Distributed Detection of Correlated Random Processes ... - IEEE Xplore

2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM)

Distributed Detection of Correlated Random Processes under Energy and Bandwidth Constraints Juan Augusto Maya† , Leonardo Rey Vega†‡ , Cecilia G. Galarza†‡ and Andrés Altieri† University of Buenos Aires† , CSC-CONICET‡ , Buenos Aires, Argentina Emails: {jmaya, lrey, cgalar, aaltieri}@fi.uba.ar

Abstract—We analyze a binary hypothesis testing problem built on a wireless sensor network (WSN). Using Large Deviation Theory (LDT), we compute the probability error exponents of a distributed scheme for detecting a correlated circularlysymmetric complex Gaussian process under the Neyman-Pearson framework. Using an analog scheme, the sensors transmit scaled versions of their measurements several times through a multiple access channel (MAC) to reach the fusion center (FC), whose task is to decide whether the process is present or not. In the analysis, we consider the energy constraint on each node transmission. We show that the proposed distributed scheme requires relatively few MAC channel uses to achieve the centralized error exponents when detecting correlated Gaussian processes.

I.

I NTRODUCTION

Distributed detection on WSN is a topic which has attracted a lot of interest in recent years [1], [2]. Design constraints such as low cost and long life cycle call for building a clever strategy to use the resources available on the network. In this regard, the system performance may be optimized to comply with constraints on the resources the nodes consume individually (energy and bandwidth mainly). In particular, we will consider a distributed detection scenario where the sensors measure locally a realization of a stochastic process and make some processing. Then, the result is transmitted to the FC that performs some further data processing and the final decision is made. The traditional approach is to consider that sensors communicate with FC using orthogonal channels as in the case of, for example, time division multiple access (TDMA) or frequency division multiple access (FDMA) [2], [3]. 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 will be necessary if sensors use the same bandwidth and transmit on different time slots. Alternatively, we can use a multiple access channel (MAC) where sensors transmit simultaneously sharing the bandwidth. In this case, the bandwidth requirement does not depend on the number of sensors [4]. However, this simple scheme is not suitable for detection of a correlated process unless certain cooperation is allowed among sensors. This is because the transmission through the MAC channel does not readily permit to construct a suitable quadratic statistic of the data. In [5] a scheme is presented that works well with low correlation processes but it is restricted to autoregressive processes of order 1 and it has low performance for highly correlated processes. This work was partially supported by the Peruilh grant of the UBA and project UBACYT 2002010200250.

978-1-4799-1481-4/14/$31.00 ©2014 IEEE

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In this work, we propose a scheme that could be seen to be in the middle of the previous ones mentioned. After the sensors measure their own local realization of the process, they transmit synchronously several times through a MAC channel. In each MAC channel use, the sensors transmit their local measurements scaled by different gains. Finally, the FC constructs an appropriate quadratic statistic with all the data received through several MAC channel uses and makes a decision. We assume that the sensors have limited energy budget, and we carefully select the gains for each channel utilization in order to comply with this budget. For that, we propose two distributed schemes that differ on how they allocate their available energy through the channel uses to communicate their measurements to the FC. We analyze the asymptotic performance (when the number of sensors approach to infinity) of both schemes. The paper is organized as follows. In Section II, we formulate the binary hypothesis testing problem. In Section III, we present first the centralized detector and then develop the distributed detector. In Section IV, we enunciate the main theorems used to compute the error exponents in Section V. In Section VI, we show numerical results and in Section VII we elaborate the main conclusions. II.

D ETECTION P ROBLEM

We consider a binary hypothesis testing problem where each one of the n sensors obtains one of the following measurements under both hypothesis:  H1 : xk = s k + v k , (1) H0 : xk = v k , k = 0, . . . , n − 1, where sk is a zero-mean circularly-symmetric complex Gaussian process with power spectral density (PSD) S(f ), and vk is zero-mean circularly-symmetric complex white Gaussian noise independent of sk with variance σv2 . Thus, xk is Gaussian distributed under both hypothesis. We define the following column vectors: sn = [s0 , . . . , sn−1 ]T , v n = [v0 , . . . , vn−1 ]T and xn = [x0 , . . . , xn−1 ]T where (·)T means transpose. The covariance matrices of sn and v n are Σs,n and σv2 In respectively, where Σs,n is assumed to be a Toeplitz matrix and In is the identity matrix of size n×n. The covariance matrices of xn under H0 is Σ0,n = σv2 In , and Σ1,n = Σs,n + σv2 In under H1 . The MAC channel is assumed to be defined by: zj,n =

n X

k=1

yj,k + wj , j = 0, . . . , m − 1,

2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM)

where zj,n is the received symbol at channel use j, yj,k is the symbol transmitted by sensor k at channel use j and wj is zero-mean circularly-symmetric complex white Gaussian noise 2 independent of everything else with variance σw . III.

D ETECTORS

In this section we first present the centralized detector and then a decentralized detector to perform the Neyman-Pearson test in the FC. A. Centralized detector Suppose that the FC has direct access to the complete measured vector xn . We will call this detector the centralized detector (CD). This will be the case when noiseless communication channels are used. Clearly, this ideal detector will provide an upper bound (not necessarily tight) on the performance of any decentralized scenario. The logarithmic likelihood ratio (LLR) of the centralized detector is  
1 |Σ1,n | −1 x − log xTn Σ−1 , (2) − Σ Tc(n) (xn ) = n 0,n 1,n n |Σ0,n | where Σ0,n and Σ1,n are the correlation matrices of the measurements under H0 and H1 respectively. The optimum (n) centralized test is to choose H1 if Tc (xn ) > τ , and choose H0 otherwise, where τ is the pre-established threshold under the Neyman-Pearson framework. The complex-valued Hermitian and non-negative definite covariance matrix Σs,n can be decomposed as Σs,n = Vn Dn VnH , where (·)H means transpose conjugate, Vn is a unitary matrix that has the eigenvectors of Σs,n in its columns, and Dn is a diagonal matrix whose elements λk,n are the (real an non-negative) eigenvalues of Σs,n . Using the transformation X n = VnH xn , we obtain the vector X n whose components Xk,n are uncorrelated and independent under both hypothesis due to the Gaussian assumption. Then, (2) can be expressed equivalently as follows: n−1 1 X ρcj,n |Xj,n |2 − log(1 + ρcj,n ), Tc(n) (xn ) = n j=0 1 + ρcj,n σv2

(3)

with ρcj,n = λj,n /σv2 . B. Distributed detector under energy and bandwidth constraints We propose a distributed detector (DD) scheme where the nodes make several uses of the MAC channel. Let the available degrees of freedom (DoF) of the communication channel be m, i.e., m is the amount of channel uses of the MAC channel. Given a vector xn , the sensors transmit m times (m ≤ n) through the MAC channel different scaled versions of their local measurement. At channel use j, zj,n =

αTj,n xn

T

+ wj , αj,n = [αj,1 αj,2 . . . αj,n ] ,

with 0 ≤ jP ≤ m − 1. In each sensor the energy is constrained m−1 to satisfy j=0 |αj,k |2 (σv2 + σs2 ) ≤ Et ∀k = 0, . . . , n − 1. The selection of vectors αj,n should be done for maximizing the detection performance subject to the energy constraint on each sensor.

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In this work we restrict ourselves to the case where αj,n are orthogonal vectors. Then, αj,n = γjn vjn , where vjn is the j-th eigenvector of Σs,n and γkn is a non-negative number. the energy constraint can be written as Pm−1 nHence, 2 n 2 2 2 |γ | |v | j k,j (σv + σs ) ≤ Et ∀k = 0, . . . , n − 1. We j=0 can exploit that the correlation matrices under both hypothesis are Toeplitz. Toeplitz matrices are asymptotically equivalent to circulant matrices, which have the interesting property of being diagonalized by the discrete Fourier transform (DFT) matrix. Therefore, we have no asymptotic loss assuming that ′ ′ vjn can be replaced by √1n [1 e2πj /n . . . e2πj (n−1)/n ] for some j ′ ∈ [0, n − 1] [6], [7]. The signal received at the FC at channel use j is then, X γjn n−1 xl e2πlj/n + wj , zj,n = √ n

j ∈ Jβ,n ,

l=0

(4)

where Jβ,n = {j ∈ [0, n−1] : S(j/n) ≥ δn (β)} with δn (β) ≥ 0 is a level such that the cardinal of Jβ,n , |Jβ,n | = m and β = limn→∞ m n (0 < β ≤ 1) is defined as the asymptotic fraction of available DoF’s of the communication Pm−1 channel per sensor. The energy constraint becomes n1 j=0 |γjn |2 (σv2 + σs2 ) ≤ Et ∀k = 0, . . . , n − 1. Finally, the FC can build the LLR of the decentralized detector which is asymptotically equivalent to the following expression when n → ∞ (n)

ρdj,n |yj,n |2 1 X − log(1 + ρdj,n ), d |γ n |2 σ 2 + σ 2 n 1 + ρ v w j j,n j∈Jβ,n (5) 2 ). = (|γjn |2 S(j/n))/(|γjn |2 σv2 + σw

Td (xn ) = where ρdj,n

IV.

P RELIMINARY T OOLS

In this section, we introduce the main tools to compute the error exponents. First, a variant of the Toeplitz distribution theorem is presented and then, the Gärtner-Ellis theorem is enunciated. Theorem 1 (Toeplitz Distribution [6], [7]): For an absolutely summable Toeplitz matrix Σn with spectral n−1 density S(ω), let {λk,n }k=0 be the eigenvalues of Σn contained on the interval [δ1 , δ2 ], let F (·)R be a continuous function defined on [δ1 , δ2 ] and assume that f :S(f )=δ F (S(f ))df = 0, then Z 1 X F (S(f ))df. F (λk,n ) = lim n→∞ n f :S(f )≥δ k:λk,n ≥δ

Theorem 2 (Gärtner-Ellis [8]): Let {T (n) } be a sequence of real random variables drawn according to the probability laws {Pn }, and define Λ(n) (t) = log E[etT

(n)

].

(6)

Assumptions: (1) For each t ∈ R, the logarithmic momentgenerating function (LMGF), defined as the limit Λ(t) = lim 1 Λ(n) (nt) exists as an extended real number. (2) The n→∞ n o interior of DΛ = {t ∈ R : Λ(t) < ∞}, denoted by DΛ , o contains the origin. (3) Λ(·) is differentiable throughout DΛ , and (4) Λ(·) is steep, i.e., limn→∞ Λ′ (tn ) = ∞ whenever o {tn } is a sequence in DΛ converging to a boundary point of o DΛ . Under the above assumptions, the large deviation principle

2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM)

(LDP) satisfied by the sequence {Pn } can be characterized by the Fenchel-Legendre transform of Λ(t): (7)

t∈R

¯ are the interior and closure of a set That is, if Go and G G ⊂ R, respectively, we say that {T (n) } satisfies the LDP with rate function Λ∗ (x) if, for any G ⊂ R we have − inf o Λ∗ (x) ≤ lim inf

n→∞

x∈G

1 log P(yn ∈ G) n

1 log P(yn ∈ G) ≤ − inf Λ∗ (x). ¯ n x∈G n→∞

≤ lim sup

(8)

In most hypothesis testing problems, and in our particular case, G satisfies the ∆-continuous property [8] where ǫG is the error exponent, therefore ǫG , inf x∈Go Λ∗ (x) = inf x∈G¯ Λ∗ (x). V.

E RROR E XPONENTS

In this section we compute the error exponents for false (n) (n) alarm and miss probabilities (Pf a and Pm ) for both centralized and decentralized detectors presented in Section III. The computation is performed using Th. 2. For that, we obtain the LMGF of the statistics (3) and (5). Notice that these detectors have similar structures because they are both constructed from complex Gaussian processes. Hence, the development of the expressions for the error exponents will follow the same steps in both cases. For simplicity, we will use T (n) to denote either (n) (n) Tc or Td . Since the detectors are LLR’s, the LMGF’s satisfy the following properties [9]: i) Λ1 (t) = Λ0 (t + 1), ii) Λ∗1 (x) = Λ∗0 (x) − x., The LMGF of T (n) under H0 is as follows: 1 1 (n) Λ (nt) = log E0 [entT ] n 0 n 1 X {log(1 + (1 − t)ρj,n ) =− n (n)

j∈Jn

−(1 − t) log(1 + ρj,n )} , t < 1 +

When ξ(f ) is chosen according to CEP, all sensors transmit their measurements using the same gain for all channel uses. In this case, we assume that the FC broadcasts through a feedback channel the set of frequencies to be used by the sensors. However, in a scenario without a feedback channel, this set could be fixed before the detection process. On the other hand, when ξ(f ) follows SEP, more energy is allocated to the frequencies where the process under H1 concentrates more power. In this case, the FC broadcasts the frequency and the common gain that sensors would use on the following MAC channel use.

The error exponents for miss and false alarm, km and kf a , are obtained through a numerical evaluation of (7). VI.

N UMERICAL R ESULTS

In this section we compute the error exponents for two complex Gaussian correlated auto-regressive moving average (ARMA) processes with PSD given by P M b e−2πf k 2 2 k=0 k S(f ) = σin PN ak e−2πf k k=0

1 ρ∗n

where E0 [·] is the expectation under H0 , Jn = {0, . . . , n − 1} for the CD and Jn = Jβ,n for the DD, and ρ∗n = maxj ρj,n . Applying Th.1, we obtain Z Λ0 (t) =− {log(1  + (1 − t)Γ(f ))−(1−t) log(1 + Γ(f ))}df, Ω 2 for CD   S(f )/σv Γ(f ) = , (9) ξ(f )S(f )  for DD,  2 ξ(f )σv2 + σw [− 21 , 21 ]

for CD, and Ω = Ωβ for DD. The interval and Ω = Ωβ is defined by Ωβ = {f ∈ [− 21 , 12 ] : Γ(f ) ≥ δ(β)} such that its Lebesgue measure is β. ξ(f ) is the common energy profile (EP) used by each sensor to transmit along the MAC channel and it is the limiting distribution of the gains |γkn |2 when n → ∞. An optimal choice for ξ(f ) could be obtained by formulating a Neyman-Pearson problem with an additional constraint on the maximum total energy spent per sensor, i.e., Z ξ(f )df ≤ Et /(σv2 + σs2 ). (10) Ωβ

135

where M and N are the degrees of the numerator and denom2 inator polynomials, respectively, and σin is selected such that the variance of the process is σs2 . The processes are plotted in Fig. 1 with coefficients shown in Table I. Let SNRM = σs2 /σv2 20 PSD (dB)

Λ∗ (x) = sup{xt − Λ(t)}.

In this paper, we will not solve the optimization problem due to lack of space. Instead, in order to analyze the interaction among ξ(f ), β, and the error exponents, we propose two different energy profiles: i) Constant EP (CEP) where ξ(f ) is constant over the set Ωβ ; ii) Spectral EP (SEP) where the energy profile reproduces the shape of the S(f ) in the set Ωβ . The energy profiles are defined as  Et  for f ∈ Ωβ 2 (σv + σs2 )β ξCEP (f ) = (11)  0 otherwise,  E S(f ) t   R for f ∈ Ωβ (σv2 + σs2 ) Ωβ S(f )df ξSEP (f ) = (12)   0 otherwise.

PSD1 PSD2

10 0 −10 −20

0

Figure 1.

0.05

0.1

0.15

0.2 0.25 0.3 Normalized frequency, f

0.35

0.4

0.45

0.5

PSD’s (in dB) with variance σs2 = 1.

2 and SNRC = Et /(βσw ) be the measurement and the commu2 nication signal-to-noise ratios, respectively and η = σv2 /σw the ratio of variance noises. In Fig. 2 we show the behavior of the error exponents against SNRC when both exponents are equal, SNRM = 5 dB and β is as indicated in the corresponding subfigure. When the process to be detected has relatively low correlation (Fig. 2(a)), both DD schemes approach CD scheme

b1 0

a0 1

a1 0

Table I.

b2 −.78 a2 −.37

b3 0

b4 .39

a3 0

a4 .19

b0 3

b1 0

a0 1

a1 0

b2 −6

a2 1.82

b3 0

b4 3

a3 0

a4 0.83

fa

b0 .39

2 Coefficients PSD2: σin = 2.367 · 10−5

Error Exponents k =k

2 Coefficients PSD1: σin = 1.702

m

2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM)

ARMA COEFFICIENTS TO GENERATE BOTH PSD’ S WITH 2 = 1 AND N = M = 4. VARIANCE σs

−1

10

CD DD−CEP DD−SEP

−2

10

−3

10

0

0.1

0.2

0.3

0.4 0.5 0.6 DoF per sensor, β

0.7

0.8

0.9

1

−1

10

CD DD−CEP DD−SEP

fa

Error Exponents k =k

m

(a) PSD1,η = −10dB.

for SNRC ≃ 10 dB. We see that the difference between allocating power uniformly (CEP) or following the process PSD (SEP) is negligible due to the flatness of the spectrum. However, if we consider processes with moderate or strong correlation (Fig. 2(b)), power allocation through the different channel uses becomes important for low SNRC .

−2

10

Error Exponents kfa=km

0 −1

0.2

0.3

0.4 0.5 0.6 DoF per sensor

0.7

0.8

0.9

1

(b) PSD2,η = −10dB.

10

−2

10

−3 −4

10

−20

Figure 3. Error exponents for CD and DD detectors against the DoF of the communication channel for process with PSD1 and PSD2.

CD DD−CEP DD−SEP

10

−15

−10

−5

0 5 SNRC (dB)

10

15

20

25

where the process presents higher power.

(a) PSD1,β = 0.8. Error Exponents kfa=km

0.1

0

10

VII.

0

C ONCLUSIONS

10

(b) PSD2,β = 0.3.

We have analyzed and computed the false alarm and miss probability exponents for a distributed scheme under the Neyman-Pearson framework where a WSN is used to detect whether a circularly-symmetric complex Gaussian process is present or not. The proposed distributed scheme with spectral energy profile SEP gives a near-optimal statistic for correlated Gaussian processes even with low fraction of DoF per sensor and a fair SNRC .

Figure 2. Error exponents for CD and DD detectors against η (dB) for process with PSD1 and PSD2.

R EFERENCES

−1

10

−2

10

−3

CD DD−CEP DD−SEP

10

−4

10

−20

−15

−10

−5

0 5 SNR (dB)

10

15

20

25

C

[1]

In Fig. 3 we see that for both process PSD1 and PSD2 only a small fraction of DoF of the communication channel is necessary to reach the peak of performance. Thus, the WSN can save bandwidth or reduce the detection delay compared to a parallel scheme without loosing performance with respect to the CD bound for a fair communication channel with SNRC about 10 dB (5 dB) for PSD1 (PSD2). It is worth to analyze the behavior of both energy profiles, for a high correlated processes as in Fig. 3(b), and low SNRC . When the limited transmitted energy Et is concentrated in a very small fraction of DoF (β < 0.05), the small amount of samples available at the FC lowers the exponents. In fact, when only one carrier is transmitted, the statistic follows an exponential distribution and the error exponents are zero despite of the transmitted energy is allocated entirely to this carrier. If β is increased up to 0.05, SNRC decreases inversely proportional to β, but the exponents increase because the statistic is built with more samples in the FC and, hence, more realizations of the communication noise are available. This decreases the tails of the distribution of (n) Td rapidly. However, in the case of DD-CEP if we continue increasing β (β > 0.1) and we allocate energy to less relevant measurements, this effect dominates the behavior and it lowers the error exponents. Thus, a trade-off is established between increasing DoF’s and allocating more energy to the frequencies

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[2]

[3]

[4]

[5]

[6] [7] [8] [9]

R. Viswanathan and P. K. Varshney, “Distributed detection with multiple sensors: Part I: Fundamentals,” in Proc. IEEE, vol. 85, Jan. 1997, pp. 54–63. J.-F. Chamberland and V. Veeravalli, “How dense should a sensor network be for detection with correlated observations?” Information Theory, IEEE Transactions on, vol. 52, no. 11, pp. 5099–5106, Nov 2006. P. Bianchi, J. Jakubowicz, and F. Roueff, “Linear precoders for the detection of a gaussian process in wireless sensors networks,” Signal Processing, IEEE Transactions on, vol. 59, no. 3, pp. 882–894, March 2011. J. Maya, L. Rey Vega, and C. Galarza, “Error exponents for bias detection of a correlated process over a mac fading channel,” in Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2013 IEEE 5th International Workshop on, Dec 2013, pp. 484–487. W. Li and H. Dai, “Distributed Detection in Wireless Sensor Networks Using A Multiple Access Channel,” IEEE Trans. Signal Process., vol. 55, no. 3, pp. 822–833, 2007. U. Grenander and G. Szego, Toeplitz Forms and Their Applications. University of Calif. Press, Berkeley and Los Angeles, 1958. R. M. Gray, “Toeplitz and circulant matrices: A review,” Tech. Rep., 2001. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed. New York: Springer, 1998. B. C. Levy, Principles of signal detection and parameter estimation. Springer, 2008.