Optimal Quantization in Energy-Constrained Sensor Networks under Imperfect Transmission

Transcription

1 Optimal Quantization in Energy-Constrained Sensor etworks under Imperfect Transmission Li Zhang l,, Tao Cui l, Tracey HOI Department of Electrical Engineering California Institute of Technology Pasadena CA 95, USA Xian-Da Zhang Department of Automation, Tsinghua University Beijing 00084, China Abstract-This paper addresses the optimization of quantization at local sensors under strict energy constraint and imperfect transmission to improve the reconstruction performance at the fusion center in the wireless sensor networks (WSs) We present optimized quantization scheme including the optimal quantization bit rate and the optimal transmission power allocation among quantization bits for BPSK signal and binary orthogonal signal with envelope detection, respectively The optimization of the quantization is formulated as a convex problem and the optimal solution is derived analytically in both cases Simulation results demonstrate the effectiveness of our proposed quantization schemes I ITRODUCTIO Parameter estimation by a set of distributed sensors and a fusion center (FC) is frequently encountered in the wireless sensor network (WS) applications [] Constrained by limited power and bandwidth resources, sensors should perform local quantization to their observations before transmission [] At the FC, sensor observations are reconstructed and combined to produce a final estimate of the unknown parameter Some previous works [3]-[5] assume error-free transmission between sensors and the FC [6]-[ 0] model the wireless links between sensors and the FC as additive white Gaussian noise (AWG) channels [6] derives an estimate scheme based on I-bit quantized observation The optimal power scheduling problems among sensors are investigated in [7] [8] with different assumptions about the sensor observation noise Training sequence is used in [9] to estimate the unknown channel Different from these works, this paper addresses the optimization of the quantization scheme per sensor including the quantization bit rate and the transmission energy allocation among quantization bits under strict energy constraint and imperfect transmission A quantization scheme is proposed in [0], which minimizes the upper bound of the reconstruction mean-absolute error (MBE), which is called the MBE quantization scheme below In contrast, we propose a quantization scheme which minimizes an upper bound of the reconstruction MSE for BPSK signal and binary orthogonal signal with envelope detection, respectively The optimal transmission power allocation among This work was supported in part by the Major Program of the ational atural Science Foundation of China under Grant and Caltech's Lee Center for Advanced etworking quantization bits is formulated as a convex problem and the optimal solution is derived analytically in both cases Simulation results show that the proposed quantization schemes achieve significant reduction in reconstruction MSE compared with the MBE quantization scheme and the quantization scheme with uniform energy allocation for either BPSK signal or binary orthogonal signal with envelope detection II PROBLEM FORMULATIO We consider that each sensor in the network makes a local observation y and wishes to transmit it to FC Suppose that the observation y is bounded to [-T, T] with variance 0', where T is known and decided by the sensor's dynamic range We can express y as 00 y = TL bn- n - T () Here, we consider the -bit uniform quantization, which is widely used in energy-constrained sensor networks Denote y as the -bit quantized version of y, where y = TL bn- n - T Due to the constrained power, only the quantization bits {b n };;=l will be transmitted to the FC We model the wireless link between the sensor and the FC as an AWG channel Here, the uncoded transmission strategy is used Denote the total transmission energy as eand the channel noise variance as O' At the FC, the total receiving signal-to-noise ratio (SR) is defined by = e/o' We assume the transmission energy for b n is a fraction X n of the total energy efor n =,,, and the quantization bits experience independent channels Denote b as the demodulated version of b n at the FC Then the SR of b is X n Denote Pn as the bit error rate of bn, that is Pn = P(b == bn) At the FC, y can be reconstructed as y' = TLb-n -T (3) Our goal in this paper is to optimize the quantization scheme under the total energy constraint e in order to minimize () /09/$ IEEE 63 Authorized licensed use limited to CALIFORIA ISTITUTE OF TECHOLOGY Downloaded on April,00 at UTC from IEEE Xplore Restrictions apply

2 the reconstruction MSE at the FC The quantization scheme includes the quantization bit rate and the transmission energy allocation among bits x = [Xl,,X] III OPTIMAL QUATIZATIO SCHEMES Firstly, we obtain an upper bound of the reconstruction MSE At the FC, the reconstruction error can be expressed as t PnPm Tn - m ;t Pn Tn (pni T m + t T m ) n#m m=l m=n+l = L [Pn - n (- n - - ) + p-n( - - n )J y-y' =(y-y)+(y-y'), (4) Substituting () into (9), we have () where the first part is the quantization error and the second part is the demodulation error Using (4) and the Cauchy-Schwartz inequality, we can bound the reconstruction MSE as E y - y ' ; E Iy - yl + E y - y ' According to () and (3), we obtain Ely - yr = 4T E It,(bn - b)tni (5) (6) Ely - y' I ; 4T L [Pn - n (- n - - ) +p-n( - - n )J () It has been shown in [5] that the quantization error satisfies T Ely - Yl ; a (3) According to () and (3), we can finally bound the reconstruction MSE as [cf (5)] ; 4T E (t, Ibn _ b-n) (7) (8) T E Iy - y'l ; a T L [Pn - n ( - n - - I ) + p-n( - - n )J = f(;x), (4) The power scheduling among quantization bits is constructed as follows where the last step is due to the assumption that the quantization bits experience independent channels Recalling that Ibn - b is a {O, I} Bernoulli random, ' variable, we have Elbn - bnl = Elbn - bnl = Pn Thus (8) can be expressed as ( ) Ely - y'l ; 4T LPn Tn + L PnPm Tn - m n#m (9) From (6) we find that the demodulation MSE is more dependent on the demodulation accuracy of the leading bits of y, and less dependent on the accuracy of its trailing bits Intuitively, it is reasonable to assume that the optimal quantization scheme allocates more transmission energy to the leading bits of y, and less energy for its trailing bits, that is X n if m ; n Then we have X m Pm ; Pn, if m; n, min f(; x), st X n 0, n =,, LXn =l The optimal solution of (5) and the minimum of the objective function f (; x) are functions of the quantization bit rate We denote the optimal solution of x as xiv = [xiv I' Xiv ',xiv ] and the minimum of the objective funtion s f(; x) The optimal quantization bit rate which minimizes the upper bound ofthe reconstruction MSE can then be obtained by opt = argminf(; xiv) (5) (6) When is fixed, to solve the convex problem (5), we can (0) write the Lagrangian function G as which can be easily shown by contradiction Using (0) and the fact Pm ;, we can further bound the second part of (9) as G(xiv, A, JL) = f(; xiv)+a (t, x,n - )-t, ILnX,n (7) /09/$ IEEE 64 Authorized licensed use limited to CALIFORIA ISTITUTE OF TECHOLOGY Downloaded on April,00 at UTC from IEEE Xplore Restrictions apply

3 A BPSK over AWG Channels The bit error rate Pn for BPSK signal over AWG channels is given by [] as where Q() is Gaussian tail function We then have T [ ( solution of (5) as f(; x') = 0" T L Q JX',n'Y) - n 8 f(; x ) _ T4-n _ xiv,n"y { ( ) ---- e x,n rx,n x,n [ - n Q (VXn) ( n - )J +!+( n _ )e- xn'"y} yrx,n > o It is easy to show 8(xn)/8xn < 0, which implies that (X,n) is strictly decresing with respect to x,n Due to (9), we can express the optimal solution as X,n = -(n-4(a - JLn)) (5) Since the domain of (x n) defined in (4) is (0, +(0), we must have JLn = 0 for all 'n to satisfy the complementary slackness conditions in () Therefore, we obtain the final ] X,n = -(n-l A), (- n - - )+Q(JX,n'Y) - n (- - n ) whre is a constant chosen to enforce the constraint In this case, the second derivative of f(; x ) with respect to x,n is (8) Thus the optimal problem (5) is convex with respect to x,n for BPSK signal over AWG channels The associated set of KKT conditions then read [] (9) (6) x,n = Since (x n) is monotonically decreasing with respect to X,n' = X,n is a monotonically decreasing function with respect to A Thus, we can use some efficient search algorithms to find the proper A which makes the constraint satisfied, such as the bisearch algorithm The optimal energy allocation in (6) intuitively shows that more energy should be allocated to the leading bits, while less should be allocated to the trailing ones, which indicates our original assumption in (0) is reasonable Finally, the optimal solution { opt ' x opt } can be obtained based on (6) B Binary Orthogonal Signal with Envelope Detection Consider the binary orthogonal signals such as binary frequency-shift keying (FSK) or pulse-position modulation (PPM), which can be demodulated using non-coherent envelope detection The bit error rate is [9, pp ] xiv,n"y Pn = -e--4 - Then we have T [ xiv n"y f(jx') = 0" T e--4-- n A(t, x,n - ) = 0 (0) L X,n = () JLnX,n = 0, n =,, () A 0 JLn 0 X,n 0 n =,, (3) We notice that if A = 0, from (9) it follows that JLn < 0 for all n Then using (), we have xn = 0 for all n, which violates the condition () Thus, it mst holds that A > O ow we proceed to solve the KKT systems Define function (X,n) as (x* )= Te-X,n'"Y[ - n - -,n rx*,n (4) +Q ( JX',n'Y) ( n - )] x* "Y ] ( - n - - ) + e- n - - n ( _ - n ) In this case, the second derivative of f (; x ) with respect to x,n is 8 f( x* ) [Xiv "Y * ' =T e------p-- - n (- n - - ) 8x,n +e- Xn'"Y -- n(- - n)] > O x* "Y - Te- 4n - n ( _ n- - ) - T x* "Y e- n - n ( - - n ) + A - JLn = 0 (7) Thus the optimal problem (5) is also convex with respect to x n for the binary orthogonal signals with envelope detection The associated set of KKT conditions are (8) /09/$ IEEE 65 Authorized licensed use limited to CALIFORIA ISTITUTE OF TECHOLOGY Downloaded on April,00 at UTC from IEEE Xplore Restrictions apply

4 * { 0, -X T-n( - - ); x,n= -In(](-X,n)), -X <T-n(--) (35) where if n >, ](-X, n) is defined as ](-X,n)= [T - n (- - n )]- [-T - n (- n - - ) +VT"(-4n(- n- - ) + 4A"(-n(- - n )], More concisely, we have Define (-X) as A(t, x,n - ) = (9) L X,n = (30) J-LnX,n=O,,, (3) -X 0 J-Ln 0 X,n 0 n =,, (3) ext we proceed to solve the KKT systems Multiplying both side of (8) by x n eliminates J-Ln, and we obtain the following equation ' x* '"Y - T e - 4n - n( _ n- - ) Then xlv,n is calculated as else X' n'"y X,n -Te--4--n( - [ n - - ) X' n'"y ] -Te----n( - - n ) + -X = O (33) If -X T-n( - - ), x n > 0 is impossible since if xlv n > 0 then -T-n( - -') + -X > 0 and (33) cannot hold Thus xlv n = 0 if -X T-n( - - ) ow we sho that when -X < T-n(--), J-Ln should be zero When -X < T-n( - - ), we have J-Ln < 0 if xlv n = 0 using (8), which contradicts (3) Thus if -X < T-n(--), xlv n should not be zero Then to get (3), we have J-Ln = 0 if -X '< T-n( - - ) Therefore (8) becomes X' n'"y - Te----n( - - n )+-X = 0 -X ry(a, ) = T"(( _ -)' x,n = max { 0, - In (ry(a, n)) } (A)= tmax{o,_iin(ry(a,n))} (34) (36) (37) (38) (39) Finally we show that -X > O If -X = 0, then from (8) we have J-Ln < 0 which contradicts (3) Therefore -X = 0, then (9) becomes and ( tx,n -) = O Taking (38) into (40), we have (-X) = 4 i Zl = --LIn (ry(t "(T(%+)( - T), n)), (40) (4) From (39), we know that (-X) is a piecewise-linear nonincreasing function of -X with a breakpoint at T-n(- - ), thus the equation has a unique solution The unique -X can be found by the following algorithm Algorithm ) Set i = ) Calculate If Zo ; and Zl, then go to 4), else i = i + and go to 3) 3) Search -X at [T -(i+l)( - - ), T-i( - - )] using the bisearch algorithm to satisfy 4 i -- LIn (](-X, n)) = 4) Calculate {xlvopt} as follows * { 0, n > i; x,n = _ In (ry(a, n)), ns; i (4) The algorithm leads to the global optimum since the optimal -X is unique Finally, the optimal solution { opt, x lvopt } can be obtained based on (6) In Section IV, we will find the optimal solution for specific system setups and show the simulation results IV SIMULATIO RESULTS This section presents some simulation results to demonstrate the effectiveness of the proposed quantization schemes In all simulation runs, the observed signal is chosen as Gaussian truncated to the range [-5,5], ie, T = 5 with mean The channel noise is zero-mean Gaussian distributed All simulation results are averaged from 50,000 independent runs Fig depicts the reconstruction MSE of different quantization schemes with the total receiving SR = 0 at the FC for BPSK signal over AWG channels With different quantization bit rate, the optimal energy allocation {xlv} is obtained by solving (6), and the optimal quantization bit rate is also found to be opt = 5 We plot the simulated /09/$ IEEE 66 Authorized licensed use limited to CALIFORIA ISTITUTE OF TECHOLOGY Downloaded on April,00 at UTC from IEEE Xplore Restrictions apply

5 -vr- Optimal quantization MBE quantization Uniform quantization -vr- Optimal quantization Fig Reconstruction MSE versus the quantization bit rate per sensor with different energy allocation schemes among bits for BPSK signal reconstruction MSEs with the optimal energy allocation {xiv}, the MBE energy allocation scheme [0], and the uniform energy allocation scheme, respectively Fig shows that the proposed optimal quantization scheme has lower MSE than either the MBE or the uniform energy allocation schemes for BPSK signal over AWG Channels Fig shows the performance for FSK signal over AWG channels with the total receiving SR ' = 0 at the FC With different quantization bit rate, the optimal energy allocation {xiv } is obtained by solving (35), and the optimal quantization bit rate is found to be opt = 4 Fig also demonstrates that the proposed optimal quantization scheme outperforms either the MBE or the uniform energy allocation schemes for FSK signal over AWG channels V COCLUSIO This paper addressed the optimization of the quantization scheme under strict energy constraint and imperfect transmission to improve the reconstruction MSE performance at the FC Optimal quantization schemes including the optimal quantization rate as well as the optimal transmission energy allocation, which minimizes the upper bound of the reconstruction MSE, have been proposed for BPSK signal and binary orthogonal signal with envelope detection, respectively The optimization problems have been proved to be convex and closed-from solutions have been derived in both cases Simulation results showed that the proposed optimal quantization schemes perform better than either the MBE or the uniform quantization schemes 4 6, 8 MBE quantization Uniform quantization ; > 0 4 Fig Reconstruction MSE versus the quantization bit rate per sensor with different energy allocation schemes among bits for FSK signal [] J -J Xiao, A Ribeiro, Z -Q Luo and G B Giannakis, "Distributed compression-estimation using wireless sensor networks," IEEE Trans Signal Process, vol 3, no 4, pp 7-4, Jul 006 [3] Z -Q Luo, "Universal decentralized estimation in a bandwidth constrained sensor network," IEEE Trans Inf Theory, vol 5, no 6, pp 0-9, Jun 005 [4] A Ribeiro and G B Giannakis, "Bandwidth-constrained distributed estimation for wireless sensor networks-part II Unknown probability density function," IEEE Trans Signal Process, vol 54, no 7, pp 343, Jul 006 [5] Y Huang and Y Hua, "Multihop progressive decentralized estimation in wireless sensor networks," IEEE Signal Process Lett, vol 4, no, pp , Dec 007 [6] T C Aysal and K E Barner, "Constrained decentralized estimation over noisy channels for sensor networks," IEEE Trans Signal Process, vol 56, no 4, pp , Apr 008 [7] J -J Xiao, S Cui, Z -Q Luo and A J Goldsmith, "Power scheduling of universal decentralized estimation in sensor networks," IEEE Trans Signal Process, vol 54, no, pp 43-4, Feb 006 [8] J-Y Wu, Q-Z Huang and T-S Lee, "Minimal energy decentralized estimation via exploiting the statistical knowledge of sensor noise variance," IEEE Trans Signal Process, vol 56, no 5, pp 7-76, May 008 [9] H Senol, and C Tepedelenlioglu, "Distributed estimation over parallel fading channels with channel estimation error," Proc IEEE Int Conf Acoust, Speech Signal Process (ICASSP) 008, Las Vegas, Mar Apr 4 008, pp [0] X Luo and G B Giannakis, "Energy-constrained optimal quantization for wireless sensor networks," Proc I st IEEE Annual Communications Society Conference on Sensor and Ad Hoc Communications and etworks (SECO '04), pp 7-78, Santa Clara, Calif, USA, Oct 004 [] J G Proakis, Digital Communications, McGraw-Hill, 4th Edition, 00 [] S Boyd and L Vandenberghe, Convex Optimization, Cambridge, UK Cambridge Unv Press, 003 REFERECES [] S Cui, J-J Xiao, A J Goldsmith, Z-Q Luo, and H V Poor, "Estimation diversity and energy efficiency in distributed sensing", IEEE Trans Signal Process, vol 55, no 9, pp , Sep /09/$ IEEE 67 Authorized licensed use limited to CALIFORIA ISTITUTE OF TECHOLOGY Downloaded on April,00 at UTC from IEEE Xplore Restrictions apply