Distributed Detection of a Random Process over a Multiple Access Channel under Energy and Bandwidth Constraints

1 Distrib uted Detection of a Random Process o ver a Multiple Access Channel under Ener gy and Bandwidth Constraints Juan Augusto Maya, Member , IEEE, Leonardo Rey V e ga, Member , IEEE, and Cecilia G. Galarza Abstract —W e analyze a binary hypothesis testing problem built on a wireless sensor network (WSN) for detecting a stationary random process distrib uted both in space and time with cir cularly-symmetric complex Gaussian distribution under the Neyman-Pearson framework. Using an analog scheme, the sensors transmit different linear combinations of their measure- ments through a multiple access channel (MA C) to reach the fusion center (FC), whose task is to decide whether the process is pr esent or not. Considering an energy constraint on each node transmission and a limited amount of channel uses, we compute the miss error exponent of the proposed scheme using Large Deviation Theory (LDT) and show that the proposed strategy is asymptotically optimal (when the number of sensors appr oaches to infinity) among linear orthogonal schemes. W e also sho w that the proposed scheme obtains significant energy saving in the low signal-to-noise ratio regime, which is the typical scenario of WSNs. Finally , a Monte Carlo simulation of a 2-dimensional process in space validates the analytical r esults. Index T erms —Distributed detection; energy and bandwidth constraints; multiple access channel; error exponent; wireless sensor networks; I . I N T R O D U C T I O N D ISTRIBUTED detection based on wireless sensor net- works (WSN) is a topic which has attracted great interest in recent years (see [2] and references therein). A typical WSN has a large number of sensor nodes which are generally low- cost battery-powered devices with limited sensing, computing, and communication capabilities. Sensors acquire noisy mea- surements, perform simple data processing and propagate the information into the WSN to reach a decision about a physical phenomenon happening in the co verage area. The data pro- cessing at each node and the propagation of the information within the network are key aspects to be considered to achiev e the required performance level. Network resources, such as energy and bandwidth, are scarce and expensi ve, but they are key variables when the design is focused on processing latency and detection per- formance. Clev er data processing strategies are required to maximize performance under resources constraints. These constraints could be imposed on a node by node basis, or on the overall network. On any matter, appropriate choices of The three authors are with the Univ ersity of Buenos Aires (UBA), Buenos Aires, Argentina. L. Rey V ega and C. Galarza are also with the CSC- CONICET , Buenos Aires, Argentina. Emails: {jmaya, lrey , cgalar}@fi.uba.ar This work is an extension of the results pre viously presented in [1] and was partially supported by the Peruilh grant (UBA) and the project UB ACYT 2002010200250. the processing strategy could largely impact on the total cost or on the life cycle of the WSN, and as such, habilitate its deployment on remote locations or not. A. Related work Distributed detection theory has been much studied in the past. Starting with the seminal w ork of T enney and Sandell [3], sev eral results ha ve been deri ved on how each node compiles the av ailable information and communicates with the fusion center (FC) where the final decision on the true state of nature is taken. Under this setup, digital transmission schemes, where appropriate quantization rules has to be designed has been dealt with [4]–[6] (see also references therein). On the other hand, analog communication schemes were also studied in the past (see for example [7]–[9]). F or Gaussian networks, with independent and identically distributed (i.i.d.) Gaussian measurements [10], [11], it is kno wn that a simple analog scheme as the scaling and transmission of the noisy measurement, is an optimal joint source-channel scheme in terms of quadratic distortion with power constraint in the sensors. Clearly , this can be viewed as a strong motiv ation, from a theoretical point of view , for further study of analog schemes for distributed detection problems. The specific communication strategy from the sensor nodes to the FC has also been extensi vely studied. The simplest approach is to consider that sensors communicate with the FC using orthogonal parallel channels, like time division multiple access (TDMA) or frequency division multiple access (FDMA) [7], [12]. Clearly , this could not be efficient for large- scale wireless sensor networks where a large bandwidth is required for simultaneous transmissions or a large detection delay is necessary if sensors use the same bandwidth and transmit in different time slots. A more sophisticated approach is implemented with a multiple access channel (MA C) where sensors transmit simul- taneously . In this case, the bandwidth requirement does not necessarily depend on the number of sensors. T ransmission ov er a MAC is appealing because, for certain distrib uted detection problems, linear mixing of the sensor measurements, naturally performed in a MAC, could be efficiently exploited at the FC. For example, in the case of Gaussian measurements and analog transmissions, this coherent mixing could provide beamforming gains with significant impact on the performance and the energy consumed by the network. The use of a 2 MA C channel for the problem of distributed detection has been studied in the past. In [13] linear mixing through a MA C is used for collaborati vely computing the network-wide type of the measurements (which are assumed to be i.i.d. giv en the state of nature) taken at each node. As the noise in the channel is asymptotically harmless when the number of sensors grows to infinity , and the type of a set of i.i.d. measurements is a suf ficient statistic , this strategy has the same optimal performance as the best centralized scheme (see also [14], [15]). The specific case in which the possible states of nature are determined by the presence or not of a deterministic signal was considered previously [16]–[18]. Howe ver , when the possible states of nature are the presence or absence of a random process, and the nodes use a MAC to communicate with the FC, the problem is more delicate. When the random process is i.i.d. across time and along the sensor nodes, the analog transmission of the log-likelihood ratio (LLR) of the measurement at each node is optimal in the sense that the asymptotic performance con verges to that of the centralized detector . This is a consequence of the fact that the network-wide global LLR is simply the sum of the marginal LLRs computed at each node. Howe ver , when the random process to be detected is correlated in space and/or time the situation is not so simple. In this case, the measurements taken at each node are not i.i.d. across time and space and the sum of the mar ginal LLRs (achieved through the coherent combining through the MAC) is clearly suboptimal, ev en when there is no fading [19]. As the network-wide LLR is in general a non- linear function of the marginal LLRs, linear combining of the MA C cannot provide the optimal statistic to the FC, as in the case of i.i.d. data. This is also e vident gi ven that when each node sends only its marginal LLR, it is neglecting the correlation with measurements of other nodes. Problems with correlated observations could be consider - ably challenging [20]. Design of distributed processing strate- gies to benefit from the correlation among data is, in general, an open problem. It is well kno wn that signal correlation can help to improv e the detection performance, specially when low quality sensor measurements are av ailable [9], [21]. Clearly , a clev er use of the correlation in space and/or time requires of cooperation among nodes. This problem has also been studied in pre vious works [22], [23]. Howe ver , the specific case of the distributed detection of a random process where the nodes communicate with the FC through a MA C still deserv es some more study . B. Contributions W e will consider a distributed detection scenario where all the sensors communicate with the FC through a MA C. Each sensor obtains a local measurement that is a realiza- tion of a Gaussian stochastic process arbitrarily correlated in space and possibly in time. Our goal will be to analyze the asymptotic performance of the detection scheme, when the number of sensors approaches to infinity . In particular, we look at the error exponents under the Neyman-Pearson scenario. W e will consider several processing strate gies that exploit correlation in space and time. These strate gies can be briefly described as follows. On a first step, each sensor takes a single measurement or a set of measurements. Then, in a synchronous manner , all the sensors transmit different analog linear combinations of their measurements using sev eral MA C uses. Finally , the FC gathers all the data and constructs an appropriate statistic to make the decision. The cooperation among nodes is achiev ed through multiple channel uses. As these multiple uses are clearly pricey , we also impose that the sensors have a limited ener gy budget to be spent on these transmissions, and we carefully select the ener gy used in each channel use in order to comply with this b udget. W e obtain then the optimal ener gy allocation polic y which depends on the statistical properties of the process to be detected. It is shown howe ver that for a large number of nodes, knowledge of these statistics is needed only at the FC. As a side result, the optimal number of MA C uses is also obtained. It is shown, that in general the required number of channel uses is not very lar ge for general correlated processes. This implies that this form of cooperation does not impose se vere penalties in bandwidth or delay , allowing for a close to optimal performance in terms of error exponents. The performance gains obtained with the proposed schemes are more important when the channel signal to noise ratio (SNR C ) is small, which is the usual scenario for WSNs designed for long life c ycle. C. Notation and Organization V ectors are written in boldface and matrices in capital letters. A n and B nm are square and rectangular matrices of sizes n × n and n × m , respectively . det A n and tr A n are the determinant and the trace of A n , respecti vely . ( · ) T and ( · ) H denote transpose and transpose conjugate. W e do not distinguish between random variables and their realization values. p ( x |H ) is the probability density function (p.d.f.) of x conditioned to H . P i ( · ) and E i ( · ) are the probability and the expectation, respectiv ely , computed under hypothesis H i . The pre-image of the set A through the function φ ( · ) is φ − 1 ( A ) = { ν ∈ dom ( φ ) : φ ( ν ) ∈ A } . 1 ( ν ∈ A ) is the indicator function, i.e., it is 1 if ν ∈ A and 0 otherwise. The set of absolutely integrable and essentially bounded functions of support [0 , 1] are, respectively , L 1 ([0 , 1]) = { f : R 1 0 | f ( ν ) | dν < ∞} and L ∞ ([0 , 1]) = { f : ess sup | f ( ν ) | < ∞} . The paper is or ganized as follo ws. In Section II, we for- mulate the binary hypothesis testing problem and describe the MA C between the sensors and the FC. In Section III, we present the centralized detector and then we dev elop two strategies for distributed detection. In Section IV, we enunciate auxiliary tools used to compute the error exponents in Section V. In Section VI, we sho w numerical results for one-dimensional networks and in Section VII we e xtend the results for multi-dimensional networks. In Section VIII we apply the results for detecting a 2D spatial random process and in Section IX we elaborate on the main conclusions. T echnical proofs are provided in the appendices. 3 I I . D E T E C T I O N P R O B L E M W e first consider a network where the nodes are distrib uted along a line and each sensor takes a single measurement. Networks with more dimensions (in space and/or time) will be considered in section VII. The measurement at the k -th sensor under each hypothesis is:  H 1 : x k = s k + v k , H 0 : x k = v k , k = 1 , . . . , n. (1) W e assume that s k is a zero-mean circularly-symmetric com- plex Gaussian stationary process with v ariance σ 2 s and power spectral density (PSD) φ ( ν ) , and v k is a zero-mean circularly- symmetric complex white Gaussian noise independent of s k with variance σ 2 v . Thus, x k is Gaussian distrib uted either under H 0 or H 1 . W e define the following column vectors: s = [ s 1 , . . . , s n ] T , v = [ v 1 , . . . , v n ] T and x = [ x 1 , . . . , x n ] T . The cov ariance matrix of v is σ 2 v I n where I n is the identity matrix of dimension n . The signal vector s has a T oeplitz cov ariance matrix Σ n , Σ n ( φ ) whose ( i, j ) -th element is completely characterized by φ ( ν ) as, (Σ n ) i,j = Z 1 0 φ ( ν ) e −  2 π ν ( i − j ) dν 1 ≤ i, j ≤ n. where ν is the normalized frequency . The cov ariance matrices of x are Σ 0 ,n = σ 2 v I n under H 0 , and Σ 1 ,n = Σ n + σ 2 v I n under H 1 . Remark 1. The T oeplitz assumption allo ws us to manage the correlation between the measurements at the nodes in a simple manner when the number of them grows unbounded. It has also been considered in [9], [17]. Physically , it can be linked to a situation in which the sensor nodes are located on a regular grid and the continuous random process in space is stationary . More general correlation models include Gaussian random fields [22], [23]. Howe ver , under the setup considered in this paper , interesting conclusions and closed form results would more in volved. For that reason, we will work with processes described by T oeplitz cov ariance matrices, which allows us to consider suf ficiently general correlation models. W e analyze a sensor network where the nodes communicate with the FC through a MA C with equal g ain on each link node- FC. This is a simplification for the general setup, ho wever it is a reasonable approximation for networks deployed in rural and remote areas. In this case, each node has a line of sight with the FC, the nodes are steady , and the surroundings do not vary much. Under this scenario, node synchronization and channel inv ersion at the nodes are feasible. This assumption has been extensi vely considered in the past [13], [24]–[28]. Consider no w n 0 channel uses 1 , with n 0 ≤ n . Each channel use is associated to either an orthogonal time slot or a frequency band. Then, the processing strategy may use n 0 time slots, n 0 frequency bands or a combination of them such that the product of time slots and frequency bands is n 0 . During a channel use, sensors communicate with the FC through a noisy MA C without any other interference. The signal collected at 1 W e will refer to channel uses or degrees of freedom (DoF) interchangeably . the FC is a noisy version of the coherent superposition of the symbols transmitted by the n sensors through the MA C, z k 0 = n X k =1 g kk 0 ( x k ) + w k 0 , k 0 = 1 , . . . , n 0 , (2) where g kk 0 ( · ) is the encoding function used by the sensor k at channel use k 0 , and w k 0 is the zero-mean communication noise with variance σ 2 w and circularly-symmetric complex Gaussian distribution independent of ev erything else. An illustration of the distributed scheme is shown in Fig. 1. Node 1 Node 2 Node n MAC A WGN n' channel uses FC x 1 x 2 x n w k' z k' k' =1 ,...,n' c 1 k x 1 c 2 k' x 2 c nk' x n LLR H 0 /H 1 H ^ Fig. 1. Decentralized detection scheme for a WSN with linear encoding functions: g kk 0 ( x k ) = c kk 0 x k . I I I . D E T E C T O R S A. Centralized detector Suppose that the FC has direct access to the complete measured vector x through n orthogonal noiseless commu- nication channels. The appropriate use of these measurements allows the FC to construct the optimal centralized detector (CD) [29]. In a distributed setting, with noisy links from the sensors to the FC, this detector provides an upper bound (not necessarily tight) on the performance of any distributed scheme. Consider the Neyman-Pearson problem for a fixed false alarm probability level α , where a false alarm ev ent occurs when H 1 is declared but H 0 is true. The associated normalized logarithmic likelihood ratio (LLR) is [29] T n c ( x ) = 1 n log p ( x |H 1 ) p ( x |H 0 ) = 1 n  x H  Σ − 1 0 ,n − Σ − 1 1 ,n  x − log det Σ 1 ,n det Σ 0 ,n  . (3) Now , according to the Neyman-Pearson theorem, the optimum centralized test chooses H 1 if T n c ( x n ) > τ n , and H 0 other- wise, where the threshold of the test τ n depends on α . When the process { s k } is spatially correlated, the cov ari- ance matrix Σ 1 ,n is not diagonal. Hence, (3) cannot be ex- pressed as the sum of the marginal LLRs from each node, and the mixing property of the MAC does not lead to the global LLR as in [19]. Howe ver , it is possible to implement simple distributed detection schemes to reconstruct, in some way , the centralized statistic at the FC through multiple channel uses. B. Distributed detector W e consider distributed detection (DD) schemes where the nodes make n 0 channel uses through a MAC. In this work, we 4 restrict ourselves to the case where g kk 0 ( · ) are linear encoding functions. Thus, in the case of one-dimensional networks, each sensor transmits scaled versions of its local measurement through the MA C and (2) results in: z k 0 = c H k 0 x + w k 0 , k 0 = 1 , . . . , n 0 , where c k 0 = [ c 1 k 0 , c 2 k 0 , . . . , c nk 0 ] T . Define the pr ecoding matrix as C nn 0 = [ c 1 , . . . , c n 0 ] and z = [ z 1 , . . . , z n 0 ] T . Then z = C H nn 0 x + w , (4) where w = [ w 1 , . . . , w n 0 ] T has covariance matrix σ 2 w I n 0 . Considering the Neyman-Pearson problem, the normalized LLR at the FC is: T n d ( z ) = 1 n log p ( z |H 1 ) p ( z |H 0 ) = 1 n  z H  Ξ − 1 0 ,n 0 − Ξ − 1 1 ,n 0  z − log det Ξ 1 ,n 0 det Ξ 0 ,n 0  , (5) where Ξ 1 ,n 0 = C H nn 0 Σ n C nn 0 + σ 2 v C H nn 0 C nn 0 + σ 2 w I n 0 , and Ξ 0 ,n 0 = σ 2 v C H nn 0 C nn 0 + σ 2 w I n 0 are the covariance matrices of z under and H 1 and H 0 , respecti vely . The false alarm and the miss error probability are defined as P n f a = P 0 ( T n d ( z ) > τ n ) and P n m = P 1 ( T n d ( z ) < τ n ) , respecti vely . The a verage energy consumed by the k -th node during n 0 transmissions is: E k = E   n 0 X k 0 =1 | c kk 0 x k | 2   = E [ | x k | 2 ] n 0 X k 0 =1 | c kk 0 | 2 , k = 1 , . . . , n, where c kk 0 is the ( k , k 0 ) -th element of C nn 0 , E [ | x k | 2 ] = σ 2 v + p 1 σ 2 s , and p 1 is the a priori probability of the state of nature H 1 . When p 1 is unknown, a natural upper -bound for E k is obtained by taking p 1 = 1 . The problem is then to obtain appropriate precoding matri- ces C nn 0 such that the constraints on energy and channel uses are satisfied in the Neyman-Pearson setting: Problem 1 (Best linear precoding strategy) . Consider that n sensors take measurements according to (1) and the commu- nication model is (4). When the per-sensor average ener gy constraint is E t , and the probability of false alarm is less than α , the best linear precoding strategy is obtained by solving inf C nn 0 ∈A nn 0 P n m , with A nn 0 = { C nn 0 ∈ C n × n 0 : P n f a ≤ α, 1 n tr ( C nn 0 C H nn 0 ) ≤ E t σ 2 s + σ 2 v } . This problem is not con vex in C nn 0 , and closed-form solutions are not readily a vailable. Therefore, we restrict the precoding matrices C nn 0 to have orthogonal columns and consider WSNs with large number of nodes where it is relev ant to compute the err or exponent of P n m when P n f a ≤ α . Then, the problem that we attack is as follows: Problem 2 (Best asymptotic orthogonal precoding strategy) . Consider that n sensors take measurements according to (1) and they communicate with the FC through a MA C as in (4). Consider also that n 0 n → β when n → ∞ for a giv en asymptotic fraction of DoF β ∈ (0 , 1] . When the per-sensor av erage energy constraint is E t , and the lev el of false alarm is limited to α , the best asymptotic orthogonal precoding strategy is obtained by solving sup { C nn 0 } ∞ n =1 ∈{A orth nn 0 } ∞ n =1 − lim n →∞ 1 n log P n m with A orth nn 0 = { C nn 0 ∈ C n × n 0 : C nn 0 = V nn 0 ∆ n 0 , V H nn 0 V nn 0 = I n 0 , ∆ n 0 = diag ( γ n 1 , . . . , γ n n 0 ) , P n f a ≤ α, 1 n P n 0 k 0 =1 ( γ n k 0 ) 2 ≤ E t σ 2 s + σ 2 v } , where the coefficients { γ n k 0 } control the energy trans- mitted by the sensors on each channel use. T o tackle this problem, we consider two different param- eterizations for the sequence of precoding matrices { C nn 0 } . Before elaborating on them, we make the following definition: Definition 1 (T ransmitted Modes Set) . Let L be the uniform probability measure defined on the interv al [0 , 1] , and denote by L [ φ − 1 ( A )] the measure of the set φ − 1 ( A ) . Define the complementary cumulate distrib ution function Ω( t ) as the measure of the set of frequencies ν such that φ ( ν ) is at least t , i.e., Ω( t ) = L [ φ − 1 (( t, + ∞ ))] . For β ∈ (0 , 1] , define Θ β := { ν ∈ [0 , 1] : Ω( φ ( ν )) ≤ β } as the transmitted modes set of Lebesgue measure β whose elements are the frequencies ν that take on the largest v alues of φ ( ν ) . Now , to solve Problem 2, we consider two strategies: Definition 2 (Principal Component Strategy in a MA C chan- nel, PCS-MA C) . Let n ∈ N and n 0 , n 0 ( n ) ≤ n . Assume that { u k } n k =1 is a basis of eigen vectors for Σ n and { λ n k } n k =1 the corresponding eigen values, where λ n 1 ≥ λ n 2 ≥ · · · ≥ λ n n . W e choose the precoding matrix for the PCS-MA C strategy as C nn 0 = U nn 0 ∆ n 0 , (6) where U nn 0 = [ u 1 , . . . , u n 0 ] , ∆ n 0 = diag ( γ n 1 , . . . , γ n n 0 ) and n 0 = max { k ∈ [1 , n ] : Ω( λ n k ) ≤ β } . T o implement this strate gy , the rows of C nn 0 are required to be known at the corresponding nodes. For that, either each node should perform a local eigen value-eigen vector decompo- sition, or the FC should communicate the corresponding row of C nn 0 to each sensor through a feedback channel. The first option entails a more expensiv e node deployment, and the sec- ond option in volv es the implementation of a high throughput multi-terminal communication link whose complexity would grow linearly as the number of nodes increases. T o reduce complexity , we replace the basis of eigen vectors in PCS-MA C by the Fourier basis, and we propose the next strate gy: Definition 3 (Principal Frequencies Strategy in a MA C chan- nel, PFS-MA C) . Consider the power spectral density (PSD) φ ( ν ) and let n ∈ N and n 0 , n 0 ( n ) ≤ n . For each n , let ( j 1 , j 2 , · · · , j n ) be a permutation of { 1 , 2 , . . . , n } such that φ  j 1 − 1 n  ≥ φ  j 2 − 1 n  ≥ · · · ≥ φ  j n − 1 n  . W e choose the precoding matrix PFS-MA C strate gy as C nn 0 = F nn 0 ∆ n 0 (7) 5 where n 0 = max { k ∈ [1 , n ] : Ω ( φ (( j n k − 1) /n )) ≤ β } , F nn 0 = [ f j 1 , . . . , f j n 0 ] is a sub-matrix of the DFT matrix of order n , i.e., f k 0 = [ f 1 k 0 , f 2 k 0 , . . . , f nk 0 ] T with f kk 0 = 1 √ n exp(  2 π ( k − 1)( k 0 − 1) /n ) for k = 1 , . . . , n , k 0 = j 1 , . . . , j n 0 and ∆ n 0 = diag ( γ n 1 , · · · , γ n n 0 ) . Under this strate gy , at each channel use, the FC receives a noisy version of a gi ven frequency bin of the DFT of the measurement v ector . For that, the nodes are required to kno w the inde x number of the DFT bin to be transmitted only . This information is common to all nodes and it may be broadcasted by the FC on a lo w rate feedback channel. Let us interpret the above definitions. For the proposed decentralized strategies, we have imposed that the number of channel uses be limited to n 0 ≤ n . Using a MA C in each channel use, we can limit the number of channel uses without discarding any measurement of the sensors and distributing the transmitted ener gy more ef ficiently . W e will see in Corollary 2 that PCS-MAC and PFS-MA C results in the the asymptotic transmission of the components of the spectrum φ ( ν ) in the set Θ β . Intuiti vely , we see that this is the best we can do, as the large components of φ ( ν ) are the more informative about the state of nature H 1 , when the fraction of channel uses approaches to β . Remark 2. For PCS-MAC, c kk 0 = u kk 0 γ k 0 where u kk 0 is the ( k , k 0 ) -th element of U nn 0 , and the av erage energy consumed in each node is E k = ( p 1 σ 2 s + σ 2 v ) P n 0 k 0 =1 | u kk 0 | 2 ( γ n k 0 ) 2 . This depends on each sensor k . If we additionally consider the av erage over all the sensors, we obtain the ener gy constraint considered in Problem 2: 1 n n 0 X k 0 =1 ( γ n k 0 ) 2 ≤ E t σ 2 s + σ 2 v . (8) On the other hand, in the case of PFS-MA C , c kk 0 = 1 √ n γ n k 0 e  2 π ( k − 1)( k 0 − 1) /n , and the av erage energy consumed in each node is E k = ( p 1 σ 2 s + σ 2 v ) 1 n P n 0 k 0 =1 ( γ n k 0 ) 2 . This is independent of the sensor k and therefore (8) is actually an energy constraint on each single sensor . It is con venient to express γ n k 0 as the sampled version of a function ξ : [0 , 1] → R + , γ n k 0 = s ξ  k 0 − 1 n  , k 0 = 1 , . . . , n 0 , (9) where ξ ( · ) is a Riemann integrable function defined as the asymptotic energy profile. The average ener gy constraint when n goes to infinity results in lim n →∞ 1 n n 0 X k =1 ( γ n k ) 2 = Z Θ β ξ ( ν ) dν ≤ E t σ 2 s + σ 2 v . (10) Remark 3. A similar general setup was previously considered in [12]. Ho wev er, the authors imposed certain restrictions to the ov erall model that were lessen in our setup. For instance, the communication channels between the nodes and the FC were orthogonal, instead of a MAC as it is analyzed here. It was also assumed that each sensor measured the same realization of a process under H 1 disturbed by different noise realizations. That would have been the case of a random process maximally correlated in space. Our analysis allows for general correlation functions both in space and/or time. In addition and more importantly , by introducing the coef ficients { γ n k 0 } , we optimize the ener gy profile for a giv en ener gy b ud- get and we find the optimum number of MA C uses to achieve the best error exponent among the orthogonal strate gies. I V . P R E L I M I N A RY T O O L S Definition 4 (W eak and Strong Norms) . Let A be a Hermitian n × n matrix with eigen values { λ k } n k =1 , its weak (normalized Frobenius) and strong (spectral) norms are, respecti vely , | A | =   1 n n X i =1 n X j =1 | a ij | 2   1 / 2 , k A k = max 1 ≤ k ≤ n | λ k | . Definition 5 (W iener Class Functions) . A function φ ( ν ) defined on the normalized frequency interv al [0 , 1] is said to be in the W iener class if it has a Fourier series with absolutely summable Fourier coef ficients a k , i.e., P ∞ k = −∞ | a k | < ∞ and φ ( ν ) = ∞ X k = −∞ a k e  2 π νk , a k = Z 1 0 φ ( ν ) e −  2 π νk dν. (11) Definition 6. The circulant matrix B n , B n ( φ ) generated by the samples of the function φ ( ν ) with ν = i − 1 n , i = 1 , . . . , n , is completely specified by its first ro w b n = [ b n 1 , . . . , b n n ] , B n ( φ ) =       b n 1 b n 2 . . . b n n b n n b n 1 . . . b n n − 1 . . . . . . b n 2 b n 3 . . . b n 1       (12) where b n k = P n i =1 φ ( i − 1 n ) e  2 π n ( i − 1)( k − 1) , k = 1 , . . . , n. The eigen values of this matrix are giv en by { φ ( i − 1 n ) } n i =1 . Definition 7 (Asymptotically Equi valent Matrices) . T wo se- quences of n × n matrices { A n } and { B n } are said to be asymptotically equiv alent, A n ∼ B n , if 1) A n and B n are uniformly bounded in strong norm, i.e.: k A n k , k B n k ≤ M < ∞ , n = 1 , 2 , . . . (13) 2) A n − B n con verges to zero in weak norm as n → ∞ , i.e.: lim n →∞ | A n − B n | = 0 . (14) Lemma 1 (Asymptotically Equi valence of T oeplitz and Cir - culant Matrices) . Let A n ( φ ) = [ a i − j ] 1 ≤ i,j ≤ n be a T oeplitz matrix with the function φ ( ν ) in the Wiener class related to a k as in (11) and let B n ( φ ) be a circulant matrix as in (12). Then, A n ( φ ) and B n ( φ ) are asymptotically equiv alent. Pr oof: See [30, p. 53]. Theorem 1 (Asymptotic T oeplitz Distrib ution Theorem) . As- sume that φ ( ν ) is a Wiener class function and Σ n the 6 Hermitian T oeplitz matrix related to φ . Let { λ n k } n k =1 be the eigen values of Σ n , and δ 2 ≥ λ n 1 ≥ · · · ≥ λ n n ≥ δ 1 . If F ( · ) is a continuous function defined on the interv al [ δ 1 , δ 2 ] , such that R ν : φ ( ν )= δ F ( φ ( ν )) dν = 0 for any δ ∈ [ δ 1 , δ 2 ] ∩ ∆ where ∆ ⊆ [0 , ∞ ) , then lim n →∞ 1 n n X k =1 F ( λ n k ) 1 ( λ n k ∈ ∆) = Z φ − 1 (∆) F ( φ ( ν )) dν. Pr oof: The proof follows from [30, Corollary 4.1]. Definition 8 (Large Deviation Principle, LDP) . If G ◦ and ¯ G are the interior and closure of a set G ⊂ R , respectively , we say that the sequence of random variables { Y n } ∞ n =1 satisfies the LDP with rate function Λ ∗ ( x ) if, for any G ⊂ R we ha ve − inf x ∈ G o Λ ∗ ( x ) ≤ lim inf n →∞ 1 n log P ( Y n ∈ G ) ≤ lim sup n →∞ 1 n log P ( Y n ∈ G ) ≤ − inf x ∈ ¯ G Λ ∗ ( x ) . (15) The set G is said to satisfy the Λ -continuous property if inf x ∈ G ◦ Λ ∗ ( x ) = inf x ∈ ¯ G Λ ∗ ( x ) . Thus, the lo wer and upper bounds in (15) coincide, and the exponent  G is defined as:  G = lim n →∞ − 1 n log P ( Y n ∈ G ) = inf x ∈ G o Λ ∗ ( x ) = inf x ∈ ¯ G Λ ∗ ( x ) . (16) The Gärtner -Ellis theorem [31] is a handy result for the computation of a good rate function for a general sequence of random variables. For this result to be true, certain conditions on the asymptotic logarithmic moment-generating function (LMGF), defined as the limit Λ( t ) = lim n →∞ 1 n Λ n ( nt ) , with Λ n ( t ) = log E [ e tY n ] need to be satisfied. A particular critical condition is the steepness of Λ( t ) at the boundary of its domain. When Y n is a sequence of Gaussian quadratic forms constructed from a stationary Gaussian random process (which is the case for T n d ( z ) and T n c ( x ) ) this steepness condition is a delicate issue [32]. The asymptotic bad behavior of some eigen values of the corresponding T oeplitz matrices of the stationary process could have a critical role in the beha vior of Λ( t ) . F or that reason, and for the particular case of the hypothesis testing problem based on likelihood ratio test, we need the following result: Theorem 2 (Modified Gärtner -Ellis theorem for Gaussian LLR) . Let { T n } ∞ n =1 be a sequence of LLRs of complex circularly symmetric Gaussian random variables drawn from a stationary Gaussian random process with spectral density h i ( ν ) under H i , i = 0 , 1 , and define its LMGF as Λ i ( t ) = lim n →∞ 1 n Λ n i ( nt ) , Λ n i ( t ) = log E i [ e tT n ] . (17) Consider the following assumptions: A 1 ) h i ( ν ) is in the Sze gö class, i.e., log h i ( ν ) ∈ L 1 ([0 , 1]) , i = 0 , 1 . A 2 ) The ratio of spectral densities is essentially bounded, i.e., h i ( ν ) h \ i ( ν ) ∈ L ∞ ([0 , 1]) , where \ i = 1 if i = 0 and \ i = 0 if i = 1 . Then, under H i , the sequence { T n } satisfies the LDP in Def. 8 with good rate function Λ ∗ i ( x ) computed through the Fenchel- Legendre transform of Λ i ( t ) , i = 0 , 1 : Λ ∗ i ( x ) = sup t ∈ R { xt − Λ i ( t ) } , (18) with Λ i ( t ) = − Z 1 0 n log  1 + t h 0 ( ν ) − h 1 ( ν ) h \ i ( ν )  + t log h 1 ( ν ) h 0 ( ν ) o dν. Pr oof: See [32, Proposition 7]. V . E R R O R E X P O N E N T S W e are no w ready to compute the error exponents for both the centralized detector and the decentralized detector considering both strategies, PCS-MA C and PFS-MAC. Before proceeding, we formulate some handy definitions. Consider a spectral density Γ , Γ( ν ) , a set of frequencies Θ , and a threshold τ . Then, we define the following functionals: m 0 (Γ) = Z Θ  Γ( ν ) 1 + Γ( ν ) − log(1 + Γ( ν ))  dν, (19) m 1 (Γ) = Z Θ { Γ( ν ) − log(1 + Γ( ν )) } dν . (20) κ f a (Γ) = Z Θ { log(1 − t ∗ Γ( ν )) + t ∗ log(1 + Γ( ν )) } dν + τ (1 + t ∗ ) , κ m (Γ) = Z Θ { log(1 − t ∗ Γ( ν )) + t ∗ log(1 + Γ( ν )) } dν + τ t ∗ , where t ∗ is the unique solution to τ + Z Θ log(1 + Γ( ν )) dν = Z Θ Γ( ν ) 1 − t ∗ Γ( ν ) dν. (21) A. Centralized Err or Exponents Theorem 3 (CD Error Exponents) . Consider the LLR test in (3) with a fixed threshold τ , the spectral density Γ CD , Γ CD ( ν ) = φ ( ν ) σ 2 v , Θ = [0 , 1] and assume that φ ( ν ) is in the W iener class. Then, the false alarm and miss error exponents for the hypothesis testing problem in (1) are: κ CD f a =  κ f a (Γ CD ) if τ > m 0 (Γ CD ) , 0 if τ ≤ m 0 (Γ CD ) , (22) κ CD m =  κ m (Γ CD ) , if τ < m 1 (Γ CD ) , 0 if τ ≥ m 1 (Γ CD ) , (23) Pr oof: See App. I. Corollary 1. Consider the LLR test in (3). The miss error exponent subject to P n f a ≤ α with α ∈ (0 , 1) is κ CD m,α = Z 1 0  1 1 + Γ CD ( ν ) + log(1 + Γ CD ( ν )) − 1  dν. (24) Pr oof: Ev aluate Th. 3 with τ = m 0 (Γ CD ) +  , where  > 0 is arbitrary small. See [33, Prop. 2] for a detailed proof. 7 B. Decentralized Err or Exponents Consider the setup in (4), and the LLR test in (5). Bearing in mind the asymptotic behavior of the covariance matrices of z under H 0 and H 1 , we formulate the following lemma: Lemma 2 (Decentralized Hypothesis T esting) . Under either PCS-MA C or PFS-MA C, the decision at the FC asymptotically consists on choosing one of the follo wing PSDs, with { γ n k 0 } as in (9), and ν ∈ Θ β :  H 1 : h ( ν ) = ( φ ( ν ) + σ 2 v ) ξ ( ν ) + σ 2 w H 0 : h ( ν ) = σ 2 v ξ ( ν ) + σ 2 w . (25) Pr oof: See App. II-A The following theorem establishes the error exponents for the decentralized strategies presented in Section III. Both strategies PCS-MA C and PFS-MAC allow to improve the performance of detection by careful selection of ξ ( ν ) . Theorem 4 (DD Error Exponents) . Consider the hypothesis testing problem in (25), where ξ ( ν ) is a fixed asymptotic energy profile that satisfies (10). Consider also that the asymp- totic fraction of DoF is limited to β . Assume that φ ( ν ) is in the Wiener class and that the threshold of the test is fixed to τ . Let Γ DD , Γ DD ( ν ) = ξ ( ν ) φ ( ν ) ξ ( ν ) σ 2 v + σ 2 w . Then, the strategies PCS- MA C and PFS-MA C have the same false alarm and miss error exponents, and they are: κ DD f a =  κ f a (Γ DD ) if τ > m 0 (Γ DD ) , 0 if τ ≤ m 0 (Γ DD ) , (26) κ DD m =  κ m (Γ DD ) , if τ < m 1 (Γ DD ) , 0 if τ ≥ m 1 (Γ DD ) , (27) where t ∗ is the unique solution to (21) and m 0 ( · ) and m 1 ( · ) are giv en by (19) and (20) with Γ = Γ DD , Θ = Θ β . Pr oof: See App. II-B. Corollary 2 (DD Miss Error Exponent and Optimum Energy Profile) . Consider the hypothesis testing problem in (25). Assume that φ ( ν ) is in the W iener class, the fraction of DoF is limited to β , and the energy constraint is gi ven by (10). The optimal miss error exponent subject to P n f a ≤ α , with α ∈ (0 , 1) fixed, considering orthogonal schemes is achiev ed with either the PCS-MA C or PFS-MA C scheme and is giv en by κ DD m,α = Z Θ β ∗  1 1 + Γ DD ( ν ) + log (1 + Γ DD ( ν )) − 1  dν, (28) where Θ β ∗ ⊆ Θ β is the support of the optimal ener gy profile ξ OEP ( ν ) = max { ˆ ξ 1 ( ν ) , ˆ ξ 2 ( ν ) , ˆ ξ 3 ( ν ) , 0 } and ˆ ξ i ( ν ) , i = 1 , 2 , 3 , are the roots of the follo wing cubic equation: a 3 ( ν ) ˆ ξ 3 + a 2 ( ν ) ˆ ξ 2 + a 1 ( ν ) ˆ ξ + a 0 = 0 (29) with coefficients a 0 = λ ∗ σ 6 w a 1 ( ν ) = σ 2 w ( − φ ( ν ) 2 + λ ∗ σ 2 w (2 φ ( ν ) + 3 σ 2 v )) (30) a 2 ( ν ) = λ ∗ σ 2 w ( φ ( ν ) 2 + 4 φ ( ν ) σ 2 v + 3 σ 4 v ) a 3 ( ν ) = λ ∗ σ 2 v ( φ ( ν ) + σ 2 v ) 2 , where λ ∗ is the Lagrange multiplier that satisfies the ener gy constraint (10) with equality . The closed-form solution of ξ OEP ( ν ) is sho wn in (31), at the top of the next page. Pr oof: See App. II-C C. Suboptimal Ener gy Profiles In this section we define three energy profiles: i) constant energy profile, ξ CEP ( ν ) , E t ( σ 2 v + σ 2 s ) , where all the sensors transmit using the same gain for all channel uses; ii) spectral energy profile, ξ SEP ( ν ) , E t φ ( ν ) σ 2 s ( σ 2 v + σ 2 s ) , that reproduces the shape of φ ( ν ) and therefore, allocates more energy to the frequencies where the process under H 1 concentrates more po wer; iii) ON/OFF energy profile, ξ ON/OFF-EP ( ν ) , E t ( σ 2 v + σ 2 s ) β ∗ 1 ( ν ∈ Θ β ∗ ) , with β ∗ = L (Θ β ∗ ) ≤ β , where each sensor transmits with constant gain if ν ∈ Θ β ∗ and stays silent otherwise. In i) and ii) we assume that the sensors do not have access to the transmitted modes set and no DoF compression is possible ( β = 1 ). In iii) and in the optimal ener gy profile, the sensors know the transmitted modes set. V I . N U M E R I C A L R E S U LT S F O R 1 D In this section, we ev aluate the miss error exponent for two complex Gaussian correlated auto-regressi ve moving av erage (ARMA) processes with PSD giv en by φ ( ν ) = σ 2 in    P M k =0 b k e −  2 πν k P N k =0 a k e −  2 πν k    2 , where M and N are the degrees of the numerator and denominator polynomials, respectiv ely , and σ 2 in is selected such that the variance of the process is σ 2 s . The simulated processes (with even PSD) are plotted in Fig. 3 with coefficients shown in T able I. W e will referred to them as PSD1 and PSD2. Let SNR M = σ 2 s /σ 2 v and SNR C = E t /σ 2 w 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −20 −10 0 10 20 Normalized frequency, f PSD (dB) PSD1 PSD2 Fig. 3. PSDs (in dB) with variance σ 2 s = 1 . Only positiv e normalized frequencies are plotted. be the measurement and communication signal-to-noise ratios, respectiv ely . W e use SNR M = 5 dB, σ 2 s = 1 and E t = 1 on all figures of this section. In Fig. 4, we sho w the behavior of the miss error exponent for processes PSD1 and PSD2 against SNR C when P n f a ≤ α . In both cases the decentralized detectors (DD) approach the centralized detector (CD) performance when SNR C is high enough. W e see that the optimum scheme DD-OEP allows to sa ve a significant amount of energy when the communication signal-to-noise ratio is low . This is the desirable operation regime of a wireless sensor network with massiv e amount of nodes since it would extend the useful life of the WSN, avoid maintenance action for changing the battery of the nodes, or e ven elude network reconfiguration when nodes run out of ener gy . 8 ξ OEP ( ν ) = { (1 / (62 1 / 3 λ ( φ 2 v + 2 φv 2 + v 3 )))((1 −  √ 3)(2 λ 3 φ 6 w 3 + 6 λ 3 φ 5 v w 3 + 6 λ 3 φ 4 v 2 w 3 + 2 λ 3 φ 3 v 3 w 3 + 9 λ 2 φ 6 v w 2 + 54 λ 2 φ 5 v 2 w 2 + 108 λ 2 φ 4 v 3 w 2 + 90 λ 2 φ 3 v 4 w 2 + 27 λ 2 φ 2 v 5 w 2 + (4(3 λw ( φ 2 v + 2 φv 2 + v 3 )(2 λφw + 3 λv w − φ 2 ) − λ 2 w 2 ( φ 2 + 4 φv + 3 v 2 ) 2 ) 3 + (2 λ 3 φ 6 w 3 + 6 λ 3 φ 5 v w 3 + 6 λ 3 φ 4 v 2 w 3 + 2 λ 3 φ 3 v 3 w 3 + 9 λ 2 φ 6 v w 2 + 54 λ 2 φ 5 v 2 w 2 + 108 λ 2 φ 4 v 3 w 2 + 90 λ 2 φ 3 v 4 w 2 + 27 λ 2 φ 2 v 5 w 2 ) 2 ) 2 / 3 ) − ((1 +  √ 3)(3 λw ( φ 2 v + 2 φv 2 + v 3 )(2 λφw + 3 λv w − φ 2 ) − λ 2 w 2 ( φ 2 + 4 φv + 3 v 2 ) 2 )) / (32 ( 2 / 3) λ ( φ 2 v + 2 φv 2 + v 3 )(2 λ 3 φ 6 w 3 + 6 λ 3 φ 5 v w 3 + 6 λ 3 φ 4 v 2 w 3 + 2 λ 3 φ 3 v 3 w 3 (31) + 9 λ 2 φ 6 v w 2 + 54 λ 2 φ 5 v 2 w 2 + 108 λ 2 φ 4 v 3 w 2 + 90 λ 2 φ 3 v 4 w 2 + 27 λ 2 φ 2 v 5 w 2 + (4(3 λw ( φ 2 v + 2 φv 2 + v 3 ) × (2 λφw + 3 λv w − φ 2 ) − λ 2 w 2 ( φ 2 + 4 φv + 3 v 2 ) 2 ) 3 + (2 λ 3 φ 6 w 3 + 6 λ 3 φ 5 v w 3 + 6 λ 3 φ 4 v 2 w 3 + 2 λ 3 φ 3 v 3 w 3 + 9 λ 2 φ 6 v w 2 + 54 λ 2 φ 5 v 2 w 2 + 108 λ 2 φ 4 v 3 w 2 + 90 λ 2 φ 3 v 4 w 2 + 27 λ 2 φ 2 v 5 w 2 ) 2 ) 2 / 3 ) − ( w ( φ 2 + 4 φv + 3 v 2 )) / (3( φ 2 v + 2 φv 2 + v 3 )) } 1 ( D ( ν ) ≥ 0) . D ( ν ) = − ( λφ 2 + 2 λφv + λv 2 )(4 λ 2 φ 5 w 5 − λφ 6 w 4 + 18 λφ 5 v w 4 + 27 λφ 4 v 2 w 4 − 4 φ 6 v w 3 ) . (32) Fig. 2. Optimal energy profile and the discriminant of (29), D ( ν ) . For readability , we define φ , φ ( ν ) , w , σ 2 w , v , σ 2 v and λ , λ ∗ . Coefficients PSD1: σ 2 in = 1 . 70 b 0 b 1 b 2 b 3 b 4 . 39 0 − . 78 0 . 39 a 0 a 1 a 2 a 3 a 4 1 0 − . 37 0 . 19 Coefficients PSD2: σ 2 in = 2 . 37 · 10 − 5 b 0 b 1 b 2 b 3 b 4 3 0 − 6 0 3 a 0 a 1 a 2 a 3 a 4 1 0 1 . 82 0 0 . 83 T ABLE I A R MA C O E FFIC I E N TS T O G E NE R A T E B OT H P S D S W I TH V A R I AN C E σ 2 s = 1 A N D N = M = 4 . T able II shows the energy savings of the optimum scheme DD-OEP and the suboptimal energy profile DD-SEP with respect to (wrt) the constant ener gy profile DD-CEP for sev eral miss error exponents, and for both PSDs. A remarkable result is obtained when the scheme DD- ON/OFF-EP is used. Comparing with the optimal strategy , DD-OEP , we observe that the differences between the e xpo- nents obtained with both strate gies are ne gligible. This sho ws Energy gap (in dB) wrt CEP for PSD1 κ m SEP OEP , β = 0 . 6 10 − 1 2 2.5 10 − 2 2 5 10 − 3 2 10 Energy gap (in dB) wrt CEP for PSD2 κ m SEP OEP , β = 0 . 2 10 − 1 10.5 11 10 − 2 14 15 10 − 3 >14 >15 T ABLE II E N ER G Y S A V I N G A L LO C A T I N G E N E RG Y I N T H E S E N S OR S . −25 −20 −15 −10 −5 0 5 10 15 20 25 10 −4 10 −3 10 −2 10 −1 10 0 SNR C [dB] Miss Error Exponent CD DD−CEP DD−SEP DD−OEP DD−ON/OFF−EP (a) PSD1, β = 0 . 6 for OEP and ON/OFF . −25 −20 −15 −10 −5 0 5 10 15 20 25 10 −4 10 −3 10 −2 10 −1 10 0 SNR C [dB] Miss Error Exponent CD DD−CEP DD−SEP DD−OEP DD−ON/OFF−EP (b) PSD2, β = 0 . 2 for OEP and ON/OFF . Fig. 4. Error e xponents for CD and DD detectors ag ainst SNR C (dB) for process PSD1 and PSD2. that knowing Θ β ∗ is much more important than knowing the optimal gains { γ n k 0 } . This observation leads to a very simple strategy: at the be ginning of the detection process, the sensors are communicated the optimal set Θ β ∗ (e.g., the FC broadcasts it through a low-rate feedback channel); then, on all channel uses, each node transmits with the same gain using PFS-MAC. In Fig. 5, we plot the optimum fraction of DoF β ∗ con- strained to the allo wed fraction of DoF β against this constraint for processes PSD1 and PSD2. F or lo w values of β , β ∗ increases linearly with slope one ( β ∗ ≤ β ) up to a certain value where β ∗ saturates. The saturation effect is observ ed for dif ferent lev els depending on SNR C and on the frequenc y 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 Constrained β Optimum β SNR C = −10 dB SNR C = 0 dB SNR C = 10 dB (a) PSD1. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 Constrained β Optimum β SNR C = −10 dB SNR C = 0 dB SNR C = 10 dB (b) PSD2. Fig. 5. Optimum constrained fraction of DoF against the constrained fraction of DoF . selectivity of the process, which is related to its correlation. Thus, a strongly correlated process (PSD2) needs relativ ely less channel uses than a weakly correlated process (PSD1). When SNR C is high, saturation occurs for high values of β . Con versely , if SNR C is low , the optimum scheme uses the av ailable energy E t in each sensor to transmit more reliably a reduced set of frequencies where the PSD of the process is high. As a remark, the asymptotically optimal orthogonal strategy PFS-MA C not only allo ws to sa ve a v aluable amount of energy in the sensors but also allows to save channel uses (i.e., bandwidth, detection delay). In Fig. 6, we plot the optimum energy profiles as a function of the normalized frequency for both processes and for se veral v alues of SNR C , when the constraint on the fraction of DoF is inactiv e ( β = 1 ). Note that these figures are closely related to Fig. 5 since the Lebesgue measure of the support of the optimum energy profile is indeed β ∗ . V I I . 2 A N D 3 D I M E N S I O N A L W S N S In this section we enlarge the model to networks with up to 2 dimensions in space and we include the time dimension. The extension to more dimensions is straightforward. The measurement taken by the sensor at coordinate ( i, j ) at time m under each hypothesis is:    H 1 : x ij ( m ) = s ij ( m ) + v ij ( m ) , i = 1 , . . . , n 1 , H 0 : x ij ( m ) = v ij ( m ) , j = 1 , . . . , n 2 , m = 1 , . . . , n 3 . where the parameters n 1 , n 2 , and n 3 are chosen for describing one, two, or three dimensional networks. In these networks, sensors are distributed along a line (1D) or in space (2D), and the y tak e se veral measurements during dif ferent time 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 1 2 3 4 5 Normalized Frequency, ν Energy Profile, ξ (f) SNR C = −10 dB SNR C = 0 dB SNR C = 10 dB (a) PSD1. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 5 10 15 Normalized Frequency, ν Energy Profile, ξ (f) SNR C = −10 dB SNR C = 0 dB SNR C = 10 dB (b) PSD2. Fig. 6. Energy profiles for DD-OEP detector against the normalized frequency for process with PSD1 and PSD2 without DoF constraint ( β = 1 ). Only [0 , 0 . 5] frequency interval is shown due to the ev en symmetry of ξ ( ν ) inherited from φ ( ν ) . instants (time dimension). W e assume that s ij ( m ) is a zero- mean circularly-symmetric complex Gaussian stationary pro- cess with v ariance σ 2 s and po wer spectral density (PSD) φ ( ν ) , where ν ∈ [0 , 1] p is the vector of frequencies normalized to the interval [0 , 1] and p is the dimension of the network. v ij ( m ) is zero-mean circularly-symmetric complex white Gaussian noise independent of s ij ( m ) with v ariance σ 2 v . Therefore, x ij ( m ) is Gaussian distributed under each hypothesis. All sensors transmit synchronously ov er a MA C, and the receiv ed signal at the FC during the k 0 -th channel use is: z n k 0 = n 1 X i =1 n 2 X j =1 g ij k 0 ( { x ij ( m ) } n 3 m =1 ) + w k 0 , k 0 = 1 , . . . , n 0 , (33) where g ij k 0 ( { x ij ( m ) } n 3 m =1 ) is the symbol transmitted by the sensor located at ( i, j ) during channel use k 0 using the encod- ing function g ij k 0 ( · ) , w k 0 is a zero-mean circularly-symmetric complex white Gaussian noise independent of ev erything else with variance σ 2 w , and n 0 is the number of MA C uses. As in the one dimensional case, we will consider linear encoding functions only . T o each 3-tuple index ( i, j, m ) , we associate the following unwound inde x k = ( i − 1) n 2 n 3 + ( j − 1) n 3 + m (34) that takes values k = 1 , . . . , n with n = n 1 n 2 n 3 . This is a one to one mapping, i.e., each 3-tuple index ( i, j, m ) can be recov ered from the index k and vice versa. W e denote this as k ↔ ( i, j, m ) . If we define the follo wing column vectors: s = [ s n 1 , . . . , s n n ] T , v = [ v n 1 , . . . , v n n ] T and x = [ x n 1 , . . . , v n n ] T , the vector of measurements at the FC are expressed as in (4) 10 and hence, its statistic results (5). Ho wever , we assume no w that the cov ariance matrix Σ n is a p -lev el T oeplitz matrix [34, Sec. 6.4] instead of a regular T oeplitz matrix. The precoding matrix for the p -dimensional ( p -D) PCS-MA C strategy has the same expression (6) although the eigenv ectors of the p -lev el T oeplitz matrix Σ n change. Ho wev er , we need to redefine the precoding matrix for the p -D PFS-MA C strategy . Definition 9 ( p -D PFS-MA C) . For each n = n 1 × · · · × n p , denote by ( j 1 , j 2 , . . . , j n ) a permutation of { 1 , 2 , . . . , n } such that φ [ j 1 ] ≥ φ [ j 2 ] ≥ · · · ≥ φ [ j n ] where j k 0 ↔ ( i 0 1 , . . . , i 0 p ) , i 0 l ∈ [1 , n l ] , l = 1 , . . . , p and φ [ j k 0 ] is defined as a sample of the p -D PSD: φ [ j k 0 ] , φ  i 0 1 − 1 n 1 , . . . , i 0 p − 1 n p  with k 0 = 1 , . . . , n 0 . The precoding matrix of the p -D PFS-MAC strategy is C nn 0 = F nn 0 ∆ n 0 , (35) where F nn 0 = [ f n j n 1 , . . . , f n j n n 0 ] is a sub-matrix of the p -D DFT matrix of size n × n , i.e., f n k 0 = [ f n 1 k 0 , f n 2 k 0 , . . . , f n nk 0 ] T with f n kk 0 = 1 √ n exp(  2 π P p l =1 ( i l − 1)( i 0 l,k 0 − 1) n l ) , k = 1 , . . . , n ; k ↔ ( i 1 , . . . , i p ) , and ∆ n 0 = diag ( γ n 1 , . . . , γ n n 0 ) . Similar to the 1D case, the objective of both p -D PCS-MA C and p -D PFS-MAC is to communicate the most important modes (frequencies) of the random process to the FC. Under both strategies, each sensor needs to know the complete vector of time measurements to transmit during each channel use a different p -tuple index ( i 0 1 ,k 0 , . . . , i 0 p,k 0 ) . Once the FC has the n 0 measurements, it builds the statistic (5) to make a decision. W e need now a multi-dimensional version of the T oeplitz theorem. Theorem 5 (T oeplitz Distrib ution for p -D processes) . For a Hermitian p -lev el T oeplitz matrix Σ n generated by the spectral density φ ( ν ) which belongs to the W iener class, with ν = ( ν 1 , . . . , ν p ) and multilevel index n = ( n 1 , . . . , n p ) , n = n 1 × · · · × n p , let { λ n k } n k =1 be the eigen values of Σ n contained on the interv al [ δ 1 , δ 2 ] , let F ( · ) be a continuous function defined on [ δ 1 , δ 2 ] and assume that R ν : φ ( ν )= δ F ( φ ( ν )) d ν = 0 for any δ ∈ [ δ 1 , δ 2 ] ∩ ∆ where ∆ ⊆ [0 , ∞ ) then lim n →∞ 1 n n X k =1 F ( λ n k ∈ ∆) = Z φ − 1 (∆) F ( φ ( ν )) d ν . where n → ∞ means that all components of n tend to infinity simultaneously . Pr oof: See [34, Th. 6.4.1] and [30, Corollary 4.1]. Using Th. 5 and Th. 2 together with the p -D PCS-MA C and p -D PFS-MA C strategies we obtain the same results given in Section V. This is summarized in the following theorem. Theorem 6 (DD Error Exponents for p -D networks) . The error exponents for the p -D PCS-MAC and p -D PFS-MA C strategies are gi ven by Th. 4 and Corollary 2 under the same hypotheses considering no w that the normalized frequency is a p -dimensional variable ν and Θ is a p -dimensional set. Pr oof: Apply Th. 4 using (34). The same energy profiles of subsection V -C are defined if we consider again that the normalized frequency is a p - dimensional variable ν and Θ is a p -dimensional set. V I I I . N U M E R I C A L E X P E R I M E N T F O R 2 D In this section we compare the theoretical results with a Monte Carlo simulation for detecting a 2D correlated process described by the following partial differential equation: ˜ a x ∂ 2 s ( x, y ) ∂ x 2 + ˜ a y ∂ 2 s ( x, y ) ∂ y 2 + ˜ a 0 s ( x, y ) = q ( x, y ) (36) where q ( x, y ) is the random source, assumed to be white and Gaussian with PSD σ 2 q , and ˜ a x , ˜ a y and ˜ a 0 are constants. Using a second order approximation, we discretize (36), ∂ 2 s ( x, y ) ∂ x 2 ≈ s ( x + h, y ) − 2 s ( x, y ) + s ( x − h, y ) h 2 (37) ∂ 2 s ( x, y ) ∂ y 2 ≈ s ( x, y + h ) − 2 s ( x, y ) + s ( x, y − h ) h 2 (38) where h is the discretization step in both directions. The discrete equation results a x ( s i +1 ,j + s i − 1 ,j ) + a y ( s i,j +1 + s i,j − 1 ) + a 0 s i,j = q i,j , i = 1 , . . . , n 1 , j = 1 , . . . , n 2 . with a x = ˜ a x /h 2 , a y = ˜ a y /h 2 and a 0 = ˜ a 0 − 2(˜ a x + ˜ a y ) /h 2 . The PSD of the process is φ ( ω x , ω y ) = σ 2 q ( a 0 + 2 a x cos( ω x ) + 2 a y cos( ω y )) 2 (39) and its variance is σ 2 s . If we organize the samples of the process in a vector as in Section VII, (39) results a linear system: A n s n = q n . In there, s n and q n are vectors of dimension n = n 1 n 2 , and A n is a two-lev el banded T oeplitz matrix of size n × n with the sub-block matrix B n 1 : A n =     B n 1 a x I n 1 a x I n 1 B n 1 . . . . . . . . .     , B n 1 =     a 0 a y a y a 0 . . . . . . . . .     , where only non-zero elements are indicated. Assuming that A n is non-singular , the co variance matrix of the process is Σ n = σ 2 q ( A n A T n ) − 1 (40) which is in general a non-T oeplitz matrix. Howe ver , A n is asymptotically a circulant matrix. The product of two circulant matrices, as well as the in verse of a circulant matrix, are circulant matrices [30, p. 50, 63 and 67]. Circulant matrices are a particular case of T oeplitz matrices. Therefore, Σ n is asymptotically a 2-level T oeplitz matrix and we apply Th. 6 to compute the miss error e xponent subject to any fix ed lev el of false alarm error probability α ∈ (0 , 1) . In Fig. 7 we sho w the estimation of the miss error exponent as a function of the number of sensors in the network using the Monte Carlo method with the following parameters: SNR C = − 10 dB, a 0 = − 5 , a x = a y = 1 , 10 6 experiments and α = 10 − 2 . W e consider that the sensors are placed in a re gular 11 square grid, i.e., n 1 = n 2 and the total amount of sensors in the network is n = n 1 n 2 . The threshold of the test is modified for each n in order to keep the f alse alarm error probability constant. W e also plot the theoretical miss error e xponent. W e observe that both estimated miss error e xponents con verge to their corresponding theoretical v alues. W e also note that both decentralized schemes PCS-MA C and PFS-MAC hav e almost the same performance for the number of sensor considered in the figure, which validates the circulant approximation of the product of T oeplitz matrices in (40). 200 400 600 800 1000 1200 1400 1600 10 −3 10 −2 n, number of sensors Miss Error Exponent CD Theoretical CD Monte Carlo DD Theoretical PCS Monte Carlo PFS Monte Carlo Fig. 7. Miss error probability vs. the amount of sensors in the network. I X . C O N C L U S I O N S W e hav e proposed se veral schemes for distributed detection of circularly-symmetric comple x Gaussian random processes with arbitrary correlation function both in space and time. These schemes tak e into account possible correlated measure- ments and use this correlation beneficially at the FC to b uild an appropriate statistic to make a decision. Considering a multiple access channel and imposing bandwidth and per-sensor energy constraints, we hav e obtained the optimal orthogonal scheme in terms of the miss error exponent. W e hav e also shown that one of the proposed schemes is particularly attractiv e for WSNs with low-cost and energy-limited nodes because it requires only the set of modes to be transmitted, obtains significant energy saving in the lo w signal-to-noise ratio regime, and achieves a close-to-optimal (with negligible loss) performance in terms of miss error exponent. A P P E N D I X I P RO O F O F T H E O R E M 3 : C D E R R O R E X P O N E N T S The asymptotic mean of the centralized LLR statistic de- fined in (3) under the hypothesis H i , i = 0 , 1 is m CD i = lim n →∞ E i ( T n c ) = lim n →∞ 1 n  tr  Σ i,n  Σ − 1 0 ,n − Σ − 1 1 ,n  − log det Σ 1 ,n det Σ 0 ,n  . Considering that the eigen v alues of Σ 0 ,n and Σ 1 ,n are { σ 2 v } and { λ n k + σ 2 v } , respectively , and defining ρ n CD ,k = λ n k σ 2 v , we obtain m CD 0 = lim n →∞ 1 n P n k =1 1 − 1 1+ ρ n CD ,k − log (1 + ρ n CD ,k ) . Using Th. 1 with ∆ = [ δ 1 , δ 2 ] , which implies φ − 1 (∆) = [0 , 1] , we finally have m CD 0 = m 0 (Γ CD ) defined (19). T o compute the error exponents we first need to v erify the assumptions of Th. 2 with h 0 , h 0 ( ν ) = σ 2 v ∀ ν and h 1 , h 1 ( ν ) = φ ( ν ) + σ 2 v : A 1 ) h 0 is a positi ve constant and log h 0 belongs to L 1 ([0 , 1]) trivially . log h 1 ∈ L 1 ([0 , 1]) is easily proved by noting that ess inf ( φ ( ν ) + σ 2 v ) > 0 and ess sup( φ ( ν ) + σ 2 v ) < ∞ because φ ( ν ) is a power spectral density in the W iener class, and therefore, it is essentially bounded. A 2 ) h 0 /h 1 = 1 / (1 + Γ CD ) ∈ L ∞ ([0 , 1]) gi ven that ess inf Γ CD ≥ 0 because Γ CD is a spectral density . h 1 /h 0 = 1 + Γ CD ∈ L ∞ ([0 , 1]) is easily proved by considering again that φ ( ν ) belongs to the W iener class. Giv en that ( A 1 ) and ( A 2 ) are satisfied, the error exponents are obtained through the Fenchel-Legendre transform Λ ∗ CD ,i ( x ) of the LMGF Λ CD ,i ( t ) = log E i  e tT n c  , i = 0 , 1 . The follo wing properties can be verified: Properties 1 (LMGF and its Fenchel Le gendre T ransform) . P 1 ) Λ CD , 1 ( − 1) = Λ CD , 1 (0) = 0 . P 2 ) Λ 0 CD , 1 ( − 1) = m CD 0 and Λ 0 CD , 1 (0) = m CD 1 . P 3 ) Λ CD , 1 ( t ) is a con vex function and then Λ CD , 1 ( t ) = Λ CD , 1 ((1 + t )0 − t ( − 1)) ≤ (1 + t )Λ CD , 1 (0) − t Λ CD , 1 ( − 1) = 0 ∀ t ∈ [ − 1 , 0] . P 4 ) As the Fenchel-Le gendre transform giv en in (18) trans- forms con ve x functions into con vex functions, Λ ∗ CD , 1 ( x ) is a con vex function. Moreover , Λ ∗ CD , 1 ( x ) has a minimum of value 0 at x = m CD 1 meaning that Λ ∗ CD , 1 ( x ) is decreasing if x < m CD 1 . P 5 ) Λ CD , 0 ( t ) = Λ CD , 1 ( t − 1) and Λ ∗ CD , 0 ( x ) = Λ ∗ CD , 1 ( x ) + x . Therefore, Λ ∗ CD , 0 ( x ) is also a con vex function with a minimum of value 0 at x = m CD 0 and it is an increasing function if x > m CD 0 . Using properties ( P 1 ) - ( P 5 ) and (16) when G = [ τ , ∞ ) , k CD f a = Λ ∗ CD , 0 ( τ ) if τ > m CD 0 and k CD f a = 0 if τ ≥ m CD 0 . Moreov er, when G = ( −∞ , τ ] , k CD m = Λ ∗ CD , 1 ( τ ) if τ < m CD 1 and k CD m = 0 if τ ≥ m CD 1 . The usual interval of interest for the threshold τ is [ m CD 0 , m CD 1 ] and because ( P 1 ) and ( P 2 ) hold, the interv al of optimization of t in (18) can be restricted to [ − 1 , 0] . Due to the conv exity of Λ CD , 1 ( t ) there exists a unique t ∗ that solv es (18) and satisfies (21), which is obtained from deriving τ t − Λ CD , 1 ( t ) . Then, the error exponents are (22) and (23). A P P E N D I X I I D E C E N T R A L I Z E D E R R O R E X P O N E N T S A. Pr oof of Lemma 2: Decentr alized Hypothesis T esting W e first consider the PCS-MAC scheme gi ven in Def. 2. The cov ariance matrices under H 0 and H 1 are, respectiv ely , Ξ 0 ,n 0 = σ 2 v ∆ 2 n 0 + σ 2 w I n 0 , Ξ 1 ,n 0 = ( diag ( λ n 1 , . . . , λ n n 0 ) + σ 2 v I n 0 )∆ 2 n 0 + σ 2 w I n 0 . (41) Let θ i,k be the k -th element of the diagonal matrix Ξ i,n 0 , i = 0 , 1 . It is easy to prove that diag ( θ i, 1 , . . . , θ i,n 0 ) ∼ diag ( h i ( 0 n ) , . . . , h i ( n 0 − 1 n )) . T o recov er the original dimen- sion of the problem, define the n -dimensional vector ˜ z as 12 the zero padding of the n 0 -dimensional vector of measure- ments z , i.e., ˜ z = [ z T 0 T ] T . The cov ariance matrix of ˜ z , diag ( θ i, 1 , . . . , θ i,n 0 , 0 , . . . , 0) , is asymptotically equiv alent to diag ( h i ( 0 n ) , . . . , h i ( n 0 − 1 n ) , 0 , . . . , 0) . Consider the follo wing transformation y = F n ˜ z , where F n is the DFT matrix of size n × n . It is well known that applying an inv ertible transforma- tion (in particular, the orthogonal DFT matrix) to the data does not modify the performance of the statistic. Because matrix multiplication preserves asymptotic equiv alence of matrices [30, Th. 2.1 (3)], we ha ve that the cov ariance matrix of y is asymptotically circulant, i.e., F n diag ( θ i, 1 , . . . , θ i,n 0 , 0 , . . . , 0) F H n ∼ F n diag ( h i (0 /n ) , . . . , h i (( n 0 − 1) /n ) , 0 , . . . , 0) F H n = B n ( ˜ h i ) , where B n ( ˜ h i ) is the circulant matrix generated by the samples of ˜ h i ( ν ) , i = 0 , 1 and ˜ h i ( ν ) =  h i ( ν ) if ν ∈ Θ β 0 if ν ∈ [0 , 1] \ Θ β . (42) Therefore, ˜ h i ( ν ) may be interpreted as the PSD of the mea- surements av ailable at the FC under H i , i = 0 , 1 and the test (25) can be formulated. In the case of PFS-MAC, the covariance matrices under H 0 and H 1 are, respectiv ely , Ξ 0 ,n 0 = σ 2 v ∆ 2 n 0 + σ 2 w I n 0 , Ξ 1 ,n 0 = ∆ n 0 ( F H nn 0 Σ n F nn 0 + σ 2 v I n 0 )∆ n 0 + σ 2 w I n 0 . (43) In this case, F H nn 0 Σ n F nn 0 is not diagonal for finite n 0 . Ho w- ev er , considering that asymptotic equiv alence between matri- ces is preserved by matrix multiplication [30, Th. 2.1 (3)], and using Lem. 1 we have that Σ n ∼ F nn 0 diag ( λ n 1 , . . . , λ n n 0 ) F H nn 0 if and only if F H nn 0 Σ n F nn 0 ∼ diag ( λ n 1 , . . . , λ n n 0 ) , (44) which makes (41) and (43) asymptotically equi valent, obtain- ing the same test of hypothesis (25). B. Pr oof of Theorem 4: DD Err or Exponents The asymptotic mean of the decentralized LLR statistic defined in (5) under H i , m DD i = lim n →∞ E i ( T n d ) , i = 0 , 1 , depends entirely on the spectral density under H i . A similar situation occurs for the centralized detector in Th. 3. Then, using Lem. 2, we hav e that m DD i = m i (Γ DD ) for both strategies PCS-MAC and PFS-MAC, where m i ( · ) , i = 0 , 1 , are defined in (19) and (20), respecti vely . T o compute the error exponents we first need to verify the assumptions of the modified version of the Gärdner-Ellis theorem with h 0 , h 0 ( ν ) = σ 2 v ξ ( ν ) + σ 2 w and h 1 , h 1 ( ν ) = ( φ ( ν ) + σ 2 v ) ξ ( ν ) + σ 2 w : A 1 ) R Θ β | log ( ξ ( ν ) σ 2 v + σ 2 w ) | dν ≤ R Θ β | log (1 + ξ ( ν ) σ 2 v σ 2 w ) | dν + | log ( σ 2 w ) | ≤ 1 + σ 2 v σ 2 w R Θ β ξ ( ν ) dν + | log( σ 2 w ) | < ∞ because ξ ( ν ) ≥ 0 , the energy constraint by (10) and | log ( x ) | < | x | if x ≥ 1 . Then, log h 0 ∈ L 1 ([0 , 1]) . h 1 ∈ L 1 ([0 , 1]) is proved similarly by additionally considering that φ ( ν ) is a po wer spectral density in the W iener class. A 2 ) h 0 /h 1 = 1 / (1 + Γ DD ) ∈ L ∞ ([0 , 1]) gi ven that ess inf Γ DD ≥ 0 since ξ ( ν ) ≥ 0 and ess inf φ ( ν ) ≥ 0 . h 1 /h 0 = 1 + Γ DD ∈ L ∞ ([0 , 1]) is easily prov ed by considering again that φ ( ν ) belongs to the W iener class and therefore ess sup φ ( ν ) = M φ is a finite constant. Then, ess sup Γ DD ≤ M φ /σ 2 v . Now , we apply Th. 2 to obtain the error exponents by comput- ing the Fenchel-Legendre transforms Λ ∗ DD ,i ( x ) of the LMGF Λ DD ,i ( t ) = log E i  e tT n d  , i = 0 , 1 . The same properties (P1)-(P5) in Prop. 1 are satisfied by Λ DD ,i ( t ) and Λ ∗ DD ,i ( x ) considering m DD i instead of m CD i , for i = 0 , 1 . Therefore, the error exponents are (26) and (27). C. DD Optimum Energy Pr ofile for the Miss Err or Exponent If the false alarm probability constraint is P n f a ≤ α < 1 , the miss error exponent is given by Th. 4 with τ = m 0 (Γ DD ) +  , where  > 0 arbitrary small and the constraint over P n f a is satisfied for n lar ge enough. See [33, Prop. 2] for a detailed proof. This case allows to find the optimality of the PCS-MAC and PFS-MA C strate gies among all orthogonal strategies for a fixed energy profile (see [12, Th. 2]) and then, to obtain a closed form solution for the optimal energy profile ξ ( ν ) . In this respect, we use v ariational calculus to solve the problem: sup κ m ( ξ ) s.t. ξ ( ν ) ≥ 0 ∀ ν ∈ Θ β , (45) Z Θ β ξ ( ν ) dν ≤ c (46) where c = E t / ( σ 2 s + σ 2 v ) . Because the error exponent κ m ( ξ ) is an increasing function of ξ (easily checked by computing the fist deriv ativ e of its integrand), it is both a quasicon vex and quasiconcav e (and thus quasilinear) function. The domain of the functional κ m ( ξ ) is all nonne gativ e functions, a con vex set. The constraints of the problem are affine functions of ξ . Therefore, we have a quasicon vex optimization problem where the solution is not unique. The Lagrangian is L ( ξ , λ, ν ) = Z Θ β {− I ( ξ ( ν )) − m ( ν ) ξ ( ν ) + λ ( ξ ( ν ) − c/β ) } dν where I ( ξ ( ν )) = ξ ( ν ) σ 2 v + σ 2 w ξ ( ν )( φ ( ν )+ σ 2 v )+ σ 2 w + log ξ ( ν )( φ ( ν )+ σ 2 v )+ σ 2 w ξ ( ν ) σ 2 v + σ 2 w − 1 is the inte grand of κ m ( ξ ) . The scalar function m ( ν ) and the scalar constant λ are the multipliers of Lagrange. The constraints (45) and (46) together with the following constraints are the Karush-K uhn-T ucker necessary conditions for local extremes: C 3 ) λ ∗ − I 0 ( ξ ∗ ( ν )) ≥ 0 , ∀ ν ∈ Θ β , C 4 ) [ λ ∗ − I 0 ( ξ ∗ ( ν ))] ξ ∗ ( ν ) = 0 , ∀ ν ∈ Θ β , C 5 ) λ ∗ ≥ 0 , C 6 ) λ ∗ R Θ β ( ξ ∗ ( ν ) − c/β ) dν = 0 . The Euler-Lagrange equation together with the complementar- ity condition of m ( ν ) and the solution to the problem ξ ∗ ( ν ) , ξ OEP ( ν ) give the constraint ( C 4 ) . The non-negati veness of m ( ν ) and λ ∗ produce ( C 3 ) and ( C 5 ) , respecti vely . Finally , ( C 6 ) is due to the complementarity of λ ∗ and (46). 13 ξ ( ν ) = 0 , ∀ ν and λ = 0 satisfies all constraints but it is not the desired solution because the error exponent is 0. Then, for a non-tri vial solution λ ∗ > 0 . It can be shown that the solutions to I 0 ( ˆ ξ ) = λ ∗ correspond to the cubic equation (29) with coefficients (30). Descartes’ rule of signs of a polynomial establishes that then number of positi ve roots of a polynomial is related with the number of sign changes of the nonzero coefficients of consecutive po wers. In the case of (30) and considering ( C 5 ) , Descartes’ rule determines that there could be at most 2 or 0 positive roots. Then, there exists a root, with a non-positiv e value, that does not satisfy (45) and it is discarded. Define I i = { ν ∈ [0 , 1] : ˆ ξ i ( ν ) > 0 } , i = 1 , 2 as the sets of frequencies corresponding to the positi ve roots. Because of Descartes’ rule, I 1 ∩ I 2 = I 1 = I 2 = Θ β ∗ ⊆ Θ β . W e hav e two cases. 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