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

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.

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 Section 2. 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, 𝐀^𝑇, 𝐀^𝐻, 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 are 𝐿 sensors for estimating 𝑝 random source signals, written in a vector form 𝐬=[𝑠₁,…,𝑠_𝑝]^𝑇∈ℂ^𝑝, as shown in Figure 1. The sensors observe the sources through a sensing matrix 𝐅∈ℂ^𝑘×𝑝 and each sensor, 𝑙th, has 𝑘 measurements given by 𝐱_𝑙=𝐅_𝑙𝐬+𝐧_𝑙,1≤𝑙≤𝐿, (1) where 𝐧_𝑙∈ℂ^𝑘 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=𝐅_𝑜𝐫^1/2, where 𝐅_𝑜 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 𝐱=𝐅𝐬+𝐧, (2) where 𝐱=[𝐱^𝑇₁,…,𝐱^𝑇_𝐿]^𝑇∈ℂ^𝐿𝑘, 𝐅=[𝐅^𝑇₁,…,𝐅^𝑇_𝐿]^𝑇∈ℂ^{𝐿𝑘×𝑝}, and 𝐧=[𝐧^𝑇₁,…,𝐧^𝑇_𝐿]^𝑇∈ℂ^𝐿𝑘 with 𝐿𝑘=∑^𝐿_𝑙=1𝑘_𝑙. The sensor observations are encoded using a linear coding matrix 𝐀_𝑙∈ℂ^𝑁×𝑘, where 𝑁 is the number of encoded messages transmitted from the 𝑙th sensor. The message vector is transmitted through channel matrix 𝐇_𝑙∈ℂ^𝑁×𝑁 using 𝑁 different frequencies. The received signal at FC can be written as 𝐲=^𝐿_𝑙=1𝐇_𝑙𝐀_𝑙𝐅_𝑙𝐬+𝐧_𝑙()+𝐯=𝐁𝐅𝐬+𝐧+𝐯, (3) where 𝐁=[𝐁_𝑙,…,𝐁_𝐿]∈ℂ^{𝑁×𝐿𝑘} with 𝐁_𝑙=𝐇_𝑙𝐀_𝑙∈ℂ^𝑁×𝑘 and 𝐯∈ℂ^𝑁 is additive Gaussian noise. FC employs LMMSE estimator to estimate parameter 𝐬 based on the received signal 𝐲 in (3) [14] 𝐬̂=𝐸𝐬𝐲^𝐻𝐸𝐲𝐲^𝐻−1𝐲, (4) and the total distortion in terms of MSE can be expressed as 𝐷_𝑡=tr𝐸(𝐬−𝐬)̂(𝐬−𝐬)̂^𝐻. (5) We can view the total distortion as a summation of noise observation distortion, 𝐷_𝑜, and channel noise distortion, 𝐷_𝑐, as follows: 𝐷_𝑡=𝐸𝐬−𝐬²=𝐸𝐬−𝐬+𝐬−𝐬²=𝐸𝐬−𝐬²_∶=𝐷𝑐+𝐸(𝐬−𝐬²_∶=𝐷𝑜)+2𝐸𝐬−𝐬(𝐬−𝐬)_∶=𝐷𝑚, (6) where 𝐷_𝑚 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 𝐷_𝑡 as follows: 𝐷_𝑡≤𝐸𝐬−𝐬²+𝐸(𝐬−𝐬)²+2𝐸𝐬−𝐬²(𝐬−𝐬)²=𝐷_𝑐+𝐷_𝑜²≤2𝐷_𝑐+𝐷_𝑜=2𝐷_𝑡. (7) To obtain (7), we have used (𝐸[𝑥𝑦])²≤𝐸[𝑥²]𝐸[𝑦²] in the first inequality and (∑^𝐾_𝑘=1𝐺_𝑘)²≤𝐾∑^𝐾_𝑘=1𝐺²_𝑘 in the second inequality. Then, we have 𝐷_𝑡=𝐷_𝑐+𝐷_𝑜. (8)

Figure 1 

Distributed estimation over a wireless sensor network.

We can calculate distortion due to noisy observation, 𝐷_𝑜, by assuming that channel noise is free, 𝐯=0. Then, the received signal can be written as 𝐲=^𝐿_𝑙=1𝐇_𝑙𝐀_𝑙𝐱_𝑙=𝐁(𝐅𝐬+𝐧). (9) Based on (9), we can derive 𝐷_𝑜 as 𝐷_𝑜=tr𝐑_𝑠−𝐑_𝑠𝑥𝐁^𝐻𝐁𝐑_𝑥𝑜𝐁^𝐻−1𝐁𝐑^𝐻_𝑠𝑥, (10) where 𝐑_𝑥𝑜=𝜎²_𝑠𝐅𝐫𝐅^𝐻+𝐑_𝑛𝑙 with 𝐫 being spatial correlation matrix. We assume covariance matrices of the targets, 𝐑_𝑠=𝜎²_𝑠𝐈_𝑝, noise observation, 𝐑_𝑛𝑙=𝜎²_𝑛𝐈_𝑘, and channel noise, 𝐑_𝑣=𝜎²_𝑣𝐈_𝑁. Introducing 𝐑_𝑠𝑥=𝐫^1/2𝐑_𝑠𝐅^𝐻 is covariance between 𝐬 and 𝐱. Moreover, we can calculate distortion due to channel noise, 𝐷_𝑐, by assuming that observation noise is free, 𝐧_𝑙=0. Then, the received signal can be written as 𝐲=^𝐿_𝑙=1𝐇_𝑙𝐀_𝑙𝐱_𝑙+𝐯=𝐁(𝐅𝐬)+𝐯. (11) Based on (11), we can derive 𝐷_𝑐 as 𝐷_𝑐=tr𝐑_𝑠−𝐑_𝑠𝑥𝐁^𝐻𝐁𝐑_𝑥𝑐𝐁^𝐻+𝐑_𝑣⁻¹𝐁𝐑^𝐻_𝑠𝑥, (12) where 𝐑_𝑥𝑐=𝜎²_𝑠𝐅𝐫𝐅^𝐻.

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: [𝐫]_𝑖,𝑗=𝜌^{|𝑑𝑖𝑗|}=𝜌;𝑖,𝑗∈1,…,𝑝_|𝑑11|⋯𝜌^|𝑑1𝑗|𝜌⋅⋅⋅⋅⋅⋅_|𝑑𝑖1|⋯𝜌^{|𝑑𝑖𝑗|}, (13) where 𝑑_𝑖𝑗 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=𝐫^1/2𝐫^1/2.

Figure 2 

The targets are placed in line with spacing distance, 𝑑.

3.2. Objective Function

According to the total distortion, we need to minimize 𝐷_𝑡 under a total power constraint, 𝑃. The total transmit power for the 𝐿 sensors is defined as tr𝐀^𝐿_𝑙=1𝑙𝐑_𝑥𝑙𝐀^𝐻_𝑙≤𝑃, (14) where 𝐑_𝑥𝑙=𝜎²_𝑠𝐅_𝑙𝐫𝐅^𝐻_𝑙+𝜎²_𝑛𝐈_𝑘.

Furthermore, based on (8) and (14), we have an objective function as follows: min𝐷_𝑡subjectto^𝐿_𝑙=1tr𝐀_𝑙𝐑_𝑥𝑙𝐀^𝐻_𝑙≤𝑃. (15)

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 matrices 𝐗 and 𝐘 with size 𝑛, it holds that tr(𝐗𝐘)≤^𝑛_𝑖=1𝛼_𝑖𝛽_𝑖, (16) where 𝛼_𝑖 and 𝛽_𝑖 are the 𝑖th eigenvalues of 𝐗 and 𝐘, respectively, in an increasing order.
Since 𝐑_𝑥𝑜 and 𝐑_𝑥𝑐 are positive definite, they can be written as 𝐑_𝑥𝑜=𝐑^1/2_𝑥𝑜𝐑^1/2_𝑥𝑜 and 𝐑_𝑥𝑐=𝐑^1/2_𝑥𝑐𝐑^1/2_𝑥𝑐, respectively. By performing SVD, we can write them as 𝐑^−1/2_𝑥𝑜𝐅𝐅^𝐻𝐑^−1/2_𝑥𝑜=𝐔_𝑓Σ𝐟𝐔^𝐻_𝑓,𝐑^−1/2_𝑥𝑐𝐅𝐅^𝐻𝐑^−1/2_𝑥𝑐=𝐔_𝑔Σ𝐠𝐔^𝐻_𝑔, (17)(18) where Σ𝐟=diag(𝑓₁,…,𝑓_𝑝) with 𝑓₁≥𝑓₂≥⋯≥𝑓_𝑝>0 and Σ𝐠=diag(𝑔₁,…,𝑔_𝑝) with 𝑔₁≥𝑔₂≥⋯≥𝑔_𝑝>0. 𝐔_𝑓∈ℂ^{𝐿𝑘×𝐿𝑘} and 𝐔_𝑔∈ℂ^{𝐿𝑘×𝐿𝑘} are unitary, respectively.

Let 𝐁𝐑^1/2_𝑥𝑜 and 𝐁𝐑^1/2_𝑥𝑐 be performed as SVD as follows: 𝐓=𝐁𝐑^1/2_𝑥𝑜=𝐔_𝑡Σ√𝐭𝐕^𝐻_𝑡𝐖=𝐁𝐑^1/2_𝑥𝑐=𝐔_𝑤Σ√𝐰𝐕^𝐻_𝑤, (19)(20) where 𝐔_𝑡∈ℂ^𝑁×𝑁 and 𝐔_𝑤∈ℂ^𝑁×𝑁 are unitary, respectively. Matrix Σ√𝐭=diag(𝑡₁,…,𝑡_𝑁) with 𝑡₁≥⋯≥𝑡_𝑁>0, Σ√𝐰=diag(√𝑤₁,…,√𝑤_𝑁) with 𝑤₁≥⋯≥𝑤_𝑁>0, 𝐕_𝑡∈ℂ^{𝐿𝑘×𝑁}, and 𝐕_𝑤∈ℂ^{𝐿𝑘×𝑁} have orthonormal columns.

Based on (17) and (19), the distortion due to observation noise, 𝐷_𝑜, can be reformed as 𝐷_𝑜=tr𝜎²_𝑠𝐈_𝑝−𝜎⁴_𝑠𝐑^−1/2_𝑥𝑜𝐅𝐅^𝐻𝐑^−1/2_𝑥𝑜𝐁^𝐻𝐑^1/2_𝑥𝑜×𝐁𝐑^1/2_𝑥𝑜𝐁^𝐻𝐑^1/2_𝑥𝑜⁻¹𝐁𝐑^1/2_𝑥𝑜=tr𝜎²_𝑠𝐈_𝑝−𝜎⁴_𝑠𝐔_𝑓𝐟𝐔^𝐻_𝑓𝐓^𝐻𝐓𝐓^𝐻−1𝐓. (21) By using Lemma 1, we have 𝐷_𝑜≤𝑝𝜎²_𝑠−𝜎⁴_𝑠^𝑝_𝑖=1𝑓_𝑖𝑏_𝑖𝑏_𝑖≤𝑝𝜎²_𝑠−𝜎⁴_𝑠^𝑝_𝑖=1𝑓_𝑖. (22) According to (18) and (20), the distortion due to channel noise, 𝐷_𝑐, can be reformed as 𝐷_𝑐=tr𝜎²_𝑠𝐈_𝑝−𝜎⁴_𝑠𝐑^−1/2_𝑥𝑐𝐅𝐅^𝐻𝐑^−1/2_𝑥𝑐𝐁^𝐻𝐑^1/2_𝑥𝑐×𝐁𝐑^1/2_𝑥𝑐𝐁^𝐻𝐑^1/2_𝑥𝑐+𝐑_𝑣⁻¹𝐁𝐑^1/2_𝑥𝑐=tr𝜎²_𝑠𝐈_𝑝−𝜎⁴_𝑠𝐔_𝑔𝐠𝐔^𝐻_𝑔𝐖^𝐻×𝐖𝐖^𝐻+𝜎²_𝑣𝐈_𝑁⁻¹𝐖. (23) By using Lemma 1, we have 𝐷_𝑐≤𝑝𝜎²_𝑠−𝜎⁴_𝑠^𝑝_𝑖=1𝑔_𝑖𝑤_𝑖𝑤_𝑖+𝜎²_𝑣. (24) We have 𝐷_𝑡 in terms of SVD as follows: 𝐷_𝑡≤𝐷_𝑜+𝐷_𝑐≤2𝑝𝜎²_𝑠−𝜎⁴₂^𝑝_𝑖=1𝑓_𝑖+𝑔_𝑖𝑤_𝑖𝑤_𝑖+𝜎²_𝑣. (25) For simplifying analysis, we set 𝐕^𝐻_𝑤𝐔_𝑔=𝐈_𝑁0, where 𝐔_𝑔=𝐔_𝑔(∶,1∶𝑁). This is intended to keep 𝐷_𝑐 diagonal. When the number of transmitters is set minimum, 𝑁=𝑝, we have 𝐕_𝑤=𝐔_𝑔;where𝐔_𝑔=𝐔_𝑔∶,1∶𝑝, (26) where 𝐔_𝑔 is taken from the first 𝑝 columns of matrix 𝐔_𝑔. Then, from (26), we can express (20) as 𝐁𝐑^1/2_𝑥𝑐=𝐔_𝑤√𝐰𝐔^𝐻_𝑔𝐁=𝐔_𝑤√𝐰𝐔^𝐻_𝑔𝐑^−1/2_{𝑥𝑐𝐔𝑅}, (27) where 𝐔_𝑤∈ℂ^𝑁×𝑁 is unitary in (27); thus, we have 𝐁=Σ√𝐰𝐔_𝑅. Because of 𝐁=[𝐁₁,…,𝐁_𝑙], we can express 𝐁_𝑙 as 𝐁_𝑙=Σ√𝐰𝐔_𝑅𝑙, (28) where 𝐔_𝑅=[𝐔_𝑅1,…,𝐔_𝑅𝐿] with 𝐔_𝑅𝑙∈ℂ^𝑝×𝑘.

Let us reform the total power constraint in terms of SVD. We have 𝐀_𝑙=𝐇_𝑙⁻¹𝐁_𝑙. We can rewrite (14) by taking into account (28) as follows: tr𝐇^𝐿_𝑙=1𝑙⁻¹𝐁_𝑙𝐑_𝑥𝑙𝐁^𝐻_𝑙𝐇_𝑙^−𝐻≤𝑃_𝐿𝑙=1tr𝐇_𝑙⁻¹Σ√𝐰𝐔_𝑅𝑙𝐑_𝑥𝑙Σ√𝐰𝐔^𝐻_𝑅𝑙𝐇_𝑙^−𝐻≤𝑃_𝐿𝑙=1tr𝐇_𝑙⁻¹𝐔_𝑅𝑙𝐑_𝑥𝑙𝐔^𝐻_𝑅𝑙𝐇_𝑙^−𝐻Σ𝐰≤𝑃_𝑝𝑖=1𝑛_𝑖𝑤_𝑖≤𝑃, (29) where ∑^𝐿_𝑙=1(𝐇_𝑙⁻¹𝐔_𝑅𝑙𝐑_𝑥𝑙𝐔^𝐻_𝑅𝑙𝐇_𝑙^−𝐻) have diagonal entries 𝑛_𝑖,1≤𝑖≤𝑝. We also have 𝐑_𝑥𝑙=𝜎²_𝑠𝐅_𝑙𝐫𝐅^𝐻_𝑙+𝜎²_𝑛𝐈_𝑘.

From (25) and (29), we can express the objective function in terms of SVD as follows: max−^𝑝_𝑖=1𝑓_𝑖+𝑔_𝑖𝑤_𝑖𝑤_𝑖+𝜎²_𝑣subjectto^𝑝_𝑖=1𝑛_𝑖𝑤_𝑖𝑤≤𝑃,_𝑖≥0,𝑖=1,…,𝑝. (30) 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 𝐿𝜇,𝑤_𝑖=−^𝑝_𝑖=1𝑓_𝑖+𝑔_𝑖𝑤_𝑖𝑤_𝑖+𝜎²_𝑣+𝜇^𝑝_𝑖=1𝑛_𝑖𝑤_𝑖−𝑃−^𝑝_𝑖=1𝑢_𝑖𝑤_𝑖. (31) We can obtain the global optimum by solving the KKT conditions [16]: 𝑔_𝑖𝜎²_𝑣𝑤_𝑖+𝜎²_𝑣²=𝜇𝑛_𝑖−𝑢_𝑖𝜇,𝑖=1,…,𝑝_𝑝𝑖=1𝑛_𝑖𝑤_𝑖𝑢−𝑃=0_𝑖𝑤_𝑖=0,𝑖=1,…,𝑝𝜇≥0. (32) The solution of the above can be expressed as follows: for 𝑖=1,…,𝐿𝑤_𝑖=𝑔_𝑖𝜇𝑛_𝑖𝜎²_𝑣−1𝜎²_𝑣,𝑔_𝑖𝑛_𝑖𝜎²_𝑣𝑔≥𝜇,0,^𝑖𝑛_𝑖𝜎²_𝑣<𝜇. (33) The parameter 𝜇 satisfies ^𝐿_𝑖=1𝑔_𝑖𝜇𝑛_𝑖𝜎²_𝑣−1⁺𝜎²_𝑣𝑛_𝑖=𝑃, (34) where (𝑥)⁺= max(0,𝑥). We solve the parameters 𝑤_𝑖 and 𝜇 using water-filling algorithm [5]:  input:𝐠_𝑖=(𝑔₁,…,𝑔_𝑝); 𝐧_𝑖=(𝑛₁,…,𝑛_𝑝); 𝜎²_𝑣; 𝑃 output:𝜇; 𝑚_𝑖, 𝑖=1,…,𝑝 (1) Reorder the sequence 𝐭_𝑖=𝐠_𝑖/𝐧_𝑖𝜎²_𝑣, in the increasing order, and set 𝑚=1 (2)do  ( 𝜇←𝑡_𝑚 𝜏←∑^𝑝_𝑖=𝑚√𝑡_𝑖𝜎²_𝑣𝑛_𝑖/𝑃+∑^𝑝_𝑖=𝑚𝜎²_𝑣𝑛_𝑖 𝑚=𝑚+1) while  (𝜇<𝜏 and 𝑚≤𝑝) (3)𝜇←𝜏 𝜔_𝑖←𝜎²_𝑣(𝑔_𝑖/𝜇𝑛_𝑖𝜎²_𝑣−1)⁺After we obtain Σ√𝐰=diag(√𝑤₁,…,𝑤_𝑝), we can write coding matrix 𝐀_𝑙 as 𝐀_𝑙=𝐇_𝑙⁻¹Σ√𝐰𝐔_𝑅𝑙. (35) We also define a lower bound from (25); as 𝑃→∞, we have 𝑤_𝑖→∞ as 𝐷_low=2𝑝𝜎²_𝑠−𝜎⁴_𝑠^𝑝_𝑖=1𝑓_𝑖+𝑔_𝑖. (36)

4.2. Equal Power Method

Under equal power strategy, the transmit power for all sensors is set to be equal as follows: 𝑃_𝑙=𝑃𝐿,1≤𝑙≤𝐿, (37) where 𝑃_𝑙 is the transmit power for the 𝑙th sensor. Then, we have the 𝑙th coding matrix, 𝐀_𝑙=𝜆_𝑙𝐈_𝑁0, where 𝜆_𝑙=𝑃/[𝐿tr(𝐑_𝑥𝑙(1∶𝑁,1∶𝑁))], so that tr(𝐀_𝑙𝐑_𝑥𝑙𝐀^𝐻_𝑙)=𝑃_𝑙, 1≤𝑙≤𝐿. The 𝐑_𝑥𝑙(1∶𝑁,1∶𝑁) denotes the first 𝑁 rows and columns of 𝐑_𝑥𝑙 and [𝐈_𝑁0] 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 use 𝑃 to denote the total transmit power constraint across the network. In all simulations, the random vectors 𝐬, 𝐧_𝑙, 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 𝐇_𝑙 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 𝐷_𝑡 due to mathematical tractability.

Figure 3 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 parameters 𝜌=0, 𝐿=10, 𝑘=8, 𝑁=5, 𝑝=5.

Figure 4 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 parameters 𝜌=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 versus 𝑘 with parameters 𝜌=0.7, 𝐿=10, 𝑁=5, 𝑝=5, 𝑃=5, 10, 20 dB.

From Figure 6, 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 versus 𝐿 with parameters 𝜌=0.7, 𝑘=8, 𝑁=5, 𝑝=5, 𝑃=1, 2, 5, 8, 15, 20 dB.

In Figure 7, 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, 𝐷_𝑜=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, 𝐷_𝑐=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 parameters 𝜌=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.