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

1 On Po wer Allocation for Distrib uted Detection with Correlated Observ ations and Linear Fusion Hamid R. Ahmadi, Member , IEEE, Nahal Maleki, Student Member , IEEE, Azadeh V osoughi, Senior Member , IEEE Abstract —W e consider a binary h ypothesis testing problem in an inhomogeneous wireless sensor network, wher e a fusion center (FC) makes a global decision on the underlying hypothesis. W e assume sensors’ observations ar e correlated Gaussian and sensors are unawar e of this correlation when making decisions. Sensors send their modulated decisions over fading channels, subject to individual and/or total transmit power constraints. For parallel-access channel (P A C) and multiple-access channel (MA C) models, we derive modified deflection coefficient (MDC) of the test statistic at the FC with coherent reception. W e pr opose a transmit power allocation scheme, which maximizes MDC of the test statistic, under thr ee different sets of transmit power constraints: total power constraint, individual and total power constraints, individual power constraints only . When analytical solutions to our constrained optimization pr oblems ar e elusive, we discuss how these problems can be con verted to con vex ones. W e study ho w correlation among sensors’ observ ations, reliability of local decisions, communication channel model and channel qualities and transmit power constraints affect the reliability of the global decision and power allocation of inhomogeneous sensors. Index T erms —Distributed detection, coherent reception, mod- ified deflection coefficient, power allocation, correlated obser- vations, linear fusion, parallel-access channel, multiple-access channel. I . I N T RO D U C T I O N The classical problem of binary distributed detection in a network consisting of multiple distributed sensors and a fusion center (FC), has a long and rich history . Each sensor (local detector) processes its single observation locally and passes its binary decision to the FC, that is tasked with fusing the binary decisions received from the individual sensors and deciding which of the two underlying hypotheses is true [1]– [3]. Motiv ated by the potential application of wireless sensor networks (WSNs) for event monitoring, researchers hav e fur- ther studied this problem and extended its setup, taking into account that bandwidth-constrained communication channels between sensors and the FC are error-prone, due to limited transmit power to combat noise and fading (so-called channel aware binary distributed detection [4]–[6]). Given each sensor makes its binary decision based on one local observation, they hav e inv estigated how the reliability of the final decision at the FC is affected by performance indices of local detectors (sensors) as well as wireless channel properties. Follo wing these works, we consider channel aware binary distributed detection in a WSN with coherent reception at the FC [7]–[9]. In this paper , our goal is to study transmit power allocation, when each sensor has an individual transmit power constraint and/or all sensors have a joint transmit power constraint, such that the reliability of the final decision at the FC is maximized. This work is supported by the National Science Foundation under grants CCF-1341966 and CCF-1319770. Power allocation for channel aware binary distributed de- tection in WSNs has been studied in [10], [11]. More specif- ically , [10] studied the power allocation that maximizes the J-div ergence between the distributions of the received signals at the FC under two different hypotheses, subject to indi vidual and total transmit power constraints on the sensors, with parallel access channel (P A C) 1 and coherent reception at the FC (i.e., channel phases are known and compensated at the sensors). Le veraging on [10], [11] studied detection outage and detection diversity , as the number of sensors goes to infinity , and sensors hav e identical performance indices. Note that [10], [11] assume the sensors hav e uncorrelated observations under each hypothesis. Power allocation in WSNs has also been studied for dis- tributed estimation [13]–[22], where some works minimized the mean square error (MSE) of an estimator subject to certain transmit po wer constraints [14]–[21], while others minimized total transmit power subject to a constraint on the MSE of an estimator [13], [22]. These works, except [13], [19], [20], mainly focus on P A C with coherent reception at the FC. In [13], [19], [20], sensors and the FC are connected differently via a multiple-access channel (MA C), where the individual sensors send their signals simultaneously , albeit after channel phases are compensated at the sensors, and the FC receiv es the coherent sum of these transmitted signals. Most of these works assume the sensors’ observ ations are uncorrelated, with the exception of [14], [16], [22]. In [19], [20] sensors collaborate with each other by linearly combining their independent observations before sending to the FC. For binary distributed detection in WSNs, [12] compared the detection performance using both P AC and MA C, with linear fusion rule and noncoherent reception at the FC (i.e., no channel phase compensation at the sensors), albeit without im- posing any transmit power constraint. Assuming the sensors’ observations are uncorrelated under each hypothesis and the FC utilizes a linear fusion rule when using P AC, [12] showed that coherent MAC outperforms coherent P A C, whereas non- coherent P A C (MAC) outperforms noncoherent MAC (P A C) when sensors’ decisions are (un)reliable. Distributed detection with correlated observations has been studied assuming error- free [3], [23], [24] and erroneous communication channels [25]. The focus of these works though is on how to design op- timal local and global decisions rules to improv e the detection reliability at the FC, assuming sensors know the correlation among their observ ations. Dif ferent from [3], [23]–[25] we focus on ho w to optimally transmit the sensors’ decisions to 1 In P AC, channels between the sensors and the FC are orthogonal (non- interfering). This can be realized by either time, frequency , or code division multiple access [12], [13]. 2 the FC within certain transmit power constraints, with a linear fusion rule at the FC and assuming sensors are unaware of the correlation among their observations. Our Contributions : W e consider a binary hypothesis test- ing problem using M sensors and a FC, where under H 0 , sen- sors’ observ ations are uncorrelated Gaussian with cov ariance matrix σ 2 0 I and under H 1 they are correlated Gaussian with a non-diagonal covariance matrix Σ . W e relax the assumption in [3], [23]–[25] that sensors know the correlation among their observations and consider a more practical scenario, where the sensors are una ware of such correlation. Sensors send their modulated binary decisions ov er nonideal fading channels, subject to individual and/or total transmit power constraints. W e consider P A C and MA C with coherent reception at the FC, assuming that channel phases are compensated at the sensors similar to [13], [19], [20]. T o curb the hardware and computational complexity and also ha ve a fair comparison between P A C and MA C, we assume that, when the sensors and the FC are connected via P A C, the FC utilizes a linear fusion rule to obtain the global test statistic T . W e propose a transmit power allocation scheme, which maximizes modified deflection coefficient (MDC) of T . W e choose MDC as the performance metric, since unlike detection probability and J- div ergence that require the probability distribution function of T , obtaining MDC only needs the first and second order statistics of T , and often renders a closed-form expression [26]–[28]. Also, an MDC-based optimization problem can lead into near-optimal solutions for its corresponding detec- tion probability-based optimization problem with much less computational complexity [7], [26], [29]. W e obtain the MDC of T for coherent P AC and MAC in closed-forms that depends on the correlation among sensors’ observations. Considering three different sets of transmit power constraints, we inv es- tigate transmit po wer allocation schemes that maximize the MDC. Under the conditions that analytical solutions to our constrained optimization problems are elusiv e, we discuss how these problems can be conv erted to con vex ones and thus can be solved numerically . Paper Organization : Section II details our system model and three different sets of transmit power constraints. Section III deriv es the MDC of T for coherent P A C and MA C in closed-form expressions. Section IV formulates three different sets of constrained optimization problems and describes our approach to solve these problems. Section V presents our numerical results for different correlation v alues, sensors’ observations and communication channel qualities. Section VI concludes the paper . Notations : Scalars, vectors and matrices are denoted by non-boldface lo wer , boldface lower , and boldface upper case letters, respectively . A Gaussian random vector x with mean vector µ and covariance matrix Σ is shown as x ∼ N ( µ , Σ ) . T ranspose and complex conjugate transpose (Hermitian) of vector a are denoted as a T and a H , respecti vely . DIA G { a } represents a diagonal matrix whose diagonal elements are the components of column vector a . A  0 ( A  0 ) indicates that A is a positiv e (semi-)definite matrix. a  b ( a  b ) indicates that each entry of a is greater than (or equal to) the corresponding entry of b . Re { x } is the real part of x . 0 = [0 , ..., 0] T and 1 = [1 , ..., 1] T are two M × 1 vectors. The ( i, j ) entry of matrix A is indicated with [ A ] ij . For vector a we hav e || a || 2 = a T a and || a || = √ a T a . I I . S Y S T E M M O D E L A N D P R O B L E M S TA T E M E N T Our system model consists of an FC and M distributed sensors with observ ation vector x = [ x 1 , x 2 , ..., x M ] T . The FC is tasked with solving the binary hypothesis testing problem H 0 : x ∼ N (0 , σ 0 I ) , H 1 : x ∼ N (0 , Σ ) , where σ 0 is the variance under H 0 and Σ is a non-diagonal cov ariance matrix under H 1 with diagonal entries different from σ 0 , i.e., under H 1 ( H 0 ) sensors’ observ ations are correlated (uncorrelated) Gaussian variables with different energy levels. Suppose sen- sor k , only based on its own observation x k , makes a binary decision [5], [10], [12] and maps it to u k = 1 ( u k = 0 ) when it decides H 1 ( H 0 ), i.e., we assume that sensor k is unaware of the correlation among sensors’ observations, Σ . W e denote p f k = P ( u k = 1 |H 0 ) and p d k = P ( u k = 1 |H 1 ) as the false alarm and detection probabilities of sensor k and assume p d k > p f k . The decision u k is communicated to the FC over a fading channel with transmit power P t k . Let h k = | h k | e j ϕ k denote the complex fading coefficient corresponding to sensor k , with | h k | and ϕ k being the channel amplitude and phase, respecti vely . Let y k and y denote the channel output corresponding to the channel input u k and ( u 1 , u 2 , ..., u M ), when the sensors and the FC are connected via P AC and MA C, respectively . Since channel phases are compensated at the sensors, we hav e [12] P A C : y k = p P k | h k | u k + n k , k = 1 , ..., M and MA C : y = M X k =1 p P k | h k | u k + n (1) where communication channel noises are n k ∼ C N (0 , σ 2 n ) and n ∼ C N (0 , σ 2 n ) . Fading coefficients h k ’ s and noises n k ’ s and n are all mutually uncorrelated and h k is assumed to be constant during a detection interval. Also P k = P t k θ k , where θ k = Gd −  c F S k , d F S k is the distance between sensor k and the FC,  c is the pathloss e xponent, and G is a constant. W e assume that the FC obtains a test statistic T from the channel output(s) and makes a global decision u 0 ∈ { 0 , 1 } where u 0 = 1 and u 0 = 0 correspond to H 1 and H 0 , respectively . In particular , the FC applies T u 0 =1 ≷ u 0 =0 τ 0 where the threshold τ 0 is chosen to maximize the total detection probability P D 0 = P ( u 0 = 1 |H 1 ) at the FC, subject to the constraint that the total false alarm probability satisfies P F 0 = P ( u 0 = 1 |H 0 ) ≤ β F at the FC, where β F ∈ (0 , 1) . In a P AC, we assume that the FC is restricted to utilize a linear fusion rule to obtain the test statistic T [5], [12]. Implementing the linear fusion rule has low complexity and allows a f air comparison between P AC and MA C. Furthermore, the authors in [5] ha ve sho wn that, when identical sensors and the FC are connected via P A C, the linear fusion rule is a good approximation to the optimal Likelihood Ratio T est (LR T) rule at lo w signal-to-noise-ratio (SNR) regime. W e let T be 3 P A C : T = M X k =1 Re ( y k ) , MA C : T = Re ( y ) . (2) W e consider coherent P A C and MA C with channel phase compensation at the sensors [5], [10]. Our goal is to find the transmit po wers at sensors such that the MDC of T is maximized, subject to different sets of power constraints. W e refer to these as the MDC-based transmit power allo- cation . W e consider three different sets of transmit power constraints: ( A ) there is a total power constraint (TPC) such that P 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 po wer constraint (IPC) for each sensor such that 0 ≤ P t k ≤ P 0 k as well as a TPC P M k =1 P t k ≤ P tot , where P tot < P M k =1 P 0 k , we refer to this set as TIPC; ( C ) there are only IPCs for sensors such that 0 ≤ P t k ≤ P 0 k , we refer to this set as IPC. Section III driv es the MDC of T for coherent P AC and MA C. The MDC-based transmit power allocations under these three dif ferent sets of po wer constraints are discussed in Section IV. I I I . D E R I V I N G M O D I FI E D D E FL E C T I O N C O E FFI C I E N T Before delving into the deri vations, we introduce the follo w- ing definitions and notations. Consider the signal model in (1) and (2). W e let a k = √ P k , w k = Re ( n k ) , w = Re ( n ) . W e define the column vectors 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 d 1 , ..., p d M ] T , p f = [ p f 1 , ..., p f M ] T , u = [ u 1 , u 2 , ..., u M ] T , P = [ P 1 , ..., P M ] T , ψ = [ ψ 1 , ..., ψ M ] , φ = [ φ 1 , ..., φ M ] , and the square matrix | H | = DIA G {| h |} . W e define the MDC of T as [26] MDC =  E { T |H 1 , h } − E { T |H 0 , h }  2 V ar { T |H 1 , h } , (3) where E { . } and V ar { . } are performed with respect to the channel inputs u k ’ s and the channel noises. T o calculate E { T |H i , h } for i = 0 , 1 and V ar { T |H 1 , h } in (3), we use the Bayes rule and the fact that H i → u k → y k ( y ) → u 0 in P A C(MAC) form Markov chains for i = 0 , 1 . Hence E { T |H i , h } = X u E { T | u , h } P ( u |H i ) , i = 0 , 1 (4) V ar { T |H 1 , h } = X u ¯ ∆ P ( u |H 1 ) , (5) where ¯ ∆ = E n ( T − E { T |H 1 , h } ) 2 | u , h o , and the sums are taken over all values of vector u . T o simplify ¯ ∆ in (6), we add and subtract E { T | u , h } to the terms inside the parenthesis in (6) and expand the products. W e have ¯ ∆ = (6) E  ( T − E { T | u , h } ) 2 | u  | {z } ∆ + ( E { T | u , h } − E { T |H 1 , h } ) 2 | {z } ∆ 0 + 2 E { ( T − E { T | u , h } )( E { T | u , h } − E { T |H 1 , h } ) | u } . W e observe that the last term in (6) is zero. Thus ¯ ∆ in (6) is simplified to ¯ ∆ = ∆ + ∆ 0 . Using (4), (6) and (6), we deriv e the MDC in the following. A. P A C Considering the signal model in (1) and (2), we have Re ( y k ) = a k | h k | u k + w k where w k ∼ N (0 , σ 2 n 2 ) . W e write T = a T | H | u + 1 T w . Therefore E { T | u , h } = a T | H | u . Substi- tuting E { T | u , h } into (4) and using the facts p d = E { u |H 1 } = P u u P ( u |H 1 ) and p f = E { u |H 0 } = P u u P ( u |H 0 ) we find E { T |H 1 , h } = a T | H | p d , and E { T |H 0 , h } = a T | H | p f . (7) Next, we deriv e ∆ , ∆ 0 for V ar { T |H 1 , h } . Since T − E { T | u , h } = 1 T w , we find ∆ = E { 1 T w w T 1 } = M σ 2 n 2 . Also, because E { T | u , h } − E { T |H 1 , h } = a T | H | ( u − p d ) , we have ∆ 0 = a T | H | ( u − p d )( u − p d ) T | H | a . Substituting ¯ ∆ = ∆ + ∆ 0 into (6) and using the facts P u P ( u |H 1 ) = 1 , P u ( u − p d )( u − p d ) T P ( u |H 1 ) = E { uu T |H 1 } − p d p T d , we reach to V ar { T |H 1 , h } = M σ 2 n 2 + a T | H | ( ¯ P d − p d p T d ) | H | a , (8) where ¯ P d = E { uu T |H 1 } is a square matrix with diago- nal entries [ ¯ P d ] ii = p d i and off-diagonal entries [ ¯ P d ] ij = P ( u i = 1 , u j = 1 |H 1 ) for i, j = 1 , ..., M , i 6 = j . Note that the correlation among sensors’ observ ations affects the off-diagonal entries of ¯ P d , i.e., for independent observations [ ¯ P d ] ij = p d i p d j for all i 6 = j and equiv alently ¯ P d = DIA G { p d } ( I − DIA G { p d } ) + p d p T d . (9) Substituting (7), (8) into (3) we hav e MDC ( a ) = a T bb T a a T K a + c (10) where b = | H | ( p d − p f ) , c = M σ 2 n 2 , K = | H | ( ¯ P d − p d p T d ) | H B. MA C Considering the signal model in (1) and (2), we have Re ( y ) = P M k =1 a k | h k | u k + w where w ∼ N (0 , σ 2 n 2 ) . W e 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 facts as stated abov e, we find E { T |H 1 , h } = a T | H | p d , and E { T |H 0 , h } = a T | H | p f . (11) Next, we find ∆ and ∆ 0 . Since T − E { T | u , h } = w , we find ∆ = E { w 2 } = σ 2 n 2 . Also, since E { T | u , h } − E { T |H 1 , h } = a T | H | ( u − p d ) , we hav e ∆ 0 = a T | H | ( u − p d )( u − p d ) T | H | a . Substituting ¯ ∆ = ∆ + ∆ 0 into (6) and using similar facts as stated abov e we reach V ar { T |H 1 , h } = σ 2 n 2 + a T | H | ( ¯ P d − p d p T d ) | H | a . (12) Substituting (11), (12) into (3) we hav e MDC ( a ) = a T bb T a a T K a + c (13) 4 where b = | H | ( p d − p f ) , c = σ 2 n 2 , K = | H | ( ¯ P d − p d p T d ) | H | Regarding the results in (10) and (13), a remark follows. Remark : For both P A C and MA C, the MDC takes the following form MDC ( a ) = a T bb T a a T K a + c . (14) V ector b and matrix K are identical for P A C and MA C, whereas scalar c , which captures the effect of the channel noises, is M times larger in P A C. Note that b and K depend on the channel amplitudes | H | and the local performance indices. Furthermore, K depends on the spatial correlation among sensors’ observations. I V . M D C - B A S E D T R A N S M I T P O W E R A L L O C AT I O N Recall P k = P t k θ k where P t k is transmit power of sensor k and θ k captures the pathloss effect. Since a k = √ P k , we define a t k = p P t k = a k √ θ k . Let a t = [ a t 1 , ..., a t M ] T , P t = [ P t 1 , ..., P t M ] T , and √ Θ be the component-wise square root of Θ = DIA G { [ θ 1 , ..., θ M ] T } . W e can rewrite (14) explicitly in terms of vector a t as MDC ( a t ) = a T t b t b T t a t a T t K t a t + c , (15) where b t = √ Θ b and K t = √ Θ K √ Θ . In this section, we maximize the MDC in (15), with respect to a t , subject to different sets of power constraints specified in Section II: ( A ) TPC, where a T t a t ≤ P tot ; ( B ) TIPC, where a T t a t ≤ P tot and 0  a t  √ P 0 . W e define v ector P 0 = [ P 0 1 , ..., P 0 M ] T and √ P 0 is the component-wise square root of P 0 ; ( C ) IPC, where 0  a t  √ P 0 . Sections IV -A, IV -B, IV -C discuss the analytical solutions for MDC-based power allocations under these different sets of power constraints. A. Maximizing MDC in (15) under TPC The MDC-based transmit power allocation under TPC is the solution to the following problem max a t . a T t b t b T t a t a T t K t a t + c ( O 1 ) s.t. a T t a t ≤ P tot a t  0 W e start with Lemma 1 which states that the solution to ( O 1 ) satisfies TPC at equality . Lemma 1. The maximum values of MDC in (15) ar e achieved when the inequality constraint a T t a t ≤ P tot turns into equality constraint. Pr oof: Suppose a t 1 maximizes MDC and a T t 1 a t 1 < P tot . Define a t 2 = a t 1 √ P tot || a t 1 || , which satisfies a T t 2 a t 2 = P tot . W e hav e MDC ( a t 2 ) = a T t 1 b t b T t a t 1 a T t 1 K t a t 1 + c ( a T t 1 a t 1 P tot ) > a T t 1 b t b T t a t 1 a T t 1 K t a t 1 + c = MDC ( a t 1 ) , which contradicts the optimality assumption of a t 1 i.e., the a t that maximizes MDC, must satisfy a T t a t = P tot . When the inequality constraint in TPC is turned into equal- ity constraint, we can rewrite MDC in (15) as MDC ( a t ) = a T t b t b T t a t a T t Q a a t , where Q a = K t + c P tot I . (16) Hence, ( O 1 ) reduces to max a t . a T t b t b T t a t a T t Q a a t ( O 0 1 ) s.t. a T t a t = P tot a t  0 T o analytically solve ( O 0 1 ) , we use the result of Lemma 2 giv en below . Lemma 2. F or Q  0 the function f ( x ) = x T b t b T t x x T Qx is maximized at x ∗ = Q − 1 b t and its non-zer o scales. Pr oof: See Appendix A. T o be able to use Lemma 2 to solve ( O 0 1 ) , we need to examine whether symmetric matrix Q a is positi ve definite. Note ¯ P d − p d p T d  0 since it is a covariance matrix. Thus K , K t  0 . Also c P tot I  0 . Therefore Q a  0 . T o solve ( O 0 1 ) , we find ˆ q = q || q || where q = Q − 1 a b t . If ˆ q  0 we let a ∗ t = ˆ q √ P tot and if − ˆ q  0 we let a ∗ t = − ˆ q √ P tot . But if all the entries of ˆ q do not have the same sign, we resort to numerical solutions. In particular , we turn the problem ( O 0 1 ) into a conv ex problem and solve it numerically . W e discuss these numerical solutions in Section IV -D. • Analytical Solution to ( O 0 1 ) with Independent Obser - vations : ¯ P d is gi ven in (9) and K simplifies to K = | H | DIA G { 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 d k (1 − p d k ) g 2 k + c P tot . Let q k be the k th entry of q = Q − 1 a b t . Then q k = ( p d k − p f k ) g k p d k (1 − p d k ) g 2 k + c P tot , k = 1 , ..., M , which is positive for p d k > p f k . W e observe q k ≈ ( p d k − p f k ) p d k (1 − p d k ) g k for large P tot c , whereas q k ≈ P tot c ( p d k − p f k ) g k for small P tot c . For homogeneous sensors where p f k = p f and p d k = p d , we find the MDC-based power allocation strategy as q k ∝ 1 g k for large P tot c (in verse water filling) and q k ∝ g k for small P tot c (water filling). B. Maximizing MDC in (15) under TIPC The MDC-based transmit power allocation is the solution to the following problem max a t . a T t b t b T t a t a T t K t a t + c ( O 2 ) s.t. a T t a t ≤ P tot 0  a t  √ P 0 While analytical solution to ( O 2 ) remains elusiv e, we find sub-optimal power allocation via solving the following opti- mization problem max a t . a T t b t b T t a t a T t Q a a t ( O 0 2 ) s.t. a T t a t = P tot 0  a t  √ P 0 where Q a is given in (16). Note that ( O 0 2 ) is identical to ( O 2 ) , except that the inequality in TPC is turned into equality , i.e., 5 the feasible set of ( O 0 2 ) is a subset of the feasible set of ( O 2 ) and the objectiv e function of ( O 2 ) is rewritten accordingly . Indeed, this sub-optimal solution is an accurate solution when κ = P tot g T g c  1 for ( O 2 ) , as we sho w in the following. Examining K and K t when κ  1 , we can establish the following inequalities a T t K t a t ( a ) ≤ a T t √ Θ | H | 11 T | H | √ Θ a t = a T t g g T a t ( b ) ≤ ( a T t a t )( g T g ) ( c ) ≤ P tot g T g ( d )  c where ( a ) is obtained noting that all entries of ¯ P d − p d p T d are less that 1 , ( b ) is found using Cauchy-Schwarz inequality , ( c ) comes from the inequality constraint in ( O 2 ), and ( d ) is due to κ  1 . This implies that when κ  1 , ( O 2 ) can be approximated as ( O l 2 ) in (17). min a t . c a T t b t b T t a t ( O l 2 ) s.t. a T t a t ≤ P tot 0  a t  √ P 0 (17) In Appendix B, we show that the solution to ( O l 2 ) satisfies the equality a T t a t = P tot . This confirms that the solution to ( O 0 2 ) (sub-optimal solution) is an accurate substitute for the solution to ( O 2 ) under the condition κ  1 . T o solve ( O 0 2 ) , we first ignore the box constraints of IPC and consider only TPC at equality . The problem solving strate gy is similar to solving ( O 0 1 ) in Section IV -A. In particular , to solve ( O 0 2 ) , we find ˆ q = q || q || where q = Q − 1 a b t . If ˆ q  0 we let a ∗ t 1 = ˆ q √ P tot and if − ˆ q  0 we let a ∗ t 1 = − ˆ q √ P tot . If a ∗ t 1 satisfies the box constraint 0  a t  √ P 0 , it is the solution to ( O 0 2 ) . Ho wever , if a ∗ t 1 does not satisfy its corresponding box constraint, follo wing Appendix A, we can easily show that f ( x ) = x T b t b T t x x T Qx does not hav e local maximum or minimum in the set { x : x  0 } . This means that, in this case, the closest feasible point to a ∗ t 1 is the solution to ( O 0 2 ) . That is, the solution to ( O 0 2 ) when 0  a ∗ t 1  √ P 0 is the solution to ( O 00 2 ) giv en below min a t . | a t − a ∗ t 1 | 2 ( O 00 2 ) s.t. a T t a t = P tot 0  a t  √ P 0 Our analytical solution to ( O 00 2 ) is presented in the appendix C. Note that ( O 00 2 ) is not con vex. In Appendix C we show that, despite this fact, the solution to Karush-Kuhn-T ucker (KKT) conditions for ( O 00 2 ) is unique. C. Maximizing MDC in (15) under IPC The MDC-based transmit power allocation is the solution to the following optimization problem max a t . a T t b t b T t a t a T t K t a t + c ( O 3 ) s.t. 0  a t  √ P 0 Similar to Section IV -B, we sho w below that, when ξ = 1 T P 0 g T g c  1 , ( O 3 ) can be approximated as ( O l 3 ) in (18). Examining K and K t when ξ  1 , we can establish the following inequalities a T t K t a t ( a ) ≤ a T t √ Θ | H | 11 T | H | √ Θ a t = a T t g g T a t ( b ) ≤ ( a T t a )( g T g ) ( c ) ≤ 1 T P 0 g T g ( d )  c, where ( a ) is because all entries of ¯ P d − p d p T d are less that 1 , ( b ) is found using Cauchy-Schwarz inequality , ( c ) comes from the inequality in IPC, and ( d ) is due to ξ  1 . This implies that when ξ  1 , ( O 3 ) can be approximated with ( O l 3 ) in (18). min a t . c a T t b t b T t a t ( O l 3 ) s.t. 0  a t  √ P 0 (18) In Appendix D we show that the solution to ( O l 3 ) is a t = √ P 0 . • Analytical Solution to ( O 3 ) with Independent Observa- tions : Suppose P 0 k = P 0 , k = 1 , ..., M . W e sho wed in Section IV -A that K , K t  0 . With independent observations, these matrices become diagonal. T o solve ( O 3 ) , we minimize 1 MDC ( a t ) under IPC. Assume ψ , φ are the Lagrange multi- pliers of the constraints a t  √ P 0 and a t  0 , respectiv ely . Then KKT conditions are 2 ( b T t a t ) 2 ([ K t ] kk a t k − b t k η ) + ψ k − φ k = 0 , k = 1 , ..., M , (19) where η = a T t K t a t + c b T t a t ψ k ( a t k − p P 0 ) = 0 , ψ k ≥ 0 , a t k ≤ p P 0 , and φ k a t k = 0 , ψ k ≥ 0 , a t k ≥ 0 , Since a T t K t a t ≥ 0 , b t  0 and c > 0 , we find η > 0 . Solving the KKT conditions yields a t k = ( b t k [ K t ] kk η , for η ≤ [ K t ] kk b t k √ P 0 √ P 0 , otherwise . (20) In Appendix E, we show that at least one of a t k s in (20) obtains its maximum √ P 0 . Suppose we sort the sensors such that b t i 1 [ K t ] i 1 i 1 ≥ ... ≥ b t i M [ K t ] i M i M , i.e., a t i 1 ≥ ... ≥ a t i M . Let a t = [ a t i 1 , ..., a t i m , a t i m +1 , ..., a t i M ] T and a t i 1 = ... = a t i m = √ P 0 , 1 ≤ m ≤ M . Solving a T t K t a t + c − η b T t a t = 0 for η , combined with (20), we find η 0 = P 0 P m j =1 [ K t ] i j i j + c √ P 0 P m j =1 b t i j . If [ K t ] i m i m b t i m √ P 0 ≤ η 0 ≤ [ K t ] i m +1 i m +1 b t i m +1 √ P 0 , then the abov e assumption is valid, and we substitute η 0 in (20) and calculate a t i m +1 , ..., a t i M and MDC in ( O 3 ) . Note that η 0 depends on m . Otherwise, we increase m by one and repeat the procedure, until we reach η 0 that lies within the proper interv al. In Appendix E we also sho w that, although ( O 3 ) is not con vex, the KKT solution in (20) is unique. D. Discussion on Maximization of MDC Using Con vex Opti- mization Pr ogram Recall that in Section IV -A we could not find a closed form solution for ( O 1 ) when all the entries of ˆ q do not hav e the same sign. Also, the analytical solution to ( O 2 ) , 6 formulated in Section IV -B, remains elusive. Hence, we hav e provided a sub-optimal solution, via solving ( O 0 2 ) that is accurate solution when κ  1 . Similarly , we hav e deriv ed a sub-optimal solution to ( O 3 ) , formulated in Section IV -C, that is accurate solution when ξ  1 . In this section, we turn ( O 1 ) , ( O 2 ) , ( O 3 ) into con ve x optimization problems, in order to solve them numerically using CVX program. W e start with ( O 2 ) , in which we wish to minimize 1 MDC ( a t ) = a T t K t a t + c ( b T t a t ) 2 , under TIPC. Let x a = a t b T t a t and t a = 1 b T t a t . Therefore b T t x a = 1 and a t = x a t a . Employing these definitions, ( O 2 ) can be rewritten in the following equiv alent form min x a ,t a . x T a K t x a + ct 2 a ( O 2 1 ) s.t. x T a x a ≤ P tot t 2 a x a  t a √ P 0 0  x a b T t x a = 1 (21) W e can reformulate ( O 2 1 ) as min z a . z T a D a z a ( O 2 2 ) s.t. z T a  I 0 0 T −P tot  z a < 1 [ I , − √ P 0 ] z a  0 [ b T t , 0] z a = 1 , z a  0 where z a = [ x T a , t a ] T , D a =  K t 0 0 T c  . Examining ( O 2 2 ) , we realize that it is a quadratic program- ming (QP) con ve x problem since D a  0 and hence it can be solved using CVX program. One can take similar steps to turn ( O 1 ) and ( O 3 ) into a problem whose optimal solution can be found using CVX program. In particular , we formulate ( O 1 1 ) , ( O 1 2 ) via deleting the second inequality con- straints corresponding to IPC and ( O 3 1 ) , ( O 3 2 ) by removing the first inequality constraints corresponding to TPC from ( O 2 1 ) , ( O 2 2 ) , respectiv ely . Since D a  0 , ( O 1 2 ) and ( O 3 2 ) are also QP con vex problems. V . N U M E R I C A L R E S U L T S In this section, through simulations, we corroborate our analytical results. W e study the ef fect of correlation between sensors’ observations on the MDC, the performance improv e- ments achiev ed by the MDC-based transmit power allocations (we refer to as “DP A ”), and the impact of dif ferent sensing and communication channels on DP A. F or our simulations, we consider the signal model H 0 : x k = z k , H 1 : x k = s k + z k , for k = 1 , ..., M , where z k ∼ N (0 , σ 2 0 ) and s k ∼ N (0 , σ 2 s k ) is a sample of an external Gaussian signal source s ∼ N (0 , σ 2 s ) . W e assume σ 2 s k = σ 2 s d  s S k , where d S k is the dis- tance between sensor k and s and  s is the pathloss exponent. W e assume z k and s k are mutually uncorrelated, ho we ver , s k ’ s are correlated. Let s = [ s 1 , s 2 , ..., s M ] T hav e cov ariance matrix K s = E { ss T } . W e assume [ K s ] ij = ρ ij q σ 2 s i σ 2 s j where ρ ij = ρ d ij , 0 ≤ ρ ≤ 1 is the correlation at unit distance and depends on the en vironment and d ij is the distance between sensors i and j [14]. Each sensor employs an energy detector that maximizes p d k , under the constraint p f k < 0 . 1 . Sensors are deplo yed at equal distances from each other , on the circumference of a circle with diameter 5 m on the x - y plane, where the coordinate of its center is (0 , 0 , 0) . For sensing part, we assume M = 8 ,  s = 2 , σ 2 s = 5 dBm, σ 2 0 = − 70 dBm, and for communication part we let σ 2 n = − 70 dBm, G = − 55 dB [10],  c = 2 , and P 0 1 = ... = P 0 M = ¯ P . P erformance of DP A when p d k ’ s and pathloss are identical : Suppose the coordinates of signal source s and the FC, respectiv ely , are (0 , 0 , 3 m ) , and (0 , 0 , − 10 m ) . W ith this configuration, p d k = 0 . 6615 , ∀ k and pathloss are identical. W e assume h k ∼ C N (0 , 1) , ∀ k and we average over 10,000 number of channel realizations to obtain the results. W e explore the MDC enhancements achiev ed by DP A and compare the MDC v alues with those of obtained by uniform power allocation (we refer to as “UP A ”), in which sensors transmit at equal powers. Fig. 1 compares optimal power allocation (OP A), which finds the sensors’ po wers that maximize P D 0 under the con- straint P F 0 < β F , DP A and UP A, for linear fusion rule and the optimal LR T rule. T o find OP A with both linear and the LR T rules and DP A with the LR T rule, we use brute force search to find the po wer values, and we simplify the network and only consider s 1 and s 5 . W e assume ρ = 0 . 1 , β F = 0 . 1 and plot P D 0 versus P tot under TPC for P A C. Fig. 1(a) compares OP A, DP A and UP A, giv en linear fusion rule. W e observe that at low P tot , they are close to each other but as P tot increases, they di ver ge and DP A outperforms UP A but performs worse than OP A. Fig. 1(b) compares OP A, DP A and UP A, giv en the LR T rule. Similarly , we see that DP A performs between UP A and OP A. W e note that at low P tot , DP A approaches OP A. Also we plot P D 0 for the LR T 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 ”Po wer alloc. by DP A with linear rule”). W e observe that, except at lo w P tot , this curve is very close to P D 0 corresponding to DP A for the LR T rule, implying that the MDC-based po wer allocation with linear fusion rule is very close to the MDC-based power allocation with LR T rule. Figs. 2 and 3 show P D 0 and maximized MDC versus P tot , respectiv ely , under TPC for P A C and MAC, ρ = 0 . 1 , 0 . 9 , and β F = 0 . 05 in Fig. 2. Comparing Figs. 2 and 3, we observe that the MDC and P D 0 follow similar trends. Hence, to make our computations faster and less complex, in the rest of this section we only calculate the MDC. From Fig. 3, we note that the MDC increases by increasing P tot or by decreasing ρ . Comparing P A C and MAC, we note that MAC outperforms P A C at low P tot , whereas P A C con verges to MAC at high P tot . These are due to the facts that, at low P tot the effect of communication channel noise characterized by c in the MDC expression of P A C is M times larger than that of MA C (see equations (10) and (13)) and thus MAC outperforms P A C. Howe ver , at high P tot this difference in c values is negligible and hence P A C conv erges to MAC. W e also observe that, at low P tot the performance gaps corresponding to DP A and UP A are negligible, despite the fact that sensors experience different communication channel fading. This is because at low P tot the dominant effect of communication channel noise 7 renders the decisions of sensors equally important to the FC, regardless of the channel realizations and the actual (different) decisions. On the other hand, at high P tot the performance gaps corresponding to DP A and UP A are significant. Note that this performance gap in MA C is wider than that of P A C. This is expected, since the larger c v alue in P AC undermines the differences between sensors and narrows the performance gap between DP A and UP A. As ρ increases, the chances that sensors make similar decisions increase and therefore the performance gaps between DP A and UP A shrink. Fig. 4 shows the MDC maximized under TIPC versus P tot for P A C and MA C, ¯ P = 30 mW, and ρ = 0 . 1 , 0 . 9 . Similar to Fig. 3, the MDC increases by increasing P tot or decreasing ρ and MA C outperforms P A C. W e also compare the MDC obtained from solving ( O 2 ) and ( O 0 2 ) , in which we hav e the inequality constraint (I) | a t | 2 ≤ P tot and the equality constraint (E) | a t | 2 = P tot , respectiv ely . W e observe that at low P tot , there is no performance gap corresponding to “DP A with E” and “DP A with I”, whereas at high P tot , the performance of “DP A with E” degrades from that of “DP A with I”. This performance degradation in MA C is due to the increasing interference of sensors’ decisions at the FC when sensors are assigned higher transmit power . At very high P tot the performance of “DP A with E” reduces to that of UP A. This is because the maximum value that P tot can assume is M ¯ P . Hence, at very high P tot we hav e | a t | 2 = M ¯ P , implying that a t k = √ ¯ P , ∀ k . Fig. 5 shows the MDC maximized under IPC versus ¯ P for P A C and MA C and ρ = 0 . 1 , 0 . 9 . W e note that at low ¯ P , the performances of DP A and UP A are similar , since a t k = √ ¯ P , ∀ k . This observation is in agreement with our analytical results in Section IV -C, where we showed for ξ  1 we ha ve a t = √ P 0 . Examining the effect of increasing ρ from 0 . 1 to 0 . 9 in Figs. 3, 4 and 5, we observe that the performance gap (i.e., the difference between the two maximized MDC values) increases as P tot or ¯ P increases. T r ends of DP A when p d k ’ s are differ ent and pathloss are identical : W e change the coordinate of signal source s to (2 . 5 m , 0 , 3 m ) , right above sensor S 1 . W ith this configuration, p d k ’ s change to p T d = [0 . 7329 , 0 . 6882 , 0 . 6083 , 0 . 5505 , 0 . 5307 , 0 . 5505 , 0 . 6083 , 0 . 6882] , while pathloss are still identical (note that p d 1 , p d 2 , p d 8 are the three largest). Assuming h k = 1 , ∀ k , we inv estigate the impact of different p d k ’ s on DP A via plotting P t k , ∀ k . Consider Fig. 6 which plots P t k for MA C. Figs. 6(a), 6(b) correspond to the case when the MDC is maximized under TPC, ρ = 0 . 1 , 0 . 9 , and P tot = 30 mW , 120 mW , 240 mW. Figs. 6(c), 6(d) correspond to the case when the MDC is maximized under TIPC, ρ = 0 . 1 , 0 . 9 , ¯ P = 30 mW, and P tot = 30 mW , 120 mW , 240 mW. Figs. 6(e), 6(f) correspond to the case when the MDC is maximized under IPC, ρ = 0 . 1 , 0 . 9 , and ¯ P = 4 mW , 15 mW , 30 mW. These figures show that for the cases when the MDC is maximized under TPC or under TIPC, sensors with higher p d k values (i.e., more reliable local decisions) are assigned higher P t k for all P tot values. For the case when the MDC is maximized under IPC, at low ¯ P , P t k = ¯ P , ∀ k (we have UP A). Howe ver , as ¯ P increases, for those sensors with smaller p d k values (i.e., less reliable local decisions) we hav e smaller P t k .W e also note that as ρ increases from 0.1 to 0.9 the variations of P t k across sensors increase: for the case when the MDC is maximized under TPC, sensors with larger and smaller p d k ’ s, respectiv ely , are assigned further more and lesser P t k ; for the case when the MDC is maximized under TIPC or IPC, sensors with smaller p d k ’ s are assigned less P t k , such that for p d i < p d j we hav e P t i < P t j ≤ ¯ P . Fig. 7 plots P t k for P A C. Comparing Figs. 6 and 7, we note that similar trends hold true, while the v ariations of P t k ’ s across sensors in MA C, especially in TIPC and IPC, are wider than those of P A C (i.e., P t k ’ s across sensors in MA C are more dif ferent than UP A), due to the fact that the c value in MAC is smaller . T r ends of DP A when p d k ’ s are identical and pathloss ar e differ ent : Suppose the coordinates of s and the FC, respec- tiv ely , are (0 , 0 , 3 m ) , and (2 . 5 m , 0 , − 3 m ) , where the FC is right belo w sensor S 1 . With this configuration, p d k = 0 . 6615 , ∀ k , whereas the pathloss are different (note that the pathloss corresponding to S 1 , S 2 and S 8 are the three small- est). W e observed that P t k ’ s for different ρ values remain the same. Hence, in this part we focus on ρ = 0 . 1 . DP A is shown in figures 8 and 9 for MAC and P A C, respecti vely . W e observe that, for both P A C and MA C under TPC or TIPC, sensors with larger pathloss are assigned higher P t k (we refer to as in verse water filling). Examining the case when the MDC is maximized under IPC and ¯ P = 4 mW, we hav e P t k = ¯ P , ∀ k (UP A) in P A C, whereas sensors with larger pathloss are assigned higher P t k (in verse water filling) in MA C. This is due to the fact that the c value in MAC is smaller and therefore, the effecti ve receiv ed signal-to-noise ratio in MA C is larger , leading to variations of P t k ’ s across sensors. T o in vestigate more the ef fect of different pathloss on DP A, we mov e the FC further from the sensors and change its coordinate to (2 . 5 m , 0 , − 10 m ) , to effecti vely increase the pathloss between all the sensors and the FC (and decrease receiv ed power at the FC), while still S 1 , S 2 and S 8 hav e the three smallest pathloss. W e observe that in TPC and TIPC sensors with smaller pathloss are assigned higher P t k (we refer to as water filling), whereas in IPC, P t k = ¯ P , ∀ k (we hav e UP A). V I . C O N C L U S I O N W e considered a channel aware binary distributed detection problem in a WSN with coherent reception and linear fusion rule at the FC, where observ ations are correlated Gaussian and sensors are unaw are of such correlation when making decisions. Assuming that the sensors and the FC are connected via P A C or MAC, we studied power allocation schemes that maximize the MDC at the FC. Our numerical results suggest that when MDC-based power allocation and optimal transmit power allocation are employed at low P tot , the resulting P D 0 is v ery close for both linear fusion rule and the LR T rule. For homogeneous sensors with identical pathloss, MAC out- performs P A C at low P tot under TPC and TIPC (lo w ¯ P under IPC), whereas P A C con verges to MA C at high P tot . Compared with equal power allocation, performance enhancement of fered by the MDC-based power allocation is more significant in MA C and this improv ement reduces as correlation increases. For inhomogeneous sensors with identical pathloss, sensors with more reliable decisions are assigned higher powers. As 8 correlation increases, the variations of power across sensors increase: sensors with more (less) reliable decisions, are as- signed further more (lesser) powers. For homogeneous sensors with dif ferent pathloss, power allocations are inv ariant as correlation changes. At lo w (high) received power at the FC, sensors with smaller (larger) pathloss are assigned higher powers under TPC and TIPC. A P P E N D I X A. Pr oof of Lemma 2 Consider Q = D Λ D T , where Λ = DIA G { [ λ 1 , ..., λ M ] T } and λ k ’ s are the positiv e eigenv alues of Q and columns of D are the eigenv ectors of Q . W e can rewrite f ( x ) as f ( x ) = ( x T ( D √ Λ )( D √ Λ ) − 1 b t ) 2 x T D √ Λ √ Λ D T x = ¯ x T ¯ b t ¯ b t T ¯ x ¯ x T ¯ x , where ¯ x = √ Λ D T x and ¯ b t = ( D √ Λ ) − 1 b t . Using the Rayleigh Ritz inequality [30], we find f ( x ) ≤ λ max ( ¯ b t ¯ b t T ) and the equality is achie ved when ¯ x is the corresponding eigen vector of λ max ( ¯ b t ¯ b t T ) . Since ¯ b t ¯ b t T is rank-one with the eigen value | ¯ b t | 2 and the eigen vector ¯ b t , we have f ( x ) ≤ ¯ b t T ¯ b t = b T t Q − 1 b t and the equality is achiev ed at ¯ x ∗ = ¯ b t or x ∗ = Q − 1 b t and its non-zero scales. B. Pr oving that solution of ( O l 2 ) satisfies TPC at the equality Consider ( O l 2 ) and let µ and ψ , φ be the Lagrange multipliers corresponding to a T t a t ≤ P tot , a t  √ P 0 and a t  0 , respectiv ely . The KKT conditions are − 2 cb t k b T t a t | a T t b t b T t a t | 2 − 2 µa t k + ψ k − φ k = 0 , k = 1 , ..., M (22) µ ( a T t a t − P tot ) = 0 , µ ≥ 0 , a T t a t ≤ P tot (23) ψ k ( a t k − p P 0 k ) = 0 , ψ k ≥ 0 , a t k ≤ p P 0 k , and φ k a t k = 0 , φ k ≥ 0 , a t k ≥ 0 (24) W e show µ 6 = 0 . Substituting µ = 0 in (22), we ha ve − 2 cb t k b T t a t | a T t b t b T t a t | 2 = φ k − ψ k . Since b t  0 , a t  0 , a t 6 = 0 , we find − 2 cb t k b T t a t | a T t b t b T t a t | 2 = φ k − ψ k < 0 . Note that ψ k and φ k cannot be both positiv e, since from (24) it is infeasible to hav e a t k = √ P 0 k and a t k = 0 . Therefore ψ k or φ k must be zero. Since φ k − ψ k < 0 , we conclude that φ k = 0 and ψ k > 0 . No w , from (24) we ha ve a t k = p P 0 k , leading to a T t a t = P M k =1 P 0 k > P tot , which contradicts (23). Therefore, µ 6 = 0 and we hav e a T t a t = P tot . C. Analytical Solution of ( O 00 2 ) Since a ∗ T t 1 a ∗ t 1 = P tot , the objective function in ( O 00 2 ) reduces to P tot − a ∗ T t 1 a t . Let µ and ψ and φ be the Lagrange multipliers corresponding to a T t a t = P tot , a t  √ P 0 and a t  0 . The KKT conditions are 2 µa t k − a ∗ t 1 k + ψ k − φ k = 0 , k = 1 , ..., M µ ( a T t a t − P tot ) = 0 , a T t a t = P tot ψ k ( a t k − p P 0 k ) = 0 , ψ k ≥ 0 , a t k ≤ p P 0 k , and φ k a t k = 0 , φ k ≥ 0 , a t k ≥ 0 Solving the KKT conditions for a ∗ t 1  0 yields a t k =      a ∗ t 1 k 2 µ , for µ ≥ a ∗ t 1 k 2 √ P 0 k p P 0 k , for µ < a ∗ t 1 k 2 √ P 0 k . (25) T o find positive µ and consequently a t k , suppose sensors are sorted such that a ∗ t 1 i 1 √ P 0 i 1 ≥ ... ≥ a ∗ t 1 i M √ P 0 i M . Assume for 1 ≤ m ≤ M we have a t i 1 = p P 0 i 1 , ..., a t i m = p P 0 i m . Substituting (25) into P M j =1 a 2 t i j = P tot and solving for µ , we find µ 2 = P M j = m +1 a ∗ 2 t 1 i j 4( P tot − P m j =1 P 0 i j ) . Note that µ depends on m . If a ∗ t 1 i m +1 2 q P 0 i m +1 ≤ µ ≤ a ∗ t 1 i m 2 √ P 0 i m , the above assumption is v alid, and we substitute µ in (25) to calculate a t i m +1 , ..., a t i M . Otherwise, we increase m by one and repeat the procedure, until we reach µ that lies within the proper interv al. Although ( O 00 3 ) is not con vex, we show below that the KKT solution in (25) is unique. Suppose the solution in (25) is not unique, i.e., there exist 1 ≤ m, m 0 ≤ M , m 0 ≥ m + 1 such that a t i j =        a ∗ t 1 i j 2 µ , for m + 1 ≤ j ≤ M , and µ ≥ a ∗ t 1 i j 2 q P 0 i j , q P 0 i j , for 1 ≤ j ≤ m, and µ < a ∗ t 1 i j 2 q P 0 i j (26) Also, a t i j 0 can be obtained from (26) by substituting j, m, µ with j 0 , m 0 , µ 0 , respectiv ely . Since m 0 ≥ m + 1 , from (26), we hav e µ 2 ≥ a ∗ 2 t 1 i m +1 4 P 0 i m +1 . On the other hand, due to sensor ordering, we hav e a ∗ 2 t 1 i m +1 4 P 0 i m +1 ≥ ... ≥ a ∗ 2 t 1 i m 0 4 P 0 i m 0 . Applying the mediant inequality , we obtain P m 0 j = m +1 a ∗ 2 t 1 i j 4 P m 0 j = m +1 P 0 i j ≥ a ∗ 2 t 1 i m +1 4 P 0 i m +1 . W e observe that two dif ferent fractions are greater than or equal to the a ∗ 2 t 1 i m +1 4 P 0 i m +1 . Hence, using the mediant inequality and definition of µ 2 , we find P M j = m +1 a ∗ 2 t 1 i j − P m 0 j = m +1 a ∗ 2 t 1 i j 4( P tot − P m j =1 P 0 i j − P m 0 j = m +1 P 0 i j ) = µ 0 2 ≥ a ∗ 2 t 1 i m +1 4 P 0 i m +1 . Howe ver , this inequality contradicts the one in (26) when j, µ are replaced with j 0 , µ 0 . Hence, our assumption regarding the existence of m, m 0 is incorrect and the solution in (25) is unique. D. Pr oving that solution of ( O l 3 ) is equal to IPC upper limit Consider ( O l 3 ) and let ψ , φ be the Lagrange multipliers corresponding to a t  √ P 0 and a t  0 , respectiv ely . The KKT conditions are − 2 cb t k b T t a t | a T t b t b T t a t | 2 + ψ k − φ k = 0 , k = 1 , ..., M ψ k ( a t k − p P 0 k ) = 0 , ψ k ≥ 0 , a t k ≤ p P 0 k , and φ k a t k = 0 , φ k ≥ 0 , a t k ≥ 0 (27) Since b t  0 , a t  0 , a t 6 = 0 , we find − 2 cb t k b T t a t | a T t b t b T t a t | 2 = φ k − ψ k < 0 . Note that ψ k and φ k cannot be both positiv e, since from (27) it is infeasible to have a t k = √ P 0 k and a t k = 0 . Therefore either ψ k or φ k must be zero. Since φ k − ψ k < 0 , we conclude that φ k = 0 and ψ k > 0 . Now , from (27) we have a t k = p P 0 k , or equiv alently , a t = √ P 0 . 9 E. Regar ding Analytical Solution to ( O 3 ) with Independent Observations First, we sho w that at least one of a t k ’ s in (20) is equal to √ P 0 . Suppose a t k < √ P 0 , ∀ k . From (20), we hav e a t k = b t k [ K t ] kk η or equi v alently η = a t k [ K t ] kk a t k b t k a t k , ∀ k . Using the mediant inequality , we rewrite η as η = P M k =1 a t k [ K t ] kk a t k P M k =1 b t k a t k = a T t K t a t b T t a t < a T t K t a t + c b T t a t , since c > 0 . Howe ver , this violates the definition of η in (19) and contradicts our initial assumption. Hence, there should be at least one a t k = √ P 0 . Next, we show that, although ( O 3 ) is not con vex, the KKT solution in (20) is unique. Suppose sensors are sorted such that b t i 1 [ K t ] i 1 i 1 ≥ ... ≥ b t i M [ K t ] i M i M , i.e., a t i 1 ≥ ... ≥ a t i M , where at least a t i 1 = √ P 0 . Assume that the solution in (20) is not unique, i.e., there exist two indices m, m 0 ∈ { 1 , ..., M } , m 0 ≥ m + 1 such that a t i j =      b t i j [ K t ] i j i j η , for m + 1 ≤ j ≤ M , η ≤ [ K t ] i j i j b t i j √ P 0 √ P 0 , for 1 ≤ j ≤ m, and η > [ K t ] i j i j b t i j √ P 0 where η = P 0 P m ` =1 [ K t ] i ` i ` + c √ P 0 P m ` =1 b t i ` . Also, a t i j 0 can be obtained from abov e by substituting j, m, η with j 0 , m 0 , η 0 , respecti vely . Since m 0 ≥ m + 1 , from (28), we find η ≤ [ K t ] i m 0 i m 0 b t i m 0 √ P 0 . On the other hand, due to sensor ordering, we hav e b t i m +1 [ K t ] i m +1 i m +1 ≥ ... ≥ b t i m 0 [ K t ] i m 0 i m 0 . Applying the mediant inequality , we obtain P 0 P m 0 ` = m +1 [ K t ] i ` i ` √ P 0 P m 0 ` = m +1 b t i ` ≤ [ K t ] i m 0 i m 0 b t i m 0 √ P 0 . W e observe that two different fractions are less than or equal to [ K t ] i m 0 i m 0 b t i m 0 √ P 0 . Thus P 0 P m ` =1 [ K t ] i ` i ` + c + P 0 P m 0 ` = m +1 [ K t ] i ` i ` √ P 0 P m ` =1 b t i ` + √ P 0 P m 0 ` = m +1 b t i ` = η 0 ≤ [ K t ] i m 0 i m 0 b t i m 0 p P 0 . Howe ver , this inequality contradicts the one in (28) when j, η are replaced with j 0 , η 0 . Hence, our assumption regarding the existence of two indices m, m 0 is incorrect and the solution in (20) is unique. R E F E R E N C E S [1] P . K. 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W iley; 2nd edition, 1999. 10 0 10 20 30 40 50 60 70 80 90 100 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Total transmit power (mW) P D0 OPA DPA UPA (a) 0 10 20 30 40 50 60 70 80 90 100 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 Total transmit power (mW) P D0 OPA Power alloc. by DPA with linear rule DPA UPA (b) Fig. 1. P D 0 under TPC versus P tot for a 2-sensor P A C with identical p d k ’ s and pathloss and ρ = 0 . 1 : (a) Linear fusion rule, (b) LR T fusion rule. 11 0 50 100 150 200 250 Total transmit power (mW) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P D0 PAC DPA =0.1 PAC UPA =0.1 PAC DPA =0.9 PAC UPA =0.9 MAC DPA =0.1 MAC UPA =0.1 MAC DPA =0.9 MAC UPA =0.9 Fig. 2. P D 0 under TPC versus P tot . 0 50 100 150 200 250 Total transmit power (mW) 0 2 4 6 8 10 12 MDC PAC DPA =0.1 PAC UPA =0.1 PAC DPA =0.9 PAC UPA =0.9 MAC DPA =0.1 MAC UPA =0.1 MAC DPA =0.9 MAC UPA =0.9 Fig. 3. Maximized MDC under TPC versus P tot . 12 50 100 150 200 250 Total transmit power (mW) 2 3 4 5 6 7 8 9 10 MDC PAC DPA I =0.1 PAC UPA =0.1 PAC DPA E =0.1 PAC DPA I =0.9 PAC UPA =0.9 PAC DPA E =0.1 MAC DPA I =0.1 MAC UPA =0.1 MAC DPA E =0.1 MAC DPA I =0.9 MAC UPA =0.9 MAC DPA E =0.9 Fig. 4. Maximized MDC under TIPC versus P tot and ¯ P = 30 mW. 0 5 10 15 20 25 30 Individual transmit power (mW) 0 1 2 3 4 5 6 7 8 9 10 MDC PAC DPA =0.1 PAC UPA =0.1 PAC DPA =0.9 PAC UPA =0.9 MAC DPA =0.1 MAC UPA =0.1 MAC DPA =0.9 MAC UPA =0.9 Fig. 5. Maximized MDC under IPC versus ¯ P . 13 12345678 Sensor index 0 10 20 30 40 50 60 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (a) 12345678 Sensor index 0 10 20 30 40 50 60 70 80 90 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (b) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (c) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (d) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Max. Individ. power: 4mW Max. Individ. power: 15mW Max. Individ. power: 30mW (e) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Max. Individ. power: 4mW Max. Individ. power: 15mW Max. Individ. power: 30mW (f) Fig. 6. DP A in MAC with dif ferent p d k ’ s and identical pathloss: (a) Maximized MDC under TPC, ρ = 0 . 1 ; (b) Maximized MDC under TPC, ρ = 0 . 9 ; (c) Maximized MDC under TIPC, ρ = 0 . 1 , ¯ P = 30 mW, (d) Maximized MDC under TIPC, ρ = 0 . 9 , ¯ P = 30 mW; (e) Maximized MDC under IPC, ρ = 0 . 1 , (f) Maximized MDC under IPC, ρ = 0 . 9 . 14 12345678 Sensor index 0 10 20 30 40 50 60 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (a) 12345678 Sensor index 0 10 20 30 40 50 60 70 80 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (b) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (c) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (d) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Max. Individ. power: 4mW Max. Individ. power: 15mW Max. Individ. power: 30mW (e) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Max. Individ. power: 4mW Max. Individ. power: 15mW Max. Individ. power: 30mW (f) Fig. 7. DP A in P AC with different p d k ’ s and identical pathloss: (a) Maximized MDC under TPC, ρ = 0 . 1 ; (b) Maximized MDC under TPC, ρ = 0 . 9 ; (c) Maximized MDC under TIPC, ρ = 0 . 1 , ¯ P = 30 mW, (d) Maximized MDC under TIPC, ρ = 0 . 9 , ¯ P = 30 mW; (e) Maximized MDC under IPC, ρ = 0 . 1 , (f) Maximized MDC under IPC, ρ = 0 . 9 . 15 12345678 Sensor index 0 5 10 15 20 25 30 35 40 45 50 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (a) 12345678 Sensor index 0 5 10 15 20 25 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (b) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Max. Individ. power: 4mW Max. Individ. power: 15mW Max. Individ. power: 30mW (c) Fig. 8. DP A in MAC with identical p d k ’ s, different pathloss and ρ = 0 . 1 : (a) Maximized MDC under TPC, (b) Maximized MDC under TIPC, ¯ P = 30 mW, (c) Maximized MDC under IPC. 12345678 Sensor index 0 5 10 15 20 25 30 35 40 45 50 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (a) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Total power: 30mW Total power: 120mW Total power: 240mW (b) 12345678 Sensor index 0 5 10 15 20 25 30 Assigned power to the sensors (mW) Max. Individ. power: 4mW Max. Individ. power: 15mW Max. Individ. power: 30mW (c) Fig. 9. DP A in P AC with identical p d k ’ s, different pathloss and ρ = 0 . 1 : (a) Maximized MDC under TPC, (b) Maximized MDC under TIPC, ¯ P = 30 mW, (c) Maximized MDC under IPC.