# Water-Filling Solution for Distributed Estimation of Correlated Data in WSN MIMO System

Authors: Ajib Setyo Arifin; Tomoaki Ohtsuki
Journal: ISRN Sensor Networks
 (2013)

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## Abstract

We consider the distributed estimation of a random vector signal in a power constraint wireless sensor network (WSN) that follows a multiple-input and multiple-output (MIMO) coherent multiple access channel model. We design linear coding matrices based on linear minimum mean-square error (LMMSE) fusion rule that accommodates spatial correlated data. We obtain a closed-form solution that follows a water-filling strategy. We also derive a lower bound to this model. Simulation results show that when the data is more correlated, the distortion in terms of mean-square error (MSE) degrades. By taking into account the effects of correlation, observation, and channel matrices, the proposed method performs better than equal power method.


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## Body

## 1. Introduction

The wireless sensor network (WSN) is a potential technology in many application areas including environmental monitoring, health, security and surveillance, and robotic exploration [1]. WSNs have many interested issues, one of them is distributed estimation. In the distributed estimation scenario, sensors observe phenomena from the target(s) and transmit to a fusion center (FC). Received signal at FC is estimated using an estimation technique.Distributed estimation by considering power consumption has attracted much attention in [2–7]. Because the sensors deployed in a certain region are difficult to change the batteries, the low power consumption is important to guarantee a lifetime of the sensors. The distributed estimation is also applied on the orthogonal multiple access channel (MAC) model [4, 6, 8] and the coherent MAC model [2, 5] that considers single-input single-output (SISO). To save energy, multiple-input and multiple-output (MIMO) system has been analyzed by involving signaling overhead [9], in which the MIMO system can offer substantial energy savings in WSN. Cooperative MIMO with data aggregation also has been investigated in [10, 11].In practical scenario, if the targets or the sensors are close to each other, the data will be potentially correlated. Such a problem has been investigated in [3, 6, 12, 13]. In this paper, we consider multiple targets that are spatially distributed. The targets are observed by multiple sensors that apply an analog forwarding scheme. This scheme will multiply the observed data with a designed coding matrix in each sensor, which results in encoded messages. The encoded messages from the sensors are transmitted to the FC over a coherent MAC. The channel follows MIMO model that has multiple antenna at transmitter and at receiver. At the FC, the received signals are estimated using linear minimum mean-square error (LMMSE) rule. To perform the estimation, we calculate a distortion in terms of mean-square error (MSE). We also consider the separation of the estimation under two conditions, that is, distortion due to noisy observation and distortion due to channel noise. Both of the distortions should be minimized by designing coding matrices under total power constraint. To derive the coding matrix, we use singular value decomposition (SVD) technique. We show that the equations can be formulated as a convex optimization problem. We derive a closed-form solution that can be solved using water-filling algorithm. The designed coding matrices will be sent back to the sensors and will optimize transmit power of them. Under the above scenario, the proposed method will be compared to the equal power method. The equal power method allocates equal transmit power to each sensor. Compared to the existing literature, the contribution of this work lies in the following aspects: we consider MIMO model based on the water-filling algorithm and a spatial correlated data as an extension of [2, 5]. We obtain a closed-form solution for the optimization problem and give the coding matrices for each sensor. We also derive a lower bound of the MSE.The rest of the paper is organized as follows. We review related work in Section2. Section 3 describes the network setup under consideration. In Section 4, we formulate the linear coding matrix based on total power constraint and correlated sources. Section 5 presents some numerical simulation examples and the conclusion is drawn in Section 6.Throughout this paper, we use the following notations. A lower case letter denotes a scalar, a boldface lower case letter denotes a vector, and a boldface upper case denotes matrix. The superscripts,AT, AH, A-1, and A-H denote the transpose, the Hermitian, the inverse, and the inverse Hermitian of matrix A, respectively. The operator diag(·), manipulates the diagonal elements of matrix or a column vector into diagonal matrix and tr(·) is the trace of a matrix.


## 2. Related Work

In [2], they proposed a closed-form solution of coding matrices design in distributed estimation that follows coherent multiple access channel model. They exploited antenna diversity to improve received signal in low transmit power. However, they did not take into account the target correlation. Fang and Li studied power allocation problem in correlated sensor observation and provided lower bound of the objective function for single-input single-output (SISO) model [3]. A best linear unbiased estimator (BLUE) was used for estimating a target in multiple wireless sensor network [4], where the system performance was analyzed by the concept of estimation outage. The power allocation strategy was also employed in the multiple-input single-output (MISO) system.In [5], Guo et al. investigated joint estimation of random vector and minimized the gap to the performance benchmark using water-filling strategy. They revealed that extra transmissions beyond the dimension of the target do not improve the estimation performance. An orthogonal multiple access channel (MAC) model for distributed estimation was studied in [6, 8]. To save energy, multiple-input and multiple-output (MIMO) system has been analyzed by involving signaling overhead [9], in which the MIMO system can offer substantial energy savings in WSN. Cooperative MIMO with data aggregation has also been investigated in [10, 11]. Compared to the existing literature, the contribution of this work lies in the following aspects: we considered MIMO model based on the water-filling algorithm and a spatial correlated data as an extension of [2, 5]. We obtained a closed-form solution for the optimization problem and gave the coding matrices for each sensor.


## 3. Problem Formulation

Assume that there areL sensors for estimating p random source signals, written in a vector form s=[s1,…,sp]T∈ℂp, as shown in Figure 1. The sensors observe the sources through a sensing matrix F∈ℂk×p and each sensor, lth, has k measurements given by
(1)xl=Fls+nl,1≤l≤L,
where nl∈ℂk is the additive noise. We assume that the targets are close to each other. They become potentially correlated. Therefore, the sensing matrix, F, can be modeled as F=For1/2, where Fo is a matrix with size k×p that has i.i.d zero mean circularly symmetric complex Gaussian (ZMCSCG) entries with unit variance and r is the p×p spatial correlation matrix. If we have L sensors, (1) can be written as
(2)x=Fs+n,
where x=[x1T,…,xLT]T∈ℂLk, F=[F1T,…,FLT]T∈ℂLk×p, and n=[n1T,…,nLT]T∈ℂLk with Lk=∑l=1Lkl. The sensor observations are encoded using a linear coding matrix Al∈ℂN×k, where N is the number of encoded messages transmitted from the lth sensor. The message vector is transmitted through channel matrix Hl∈ℂN×N using N different frequencies. The received signal at FC can be written as
(3)y=∑l=1L‍HlAl(Fls+nl)+v=B(Fs+n)+v,
where B=[Bl,…,BL]∈ℂN×Lk with Bl=HlAl∈ℂN×k and v∈ℂN is additive Gaussian noise. FC employs LMMSE estimator to estimate parameter s based on the received signal y in (3) [14]
(4)s^=E[syH](E[yyH])-1y,
and the total distortion in terms of MSE can be expressed as
(5)Dt=tr(E[(s-s^)(s-s^)H]).
We can view the total distortion as a summation of noise observation distortion, Do, and channel noise distortion, Dc, as follows:
(6)Dt=E[(s---s)2]=E[(s---s-+s--s)2]=E[(s---s-)2]︸∶=Dc+E[(s--s)2]︸∶=Do+2E[(s---s-)(s--s)]︸∶=Dm,
where Dm characterizes the mutual term. The s-- and s- are estimated parameters due to channel noise and due to observation noise, respectively. Based on Cauchy-Schwarz inequality, we can have the upper bound distortion Dt as follows:
(7)Dt≤E[(s---s-)2]+E[(s--s)2]+2E[(s---s-)2(s--s)2]=(Dc+Do)2≤2(Dc+Do)=2D~t.
To obtain (7), we have used (E[xy])2≤E[x2]E[y2] in the first inequality and (∑k=1KGk)2≤K∑k=1KGk2 in the second inequality. Then, we have
(8)D~t=Dc+Do.Figure 1
Distributed estimation over a wireless sensor network.We can calculate distortion due to noisy observation,Do, by assuming that channel noise is free, v=0. Then, the received signal can be written as
(9)y=∑l=1LHlAlxl=B(Fs+n).
Based on (9), we can derive Do as
(10)Do=tr(Rs-RsxBH[BRxoBH]-1BRsxH),
where Rxo=σs2FrFH+Rnl with r being spatial correlation matrix. We assume covariance matrices of the targets, Rs=σs2Ip, noise observation, Rnl=σn2Ik, and channel noise, Rv=σv2IN. Introducing Rsx=r1/2RsFH is covariance between s and x. Moreover, we can calculate distortion due to channel noise, Dc, by assuming that observation noise is free, nl=0. Then, the received signal can be written as
(11)y=∑l=1L‍HlAlxl+v=B(Fs)+v.
Based on (11), we can derive Dc as
(12)Dc=tr(Rs-RsxBH[BRxcBH+Rv]-1BRsxH),
where Rxc=σs2FrFH.


### 3.1. Correlated Sources

We assume that data observed by sensors are spatially correlated. The targets are placed in line with spacing distance,d, as shown in Figure 2 [13]. The model of correlation is as follows:
(13)[r]i,j=ρ|dij|;i,j∈1,…,p=[ρ|d11|⋯ρ|d1j|······ρ|di1|⋯ρ|dij|],
where dij is a distance between the ith and the jth targets and ρ is the spatial correlation coefficient. In general, r is Hermitian and positive definite, so we can write r=r1/2r1/2.Figure 2
The targets are placed in line with spacing distance,d.


### 3.2. Objective Function

According to the total distortion, we need to minimizeD~t under a total power constraint, P. The total transmit power for the L sensors is defined as
(14)∑l=1L‍tr(AlRxlAlH)≤P,
where Rxl=σs2FlrFlH+σn2Ik.Furthermore, based on (8) and (14), we have an objective function as follows:
(15)minD~tsubjectto∑l=1L‍tr(AlRxlAlH)≤P.


## 3.1. Correlated Sources

We assume that data observed by sensors are spatially correlated. The targets are placed in line with spacing distance,d, as shown in Figure 2 [13]. The model of correlation is as follows:
(13)[r]i,j=ρ|dij|;i,j∈1,…,p=[ρ|d11|⋯ρ|d1j|······ρ|di1|⋯ρ|dij|],
where dij is a distance between the ith and the jth targets and ρ is the spatial correlation coefficient. In general, r is Hermitian and positive definite, so we can write r=r1/2r1/2.Figure 2
The targets are placed in line with spacing distance,d.


## 3.2. Objective Function

According to the total distortion, we need to minimizeD~t under a total power constraint, P. The total transmit power for the L sensors is defined as
(14)∑l=1L‍tr(AlRxlAlH)≤P,
where Rxl=σs2FlrFlH+σn2Ik.Furthermore, based on (8) and (14), we have an objective function as follows:
(15)minD~tsubjectto∑l=1L‍tr(AlRxlAlH)≤P.


## 4. Proposed Approach

### 4.1. Proposed Method

In this section, we aim to solve the objective function in (15). First, we need to minimize it through SVD technique. Second, we formulate a convex optimization function by considering total power constraint. Afterwards, we further show that the optimal solution can be obtained by a water filling algorithm.Let us introduce a lemma (cf. [15]).Lemma 1.
For any two positive semidefinite matricesX and Y with size n, it holds that
(16)tr(XY)≤∑i=1n‍αiβi,
where αi and βi are the ith eigenvalues of X and Y, respectively, in an increasing order.
SinceRxo and Rxc are positive definite, they can be written as Rxo=Rxo1/2Rxo1/2 and Rxc=Rxc1/2Rxc1/2, respectively. By performing SVD, we can write them as
(17)Rxo-1/2FFHRxo-1/2=UfΣfUfH,(18)Rxc-1/2FFHRxc-1/2=UgΣgUgH,
where Σf=diag(f1,…,fp) with f1≥f2≥⋯≥fp>0 and Σg=diag(g1,…,gp) with g1≥g2≥⋯≥gp>0. Uf∈ℂLk×Lk and Ug∈ℂLk×Lk are unitary, respectively.LetBRxo1/2 and BRxc1/2 be performed as SVD as follows:
(19)T=BRxo1/2=UtΣtVtH(20)W=BRxc1/2=UwΣwVwH,
where Ut∈ℂN×N and Uw∈ℂN×N are unitary, respectively. Matrix Σt=diag(t1,…,tN) with t1≥⋯≥tN>0, Σw=diag(w1,…,wN) with w1≥⋯≥wN>0, Vt∈ℂLk×N, and Vw∈ℂLk×N have orthonormal columns.Based on (17) and (19), the distortion due to observation noise, Do, can be reformed as
(21)Do=tr([BRxo1/2BHRxo1/2]-1σs2Ip-σs4Rxo-1/2FFHRxo-1/2BHRxo1/2×[BRxo1/2BHRxo1/2]-1BRxo1/2)=tr(σs2Ip-σs4Uf∑‍fUfHTH[TTH]-1T).
By using Lemma 1, we have
(22)Do≤pσs2-σs4∑i=1p‍fibibi≤pσs2-σs4∑i=1p‍fi.
According to (18) and (20), the distortion due to channel noise, Dc, can be reformed as
(23)Dc=tr([BRxc1/2BHRxc1/2+Rv]-1σs2Ip-σs4Rxc-1/2FFHRxc-1/2BHRxc1/2×[BRxc1/2BHRxc1/2+Rv]-1BRxc1/2)=tr([WWH+σv2IN]-1σs2Ip-σs4Ug∑‍gUgHWH×[WWH+σv2IN]-1W∑‍).
By using Lemma 1, we have
(24)Dc≤pσs2-σs4∑i=1p‍giwiwi+σv2.
We have D~t in terms of SVD as follows:
(25)D~t≤Do+Dc≤2pσs2-σ24(∑i=1p‍fi+giwiwi+σv2).
For simplifying analysis, we set VwHU^g=[IN0], where U^g=Ug(:,1:N). This is intended to keep Dc diagonal. When the number of transmitters is set minimum, N=p, we have
(26)Vw=U^g;whereU^g=Ug(:,1:p),
where U^g is taken from the first p columns of matrix Ug. Then, from (26), we can express (20) as
(27)BRxc1/2=Uw∑wU^gHB=Uw∑wU^gHRxc-1/2︸UR,
where Uw∈ℂN×N is unitary in (27); thus, we have B=ΣwUR. Because of B=[B1,…,Bl], we can express Bl as
(28)Bl=ΣwURl,
where UR=[UR1,…,URL] with URl∈ℂp×k.Let us reform the total power constraint in terms of SVD. We haveAl=Hl-1Bl. We can rewrite (14) by taking into account (28) as follows:
(29)∑l=1Ltr(Hl-1BlRxlBlHHl-H)≤P∑l=1Ltr(Hl-1ΣwURlRxlΣwURlHHl-H)≤P∑l=1Ltr(Hl-1URlRxlURlHHl-HΣw)≤P∑i=1pniwi≤P,
where ∑l=1L‍(Hl-1URlRxlURlHHl-H) have diagonal entries ni,1≤i≤p. We also have Rxl=σs2FlrFlH+σn2Ik.From (25) and (29), we can express the objective function in terms of SVD as follows:
(30)max-(∑i=1p‍fi+giwiwi+σv2)subjectto∑i=1p‍niwi≤P,wi≥0,i=1,…,p.
It is a convex problem since the objective function is a linear combination of convex functions and the constraints are linear. To solve (30), it can be written as the Lagrangian equation as
(31)L(μ,wi)=-(∑i=1p‍fi+giwiwi+σv2)+μ(∑i=1p‍niwi-P)-∑i=1p‍uiwi.
We can obtain the global optimum by solving the KKT conditions [16]:
(32)giσv2(wi+σv2)2=μni-ui,i=1,…,pgiσv2(wi+σv2)2i=μ(∑i=1p‍niwi-P)=0giσv2(wi+σv2)2=uiwi=0,i=1,…,pgiσv2(wi+σv2)2i=μni-uii=1μ≥0.
The solution of the above can be expressed as follows: for i=1,…,L(33)wi={(giμniσv2-1)σv2,giniσv2≥μ,0,giniσv2<μ.
The parameter μ satisfies
(34)∑i=1L‍(giμniσv2-1)+σv2ni=P,
where (x)+= max(0,x). We solve the parameters wi and μ using water-filling algorithm [5]: input:gi=(g1,…,gp); ni=(n1,…,np); σv2; Poutput:μ; mi, i=1,…,p(1) Reorder the sequence ti=gi/niσv2, in the increasing order, and set m=1(2)do  (μ←tmτ←∑i=mp‍tiσv2ni/P+∑i=mp‍σv2nim=m+1)while  (μ<τ and m≤p)(3)μ←τωi←σv2(gi/μniσv2-1)+After we obtain Σw=diag(w1,…,wp), we can write coding matrix Al as
(35)Al=Hl-1ΣwURl.
We also define a lower bound from (25); as P→∞, we have wi→∞ as
(36)Dlow=2pσs2-σs4(∑i=1p‍fi+gi).


### 4.2. Equal Power Method

Under equal power strategy, the transmit power for all sensors is set to be equal as follows:(37)Pl=PL,1≤l≤L,
where Pl is the transmit power for the lth sensor. Then, we have the lth coding matrix, Al=λl[IN0], where λl=P/[Ltr(Rxl(1:N,1:N))], so that tr(AlRxlAlH)=Pl, 1≤l≤L. The Rxl(1:N,1:N) denotes the first N rows and columns of Rxl and [IN0] is a N×k matrix with its diagonal entries equal to 1 and other entries equal to 0.


## 4.1. Proposed Method

In this section, we aim to solve the objective function in (15). First, we need to minimize it through SVD technique. Second, we formulate a convex optimization function by considering total power constraint. Afterwards, we further show that the optimal solution can be obtained by a water filling algorithm.Let us introduce a lemma (cf. [15]).Lemma 1.
For any two positive semidefinite matricesX and Y with size n, it holds that
(16)tr(XY)≤∑i=1n‍αiβi,
where αi and βi are the ith eigenvalues of X and Y, respectively, in an increasing order.
SinceRxo and Rxc are positive definite, they can be written as Rxo=Rxo1/2Rxo1/2 and Rxc=Rxc1/2Rxc1/2, respectively. By performing SVD, we can write them as
(17)Rxo-1/2FFHRxo-1/2=UfΣfUfH,(18)Rxc-1/2FFHRxc-1/2=UgΣgUgH,
where Σf=diag(f1,…,fp) with f1≥f2≥⋯≥fp>0 and Σg=diag(g1,…,gp) with g1≥g2≥⋯≥gp>0. Uf∈ℂLk×Lk and Ug∈ℂLk×Lk are unitary, respectively.LetBRxo1/2 and BRxc1/2 be performed as SVD as follows:
(19)T=BRxo1/2=UtΣtVtH(20)W=BRxc1/2=UwΣwVwH,
where Ut∈ℂN×N and Uw∈ℂN×N are unitary, respectively. Matrix Σt=diag(t1,…,tN) with t1≥⋯≥tN>0, Σw=diag(w1,…,wN) with w1≥⋯≥wN>0, Vt∈ℂLk×N, and Vw∈ℂLk×N have orthonormal columns.Based on (17) and (19), the distortion due to observation noise, Do, can be reformed as
(21)Do=tr([BRxo1/2BHRxo1/2]-1σs2Ip-σs4Rxo-1/2FFHRxo-1/2BHRxo1/2×[BRxo1/2BHRxo1/2]-1BRxo1/2)=tr(σs2Ip-σs4Uf∑‍fUfHTH[TTH]-1T).
By using Lemma 1, we have
(22)Do≤pσs2-σs4∑i=1p‍fibibi≤pσs2-σs4∑i=1p‍fi.
According to (18) and (20), the distortion due to channel noise, Dc, can be reformed as
(23)Dc=tr([BRxc1/2BHRxc1/2+Rv]-1σs2Ip-σs4Rxc-1/2FFHRxc-1/2BHRxc1/2×[BRxc1/2BHRxc1/2+Rv]-1BRxc1/2)=tr([WWH+σv2IN]-1σs2Ip-σs4Ug∑‍gUgHWH×[WWH+σv2IN]-1W∑‍).
By using Lemma 1, we have
(24)Dc≤pσs2-σs4∑i=1p‍giwiwi+σv2.
We have D~t in terms of SVD as follows:
(25)D~t≤Do+Dc≤2pσs2-σ24(∑i=1p‍fi+giwiwi+σv2).
For simplifying analysis, we set VwHU^g=[IN0], where U^g=Ug(:,1:N). This is intended to keep Dc diagonal. When the number of transmitters is set minimum, N=p, we have
(26)Vw=U^g;whereU^g=Ug(:,1:p),
where U^g is taken from the first p columns of matrix Ug. Then, from (26), we can express (20) as
(27)BRxc1/2=Uw∑wU^gHB=Uw∑wU^gHRxc-1/2︸UR,
where Uw∈ℂN×N is unitary in (27); thus, we have B=ΣwUR. Because of B=[B1,…,Bl], we can express Bl as
(28)Bl=ΣwURl,
where UR=[UR1,…,URL] with URl∈ℂp×k.Let us reform the total power constraint in terms of SVD. We haveAl=Hl-1Bl. We can rewrite (14) by taking into account (28) as follows:
(29)∑l=1Ltr(Hl-1BlRxlBlHHl-H)≤P∑l=1Ltr(Hl-1ΣwURlRxlΣwURlHHl-H)≤P∑l=1Ltr(Hl-1URlRxlURlHHl-HΣw)≤P∑i=1pniwi≤P,
where ∑l=1L‍(Hl-1URlRxlURlHHl-H) have diagonal entries ni,1≤i≤p. We also have Rxl=σs2FlrFlH+σn2Ik.From (25) and (29), we can express the objective function in terms of SVD as follows:
(30)max-(∑i=1p‍fi+giwiwi+σv2)subjectto∑i=1p‍niwi≤P,wi≥0,i=1,…,p.
It is a convex problem since the objective function is a linear combination of convex functions and the constraints are linear. To solve (30), it can be written as the Lagrangian equation as
(31)L(μ,wi)=-(∑i=1p‍fi+giwiwi+σv2)+μ(∑i=1p‍niwi-P)-∑i=1p‍uiwi.
We can obtain the global optimum by solving the KKT conditions [16]:
(32)giσv2(wi+σv2)2=μni-ui,i=1,…,pgiσv2(wi+σv2)2i=μ(∑i=1p‍niwi-P)=0giσv2(wi+σv2)2=uiwi=0,i=1,…,pgiσv2(wi+σv2)2i=μni-uii=1μ≥0.
The solution of the above can be expressed as follows: for i=1,…,L(33)wi={(giμniσv2-1)σv2,giniσv2≥μ,0,giniσv2<μ.
The parameter μ satisfies
(34)∑i=1L‍(giμniσv2-1)+σv2ni=P,
where (x)+= max(0,x). We solve the parameters wi and μ using water-filling algorithm [5]: input:gi=(g1,…,gp); ni=(n1,…,np); σv2; Poutput:μ; mi, i=1,…,p(1) Reorder the sequence ti=gi/niσv2, in the increasing order, and set m=1(2)do  (μ←tmτ←∑i=mp‍tiσv2ni/P+∑i=mp‍σv2nim=m+1)while  (μ<τ and m≤p)(3)μ←τωi←σv2(gi/μniσv2-1)+After we obtain Σw=diag(w1,…,wp), we can write coding matrix Al as
(35)Al=Hl-1ΣwURl.
We also define a lower bound from (25); as P→∞, we have wi→∞ as
(36)Dlow=2pσs2-σs4(∑i=1p‍fi+gi).


## 4.2. Equal Power Method

Under equal power strategy, the transmit power for all sensors is set to be equal as follows:(37)Pl=PL,1≤l≤L,
where Pl is the transmit power for the lth sensor. Then, we have the lth coding matrix, Al=λl[IN0], where λl=P/[Ltr(Rxl(1:N,1:N))], so that tr(AlRxlAlH)=Pl, 1≤l≤L. The Rxl(1:N,1:N) denotes the first N rows and columns of Rxl and [IN0] is a N×k matrix with its diagonal entries equal to 1 and other entries equal to 0.


## 5. Simulation Results

We present simulation results to illustrate the estimation performance of the previous section. We useP to denote the total transmit power constraint across the network. In all simulations, the random vectors s, nl, and v are complex Gaussian with zero mean and unit variance. Note that power P is taken relative to the channel noise power. Since the channel noise has unitary variance, thus we label the total transmit power in unit of dB. The channel matrix Hl is also Gaussian random variable with zero mean and unit variance. We set the number of encoding matrix, N=5, equal to that of sources, p = 5. This is because distortion performance does not degrade when N>p [17]. Assuming that targets are close to each other so that we set distance among them, d=1. The channels between sensors and the fusion center are chosen to be independent and the average LMMSE is calculated over 500 times. Therefore, we calculate the total distortion, D~t, instead of Dt due to mathematical tractability.Figure3 plots distortion in terms of average MSE performance comparison between the proposed method, the lower bound, and the equal power method, in which we take L=10, k=8, and the correlation coefficient, ρ=0. From Figure 3, we can see that the proposed method performs better than the equal power method. The proposed method converges to the lower bound as P increases. This is because the proposed method allocates power by taking into account the effects of observation and channel noise, but the equal power method does not.Figure 3
Distortion comparison between the proposed method, equal power method, and the lower bound with parametersρ=0, L=10, k=8, N=5, p=5.Figure4 shows distortion for uncorrelated, ρ=0, and correlated, ρ=0.9, data. We can see that the distortion of both methods becomes worse as the data is being correlated. If the data is more correlated, sensors provide redundant information about the targets. It becomes difficult to estimate the targets. Once ρ=0.9, the distortion of the proposed method remains constant for P<25 dB. Moreover, the equal power method performs better than the proposed method particularly for P between 10 and 25 dB because the proposed method does not have enough power to accomplish the threshold of water filling. However, the distortion of the proposed method becomes smaller than that of the equal power method for P>25 dB, because it allocates power by taking into account the effects of the correlation, observation, and channel noise.Figure 4
Distortion comparison between different correlation coefficients with parametersρ=0, 0.9, L=10, k=8, N=5, p=5.The distortion of the proposed method and the lower bound with different power levels (P=5, 10, and 20 dB), and spatial correlation coefficient, ρ=0.7, are shown in Figure 5. Both distortions become smaller as the number of measurements, k, increases. This is because increasing the number of k leads to an increase of measurement power. In Figure 5, we can also see that the gap of the distortion between the proposed method and the lower bound becomes larger. This is because the proposed method has a power constraint compared with the lower bound that does not have a power constraint.Figure 5
Distortion versusk with parameters ρ=0.7, L=10, N=5, p=5, P=5, 10, 20 dB.From Figure6, we simulate the proposed method with different power levels (P = 1, 2, 5, 8, 15, 20 dB) and correlation coefficient, ρ=0.7. The distortions of the proposed method become smaller as the number of sensors, L, increases. However, they rise slightly after reaching the optimum number of sensors. For example, the total power constraint is P=1 dB; the distortion rises when the number of sensors is L>3. This is owing to the fact that the power allocated to each sensor becomes smaller as the number of sensors increases.Figure 6
Distortion versusL with parameters ρ=0.7, k=8, N=5, p=5, P=1, 2, 5, 8, 15, 20 dB.In Figure7, we show the simulation results of the proposed method with an ideal sensor condition and an ideal channel condition. The simulation with ideal sensor condition is taken by assuming that the sensor observations have no distortion. To accommodate the assumption, we set the distortion due to observation noise, Do=0, in (8). Then, the simulation with ideal channel condition is taken by assuming that the fusion center has a perfect channel knowledge between the sensors and the fusion center. In a similar way to the ideal sensor condition, we set the distortion due to channel noise, Dc=0, in (8). We use a complex Gaussian random variable with zero mean and unit variance for observation noise, n, and noisy channel, v, respectively. The distortion with ideal sensor becomes smaller as P increases, but the distortion with ideal channel still remains constant. It means that the water-filling method is more effective to combat the noisy channel rather than the observation noise.Figure 7
Distortion comparison of the proposed method with ideal sensor and ideal channel condition with parametersρ=0.5, k=8, N=5, p=5.


## 6. Conclusion

We studied distributed estimation of a random vector in MIMO sensor network by considering total power constraint and spatial correlated data. For the spatial correlated data, we obtained a water-filling-based closed-form solution that follows water-filling strategy. We also derived lower bound of distortion to this system. From the simulation results, we showed that the distortion increases as the data becomes more correlated, because the sensors become difficult to estimate the targets. Moreover, we showed that the noisy channel was more harmful than the observation noise. By taking into account the effects of correlation, observation, and channel noise, the proposed method has a better distortion performance than the equal power method.



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*Source: 345457-2013-09-18.xml*