On the Optimal Performance in Asymmetric Gaussian Wireless Sensor Networks With Fading

Transcription

1 436 IEEE TRANSACTIONS ON SIGNA ROCESSING, O. 58, NO. 4, ARI 00 [9] B. Chen and. K. Willett, On the optimality of the likelihood-ratio test for local sensor decision rules in the presence of nonideal channels, IEEE Trans. Inf. Theory, vol. 5, pp , Fe [0] M. Kam, Q. Zhu, and W. S. Gray, Optimal data fusion of correlated local decisions in multiple sensor detection systems, IEEE Trans. Aerosp. Electron. Syst., vol. 8, pp , Jul. 99. [] S. Boyd and. andenerghe, Convex Optimization. Camridge, U.K. Camridge Univ. ress, 003. [] J. F. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Math. Oper. Res., vol. 8, no., pp , May 003. Fig.. A Gaussian wireless sensor network (WSN) with fading. On the Optimal erformance in Asymmetric Gaussian Wireless Sensor Networks With Fading Hamid Behroozi, Fady Alajaji, and Tamás inder Astract We study the estimation of a Gaussian source y a Gaussian wireless sensor network (WSN) distriuted sensors transmit noisy oservations of the source through a fading Gaussian multiple access channel to a fusion center. In a recent work Gastpar, [ Uncoded transmission is exactly optimal for a Simple Gaussian Sensor Network, IEEE Trans. Inf. Theory, vol. 54, no., pp , Nov. 008] showed that for a symmetric Gaussian WSN with no fading, uncoded (analog) transmission achieves the optimal performance theoretically attainale (OTA). In this correspondence, we consider an asymmetric fading WSN in which the sensors have differing noise and transmission powers. We first present lower and upper ounds on the system s OTA under random fading. We next focus on asymmetric networks with deterministic fading. By comparing the otained lower and upper OTA ounds under deterministic fading, we provide a sufficient condition for the optimality of the uncoded transmission scheme for a given power tuple (... ). Then, allowing the sensor powers to vary under a weighted sum constraint (this includes the sum-power constraint as a special case), we otain a sufficient condition for the optimality of uncoded transmission and provide the system s corresponding OTA. Index Terms Gaussian multiple access channel with fading, joint source-channel coding, power-distortion tradeoff, remote source coding, sensor networks, uncoded transmission. I. INTRODUCTION We consider the estimation of a memoryless Gaussian source y a Gaussian wireless sensor network (WSN) sensors oserve the source signal X corrupted y additive independent noise. The overall Manuscript received June 8, 009; accepted Decemer 9, 009. First pulished January 5, 00; current version pulished March 0, 00. The associate editor coordinating the review of this manuscript and approving it for pulication was rof. Huaiyu Dai. This work was supported in part y a ostdoctoral Fellowship from the Ontario Ministry of Research and Innovation (MRI) and y the Natural Sciences and Engineering Research Council (NSERC) of Canada. The material in this correspondence was presented in part at the IEEE International Symposium on Information Theory (ISIT), Toronto, ON, Canada, July 008, and at the IEEE Canadian Workshop on Information Theory (CWIT), Ottawa, ON, Canada, May 009. H. Behroozi was with the Department of Mathematics and Statistics, Queen s University, Kingston, ON, Canada. He is now with the Electrical Engineering Department, Sharif University of Technology, Tehran, Iran ( ehroozi@sharif.edu). F. Alajaji and T. inder are with the Department of Mathematics and Statistics, Queen s University, Kingston, ON K7 3N6, Canada ( fady@mast. queensu.ca; linder@mast.queensu.ca). Color versions of one or more of the figures in this correspondence are availale online at http//ieeexplore.ieee.org. Digital Oject Identifier 0.09/TS system is depicted in Fig.. The sensors communicate information aout their oservations through a fading Gaussian multiple access channel (MAC) to a single fusion center (FC). The fading coefficients are not known y the encoders ut are availale at the FC. The encoders are distriuted and cannot cooperate to exploit the correlation etween their inputs. Each encoder is suject to a transmission cost constraint. The FC aims to reconstruct the main source X at the smallest cost in the communication link. Our interest lies in determining the optimal power-distortion region, with the fidelity of estimation at the FC measured y the mean squared-error (MSE) distortion. Specifically, for a given -tuple of sensor powers ( ; ;...;), we seek to determine the system s minimum achievale distortion which we refer to as the optimal performance theoretically attainale (OTA). In [] and [], it is proved that uncoded transmission is exactly optimal for symmetric Gaussian WSNs with a finite numer of sensors and no fading. Uncoded transmission in this case (and in the rest of this correspondence) means scaling the encoder input suject to the channel power constraint and transmitting without explicit channel coding. Note that the separate source and channel coding theorem of Shannon [3] does not hold for this prolem [], []. In the case of deterministic fading, lower and upper ounds on the minimum distortion are presented in [], [], and [4], and for random fading, ounds are also presented in [4] and [5]. The minimum achievale distortion under a sum-power constraint for the uncoded transmission scheme in the WSN with deterministic fading is presented in [6]. The optimality of uncoded transmission in some other multiuser communication systems was recently shown in [7] and [8]. For the asymmetric fading Gaussian WSN, the following important issues remain unknown Under either random or deterministic fading, what is the system s OTA? Also, What is the optimal coding strategy that achieves OTA? Our main contriutions in this correspondence are as follows First, y applying the idea of maximum correlation coefficient, illustrated in [9] [], we generalize the OTA lower ound in [] to an asymmetric Gaussian WSN with random fading. We show that the new ound is a tighter lower ound on the OTA than that of [5] for a Gaussian WSN with random fading. We also analyze the uncoded transmission scheme and provide an upper ound on the OTA for a given set of sensor powers. These two ounds constitute an extension of the ounds given for deterministic fading case in [] and []. We next specialize the results to the case of deterministic fading. We estalish a condition under which the lower and upper ounds on the system s OTA coincide, hence making the uncoded transmission scheme optimal. We next allow the sensor powers to vary under a linear comination of powers (C) constraint. Aside from eing a natural generalization of the sum-power constraint, the C constraint explicitly allows to introduce weight coefficients that reflect the potentially differing costs of supplying power to individual sensors. Our final contriution is to provide sufficient conditions for the optimality of un X/$ IEEE Authorized licensed use limited to Queens University. Downloaded on March 5,00 at 040 EDT from IEEE Xplore. Restrictions apply.

2 IEEE TRANSACTIONS ON SIGNA ROCESSING, O. 58, NO. 4, ARI coded transmission under a given C constraint and determine the system s corresponding OTA. The remainder of this correspondence is organized as follows. In Section II, we present the system model and prolem statement. Section III provides lower and upper ounds on the OTA in an asymmetric Gaussian WSN with random fading. In Section I, we consider deterministic fading and provide matching conditions y comparing lower and upper ounds on the system s OTA. For uncoded transmission, an optimal power allocation under an C constraint and a sufficient condition for its optimality are also otained. Conclusions are presented in Section. II. ROBEM STATEMENT We consider a simple Gaussian WSN, illustrated in Fig., a team of sensors oserve independent noisy versions of the memoryless Gaussian source fx[k]g n k. The underlying source fx[k]gn k is a sequence of independent and identically distriuted (i.i.d.) realvalued Gaussian random variales of mean zero and variance X.For each oservation time k ; ; 3;..., the noisy oservations are given y Y l [k] X[k] l [k]; l ;...; () f l [k]g n k is a sequence of i.i.d. Gaussian random variales of mean zero and variance which is independent of fx[k]g n k. We represent the first n instances of fx[k]g k and fy l [k]g k y the data sequences X n (X[];X[];...;X[n]) and Yl n (Y l [];Y l [];...;Y l [n]), respectively. The correlated sources Y l are not co-located and their oservers cannot cooperate to directly exploit their correlation. Instead, the sequences Yl n are separately encoded to ' l (Yl n )Ul n the ' l are encoder functions in the form ' l n! n ; l ; ;...; () Each transmitted sequence Ul n is assumed to e average-power limited to l, i.e., n n k E ju l [k]j l ; l ; ;...; (3) The sensors communicate the coded sequences to the decoder through a fading MAC. In fact, each transmitted signal U l is multiplied y a real-valued fading random variale l, l ; ;...;. The l are not known to the encoders ut are availale to the decoder. The fading coefficients have non-zero mean and are independent of each other and of the U l random variales. The time-k output of the channel is given y W [k] l [k]u l [k] Z[k] (4) the channel noise fz[k]g n k is an i.i.d. sequence of Gaussian random variales of mean zero and variance Z that is independent of X n, n, and the fading coefficients. Based on the channel output W n (W [];...;W[n]), the FC forms an estimate ^X n of the main source X n. Fidelity etween X n and ^X n is measured y the average squared error distortion, (n)e X[j] 0 ^X[j]. The n j reconstructed signal can e descried as ^X n (W n ; n ), the decoder function is a mapping n n! n (5) et F (n) ( ; ;...; ) denote all encoder and decoder functions (' ;...;' ; ) that satisfy () (5). For a particular coding scheme (' ;...;' ; ), the performance is determined y the cost vector ( ; ;...; ) and the incurred distortion. For any target distortion D 0, the power-distortion region (D) is defined as the convex closure of the set of all achievale power-distortion pairs (;D), a power-distortion pair (;D) is achievale if for any > 0, there is an n 0 () such that for all n n 0 () there exists (' ;...;' ; ) F (n) ( ; ;...; ) with distortion D. Our aim is to investigate the power-distortion region of this fading Gaussian WSN and present lower and upper ounds on its OTA, which is defined for a fixed as D min( )inffd j (;D) (D)g (6) We also aim to derive optimality conditions for uncoded transmission in the sense of achieving the OTA. In addition, we want to minimize the MSE distortion given a linear comination of powers (C) constraint, i.e., minimize D( ) suject to l l D( ) is the distortion of the uncoded transmission scheme using power allocation ( ;...; ) and l > 0, l ;...;. This form of constraint is a slight generalization of the sum-power constraint which explicitly allows to introduce weight coefficients for the potentially differing costs of supplying power to individual sensors. III. INFORMATION-THEORETIC OTA BOUNDS IN THE RESENCE OF RANDOM FADING We present lower and upper ounds on the OTA in an asymmetric Gaussian WSN with random fading. The lower ound is ased on analyzing the remote source coding scenario, the sensor oservations are given to one common encoder; then applying the data processing as well as Jensen s inequalities and finally using the idea of maximum correlation coefficient, illustrated in [9] []. The upper ound is ased on analyzing uncoded transmission which is the transmission of scaled versions of the sensors oservations. A. ower Bound roposition A lower ound on the OTA in an asymmetric Gaussian WSN with random fading is D 3 0 r X Z D min D` D0 3 Z r X X l E[j l j ] j>l 0 E [ l ] E [ j] (7) (8) ; (9) (0) and the expectations are taken with respect to the distriution of the fading random variales l. roof See the proof for [4, Theorem ]. This ound is an extension of [, Theorem 4] to the random fading case. Next we compare the ound of roposition with the lower ound presented in [5] and demonstrate that our ound is strictly tighter as Authorized licensed use limited to Queens University. 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3 438 IEEE TRANSACTIONS ON SIGNA ROCESSING, O. 58, NO. 4, ARI 00 long as at least one of the sensor oservation noise variances is nonzero. The lower ound presented in [5], which is for the case the sensors oservations have the same noise level, can e expressed for a general asymmetric WSN as D lower (; tot )D 3 X Z 0 Z tot E[j l j ] () it is assumed that there is a total power constraint in the communication channel, i.e., n k E[jU l[k]j ] tot ; and tot n denotes the average total sensor power availale per oservation vector (U n ;...;U n ). et us denote Z l E[j l j ] X E [ l ] E [ j ] j>l and Z tot E[j l j ] Fig.. The estimation distortion versus the transmission power,. We assume that 5, 00and 30. We have le[j j j ] je[j l j ] E[j l j ]E[j j j ] E[ l ]E[ j ] tot l. By comparing (8) and (), we oserve that the only difference is in the denominator. The lower ound of (8) is in the form D` D0 3 (X Z ( )) while D lower (; tot) D0 3 (X Z ( )). We want to show that. Since the are non-negative, we have X Thus, j>l E [ l ] E [ j ] 0 tot 0 j>l j 0 j>l j>l 0 j>l E[j l j ] 0 E [ l ] E [ j] () E [ l ] E [ j ] je[j l j ] 0 j>l l E[j l j ] E [ l ] E [ j] l E[j l j ] l E[j j j ] je[j l j ] E [ l ] E [ j ] (3) the first inequality is due to the arithmetic-geometric mean inequality and the second inequality follows from the Cauchy Schwarz inequality. Thus, a term-y-term comparison in (3) shows that the difference is non-negative, proving that. Moreover, the inequality in () implies that > as long as at least one of the is positive. Therefore, the lower ound of (8) on D min is tighter than (); the improvement is mainly due to the maximization of the correlation coefficients. ) Numerical Examples ) We consider a symmetric WSN consisting of two sensors and evaluate the resulting lower ounds numerically. We assume that E[j j ] E[j j ] 4, E[ ] E[ ]. The estimation distortion ound is plotted as a function of the power level in Fig.. We oserve that the distortion is a decreasing function of and that the lower ound D` of (8) is tighter than the lower ound D lower of (). Specifically, if we calculate the percentage of the relative distortion gap (i.e., (D` 0 D lower )D lower %) versus the value of, we oserve that D` performs 0% etter (tighter) than D lower for > 5. ) Now, we assume that the channel is not symmetric and plot the distortion ounds as a function of the power level in Fig. 3 for two different fading parameters E[j j ]8;E[j j ]4;E[ ]0; E[ ]5 E[j j ]35;E[j j ]0;E[ ];E[ ]4 Again, we oserve that the lower ound D` is tighter than the lower ound D lower. 3) We next consider an asymmetric WSN. We assume that, 0, E[j j ]E[j j ]4, and E[ ] E[ ]and plot the estimation distortion ounds as a function of in Fig. 4. There is almost a fixed gap etween D` and D lower. Authorized licensed use limited to Queens University. Downloaded on March 5,00 at 040 EDT from IEEE Xplore. Restrictions apply.

4 IEEE TRANSACTIONS ON SIGNA ROCESSING, O. 58, NO. 4, ARI Fig. 3. The estimation distortion versus the transmission power, in a symmetric WSN with asymmetric fading MAC. We assume that 5, 00and 30. Fig. 5. ercentage of the relative distortion gap versus the sum-power.we assume that,,,. The fading coefficients are taken as k d, k is a normalization constant satisfying E[ ] and d is drawn uniformly from the interval [; 5]. Fig. 4. The estimation distortion versus the value of 0. We assume that 0, 0, 00 and 5. B. Upper Bound Analyzing Uncoded Transmission By analyzing the uncoded transmission in our Gaussian WSN, we next present an upper ound on the OTA. In this approach, each sensor transmits its oservation y simply scaling it to its power constraint, i.e., U l [k] l (X )Y l[k] The received signal at the FC is then given y W [k] l l [k](x[k] l [k]) Z[k] Since the encoding is memoryless, the optimal (minimum mean-squared error) estimator is easy to otain. By evaluating the resulting MSE distortion, we otain an upper ound on the minimum achievale distortion, which is summarized in the next lemma. emma An upper ound on the OTA in a Gaussian WSN with random fading can e expressed as D min D uncoded XE Z j j Z rj (4) Fig. 6. The MSE estimation distortion versus the total numer of sensors,, under a uniform power schedule. The fading coefficients are taken as kd, d is drawn uniformly from the interval [; 6] and k is a normalization constant satisfying E[ ]. We assume that,,, (for l ;...;). rj X l j l j l j j>l X X (5) and the expectation is with respect to the fading random variales, l. In the following, we provide some examples in order to compare the lower and upper ounds numerically. First, we consider a Gaussian WSN with sensors, (for l ;...;). The fading coefficients of the channels are taken as k d 035, k is a normalization constant chosen to satisfy E[ l ]and d is a random variale uniformly distriuted on the interval [; 5]. In Fig. 5, we plot the percentage of the relative distortion gap (i.e., (D uncoded 0 D`)D uncoded %) versus the sumpower tot assuming a uniform power schedule. We note that the gap is less than 6% for all values of tot. Fig. 6 shows the lower and upper ounds on the OTA versus the total numer of sensors, under a uniform power schedule and a sumpower tot. The fading coefficients are taken as kd 035, d Authorized licensed use limited to Queens University. Downloaded on March 5,00 at 040 EDT from IEEE Xplore. Restrictions apply.

5 440 IEEE TRANSACTIONS ON SIGNA ROCESSING, O. 58, NO. 4, ARI 00 is drawn uniformly from the interval [; 6] and k is a normalization constant satisfying E[ l ]. We oserve that the gap etween the lower and upper ounds decreases as the total numer of sensors increases. Before closing this section, we point out that, using some algeraic manipulations, it is not hard to show that if 0, the lower ound of (8) and the upper ound of (4) agree under some conditions (see the proof of Corollary in Section I-A, presented in Appendix A). This means that uncoded transmission can achieve the optimal performance if the fading coefficients stay constant over the duration of transmission. This may apply to situations the network conditions change very slowly. In the remainder of this work, we investigate the Gaussian WSN with such deterministic channel gains. I. DETERMINISTIC FADING SYSTEM A. Optimality Condition for Uncoded Transmission Assume that the fading coefficients l are fixed non-zero constants known at the fusion center. The lower ound of (8) and the upper ound of (4) can e expressed as follows D min D` D 3 X Z 0 Z r D min D uncoded X Z r r l j l j (X ) Z l j l j (6) (7) l l X ( ) In the symmetric case, and, the lower ound (6) and the upper ound (7) coincide if. Hence, we otain the OTA for the symmetric Gaussian WSN under deterministic and identical fading. This is the same result as recently estalished y Gastpar in [] with. Corollary For the asymmetric network with deterministic fading, the lower ound (6) and the upper ound (7) coincide if and only if X X (8) roof See Appendix A. Hence, for a given set of powers,, if (8) is satisfied then the distortion achieved y the uncoded transmission is the smallest possile achievale distortion. Note that (8) is oth necessary and sufficient for the upper and lower ounds to coincide, ut it is only a sufficient condition for the optimality of uncoded transmission. B. Optimal ower Allocation for Uncoded Transmission Under an C Constraint In a WSN, many sensor devices are attery-powered and thus power constrained. Therefore, minimizing power consumption is critical in extending the lifetime of the individual sensor nodes and the entire network. A sum-power constraint has a direct impact on the network lifetime since it is imposed on the power consumption of all sensors taken together. However in some applications, in order to prevent excessive power consumption for individual sensors, or due to differing power supply capailities at the sensors, individual power constraints for each sensor may also e desirale. This motivates us to comine cost coefficients with the sum-power constraint and consider a complexity constraint consisting of a an C. These cost coefficients might depend on the sensors location and attery lifetime and can e assigned y the fusion center. We thus herein consider an optimal power allocation for the uncoded transmission in order to minimize the MSE distortion under an C constraint, i.e., minimize D( ) suject to l l (9) D( ) is the distortion of the uncoded transmission scheme using power allocation ( ;...; ) and l > 0, l ;...;. In [6], y applying the agrange multiplier method, it was shown that the optimal distortion sum-power tradeoff D uncoded ; for the uncoded scheme can e expressed as follows D uncoded ( tot ) X X l 0 (0) tot l is the sum-power. This tradeoff is also given in [, eq. (6)], for the case of equal variance oservation noises and equal channel gains. Since we want to compare lower and upper ounds on OTA under an C constraint (which susumes the sum-power constraint), we first need to provide closed form expressions of the optimal power allocation for the C case. The proof of the next lemma, which we omit, is ased on an application of the agrange multiplier method. Note that as a special case, the lemma also yields the formula (0) for the minimum distortion under the sum-power constraint derived in [6]. emma (Optimal ower Allocation for Uncoded Transmission under an C Constraint) An optimal power allocation scheme for the constraint l l is given y l l ( X ) l Z l l Z l l ( X ) ; l ;...; () Sustituting the optimal l given in () into (7) yields an upper ound on OTA under the C constraint D uncoded () X Corollary (Uncoded Transmission Optimality Conditions Under an C Constraint) In an asymmetric Gaussian WSN with deterministic fading, uncoded transmission with the optimal power allocation given in () is optimal in the sense of achieving the OTA under an C constraint if the following conditions hold ( ) roof See Appendix B.. CONCUSION 0 ( ) () We considered a distriuted WSN noisy oservations of a memoryless Gaussian source are transmitted through a fading Gaussian MAC to a decoder. The decoder wants to reconstruct the Authorized licensed use limited to Queens University. Downloaded on March 5,00 at 040 EDT from IEEE Xplore. Restrictions apply.

6 IEEE TRANSACTIONS ON SIGNA ROCESSING, O. 58, NO. 4, ARI main source with an average distortion D at the smallest possile power consumption in the communication link. Our goal was to characterize the power-distortion region achievale y any coding strategy regardless of delay and complexity. We otained a lower ound on the system s OTA, i.e., its minimum achievale distortion for a given set of powers (;;...;). Also, y analyzing the uncoded transmission scheme, we provided an upper ound on the OTA. When specialized to the network with deterministic fading, we provided sufficient conditions for the optimality of the uncoded transmission. We also otained an optimal power allocation in order to minimize distortion for a given C constraint as well as an explicit condition for the optimality of uncoded transmission under an C constraint. AENDIX A ROOF OF COROARY roof We first rewrite the lower ound in (6) as D` X Z D0 3 Z j j j j D 3 0 l l (3) Comparing (3) with the upper ound in (7) reveals that the lower ound and the upper ound coincide if and only if X 0 D 3 0 l j l j X D0 3 X l l (4) From the definition of D 3 0 we have ( X 0 D 3 0)( XD 3 0). As a result, the equality condition in (4) is equivalent to Setting g l l l j l j X g l l l (5) l ( ), we rewrite and simplify (5) as j6l g l g j j>l (6) Comparing the coefficients of ( )(l 6 j) on oth sides of (6) shows that the equality holds if and only if g l g j g l g j ; this is equivalent to g l g j which is our condition in (8). sgn(x) xjxj is the sign of x. Sutracting oth sides in (7), we get f l j6l f l f j j>l f l from sgn( l ) sgn( j ) (8) Comparing the coefficients of ( ) in oth sides of (8) gives that the equality holds if and only if f l f j f l f j sgn( l ) sgn( j ) Since f l 0, the equality holds if and only if f l f j and sgn( l ) sgn( j ). Solving f l f j implies that l X j l X (9) j Comining (9) with sgn( l ) sgn( j), we otain () which completes the proof. REFERENCES [] M. Gastpar, Uncoded transmission is exactly optimal for a simple Gaussian sensor network, IEEE Trans. Inf. Theory, vol. 54, no., pp , Nov [] M. Gastpar, Uncoded transmission is exactly optimal for a simple Gaussian sensor network, presented at the 007 Inf. Theory Applications Workshop, San Diego, CA, Jan [3] C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., vol. 7, pp , 948. [4] H. Behroozi, F. Alajaji, and T. inder, On the optimal power-distortion region for asymmetric Gaussian sensor networks with fading, presented at the IEEE ISIT, Toronto, ON, Canada, Jul [5] M. Gastpar and M. etterli, ower, spatio-temporal andwidth, and distortion in large sensor networks, IEEE J. Sel. Areas Commun., vol. 3, no. 4, pp , Apr [6] J. Xiao, Z.-Q. uo, S. Cui, and A. Goldsmith, ower-efficient analog forwarding transmission in an inhomogeneous Gaussian sensor network, in roc. 005 IEEE 6th Workshop Signal rocessing Advances Wireless Communications, Jun. 005, pp. 5. [7] A. apidoth and S. Tinguely, Sending a i-variate Gaussian source over a Gaussian MAC, in roc. IEEE ISIT, Seattle, WA, Jul. 006, pp [8] S. Bross, A. apidoth, and S. Tinguely, Broadcasting correlated Gaussians, presented at the IEEE ISIT, Toronto, ON, Canada, Jul [9] H. O. Hirschfeld, A connection etween correlation and contingency, roc. Camridge hilos. Soc., vol. 3, pp , 935. [0] H. Geelein, Das statistische prolem der korrelation als variations-und eigenwertprolem und sein zusammenhang mit der ausgleichungsrechnung, (in German) Z. Angew. Math. Mech., vol., pp , 94. [] H. S. Witsenhausen, On sequences of pairs of dependent random variales, SIAM J. Appl. Math., vol. 8, no., pp. 00 3, Jan AENDIX B ROOF OF COROARY roof From the proof of Corollary, a necessary condition for the upper and the lower ound to coincide is that the equality in (5) holds. Sustituting the optimal l s in this condition and setting f l Z l ( X ) l, we otain f l j f l f l f j j>l sgn( l ) sgn( j ) (7) Authorized licensed use limited to Queens University. Downloaded on March 5,00 at 040 EDT from IEEE Xplore. Restrictions apply.