Optimal Power Allocation for Distributed BLUE Estimation with Linear Spatial Collaboration Mohammad Fanaei, Matthew C. Valenti Abbas Jamalipour, and Natalia A. Schmid Dept. of Computer Science and Electrical Engineering West Virginia University, Morgantown, WV, U.S.A. School of Electrical and Information Engineering University of Sydney, NSW, Australia IEEE ICASSP 04 Track : Waveforms and Signal Processing May 8, 04 M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration /
Outline Introduction and Problem Statement System Model Description Optimal Spatial-Collaboration Scheme 4 Numerical Results 5 Conclusions M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration /
Outline Introduction and Problem Statement Introduction and Problem Statement System Model Description Optimal Spatial-Collaboration Scheme 4 Numerical Results 5 Conclusions M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration /
Introduction and Problem Statement What Is Distributed Estimation in WSNs? Spatially distributed sensors Observe their surrounding environment. Process their local observations. Send their processed data to a fusion center (FC). FC performs the ultimate global estimation. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 4/
Introduction and Problem Statement Background and Previous Works I n g v n w y g v r θ n n w w y r g v y r g v Fusion Center ˆ w y r Cui, Xiao, Goldsmith, Luo, and Poor, IEEE T-SP, 55 (9), September 007. Maşazade, Niu, Varshney, and eskinoz, IEEE T-SP, 58 (9), Sept. 00. n Ribeiro and Giannakis, IEEE T-SP, 54 (), March 006. w r Fanaei, Valenti, and Schmid, MILCOM, November 0. n y w g v M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 5/ g v
Introduction and Problem Statement Background and Previous Works I n g v n w y g v r θ n n w w y r g v y r g v Fusion Center ˆ w y r How can we etend this system model? Inter-sensor collaboration for signal processing. n Estimation of a vector of signals at the FC. w r n y w g v M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 5/ g v
Introduction and Problem Statement Background and Previous Works II n g v w y r n w g v θ n n w w w w w y r g v y r g v Fusion Center ˆ w y r S. ar and P. Varshney, Linear coherent estimation with spatial collaboration, IEEE Transactions on Information Theory, vol. 59, no. 6, pp. 5 55, June 0. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 6/
Introduction and Problem Statement Background and Previous Works II n g v θ n w θ n θ n w g v y r g v y r g v Fusion Center ˆ ˆ ˆ ˆ w y r θ w y r n I. Bahçeci and A. handani, Linear Linear Spatialestimation of correlated data in wireless sensor networks with Collaboration θ optimum power w allocation y and analogr modulation, IEEE Transactions on Communications, vol. 56, no. 7, pp. 46 56, July 008. n w θ w y r ˆ M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 6/ g g v v ˆ
.. θ Introduction and Problem Statement w y...... Our Proposed Research Problem n Linear Spatial Collaboration w θ n w θ w n θ n w w w w θ Goal of This Paper w. y y y g g g g v v v v r. r r y r r Fusion Center ˆ ˆ ˆ ˆ Adaptive power allocation to local sensors for estimation of the vector of signals observed by them at the FC of a WSN, when sensors perform spatial linear collaboration and communicate with the FC through parallel fading channels. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 7/
Introduction and Problem Statement Our Proposed Research Problem n Linear Spatial Collaboration w θ n w M θ w n θ n w w w M w θ Goal of This Paper w M... M y y. y M g g. g M v v v v M r r. r M Fusion Center ˆ ˆ ˆ ˆ Adaptive power allocation to local sensors for estimation of the vector of signals observed by them at the FC of a WSN, when sensors perform spatial linear collaboration and communicate with the FC through parallel fading channels. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 7/
Outline System Model Description Introduction and Problem Statement System Model Description Optimal Spatial-Collaboration Scheme 4 Numerical Results 5 Conclusions M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 8/
System Model Description System Model Description n Linear Spatial Collaboration θ w n w M θ w n θ n w w M M... y y.... y M g g g M v r v r v M... r M Fusion Center ˆ ˆ ˆ ˆ θ w M Linear noisy observation of signals of interest θ [θ,θ,...,θ ] T with auto-correlation matri R θ E [ θθ T ]. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 9/
System Model Description System Model Description n Linear Spatial Collaboration θ w n w M θ w n θ n w w M M... y y.... y M g g g M v r v r v M... r M Fusion Center ˆ ˆ ˆ ˆ θ w M Sensors share their observations with each other through low cost, error-free links. Inter-sensor connectivity is modeled by an M-by- adjacency matri A. 0 0 0 A = 0 0 0 M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 9/
System Model Description System Model Description n Linear Spatial Collaboration θ w n w M θ w n θ n w w M M... y y.... y M g g g M v r v r v M... r M Fusion Center ˆ ˆ ˆ ˆ θ w M Sensors share their observations with each other through low cost, error-free links. Inter-sensor connectivity is modeled by an M-by- adjacency matri A. A is not necessarily symmetric. A j, j =. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 9/
System Model Description System Model Description n Linear Spatial Collaboration θ w n w M θ w n θ n w w M M... y y.... y M g g g M v r v r v M... r M Fusion Center ˆ ˆ ˆ ˆ θ w M Each connected sensor to the FC forms a linear combination of all local observations to which it has access: y j = w j,i i i= A j,i = Linear local processing (LINEAR COLLABORATION) as y = W Miing matri: W j,i = 0 if A j,i = 0 and W j,i = w j,i if A j,i =. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 9/
System Model Description System Model Description n Linear Spatial Collaboration θ w n w M θ w n θ n w w M M... y y.... y M g g g M v r v r v M... r M Fusion Center ˆ ˆ ˆ ˆ θ w M Average cumulative transmit power of the entire network is P Total = E [ y T y ] = E [ T W T W ] = Tr [ E [ yy T ]] = Tr [ W(R θ + R n )W T ] M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 9/
System Model Description System Model Description n Linear Spatial Collaboration θ w n w M θ w n θ n w w M M... y y.... y M g g g M v r v r v M... r M Fusion Center ˆ ˆ ˆ ˆ θ w M Orthogonal channels corrupted by fading and (potentially correlated) AWGN. r = Gy + v = GW + v = GWθ + GWn + v G diag(g,g,...,g ) The FC has perfect knowledge of the channel fading coefficients. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 9/
System Model Description System Model Description n Linear Spatial Collaboration θ w n w M θ w n θ n w w M M... y y.... y M g g g M v r v r v M... r M Fusion Center ˆ ˆ ˆ ˆ θ w M The FC finds the best linear unbiased estimator (BLUE) for θ. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 9/
System Model Description BLUE Estimation of θ at the FC r = Gy + v = GW + v = GWθ + GWn + v Since n v, given a realization of θ [θ,θ,...,θ ] T and G diag(g,g,...,g ): r {θ,g} N ( GWθ,GWR n W T G T ) + R v The BLUE estimator of θ at the FC is θ = (W T G T ( GWR n W T G T ) ) + R v GW W T G T ( GWR n W T G T + R v ) r The covariance matri of the BLUE estimator is R θ = (W T G T ( GWR n W T G T ) ) + R v GW S. M. ay, Fundamentals of Statistical Signal Processing: Estimation Theory, First Edition, NJ: Prentice Hall, 99 (Chapter 6). M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 0/
Outline Optimal Spatial-Collaboration Scheme Introduction and Problem Statement System Model Description Optimal Spatial-Collaboration Scheme 4 Numerical Results 5 Conclusions M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration /
Optimal Spatial-Collaboration Scheme Problem Formulation Problem Statement for Our Optimal Power-Allocation Scheme Derive the optimal miing matri W that minimizes the total distortion in the estimation of θ at the FC, given a constraint on the average cumulative transmit power of local sensors. minimize W Tr [W T G T ( GWR n W T G T ) ] + R v GW subject to Tr [ W(R θ + R n )W T ] P 0 M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration /
Optimal Spatial-Collaboration Scheme Derivation of the Optimal Power Allocation I Lemma minimize Tr [W T G T ( GWR n W T G T ) ] + R v GW W subject to Tr [ W(R θ + R n )W T ] P 0 A lower bound on Tr [ R θ] can be found as Tr [ ] R θ [ ] Tr W T G T (GWR n W T G T + R v ) GW M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration /
Optimal Spatial-Collaboration Scheme Derivation of the Optimal Power Allocation I Lemma minimize Tr [W T G T ( GWR n W T G T ) ] + R v GW W subject to Tr [ W(R θ + R n )W T ] P 0 A lower bound on Tr [ R θ] can be found as Tr [ ] R θ [ ] Tr W T G T (GWR n W T G T + R v ) GW maimize W Tr [W T G T ( GWR n W T G T ) ] + R v GW subject to Tr [ W(R θ + R n )W T ] P 0 M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration /
Optimal Spatial-Collaboration Scheme Derivation of the Optimal Power Allocation II maimize Tr [W T G T ( GWR n W T G T ) ] + R v GW W subject to Tr [ W(R θ + R n )W T ] P 0 Lemma The above optimization problem is equivalent to minimize γ W,γ,Γ subject to Tr [ W(R θ + R n )W T ] P 0 ( Γ R n R n Tr[Γ] γ W T G T R v ) GW + R 0 where γ is a real scalar, and Γ is a symmetric -by- real matri. n M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 4/
Optimal Spatial-Collaboration Scheme Derivation of the Optimal Power Allocation III Lemma The above optimization problem is equivalent to minimize γ W,γ,Γ subject to Tr [ W(R θ + R n )W T ] P 0 ( Γ R n R n Tr[Γ] γ W T G T R v ) GW + R 0 where γ is a real scalar, and Γ is a symmetric -by- real matri. n IDEAS FOR PROOF: Woodbury matri inversion lemma. Schur s complement theorem. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 5/
Optimal Spatial-Collaboration Scheme Derivation of the Optimal Power Allocation III Lemma The above optimization problem is equivalent to minimize γ W,γ,Γ subject to Tr [ W(R θ + R n )W T ] P 0 ( Γ R n R n Tr[Γ] γ W T G T R v ) GW + R 0 where γ is a real scalar, and Γ is a symmetric -by- real matri. Linear programming with bi-linear matri-inequality constraints. Solved using numerical solvers such as PENBMI. n M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 5/
Outline Numerical Results Introduction and Problem Statement System Model Description Optimal Spatial-Collaboration Scheme 4 Numerical Results 5 Conclusions M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 6/
Numerical Results Simulation Setup = 6 sensors are randomly and uniformly distributed in the twodimensional rectangle of [ 0, 0] [ 5, 5]. All sensors are connected to the FC, i.e., M =. σθ = : The variance of each component of θ. ρ i, j : Inter-sensor correlation coefficient. Monotonically decreases with the increase of the distance between sensors ( ) β di, ρ i, j e j β d i, j : Distance between sensors i and j. β = 6: Normalizing factor of the distances. β = : Controls the rate of the decay of the correlation coefficients. The covariance between the signals observed by sensors i and j R θi, j E[θ i θ j ] = σ θ ρ i, j M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 7/
Numerical Results Simulation Setup Homogeneous and equi-correlated vector of observation and channel noises with covariance matri defined as R n(v) = σ n(v) [( λn(v) ) I(M) + λ n(v) T ], σn(v) : Variance of each component of the vector of observation (channel) noises. λ n(v) : Constant correlation coefficients between each pair of distinct components of n (v). : Column vector of all ones with appropriate length. σ n = 0., σ v = 0.0, and λ n = λ v = 0.. The communication channels between local sensors and the FC have unit gain (g j = ). Each sensor collaborates with its q closest neighbors by sharing its local noisy observations through error-free, low cost links. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 8/
Numerical Results Effect of Linear Spatial Collaboration on Performance Total Estimation Distortion 0.9 0.8 0.7 5 0 Network -5 5 0 Network -5-0 -5 0 5 0 q=0 No Collaboration q= (Network ) q= (Network ) q= (Network ) q= (Network ) 0 4 6 8 0 4 Average Cumulative Transmission Power (P ) in db 0 Each sensor collaborates with its q closest neighbors by sharing its local noisy observations through error-free, low cost links. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 9/
Outline Conclusions Introduction and Problem Statement System Model Description Optimal Spatial-Collaboration Scheme 4 Numerical Results 5 Conclusions M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration 0/
Summary Conclusions We studied the effect of spatial linear collaboration on the performance of the BLUE estimator. The FC estimates the vector of spatially correlated signals observed by sensors. Even a small degree of connectivity and spatial collaboration could improve the quality of the estimators at the FC. The collaboration gain is more significant when the signals to be estimated have higher correlation. M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration /
Conclusions Questions Thank You for Your Attention. Questions and/or Comments? M. Fanaei et al. Power Allocation for () Distributed BLUE Estimation with Linear Spatial Collaboration /