IEEE CONECCT2014 1569821995

Optimal Transmit Precoding for Distributed

Estimation in Correlated Wireless Sensor Networks

Abhilash Goyal, Koustav Dey, Aditya K. Jagannatham Department of Electrical Engineering Indian Institute of Technology Kanpur

Email: {akgoyal. koustav. ajaganna} @iitk.ac . in

Abstract-In this paper we present optimal transmit pre­coding schemes for distributed estimation in a correlated wireless sensor network. The presented schemes employ linear pre­processing with multiple transmit antennas and are based on a coherent MAC between the multiple sensors and the fusion center. In this context, we derive the closed form expression for the optimal transmit processing scheme which minimizes the variance of the estimate at the fusion center. Further, we also develop a low complexity transmit scheme for scenarios with partial Channel State Information (CSI) availability. Closed form expressions are also derived to characterize the asymptotic variance of the optimal full CSI, partial CSI and other suboptimal transmit processing schemes. Simulation results are presented to demonstrate the performance of the proposed schemes for distributed estimation in a wireless sensor network.

I . INTRODUCTION

In the recent years, Wireless Sensor Networks (WSNs) have attracted a significant interest due to their wide-spread applicability in defense, civilian and environment monitoring scenarios. A typical WSN consists of a dense network of sensor nodes interconnected by wireless links that are used to measure several physical phenomena such as temperature, pressure, humidity etc . One of the key aspects in a WSN is the significant degree of spatial and temporal correlation in the observations across the sensor nodes. This correlated nature of WSN measurements can be successfully employed to derive optimal estimation schemes for efficient utilization of the physical resources such as bandwidth, power etc . A popular architecture to use this intrinsic correlation across the sensor network is the hierarchical WSN in which the nodes transmit their measurements over the wireless channel for joint estimation of the physical parameter at the fusion center, which significantly improves the accuracy of estimation in such WSNs. This paradigm of cooperative estimation of the parameter at the fusion center from the collective measure­ments of the constituent sensor nodes is termed as Distributed Estimation. Also, a typical wireless link is highly error-prone due to the fading nature of the wireless channel resulting from the mUltipath scatter environment. Therefore, multiple antenna systems such as Multiple-Input Single-Output (MISO) and Multiple-Input Multiple-Output (MIMO) have become popular for wireless applications due to their robustness to the channel fading arising from the diversity properties of such systems.

Several works in existing literature have focused on efficient distributed! cooperative estimation schemes in WSNs subject

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to various constraints such as limited channel state information (CSI), fading channel statistics , bandwidth, cost, limited sensor node power etc. In [ 1 ] the authors consider a WSN with MISO sensor nodes measuring scalar parameters and derive expressions for the minimum variance of the estimate in terms of the amplification factors and channel coefficients of the sensors, and contrast it with the variance corresponding to equal power allocation amongst the sensors . In [2] the optimal Maximum Likelihood (ML) parameter estimate is presented for a quantization based WSN wherein each sensor transmission is restricted to a single bit. A scheme for MSE minimization in distributed WSN estimation scenarios for scalar and vector signal sources with full CSI is presented in [3] . The paper [4] develops several optimal WSN estimation schemes and characterizes their performance for different levels of CSI availability such as full, partial and no CSI. However, one fundamental advantage of [4] over [3] is that schemes in the former lead to an unbiased estimate of the signal at the fusion center. A drawback of the works above is that the presented estimation algorithms involve processing at both the sensor nodes as well as the fusion center. In [5] , the authors consider a WSN system model similar to the ones in [3] and [4] , but focus on developing the optimal distributed compression and estimation schemes towards maximizing the bandwidth efficiency, scalability, and adaptability to network changes. Similarly, [6] computes the ML estimate of the parameter vector in a WSN, for both analog as well as digital pre-processing at the sensor nodes.

In contrast to the above schemes which require processing at both the transmitter and receiver in a WSN, we propose schemes for multi-antenna wireless sensor nodes which require only transmit processing . Further, while the work in [4] considers identical noisy measurements across the wireless sensor nodes, our framework considers the realistic scenario of correlated sensor measurements. Moreover, while the work in [7] employs the WSN spatial and temporal correlation to over­come the wireless bandwidth and power constraints towards efficient communication, [8] focuses on prediction of missing sensor measurements, thereby avoiding retransmission which in turn leads to increased bandwidth efficiency. In our work we consider an arbitrary correlation amongst the wireless sensor nodes and derive the optimal transmit pre-coding schemes with full and partial CSI for minimum variance estimation. We derive closed form expressions to characterize the asymptotic

variance of the proposed estimators . In addition, expressions are derived for the estimation variance with uniform power allocation to illustrate the performance enhancement with the optimal transmission scheme. Towards the end we present simulation results to demonstrate the performance of the proposed WSN distributed estimation schemes.

The rest of the paper is organized as follows. In Section II we present the system model and the associated framework for distributed transmit precoding in the wireless sensor network. In sections III and IV we derive the optimal transmit precoders for the full and partial CSI scenarios respectively, along with the expressions for the variance of the estimates. For comparison, the performance of the estimates with uniform power allocation for full and partial CSI is derived in sections V and VI respectively. Simulation results are presented in section VII followed by the conclusion in VIII.

II . SYSTEM MODEL

We consider a WSN comprising of L sensors, each having t transmit antennas . Let the lh correlated measurement Yji at the ith sensor for the physical parameter e be given as,

where, aji denotes the correlation degree and 1 ::; i ::; L and 1 ::; j ::; t . The quantity nji denotes the additive white Gaus­sian noise (AWGN) with E{ l nji I 2 } = a"J;. Let ,Bjidenote the pre-coding coefficient and hji denote the channel coefficient corresponding to the lh antenna of the ith sensor. Hence, the net received signal Y at the fusion center can be expressed as,

L t Y = L L h;i (,Bji (ajie + nji ) ) + u, ( 1 )

i= l j=l where u denotes the AWGN of variance a; at the fusion center. Let hi E Ct x l be defined as hi = [hil , hi2 , . . . , hit ] T denote the vector of fading coefficients between the ith sensor and the fusion center. Let the precoding matrix ID (f3i ) E ct x t be defined as, [ ,Bil

ll) (13, ) � I o

,Bi2

o

o o

where f3t denotes the precoding vector f3i [,Bd , ,B,2 , . . . , ,Btt ] T and ID (v) denotes the diagonal matrix with vector v as the principal diagonal . Hence, the signal received at the fusion center Y given in ( 1 ) can be expressed as,

L Y = L h{iID (f3i ) (Oie + lli ) + U ,

i= l T ]T where 0i = [ai l , ai2 , · · . , ait ] and lli = [nil , ni2 , · · . , nit is the concatenated noise vector. The distributed sensing sys­tem model and the operation at each sensor are schematically described in Fig . l and Fig.2 respectively. Observe that for

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any two vectors v, u we have, ID (v) u = ID (u) v. Hence, the expression for Y above can be recast as,

Y = (t, h{iID (Oi ) f3i) e + t, h{iID (lli ) f3i + u.

Defining Wi = ID (Oi ) hi and Ni = ID (lli ) ' we have,

L L Y = L w{i f3ie + L h{iNif3i + u. (2)

i= l i= l Finally, defining the concatenated [ T T T] T rrLt x l h WI , W2 , . . . , W L E \L- , = [ T T T] T f3 = f3l , f32 , . . . , f3 L and,

N

� [ ll)r ll) (�, )

vectors, W [ T T T] T hI , h2 , . . . , hL '

the expression for Y can be succinctly expressed as,

Y = wH f3e + hHNf3 + u. (3)

In the subsequent sections, we derive the optimal precoding vector f3 considering full and partial CSI availability at the wireless sensor nodes. In the discussion that follows we refer to the components of the vectors w, h, f3 , 0 with a single index k , 1 ::; k ::; Lt for convenience of notation. The indices i , j such that hji = hk etc . can be derived as i = I � l , where i · l denotes the ceiling function and j = k - (i - 1 ) t .

III . OPTIMAL PRECODING FOR FULL CSI

To derive the optimal precoding vector f3, we begin by deriving an expression for the effective noise covariance R = E {NHhhHN} as,

= E {NHhhHN}

h2hin'2nl = E

hl h'2n'jn2 I h2 1 2 1 n2 1 2

{ [ I h l l 2 1 n l l 2

hLt hi:n'Ltnl hLt h'2n'Ltn2

= ID (h 8 h* ) � ,

hl h'Ltn'jnLt 1 } h2h'Ltn'2nLt

I hLt l 2 1 nLt l 2 (4)

where 8 denotes the element wise product of vectors, and � = ID ( [ar , ar , . . . , alt ] T) . From (2) it can be seen that the noise

power at the receiver is given as f3H Rf3 + a; . Considering a signal power of P at the fusion center, the optimal precoding vector which minimizes the noise power is given as,

min f3HRf3 s.t . wH f3 = VP

From [9] it can be seen that the optimal precoding vector f3 which is a solution of the above optimization problem is given as,

,B = .J]5.R- lw wHR- lw (5)

Sensor 1

Sensor 2 Source

Sensor L

Fusion Center 1-1 ----1�� () Estimate

Fig. 1 . WSN Distributed Estimation Over Fading Coherent MAC.

Zih sensor with t - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - --antenn-as--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Precode �i)(a/lj+n) i ...

S

D(II,)(.,D,+� our;: Y ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Fusion Center

Fig. 2. Schematic Model of Processing at ith Sensor.

Below we compute the asymptotic variance of the parameter estimate corresponding to the optimal precoding vector {3. Observe that /j, the unbiased estimate of the parameter e is given as,

Hence the variance of the estimate /j is given as,

(6)

Observe that the expression for w can be simplified as

Employing the expression of w above along with the expres­sion for R from (4) in (5), the optimal vector {3 can be simplified as,

Q[A h[A I hLt l 2 o-L,

(7)

3

Employing the expression for {3 from above in (6), the variance of the estimate /j at the fusion center can be simplified as,

(8)

IV. OPTIMAL PRECODING FOR PARTIAL CSI

For the partial CSI scenario, similar to [4] , we consider the availability of the phase 'l/Ji of the channel coefficient hi between the ith antenna and the fusion center. Hence, the channel coefficient can be expressed as hi = 1 hi 1 eNi . In

addition, let E ( I hi I ) = Pi and E ( I hi 1 2 ) = 6r . Considering an

average power P at the fusion center, the optimization problem to compute the optimal precoding vector {3 in the presence of partial CSI is given as,

min {3HR{3 S.t . E (wH (3) = VP,

where the effective noise covariance matrix with partial CSI ii is defined as,

ii = E {NHhhHN} = E (ID (h 8 h* ) ) � [ 8'a' 0 0

1 1 1 0 6§a§ 0

(9)

0 0 6Italt Further the quantity E (w) can be simplified as,

V. UNIFORM POWER ALLOCATION WITH FULL CSI

Consider a total power of P for the distributed estimation system. For uniform transmit power allocation, since the WSN comprises of a total of L sensors and t transmit antennas per node, the individual power Pi , 1 � i � Lt is given as, Pi = E. Thus we have 13 · = f3- eNi where Lt · t t ,

E (w) =E (ID (a ) h) =

[ alPl eJ'P l a2P2eJ'P2 aLtPL:t eJ'PLt

( 10) Employing the above expression for the precoding coefficient f3i in (3) we have,

From [9] it can be seen that the optimal precoding vector (3, which is a solution of the above optimization problem is given as,

13 = VPii-lE (w)

E (wHii-1 ) E (w) ( 1 1 )

Substituting the expression for ii from (9) and E (w) from ( 10) in the above equation, the resulting expression for (3 can be simplified as,

Using the expression of (3 from above, the variance of the estimate e at the fusion center with partial CSI can be derived as,

Next we consider the performance of distributed WSN esti­mation of the parameter 8 with uniform power allocation.

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1ft Lt 1ft Lt ni y = - 2: 8 + - 2: - + u

Lt Lt ai i= l i= l 1ft Lt = V P LiB + - '" ni

+ u. Lt � a i= l t

Hence, the estimate e so obtained is,

Finally, the variance of the estimate e is given as,

' 2 1 a · a ( Lt 2 ) 2 E { 1 8 - 8 1 } = (Lt) 2 8 a� + pi

t · ( 1 4)

VI . UNIFORM POWER ALLOCATION WITH PARTIAL CSI

In this section we derive the expressions for the variance of distributed WSN estimation with uniform power allocation and partial CSI availability. Similar to the previous section, considering a total power of P, we have f3i = [JieN" where,

Substituting the above expression for f3i in (3) , the received signal y at the fusion center is given as, 1ft Lt 1ft Lt

Y = P '" � 8 +

P '" � ni + u.

Lt � 6 Lt � 6 a i= l t i= l t t

Hence the estimate 8 at the fusion center is given as,

8 = ---,-Y_-/P � 0il V Lt ·� 6 i=1 ' Lt L 0il ni i= 1 6i ai {Lt u

= e + Lt + V P Lt . " 0::J. " 0il � � � 6 · i=1 i=1 "

Calculating the variance of the estimate 8, E { 1 8 - e 1 2 } Lt E ( l hi I 2 ) E (n7 ) L 62a2 Lt E (u2) ..:.i=---"-1 ___ '_"----,, __ + _ __ -'------'------,;-( Lt 0il) 2 P ( Lt 0il) 2

L6

L6 i=1 ' i=1 I

( 1 5)

Lt Using the weak law of large numbers, we have L I�, I --+

i=1 " LtJ , where E ( l hi l ) = p and E ( l hi I 2 ) = 62 for ali I :s; i :s; Lt. Substituting in ( 1 5) , we have, ( 6 ) 2 ( Lt 2 ) 1 62

E { 1 8 - e 1 2 } = Ltp � :} + LtP p2 · ( 1 6)

VII . S IMULATION RESULTS

We consider a WSN with L = 20 sensors and t = 5 transmit antennas at each sensor. The channel coefficients hi between the sensor nodes and the fusion center are Rayleigh fading with average power E { I hi 1 2 J = 6; = 1 . The noise power at the fusion center is set as au = 1 . Similarly, the observation noise power considered is a; = 1 . The coefficients ai are generated uniform randomly in the interval [0 , 1 ] . The power P at the fusion center is varied over the range -5 to 1 5 dB . The Mean-Squared Error (MSE) of the estimate 8 achieved by the optimal transmit precoding with full CSI and partial CSI at the sensor nodes is compared with that of the uniform power allocation amongst the transmit antennas .

Fig .3 illustrates the MSE performance of the proposed optimal transmit precoding and uniform power allocation with full CSI. It can be observed that the proposed optimal precoding scheme has a superior performance in comparison to the uniform power allocation scheme. Further, the theoretical MSE values corresponding to the optimal and uniform power allocation derived in (8) and ( 14) respectively, are plotted in the figure. The plots clearly show that the analytical results match the simulated results . Also, as the power P --+ 00, the MSE attains a constant error floor because the impact of the 2 noise at the fusion center, j; --+ O.

Fig .3 also shows the MSE performance of the optimal transmit precoding scheme and uniform power allocation for the case of partial CSI at the sensor nodes. The optimal transmit precoding scheme achieves a lower MSE compared

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�lOlorm Parti.eSI Sin'ated

-+- lOlorm Parti. eSIAMI,Ic'

� lOlorm F� CSI �"'ated

� lOlorm F� eSI M1Ijtit'

�Optimai Parti' CSI Si"'ated

- Optimai Parti, CSI Maljtit'

-+- OptimaI F� eSI Sin'ated

-+- Optimal F� eSI M8\lt'

Fig. 3. Analytical and simulated MSE of distributed WSN estimation for optimal transmit precoding and uniform power allocation with full and partial CSl versus power P.

to the uniform power allocation scheme. It can be observed that the asymptotic MSE is higher due to the partial channel information at the sensor nodes. The plot also shows that the analytical MSE values computed using the expression ( 1 3) for optimal transmit precoding match the values from simulation. The analytical values for the uniform power al­location, computed from the expression ( 1 6) , are close to the values obtained from simulation. The minor deviation of the analytical results from the simulated values is due to the approximation involving the weak law of large numbers employed in deriving the expression.

Finally, in the same figure one can compare the performance of the optimal transmit precoding scheme with full and partial CSI at the sensor nodes. The plot clearly shows that for a given transmit power P, the optimal transmit precoding performs better than uniform power allocation with full CSI, validating the claim that the proposed schemes exploit all the available information in contrast to the uniform power allocation scheme.

Fig .4 compares the performance of the proposed scheme and the uniform power allocation scheme with varying number of total transmit antennas Lt in the range, 1 :s; Lt :s; 25, in the WSN. The total power P is taken to be lOdB for the purpose of this simulation. It can be observed that a lower MSE can be achieved as the total number of transmit antennas in the WSN increases . Further, the performance gap between the optimal transmit precoding and uniform power allocation increases as we add more sensors or transmit antennas to estimate the sensing parameter. Also, similar to the result in Fig .3 , the performance with full CSI is better compared to that with partial CSI for both uniform and optimal precoding. As expected, the best performance corresponds to the optimal full CSI scenario while the worst performance to that of uniform power allocation with partial CSI.

--+-Optimal Full e51 -+-Uniform Full e51 -+-Uniform Partial eSI -+- Optimal Partial e51

10 It ·· · · · · · · · · · · . . . .•••• •• •• . .. . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . • . .•.• . . .•. .","""' • . . . . . . , • .. . . . . . . . . . "''''''t-..... , .. ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . �

10·'L------'-------':--------':--------L----25

Fig. 4. MSE comparison of optimal transmit precoding and uniform power allocation with full and partial CSI versus the total number of transmit antennas Lt.

VIII . CONCLUS ION

In this work, optimal transmit pre-coding schemes were pre­sented for distributed estimation in a correlated wireless sensor network. The proposed schemes minimize the variance of the estimate at the fusion center employing linear pre-processing at the sensor nodes having multiple transmit antennas and a coherent MAC between the multiple sensors and the fusion center. A low complexity transmit scheme was developed for the scenarios when only partial channel State information (CSI) is available at the sensor nodes. The closed form

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expressions derived to characterize the asymptotic variance of the estimate for the optimal precoding scheme and uniform power allocation with full and partial CSI are in agreement with the simulation results. The results also demonstrate the superiority of the proposed distributed estimation schemes.

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