Research Article Decentralized Detection in Wireless Sensor Networks with Channel Fading Statistics

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1 Hindawi Publishin Corporation EURASIP Journal on Wireless Communications and Networkin Volume 7, Article ID 695, 8 paes doi:.55/7/695 Research Article Decentralized Detection in Wireless Sensor Networks with Channel Fadin Statistics BinLiuandBiaoChen Department of Electrical Enineerin and Computer Science EECS, Sracuse Universit, 3 Link Hall, Sracuse, NY 344-4, USA Received 5 Auust 6; Revised 6 November 6; Accepted 9 November 6 Recommended b C. C. Ko Existin channel aware sinal processin desin for decentralized detection in wireless sensor networks tpicall assumes the clairvoant case, that is, lobal channel state information CSI is known at the desin stae. In this paper, we consider the distributed detection problem where onl the channel fadin statistics, instead of the instantaneous CSI, are available to the desiner. We investiate the desin of local decision rules for the followin two cases: fusion center has access to the instantaneous CSI; fusion center does not have access to the instantaneous CSI. As expected, in both cases, the optimal local decision rules that minimize the error probabilit at the fusion center amount to a likelihood ratio test LRT. Numerical analsis reveals that the detection performance appears to be more sensitive to the knowlede of CSI at the fusion center. The proposed desin framework that utilizes onl partial channel knowlede will enable distributed desin of a decentralized detection wireless sensor sstem. Copriht 7 B. Liu and B. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the oriinal work is properl cited.. INTRODUCTION While stud of decentralized decision makin can be traced back to the earl 96s in the context of team decision problems see, e.., [], the effort sinificantl intensified since the seminal work of []. Classical distributed detection [3 7], however, tpicall assumes error-free transmission between the local sensors and the fusion center. This is overl idealistic in the emerin sstems with strinent resource and dela constraints, such as the wireless sensor network WSN with eoraphicall dispersed lower-power low-cost sensor nodes. Accountin for nonideal transmission channels, channel aware sinal processin for distributed detection problem has been developed in [8 ]. The optimal local decision rule was still shown to be a monotone likelihood ratio partition of its observation space, provided the observations were conditionall independent across the sensors. It was noted recentl that such optimalit is preserved for a more eneral settin []. The work in[8 ] assumed a clairvoant case, that is, lobal information reardin the transmission channels between the local sensors and the fusion center is available at the desin stae. This approach is theoreticall sinificant as it provides the best achievable detection performance to which an suboptimal approach needs to be compared. However, its implementation requires the exact knowlede of lobal channel state information CSI which ma be costl to acquire. In the case of fast fadin channel, the sensor decision rules need to be snchronousl updated for different channel states; this adds considerable overhead which ma not be affordable in resource constrained sstems. To make the channel aware desin more practical, the requirement of the lobal CSI in the distributed sinalin desin needs to be relaxed. In the present work, onl partial channel knowlede instead of the lobal CSI is assumed to be available. In the context of WSN, a reasonable assumption is the availabilit of channel fadin statistics, which ma remain stationar for a sufficientl lon period of time. Therefore, the updatin rate of the decision rules is more realistic. In this paper, we consider the distributed detection problem where the desiner onl has the channel fadin statistics instead of the instantaneous CSI. In this case, a sensible performance measure is to use the averae error probabilit at the fusion center where the averain is performed with respect to the channel state. We restrict ourselves to binar local sensor outputs and derive the necessar conditions for optimal local decision rules that minimize the averae error probabilit at the fusion center for the followin two

2 EURASIP Journal on Wireless Communications and Networkin H /H X X K Sensor γ. Sensor K γ K U U K Channel. Channel K Y Y K U Fusion center γ Fiure : A block diaram for a wireless sensor network tasked for binar hpothesis testin with channel fadin statistics. cases: CSIF: the fusion center has access to the instantaneous CSI. NOCSIF: the fusion center does not have access to the instantaneous CSI. We note here that for the CSIF case, the desin of sensor decision rules does not require CSI; the CSI is onl used in the fusion rule desin, which is reasonable due to the tpical enerous resource constraint at the fusion center. Its computation, however, is ver involved and has to resort to exhaustive search. On the other hand, as to be elaborated, the NOCSIF case can be reduced to the channel aware desin where one averaes the channel transition probabilit with respect to the fadin channel usin the known fadin statistics. We show that the sensor decision rules amount to local LRTs for both cases. Compared with the existin channel aware desin based on CSI, the proposed approaches have an important practical advantae: the sensor decision rules remain the same for different CSI, as lon as the fadin statistics remain unchaned. This enables distributed desin as no lobal CSI is used in determinin the local decision rules. We also demonstrate throuh numerical examples that the proposed schemes suffer small performance loss compared with the CSI-based approach, as lon as the CSI is available at the fusion center, that is, used in the fusion rule desin. The paper is oranized as follows. Section describes the sstem model and problem formulation. In Section 3,we establish, for both the CSIF and NOCSIF cases, the optimalit of LRTs at local sensors for minimum averae error probabilit at the fusion center. Numerical examples are presented in Section 4 to evaluate the performance of these two cases. Finall, we conclude in Section 5.. STATEMENT OF THE PROBLEM Consider the problem of testin two hpotheses, denoted b H and H, with respective prior probabilities π and π.a total number of K sensors are used to collect observations X k,fork,..., K. We assume throuhout this paper that the observations are conditionall independent, that is, p K X,..., X K H i p X k H i, i,. k Without this assumption, the distributed detection desin becomes much more complicated from an alorithmic viewpoint []. Theoptimal desinisnotcompletelunderstood even in the simplest case [3]. Upon observin X k,eachlocal sensor makes a binar decision U k γ k, k,..., K. The decisions U k are sent to a fusion center throuh parallel transmission channels characterized b p K Y,..., Y K U,..., U K,,..., K p Y k U k, k, where {,..., K } represents the CSI. Thus, from 3, the channels are orthoonal to each other, which can be achieved throuh, for example, partitionin time, frequenc, or combinations thereof. For the CSIF case, the fusion center takes both the channel output {Y,..., Y K } and the CSI,, and makes a final decision usin the optimal fusion rule to obtain U {H, H }, k 3 U γ,. 4 For the case of NOCSIF, the fusion output depends on the channel output and the channel fadin statistics, U γ, 5 where the dependence of fadin channel statistics is implicit in the above expression. An error happens if U differs from the true hpothesis. Thus, the error probabilit at the fusion center, conditioned on a iven, is P e γ,..., γ K P r U H γ,..., γ K,, 6 where H is the true hpothesis. Our oal is, therefore, to desin the optimal mappin γ k for each sensor and the fusion center that minimizes the averae error probabilit, defined as min P e γ,..., γ K pd, 7 γ,...,γ K where p is the distribution of CSI. A simple diaram illustratin the model is iven in Fiure. The extension to the case with multibit local decision is straihtforward b followin the same spirit as in [].

3 B. Liu and B. Chen 3 We point out here that interatin the transmission channels into the fusion rule desin has been investiated before [4, 5]. The optimal fusion rule in the Baesian sense amounts to the maximum a posteriori probabilit MAP decision, that is, f Y,..., Y K H H f > π Y,..., Y K H <. 8 π H Given specified local sensor sinalin schemes and the channel characterization, this MAP decision rule can be obtained in a straihtforward manner. As such, we will focus on the local decision rule desin without further elaboratin on the optimal fusion rule desin. 3. DESIGN OF OPTIMAL LOCAL DECISION RULES As in [9, ], we adopt in the followin a person-b-person optimization PBPO approach, that is, we optimize the local decision rule for the kth sensor iven fixed decision rules at all other sensors and a fixed fusion rule. As such, the conditions obtained are necessar, but not sufficient, for optimalit. Denote u [ U, U,..., U K ], x [ X, X,..., X K ], 9 the averae error probabilit at the fusion center is P e π i P U i H i, pd i π i P U i, p u, i u Pu xp x H i p dx d d, x where, different from the CSI-based channel aware desin, the local decision rules do not depend on the instantaneous CSI. Next, we will derive the optimal decision rules b further expandin the error probabilit with respect to the kth decision rule γ k for the two different cases. 3.. The CSIF case We first consider the case where the fusion center knows the instantaneous CSI. Define, for k,..., K and i,, [ U,..., U k, U k+,..., U K ], i [ U,..., U k, U k i, U k+,..., U K ], k [ Y,..., Y k, Y k+,..., Y K ]. We can expand the averae error probabilit in withrespect to the kth decision rule γ k, and we et P e P U k X k X k [ π p X k H Ak π p X k H Bk ] d + C, where [ C π i p X k H i X k i P U i, p, pp ] H i d d dx k 3 is a constant with reard to U k,and A k P U, [ p, p, B k pp H ] d d, 4 P U, [ p, p, pp H ] d d. 5 To minimize P e, one can see from that the optimal decision rule for the kth sensor is P, π p X k H Ak >π p X k H Bk, U k X k, otherwise. 6 Let us further take a look at A k in 4. We can rewrite it as [ A k P U k, U k k P U k, U k ] p 7 k H d k. Then, as shown in Appendix A, A k > aslonas L U k P U k H P U k H 8 is a monotone increasin function of U k monotone LR index assinment, that is, L U k >L U k. 9 Similarl, B k > if condition 9 is satisfied. This immediatel leads to the followin result. Theorem. For the distributed detection problem with unknown CSI onl at local sensors, the optimal local decision rule for the kth sensor amounts to the followin LRT assumin condition 9 is satisfied, p X k H P, p π A k, X k H π B k U k X k p X k H, p < π A k, X k H π B k where A k and B k are defined in 4 and 5,respectivel.

4 4 EURASIP Journal on Wireless Communications and Networkin An alternative derivation, alon the line of [], is iven in Appendix B. Althouh the optimal local decision rule for each local sensor is explicitl formulated in, it is not amenable to direct numerical evaluation: in 4 and5, the interand involves the fusion rule that is a hihl nonlinear function of the CSI, makin the interation formidable. The onl possible wa of findin the optimal local decision rules appears to be an exhaustive search, whose complexit becomes prohibitive when K is lare. This numerical challene motivates an alternative approach: instead of minimizin the averae error probabilit, one can first marinalize the channel transition probabilit followed b the application of standard channel aware desin [8 ]. That is, we first compute p u b marinalizin out the channel usin the channel fadin statistics: p u p u, pd. With this marinalization, we can use the channel aware desin approach [9] that tends to the averaed transmission channel. The difference between this alternative approach and that of minimizin 7 is in the wa that the channel fadin statistics, p, is utilized. In usin 7, the variable to be averaed is P e γ,..., γ K, which is a hihl nonlinear function of the decision rules, whereas the alternative approach uses p to obtain the marinalized channel transition probabilit, thus enablin the direct application of the channel aware approach. This difference can also be explained usin Fiure. The alternative approach averaes each transmission channel over respective channel statistics i to obtain py i U i, while the CSIF case averaes all the transmission channels and the fusion center the part in the dashedbox in Fiure over channel statistics to obtain P U u P U, p u, pd d. It turns out that this alternative approach is a direct consequence of the NOCSIF, as elaborated below. 3.. The NOCSIF case Here we consider the case where the fusion center does not know the instantaneous CSI. Therefore, we have P U i, P U i. 3 As the fusion rule no loner depends on, the averae error probabilit in canberewrittenas P e π i P U i p u, pd i u P u x p x H i dx d, x 4 where interation with respect to is carried out first. Notice that the term p u, pd precisel describes the marinalized transmission channels cf.. This leads to the standard channel aware desin where the transmission channels are characterized b p u. From [9], we have a result resemblin that of Theorem except that A k, B k,and C are replaced b A k, B k,andc, A k P U [ p u k p B k P H ] d, P U [ p u k p P ] H d, [ C π i p X k H i P U i X k i p H i d ]dx k. p 5 Contrar to the CSIF case, A k, B k for the NOCSIF case are much easier to evaluate. 4. PERFORMANCE EVALUATION In this section, we first use a two-sensor example to evaluate the performance of both the CSIF and NOCSIF cases and compare them with the clairvoant case where the lobal channel information is known to the desiner. For convenience, we call the clairvoant case the CSI case. Consider the detection of a known sinal S in zero-mean complex Gaussian noises that are independent and identicall distributed i.i.d. for the two sensors, that is, for k,, H : X k N k, H : X k S + N k, 6 with N and N bein i.i.d. CN, σ. Without loss of eneralit, we assume S andσ. Each sensor makes a binar decision based on its observation X k, U k γ k, Uk {, }, 7 and then transmits it throuh a Raleih fadin channel to the fusion center. Notice that U k {, } implies an on-off sinalin, thus enablin the detection at the fusion center in the absence of CSI i.e., the NOCSIF case. The channel output is Y k k U k + W k, 8 where, are independentl distributed with zero-mean complex Gaussian distributions CN, σ andcn, σ,

5 B. Liu and B. Chen 5 Averae error probabilit CSI CSIF 4 6 Sinal-to-noise ratio db CSIF NOCSI Fiure : Averae error probabilit versus channel SNR for identical channel distribution. respectivel. W, W are i.i.d. zero-mean complex Gaussian noises with distribution CN, σ. We first consider a smmetric case where the CSI is identicall distributed, that is, σ σ. Without loss of eneralit, we assume σ σ. In Fiure, with the assumption of equal prior probabilit, the averae error probabilities as a function of the averae sinal-to-noise ratio SNR of the received sinal at the fusion center are plotted for the CSIF and NOCSIF cases, alon with the CSI case. We also plot a curve, leended with CSIF in Fiure, where the local sensors use thresholds obtained via the NOC- SIF approach but the fusion center implements a fusion rule that utilizes the CSI. The motivation is twofold. First, estimatin at the fusion center is tpicall feasible. Second and more importantl, the threshold desin for NOC- SIF is much simpler compared with CSIF, as explained in Section 3.. In evaluatin the performance of CSIF case, we consider the followin two methods to et the optimal thresholds for local sensors. Exhaustive search method We first partition the space of likelihood ratio of observations into man small disjoint cells. For each cell, we compute the averae error probabilit at the fusion center b usin the center point of the cell for small enouh cell size, the center point can be considered representive of the whole cell as the local thresholds. After evaluatin the performance for all the cells, we choose the one with smallest averae error probabilit as our optimal thresholds. Intuitivel, one can et arbitraril close to the optimal thresholds b decreasin the cell size, which proportionall increases the computational complexit. 8 Greed search choose initial thresholds t, for example, the thresholds obtained via the NOCSIF case and set r iteration index; compute the averae error probabilit usin t as local thresholds; 3 choose several directions. For example, for two-sensor case, we can choose,,,,,, and, as our directions; 4 choose a small stepsize ɛ; 5 move the current thresholds t r one stepsize alon each direction and compute the averae error probabilit usin the new thresholds; 6 compare the error probabilit of usin current thresholds t r and new thresholds and assin the thresholds correspondin to the smallest one as t r+ ; 7 if t r+ t r, stop; otherwise, set r r +andoto step 5. Greed search method is much faster than exhaustive search method since we do not need to compute the averae error probabilit correspondin to ever point. But it can onl uarantee converence to a local minimum point. As seen from Fiure, the CSI case has the best performance and NOCSIF case has the worst performance. This is true because the desiner has the most information in the clairvoant case and the least information in NOCSIF case. The CSIF case is onl slihtl worse than the CSI case but is much better than the NOCSIF case. The difference of CSIF and CSIF is almost indistinuishable. The explanation is that the performance is much more sensitive to the fusion rule than to the local sensor thresholds. This phenomenon has been observed before: the error probabilit versus threshold plot is rather flat near the optimum point, hence is robust to small chanes in thresholds. We also consider an asmmetric case where σ σ. As plotted in Fiure 3 where σ andσ 3, all three cases with known CSI at the fusion center have similar performance and are much better than the NOCSIF case. This is consistent with the smmetric case. As we stated above, for the sstem with onl two local sensors, a mixed approach CSIF achieves almost same performance to that of the CSIF case. In the followin, we show that the same holds true even in the lare sstem reime, that is,as K increasin. We first consider the Baesian framework. As K oes to infinit, all the local sensors use the same local decision rule [6] and the optimal local thresholds are determined b maximizin the Chernoff information to achieve the best error exponent, C p Y H, p Y H [ min lo p s sdy Y H p Y H ]. s 9 For simplicit, we consider another performance measure, Bhattachara s distance, which is an approximation of

6 6 EURASIP Journal on Wireless Communications and Networkin Averae error probabilit KL distance Sinal-to-noise ratio db Local threshold.5 CSI CSIF CSIF NOCSI NOCSIF CSIF Fiure 3: Averae error probabilit versus channel SNR for nonidentical channel distribution. Fiure 5: KL distance versus the local threshold at the local sensors. Bhattachara distance NOCSIF CSIF.5.5 Local threshold Fiure 4: Bhattachara s distance versus the local threshold at the local sensors. Chernoff information b settin s /, B p Y H, p Y H [ lo p ] / /dy Y H p Y H [ p Y / H lo p p ] Y H dy. Y H.5 3 From Fiure 4, which ives Bhattachara s distance as a function of local threshold for both CSIF and NOCSIF cases with σ andσ, we can observe that the optimal threshold obtained in NOCSIF case is close to that of the CSIF case. Alternativel, under the Neman-Pearson framework, the Kullack-Leibler KL distance ives the asmptotic error exponent, KL p Y H, p Y H [ ] p Y H E H lo p Y H p p Y H Y H lo p dy. Y H 3 In Fiure 5, with the same settin as in Fiure 4, the KL distance as a function of local threshold for both CSIF and NOCSIF cases is plotted and we can also observe that the optimal local thresholds obtained in NOCSIF and CSIF cases are close to each other. 5. CONCLUSIONS In this paper we investiated the desin of the distributed detection problem with onl channel fadin statistics available to the desiner. Restricted to conditional independent observations and binar local sensor decisions, we derive the necessar conditions for optimal local sensor decision rules that minimize the averae error probabilit for the CSIF and NOCSIF cases. Numerical results indicate that a mixed approach where the sensors use the decision rules from the NOCSIF approach while the fusion center implements a fusion rule usin the CSI achieves almost identical performance to that of the CSIF case.

7 B. Liu and B. Chen 7 APPENDICES A. PROOF OF A K > AND B K > From A.5andA.6, A.7implies P H k, U k >P H k, U k. A.8 P H k, U k p H, k, U k p k, U k p k, U k H P H p k, U k H P H + p k, U k H P H π p k U k, H P Uk H π p k U k, H P Uk H +π p k. U k, H P Uk H A. Since the observations, X,..., X K, are conditionall independent and the local decision in each sensor depends onl on its own observation, the local decisions are also conditionall independent for a iven hpothesis. In addition, the local decisions are transmitted throuh orthoonal channels, thus the channel output for one sensor is conditionall independent to the channel input from another sensor. Therefore, p k U k, H p k H. A. Definin the likelihood ratio function for the local decision atthe kth sensor, L P U k H U k P. A.3 U k H Then P H k, U k π p k H P Uk H π p k H P Uk H + π p k H P Uk H π p k H L Uk π p k H + π p k H L Uk A.4 which is a monotone increasin function of LU k. Then, iven a monotone LR index assinment for the local output, that is, LU k >LU k, PH k, U k is a monotone increasin function of U k.thus, P H k, U k <P H k, U k. Similarl, we can et P H k, U k >P H k, U k. A.5 A.6 The optimum fusion rule is a maximum a posteriori rule i.e., to minimize error probabilit. Thus decidin U for a iven k and U k requires P H k, U k >P H k, U k. A.7 Therefore, we must have U for the same k with U k. Thus P U k, U k <P U k, U k A.9 and further, A k >. Similarl, one can show P U k, U k >P U k, U k and further, B k >. B. ALTERNATIVE DERIVATION FOR OPTIMAL LOCAL DECISION RULES A. A. A. From [7, Proposition.], with a cost function F : {, } Z {H, H } R and a random variable z which takes values in the set Z and is independent of X conditioned on an hpothesis, the optimal decision rule that minimizes the cost E[FγX, z, H] is where γx ar min P H j X α H j, U, U, j α H j, d E [ F U, z, H j Hj ]. B. B. Assumin the cost that we want to minimize at the fusion center can be written as J E [ C γ,, u, H ] B.3 and we further have J E [ C γ,, u, H ] E [ C γ Yk, k, k, k, γ k,, H ] E [ E [ C γ Yk, k, k, k, γ k,, H, k, k, k, X k, H ]] E [ E [ C γ Yk, k, k, k, γ k,, H X k, k ]] [ E C γ Yk, k, k, k, γ k,, H Y k p Y k γ k, k dyk ], B.4 where k [,..., k, k+, K ]andb.4 follows that conditioned on γ k X k and k, the channel output Y k is independent of everthin else.

8 8 EURASIP Journal on Wireless Communications and Networkin Definin X X k, z k,,, F U k, z, H C γ Yk, k,, U k,, H p Y k U k, k dyk Y k B.5 and substitutin them into B., weobtain the optimal local decision rule for kth sensor: γ k ar min P H j X k αk Hj, U k, B.6 U k, j where α k Hj, U k [ E C γ Yk, k, ], U k,, H j p Yk U k, k dyk H j. Y k B.7 Since P H j X k B.6isequivalentto where p X k H j πj i p X k H i πi, γ k ar min π j p X k H j αk Hj, U k U k, j ar min p X k H j bk Hj, U k, U k, j b k Hj, U k πj α k Hj, U k. B.8 B.9 B. Thus, the optimal local decision rule for the kth sensor is, P X k H bk H, b k H, γ k P X k H bk H, b k H,,, otherwise. B. In the approach proposed in this paper, the objective is to minimize the averae error probabilit at the fusion center, J P γ, H p d. B. Thus we have α k Hj, U k [ P U j, p, U k, pp ] H j d d. B.3 It is straihtforward to see that α k H, α k H, is the same as A k in 4 andα k H, α k H, is the same as B k in 5. Therefore, B. isequivalentto in Theorem. ACKNOWLEDGMENT This work was supported in part b the National Science Foundation under Grant REFERENCES [] R. Radner, Team decision problems, Annals of Mathematical Statistics, vol. 33, no. 3, pp , 96. [] R. R. Tenne and N. R. Sandell Jr., Detection with distributed sensors, IEEE Transactions on Aerospace and Electronic Sstems, vol. 7, no. 4, pp. 5 5, 98. [3] P. K. Varshne, Distributed Detection and Data Fusion, Spriner, New York, NY, USA, 997. [4] A. R. Reibman and L. W. Nolte, Optimal detection and performance of distributed sensor sstems, IEEE Transactions on Aerospace and Electronic Sstems, vol. 3, no., pp. 4 3, 987. [5] I. Y. Hoballah and P. K. Varshne, Distributed Baesian sinal detection, IEEE Transactions on Information Theor, vol. 35, no. 5, pp. 995, 989. [6] S. C. A. Thomopoulos, R. Viswanathan, and D. K. Bououlias, Optimal distributed decision fusion, IEEE Transactions on Aerospace and Electronic Sstems, vol. 5, no. 5, pp , 989. [7] J. N. Tsitsiklis, Decentralized detection, in Advances in Statistical Sinal Processin,H.V.PoorandJ.B.Thomas,Eds.,pp , JAI Press, Greenwich, Conn, USA, 993. [8] T. M. Duman and M. Salehi, Decentralized detection over multiple-access channels, IEEE Transactions on Aerospace and Electronic Sstems, vol. 34, no., pp , 998. [9] B. Chen and P. K. Willett, On the optimalit of the likelihoodratio test for local sensor decision rules in the presence of nonideal channels, IEEE Transactions on Information Theor, vol. 5, no., pp , 5. [] B. Liu and B. Chen, Channel-optimized quantizers for decentralized detection in sensor networks, IEEE Transactions on Information Theor, vol. 5, no. 7, pp , 6. [] A. Kashap, Comments on on the optimalit of the likelihood-ratio test for local sensor decision rules in the presence of nonideal channels, IEEE Transactions on Information Theor, vol. 5, no. 3, pp , 6. [] J. N. Tsitsiklis and M. Athans, On the complexit of decentralized decision makin and detection problems, IEEE Transactions on Automatic Control, vol. 3, no. 5, pp , 985. [3] P. K. Willett, P. F. Swaszek, and R. S. Blum, The ood, bad, and ul: distributed detection of a known sinal in dependent aussian noise, IEEE Transactions on Sinal Processin, vol. 48, no., pp ,. [4] B. Chen, R. Jian, T. Kasetkasem, and P. K. Varshne, Fusion of decisions transmitted over fadin channels in wireless sensor networks, in Proceedins of the 36th Asilomar Conference on Sinals, Sstems and Computers, vol., pp , Pacific Grove, Calif, USA, November. [5] R. Niu, B. Chen, and P. K. Varshne, Decision fusion rules in wireless sensor networks usin fadin channel statistics, in Proceedins of the 37th Annual Conference on Information Sciences and Sstems CISS 3, Baltimore,Md,USA,March 3. [6] J. N. Tsitsiklis, Extremal properties of likelihood-ratio quantizers, IEEE Transactions on Communications, vol.4,no.4, pp , 993.