 b t and its nonzero sales B Proving that solution of (O l ) satisfies TPC at the equality Consider (O) l and let µ and ψ, φ be the Lagrange multipliers orresponding to a T t P tot, P and, respetively The KKT onditions are b tk b T t a T t b t b T t µ k + ψ k φ k =, k = 1,, M () µ(a T t P tot ) =, µ, a T t P tot (3) ψ k (k P k ) =, ψ k, k P k, and φ k k =, φ k, k (4) We show µ Substituting µ = in (), we have b tk b T t at a T t btbt t at = φ k ψ k Sine b t,,, we find bt k bt t at = φ a T k ψ k < Note that ψ k and φ k t btbt t annot be both at positive, sine from (4) it is infeasible to have k = P k and k = Therefore ψ k or φ k must be zero Sine φ k ψ k <, we onlude that φ k = and ψ k > Now, from (4) we have k = P k, leading to a T t = M k=1 P k > P tot, whih ontradits (3) Therefore, µ and we have a T t =P tot C Analytial Solution of (O ) Sine a T t1 a t1 =P tot, the objetive funtion in (O ) redues to P tot a T t1 Let µ and ψ and φ be the Lagrange multipliers orresponding to a T t = P tot, P and The KKT onditions are µk a t1 k + ψ k φ k =, k = 1,, M µ(a T t P tot ) =, a T t = P tot ψ k (k P k ) =, ψ k, k P k, and φ k k =, φ k, k Solving the KKT onditions for a t1 yields a t1 k µ, for µ a t1 k P k k = Pk, for µ < a t1 k P k () To find positive µ and onsequently k, suppose sensors are a t1 sorted suh that i1 Pi1 a t1 im PiM Assume for 1 m M we have i1 = P i1,, im = P im Substituting () into M j=1 ij = P tot and solving for µ, we find µ = M j=m+1 a t1 ij 4(P tot m j=1 P i j ) Note that µ depends on m If a t1 im+1 P im+1 µ a t1 im, the above assumption is valid, and we substitute P im µ in () to alulate im+1,, im Otherwise, we inrease m by one and repeat the proedure, until we reah µ that lies within the proper interval Although (O 3 ) is not onvex, we show below that the KKT solution in () is unique Suppose the solution in () is not unique, ie, there exist 1 m, m M, m m + 1 suh that ij = a t1 ij µ at1 ij, for m + 1 j M, and µ, P ij Pij, for 1 j m, and µ < a t1 ij P ij (6) Also, ij an be obtained from (6) by substituting j, m, µ with j, m, µ, respetively Sine m m + 1, from (6), we have µ a t1 im+1 4P im+1 On the other hand, due to sensor ordering, we have a t1 im+1 4P im+1 a inequality, we obtain t1 im 4P im m j=m+1 a t1 ij 4 m j=m+1 P i j Applying the mediant a t1 im+1 4P im+1 We observe that two different frations are greater than or equal to the a t1 im+1 4P im+1 Hene, using the mediant inequality and definition of µ, we find M j=m+1 a t1 ij m j=m+1 a t1 ij 4(P tot m j=1 P ij m j=m+1 P ij ) = µ a t1 im+1 4P im+1 However, this inequality ontradits the one in (6) when j, µ are replaed with j, µ Hene, our assumption regarding the existene of m, m is inorret and the solution in () is unique D Proving that solution of (O l 3) is equal to IPC upper limit Consider (O l 3) and let ψ, φ be the Lagrange multipliers orresponding to P and, respetively The KKT onditions are b tk b T t a T t b t b T t + ψ k φ k =, k = 1,, M ψ k (k P k ) =, ψ k, k P k, and φ k k =, φ k, k (7) Sine b t,,, we find bt k bt t at a T t btbt t at =φ k ψ k < Note that ψ k and φ k annot be both positive, sine from (7) it is infeasible to have k = P k and k = Therefore either ψ k or φ k must be zero Sine φ k ψ k <, we onlude that φ k = and ψ k > Now, from (7) we have k = P k, or equivalently, = P
9 9 E Regarding Analytial Solution to (O 3 ) with Independent Observations First, we show that at least one of k s in () is equal to P Suppose k < P, k From (), we have k = bt k [K t] kk η or equivalently η = at k [Kt] kkk, k Using the b tk k M k=1 mediant inequality, we rewrite η as η = at k [Kt] kkk M = k=1 bt k at k a T t Ktat < at b T t Ktat+, sine > However, this violates the t at b T t definition of η in at (19) and ontradits our initial assumption Hene, there should be at least one k = P Next, we show that, although (O 3 ) is not onvex, the KKT solution in () is unique Suppose sensors are sorted suh that b ti1 [K t] i1 bt i M i 1 [K t] im, ie, i1 im, where at least i M i1 = P Assume that the solution in () is not unique, ie, there exist two indies m, m {1,, M}, m m + 1 suh that ij = b tij [K t] ij η, for m + 1 j M, η [Kt]i j i j i j b tij P, for 1 j m, and η > [Kt]i j i j b tij P P where η = P m l=1 [Kt]i l i l + P m l=1 bt i l Also, ij an be obtained from above by substituting j, m, η with j, m, η, respetively Sine m m + 1, from (8), we find η [Kt]i m i m b P tim On the other hand, due to sensor ordering, we have bt i m [K t] im i m P m l=m+1 [Kt]i l i l P m l=m+1 bt i l b tim+1 [K t] im+1 i m+1 Applying the mediant inequality, we obtain P We observe that two [K t] i m i m b tim different frations are less than or equal to [Kt]i m i m b tim Thus m P l=1 [K m t] il i l + + P l=m+1 [K t] il i l m P l=1 b t + m =η il P [K t ] im i m b tim P l=m+1 b t il P However, this inequality ontradits the one in (8) when j, η are replaed with j, η Hene, our assumption regarding the existene of two indies m, m is inorret and the solution in () is unique REFERENCES [1] P K Varshney, Distributed Detetion and Data Fusion New York: Springer, 1997 [] R Viswanathan and P K Varshney, Distributed detetion with multiple sensors: part Ifundamentals, Proeedings of the IEEE, vol 8, pp 64 79, Jan 1997 [3] Q Yan and R S Blum, Distributed signal detetion under the neymanpearson riterion, 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10 OPA DPA UPA 6 P D Total transmit power (mw) (a) P D OPA Power allo by DPA with linear rule DPA UPA Total transmit power (mw) Fig 1 P D under TPC versus P tot for a sensor PAC with idential p dk s and pathloss and ρ=1: (a) Linear fusion rule, (b) LRT fusion rule (b)
11 P D PAC DPA =1 PAC UPA =1 PAC DPA =9 PAC UPA =9 MAC DPA =1 MAC UPA =1 MAC DPA =9 MAC UPA =9 1 Total transmit power (mw) Fig P D under TPC versus P tot 1 8 MDC 6 4 PAC DPA =1 PAC UPA =1 PAC DPA =9 PAC UPA =9 MAC DPA =1 MAC UPA =1 MAC DPA =9 MAC UPA =9 1 Total transmit power (mw) Fig 3 Maximized MDC under TPC versus P tot
12 1 9 MDC PAC DPA I =1 PAC UPA =1 PAC DPA E =1 PAC DPA I =9 PAC UPA =9 PAC DPA E =1 MAC DPA I =1 MAC UPA =1 MAC DPA E =1 MAC DPA I =9 MAC UPA =9 MAC DPA E =9 3 1 Total transmit power (mw) Fig 4 Maximized MDC under TIPC versus P tot and P = 3 mw MDC PAC DPA =1 PAC UPA =1 PAC DPA =9 PAC UPA =9 MAC DPA =1 MAC UPA =1 MAC DPA =9 MAC UPA =9 1 3 Individual transmit power (mw) Fig Maximized MDC under IPC versus P
13 Total power: 3mW Total power: 1mW Total power: 4mW Total power: 3mW Total power: 1mW Total power: 4mW (a) (b) Total power: 3mW Total power: 1mW Total power: 4mW 1 Total power: 3mW Total power: 1mW Total power: 4mW () (d) 3 3 Max Individ power: 4mW Max Individ power: 4mW 1 Max Individ power: 1mW Max Individ power: 3mW 1 Max Individ power: 1mW Max Individ power: 3mW (e) (f) Fig 6 DPA in MAC with different p dk s and idential pathloss: (a) Maximized MDC under TPC, ρ = 1; (b) Maximized MDC under TPC, ρ = 9; () Maximized MDC under TIPC, ρ = 1, P = 3 mw, (d) Maximized MDC under TIPC, ρ = 9, P = 3 mw; (e) Maximized MDC under IPC, ρ = 1, (f) Maximized MDC under IPC, ρ = 9
14 Total power: 3mW Total power: 1mW Total power: 4mW Total power: 3mW Total power: 1mW Total power: 4mW (a) (b) Total power: 3mW Total power: 1mW Total power: 4mW 1 Total power: 3mW Total power: 1mW Total power: 4mW () (d) 3 3 Max Individ power: 4mW 1 Max Individ power: 4mW Max Individ power: 1mW Max Individ power: 3mW 1 Max Individ power: 1mW Max Individ power: 3mW (e) (f) Fig 7 DPA in PAC with different p dk s and idential pathloss: (a) Maximized MDC under TPC, ρ = 1; (b) Maximized MDC under TPC, ρ = 9; () Maximized MDC under TIPC, ρ = 1, P = 3 mw, (d) Maximized MDC under TIPC, ρ = 9, P = 3 mw; (e) Maximized MDC under IPC, ρ = 1, (f) Maximized MDC under IPC, ρ = 9
15 Total power: 3mW Total power: 1mW Total power: 4mW 1 Total power: 3mW Total power: 1mW Total power: 4mW (a) (b) 3 Max Individ power: 4mW 1 Max Individ power: 1mW Max Individ power: 3mW () Fig 8 DPA in MAC with idential p dk s, different pathloss and ρ=1: (a) Maximized MDC under TPC, (b) Maximized MDC under TIPC, P =3 mw, () Maximized MDC under IPC Total power: 3mW Total power: 1mW Total power: 4mW 1 Total power: 3mW Total power: 1mW Total power: 4mW (a) (b) 3 1 Max Individ power: 4mW Max Individ power: 1mW Max Individ power: 3mW () Fig 9 DPA in PAC with idential p dk s, different pathloss and ρ=1: (a) Maximized MDC under TPC, (b) Maximized MDC under TIPC, P =3 mw, () Maximized MDC under IPC