Sensor Network Localisation with Wrapped Phase Measurements

Transcription

1 Sensor Network Loalisation with Wrapped Phase Measurements Wenhao Li #1, Xuezhi Wang 2, Bill Moran 2 # Shool of Automation, Northwestern Polytehnial University, Xian, P.R.China. 1. wenhao23@mail.nwpu.edu.n Melbourne Systems Laboratory, Dept. Eletrial & Eletroni Engineering, University of Melbourne, Australia. 2. {xwang,wmoran}@unimelb.edu.au Abstrat One of the major issues with the appliation of Radio Interferometri Positioning System RIPS) is the ambiguous measurement aused by a wrapped signal phase. Multiple transmissions at different frequenies have been used to resolve the phase ambiguity of a RIPS measurement in the existing approahes. In pratie, the ways to solve the ambiguity problem are heavily dominated by the power onsumption whih is vital for extending the lift time of sensor networks. This paper onsiders an approah whih loalizes a free mote by diretly using the ambiguous RIPS measurements in an attempt to redue the number of transmissions. The latter roughly indiates the energy ost whih has to be paid for the loalization of a sensor in a low ost sensor network. In this work, we derive a ondition for the hoie of wavelengths suh that a free mote an be loalized with the minimal number of transmissions as little as three distint frequenies. Our simulation results demonstrate the effetiveness of the proposed algorithm. Index Terms Radio Interferometri Positioning System RIPS), phase ambiguity, mote network loalization, self ooperated sensor networks, wireless ommuniations. and C are anhor motes i.e., their loation are known) and D is a free mote with unknown loation. Two motes at as transmitters and send pure sine wave at slightly different frequenies. This results in an interferometri signal at a low beat frequeny whih an be reeived by the other two motes ating as reeivers). The distane differenes between four motes an be obtained from the phase differene of the reeived interferene signals at the two reeiver loations. If motes A and B serve as transmitters and motes C and D form the reeiver pair, the orresponding RIPS measurement an be obtained by d ABCD = d AD d BD + d AC d BC, where d XY is the distane between X and Y. Similarly, 6 5 A B I. INTRODUCTION The appliation of Radio Interferometri Positioning System RIPS) to the loalisation of a wireless Mote network was proposed in [1], followed by the disussions in [2], [3], [4]. With the integrated wireless ommuniation failities onboard of a Mote, it beomes feasible to measure and thus loalize a free Mote using the RIPS measurements. The RIPS tehnique yields enormous potential for appliations involving the sensor networks equipped with Motes. In the same priniple, the RIPS measurement may also be used to loalize a sensor in 3D. Furthermore, by utilizing Doppler effet of RIPS measurement, the movement of a sensor node an also be estimated. Based on the RIPS measurement model, a hyperboli positioning tehnique was identified. The loalization errors are analyzed in [5] and several estimators for the estimation of the loation of a free Mote using RIPS measurements were presented in the literature [6], [7]. Fig. 1 shows the basi onept of loalizing a free mote using RIPS measurements based on a simple sensor network involving a olletion of four motes A, B, C, D, where A, B Y position C D Anhor Unknown X position 5 6 Fig. 1. Illustration of a RIPS. The sensor network involves 4 Motes labeled as A, B, C and D, where nodes A, B, C are anhors and D is a free node. by assigning different sensors as transmitter and reeiver, up to three RIPS measurements, d ABCD, d ACBD and d BCAD, will be obtained from the quartet and the third measurement an be derived from the first two. The free mote D an be loalized using two independent) RIPS measurements via the 68

2 hyperboli positioning tehniques [5]. Fundamentally, a RIPS measures the distane differene d ABCD by the wavelength of the interferometri signal. Beause a feasible wavelength is muh shorter than the distane to be measured, measurement ambiguity aused by signal phase wrapping is inevitable. Therefore, a RIPS measurement is ambiguous in a single frequeny. The RIPS measurement ambiguity an be addressed by transmitting multiple frequenies. As disussed in [1], resolving the phase ambiguity of a set of RIPS measurements results in seeking a solution from a group of onstrained diophantine equations. The latter is traditionally addressed by the well known Chinese Remainder Theory CRT). Various CRT algorithms an be found in the literature suh as [8], [9]. As exhaustive searh proesses are involved, the omputational omplexities of these algorithms in terms of the number of transmissions 1 are expensive, in partiular, when measurement noise is presented. In [1], a maximum likelihood proedure is proposed to estimate the true RIPS measurement, where five or six frequenies have to be transmitted to reasonably remove the ambiguity of a RIPS measurement in a realisti senario. In [11], a Closed-form CRT is presented to solve the ambiguity problem, however, a large number of transmissions are required to resolve the ambiguity problem. As a powerful mathematis tool, the lattie theory an be used to solve the ambiguity problem. In [13], it is suessfully applied to solve the phase unwrapping problem in the frequeny estimation area. A least square phase unwrapping estimator LSPUE) is proposed in [14] for unwrapping the phase of unknown frequeny signal using a finite number of samples. It has been shown in [14] that the LSPUE algorithm is essentially a solution of the nearest point problem in lattie theory, where a speial generator basis is used and therefore, the algorithm an be performed in polynomial-time. While the sphere deoder algorithm in [15] is ommonly used to solve the nearest point problem in a lattie, it is omputationally feasible only when the sample size is small. A omputational effiient algorithm for LSPUE is presented in [16], [17]. In [12], the methods for traking and loalizing a moving target using ambiguous phase measurements were reported. Partiularly, the loation of a miro-motion target an be found by the estimation of the wrapped phase measurement distributions from the measurements of three phase-only sensors. In this paper, we onsider an approah whih addresses the free mote loalisation problem in the loation spae without diret resolving the phase ambiguity of RIPS measurements. The ambiguous RIPS measurements are mapped into the loation spae and possible loations of the free mote are then identified at the triple intersetions of hyperbolas. Within a speified region of interest, a ondition for the hoie of wavelengths is derived, whih guarantees that an unique triple intersetion of hyperbolas is able to be found. A node loalization proedure established based on the proposed triple 1 The number of transmissions in the RIPS in the ontext of this paper refers to the number of frequenies/wavelengths used for loalizing a free node. intersetion of hyperbolas is presented. Simulation results demonstrate the effetiveness and effiieny of the proposed method. II. PROBLEM A RIPS measurement involves four nodes. As shown in Fig. 1, suppose that the nodes A and B transmit pure sine waves at losed frequenies f A and f B normally, f A f B 1 KHz). The transmitted signal at time t is of the form st) =a A os 2πf A t + ϕ A )+a B os 2πf B t + ϕ B )+νt) 1) where νt) is the noise term of Gaussian distribution, a A and a B are the amplitudes, the ϕ A and ϕ B are the phase offset of the two sine waves respetively. The reeived signal at node C is s C t) =a AC os 2πf A t t A d )) AC + a BC os 2πf B t t B d BC )) + ν 1 t) 2) Similarly, the reeived signal at node D is s D t) =a AD os 2πf A t t A d )) AD + a BD os 2πf B t t B d )) BD + ν 2 t) 3) where is light speed. It an be shown that the relative phase offset of these two signals is of the form [1] φ = 2πf A t A + d ) AC +2πf B t B + d ) BC +2πf A t A + d AD ) 2πf B t B + d BD = f 2π d AD d AC + d BC d BD ) ) mod 2π) + f 2π d AD d AC + d BC d BD ) mod 2π) 4) where f = f A+f B 2, f = f A f B 2. Sine f A f B 1kHz, thus f/, then the seond term in 4) approximately vanishes and therefore, the phase offset of the nodes C and D an be represented by following equation: φ =2π d AD d BD + d BC d AC λ mod 2π) 5) where λ is the wavelength of the signal. Then the RIPS measurement the sum of distane differenes is defined as: d ABCD = d AD d BD + d AC d BC 6) From 5), it is lear that the RIPS measurement 6) observed by the system is wrapped by the wavelength λ. As stated 69

3 in introdution setion, multiple arrier frequenies should be used solve the ambiguity problem. In multiple arrier frequenies {f i },i=1,,m >, the orresponding RIPS measurements {φ i } an be generated as φ i =2π d ABCD λ i mod 2π) 7) y i = d ABCD mod λ i ) 8) where y i = λ i φ i /2π is the measurement measured by the RIPS. Equivalently, 8) an be written as i = n i λ i + y i 9) where n i Z, i = 1,,m are integers and i are the possible values of d ABCD aording to different n i and λ i. For ooperative sensor networks as disussed in this paper, aquiring eah φ i is deemed as a single transmission. Therefore, the number of required transmissions for loalizing a free mote D serve as a measure of system energy onsumption. It is desirable to develop a loalization sheme whih uses the least number of transmissions for loalizing a free node. The sensor loalization problem is to estimate the loation of a free node using a set of RIPS measurements desribed by 9), whih is traditionally addressed by firstly estimating the RIPS measurement d ABCD via 9) using multiple transmissions and the loation of the free node is then estimated based on several independent RIPS measurements d ABCD ), for example, two independent measurements d ABCD and d ACBD an loalize a free node in a 2D plane. From a pratial point of view, both the original work in [1] and the later development in [9], [11], [1] laimed that at least six or more transmissions will be required to resolve the RIPS measurement ambiguity in the measurement domain and therefore loalize a free node in a 2D plane. In the presene of measurement noise, the required number of transmissions inrease onsiderably to aount for a statistial solution. In this work, the problem of free node loalization is addressed in the loation spae using ambiguous RIPS measurements without expliitly resolving phase ambiguity. III. TRIPLE INTERSECTION In the senario illustrated in Fig. 1, without phase ambiguity, two independent RIPS measurements e.g. d ABCD and d ACBD ) an be found from three possible measurements and the third RIPS measurement e.g. d BCAD ) an be derived from the other two. i.e., d ABCD = d AD d BD + d AC d BC d ACBD = d AD d CD + d AB d BC d BCAD = d BD d CD + d AB d AC 1) and d BCAD = d ACBD d ABCD 11) Eah measurement in 1) represents a hyperbola in the loation plane and the loation of the free node is determined by the intersetion of two independent RIPS measurements. In the presene of phase ambiguity, the first two RIPS measurements beome two sets of hyperbolas in the loation plane and many intersetions an be found. By using a different wavelength, the third RIPS measurement, whih results in another set of hyperbolas in the loation plane, an be used to redue the loalization ambiguity. Possible loations of the free node are given by the triple hyperboli intersetions of the three RIPS measurements. By arefully seleting the wavelengths for all transmissions, the loation of the free node may be uniquely determined. This is main idea of this work. In the presene of phase ambiguity, 1) is given by 9): d λ1 ABCD,i = n 1λ 1 + y 1 ACBD,j = n 2λ 2 + y 2 12) BCAD,k = n 3λ 3 + y 3 where {n 1,n 2,n 3 } is a set of integers, d λ1 ABCD,i, dλ2 ACBD,j and BCAD,k are the possible value of the d ABCD, d ACBD and d BCAD based on the wavelengths λ 1, λ 2 and λ 3 respetively. It is pratial to assume that the loations of the free mote are onstrained in a region and the feasible values of d ABCD, d ACBD and d BCAD are bounded by [ Λ p, Λ p ],p=1, 2, 3, whih an be omputed from 1). The assoiated integers are n p [n p,min,n p,max ], where n p,min and n p,max are the minimum and maximum value of n p. n p,min and n p,max an be omputed using the following equations: n p,min = Λ p λ p, n p,max = Λ p λ p 13) Note that the values of both the real sum of distane differenes d ABCD and measured one y i are symmetri about. From 1) it is followed that d ABCD = d BACD 14) Suppose that y i < and d ABCD <, thus y i = d ABCD mod λ i ) y i = d AD d CD + d BC d AB ) mod λ i ) y i = d BACD mod λ i ) 15) The above onlusion means that we may onsider a negative d ABCD as a positive d BACD by swapping the order of transmitters i.e., A, B B,A). Therefore, in this paper, we will only onsider the ase where the real distane d ABCD and d ACBD are in [, Λ p ], i.e. n p [,n p,max ]. As we mentioned, if λ 1 = λ 2 = λ 3, d λ1 BCAD,k dependent on d λ1 ABCD,i and λ 2 is totally and dλ1 ACBD,j. However, if λ 1 λ 3 λ 3, the dependeny between d λ1 ABCD,i, dλ2 ACBD,j an only our at some values whih satisfy the and BCAD,k following ondition: ACBD,j dλ1 ABCD,i = dλ3 BCAD,k 16) Note that d λ1 ABCD,i, dλ2 ACBD,j and BCAD,k orrespond to a family of hyperbolas whih have at least one triple intersetion. In these intersetion points, there must exist one 7

4 intersetion whih is the true loation of free mote and all others are false loations. These triple intersetions represent possible loations of the free node and an be solved using the trilateration method in [5] or a nonlinear estimator [6]. Therefore, a triple intersetion point an only our when 16) is satisfied. Furthermore, if 16) holds only for d ABCD, d ACBD and d BCAD i.e. the ground truth), there will be an unique triple intersetion the true loation of node D. The idea is that the λ 1, λ 2 and λ 3 may be seleted smartly to result in an unique triple intersetion in the region of interest. unique triple intersetion an be found in the loation plane when the wavelengths are hosen as λ 1 = 1.115m, λ 2 = 1.125m and λ 3 =1.397m. In general, the hoie of wavelengths λ 3 for an unique triple intersetion is desribed by the following theorem: Theorem 1. For the senario in Fig. 1, an unique triple intersetion will our if λ 3 = {λ 3 R LCMλ 3,λ 1 ) > Λ 1, LCMλ 3,λ 2 ) > Λ 2, and y p y q for y p, y q Y} 17) where the LCMa, b) is the least ommon multiple of a and b, Y is a vetor with entries: y p = {modn 2 λ 2 n 1 λ 1,λ 3)} 18) Proof: λ R, we know that the rewritten as: b,,a,d,k an be Fig. 2. Multiple triple intersetions are presented with wavelengths λ 1 = 1.115m, λ 2 =1.125m and λ 3 =1.25m in the senario shown in Fig. 1. b,,a,d,k = d λ a,b,,d,i d λ a,,b,d,i where the d is the fititious measurement aording to a arbitrary wavelength. Thus, when the following ondition holds, d λ1 a,b,,d,i = d λ a,b,,d,i a,,b,d,j = d λ a,,b,d,j b,,a,d,k = d λ a,b,,d,i d λ a,,b,d,i 19) there will be a triple intersetion. λ1 Then, if multiple ombination { d a,b,,d,i, dλ 2 a,,b,d,j } satisfy 19), there will be many intersetions. 1) Let λ = λ 3,if λ 3 = {λ R LCMλ 1,λ ) < Λ 1, LCMλ 2,λ ) < Λ 2 } 2) thus, from 19), the number of d λ1 a,b,,d,i = d λ a,b,,d,i and a,,b,d,j = d λ a,,b,d,j will be greater than 1. If λ 3 = {λ R LCMλ 1,λ ) > Λ 1, LCMλ 2,λ ) > Λ 2 } 21) thus, the unique triple point may our when ȳ p = ȳ q holds, where {y p, y q } {modn 2 λ 2 n 1 λ 1,λ 3)}. 2) Let λ λ 3, we also have the same onlusion. Therefore, for the unique triple intersetion, if the λ 3 an satisfy following ondition: Fig. 3. Only a single triple intersetion is presented for the wavelengths λ 1 =1.115m, λ 2 =1.125m and λ 3 =1.397m, in the senario shown in Fig. 1. Fig. 2 shows that many triple intersetions are presented in the loation plane when the wavelengths are λ 1 =1.115m, λ 2 = 1.125m and λ 3 = 1.25m. Fig. 3 demonstrates that λ 3 = {λ 3 R LCMλ 3,λ 1 ) > Λ 1, LCMλ 3,λ 2 ) > Λ 2, and y p y q for y p, y q Y} 22) There will be only one triple point, i.e. the true loation of the target, d λ1 λ3 a,b,,d,i = d a,b,,d,i = d a,b,,d and a,,b,d,j = a,,b,d,j = d a,,b,d. 71

5 In pratial situation, the available wavelengths of a mote is limited and λ 3 an not be seleted arbitrarily. The seletion of the λ 1 and λ 2 an be ruial to the availability of λ 3. From Theorem 1, we notie that λ 1 λ 2 and, moreover, it an be shown that the optimum λ 2 given λ 1 is: λ 2 =argmax{lcmλ 1,λ 2)} 23) λ 2 For example, for mote appliation, the radio frequeny is in the range of 3MHz 5MHz. Ifλ 1 =.65m and λ 2 =.655m, the λ 3 is found as 1.455m whih is out of the mote feasible radio frequeny range. Aording to 23), if λ 2 =.955m is hosen, λ 3 =.985m will guarantee to have an unique triple intersetion. As another example, if λ 1 =.98m is seleted, we obtain λ 2 =.985m and λ 3 =.965m aording to 18) and 23). IV. NOISE EFFECT AND LOCALIZATION PROCEDURE In the presene of measurement noise, the wrapped phase 8) may be expressed as ỹ i = y i + ω i i =1, 2, 24) ω i N, σi 2 ) 25) where y i signifies the noiseless wrapped measurement orresponding to 8) and ω i signifies the noise of Gaussian distribution with zero mean and variane σ 2. Clearly, the problem beomes more ompliated sine it is hard to find an exat triple intersetion, or the alleged triple intersetion simply does not exist. A robust loalization proedure, whih an effetively address the above issues, is desribed below based on the senario illustrated in Fig. 1. Firstly, the noisy RIPS measurements are obtained using the wavelengths λ i,i = 1, 2, 3 seleted based on the role presented in Setion III and the values of ˆd ABCD, ˆdACBD and ˆd BCAD are then estimated from the RIPS measurements, where the integers ˆn 1, ˆn 2, and ˆn 3 are obtained under the following riterion: {ˆn 1, ˆn 2, ˆn 3 } = arg min{n 2 λ 2 n 1 λ 1 n 3 λ 3 +ỹ 2 ỹ 1 ỹ 3 )} 26) and in turn the values of ˆd ABCD, ˆd ACBD and ˆd BCAD an be found from the {ˆn i, ỹ i } via 12). The free node an then be loalized via a trilateration or nonlinear estimation method, i. e., determining the intersetion of the two hyperbolas defined by any two of ˆd ABCD, ˆd ACBD and ˆd BCAD. The loalization proedure is summarized as the following: Step 1: Determine the optimum λ 2 aording to the λ 1 and 18); Step 2: Find the λ 3 by the ondition 22) over n 1 and n 2 ; Step 3: Obtain the measurements ỹ i aording to λ i ; Step 4: Estimate the ˆd ABCD, ˆd ACBD and ˆd BCAD via 12); Step 5: Compute the double intersetions using the trilateration method [5] or a nonlinear estimator [6]. The proposed algorithm is tested via Monte Carlo multiple runs using the senario as shown in Fig. 4, where the loations of three anhor motes A, B and C are given as marked in the figure. At eah run, the loation [x D,y D ] of the free mote D is drawn from a uniform distribution over the area of 6 6m 2. The wavelength λ 1 =1.125m and the other two wavelengths are determined by the ondition 22), i. e., λ 2 =1.115m and λ 3 =1.397m. Y position A 1,5) B 1, 1) C 5,3) X position Fig. 4. Illustration of simulation senario, where the anhor nodes A, B and C loate at [1, 5]m, [1, 1]m and [5, 3]m, respetively. The loalization performane is evaluated in terms of the probability of a given root mean squared) estimation error and the variane of the measurement noise σ i σ via Monte Carlo statistis. The estimation error under onsideration is ˆxD x D ) 2 +ŷ D y D ) 2 1m 27) whih is ounted over 1 runs for eah value of σ. The statistial result is shown in Fig. 5. It is observed that the loalization performane an be greatly improved when additional ambiguous RIPS measurements an be taken. For example, if a forth measurement ỹ 4 about d ABCD an be taken, i.e. d λ4 ABCD,m = n 4λ 4 +ỹ 4, even with λ 4 = λ 3, a better loalization suess probability shown in Fig. 6 ompared to that shown in Fig. 5) an be observed. For a omparison, the result produed by a maximum likelihood estimation MLE) algorithm presented in [1] is also plotted in Fig. 6), where the wavelengths used are λ 1 =.56m, λ 2 =.61m, λ 3 =.63m and λ 4 =.65. To generate 2 independent RIPS measurements and therefore loalize a free node, the total number of transmissions required is 8. Note that the MLE algorithm is a CRT based approah and generally requires more transmissions than the proposed 72

6 Probability log 1 σ Fig. 5. Probability of a suessful loalization for error satisfying 27) versus the standard deviation of RIPS measurement error σ m) using three transmissions. Probability MLE algorthm New algorithm log 1 σ 2 Fig. 6. Probability of a suessful loalization for error satisfying 27) versus the standard deviation of RIPS measurement error σ m) using four transmissions. method for ahieving a similar loalization performane. The latter takes the advantage of onstraint on wavelength Theorem 1). V. CONCLUSION In this paper, the issue of RIPS measurement ambiguity due to a wrapped phase is addressed in the appliation of mote network loalization. Unlike existing approahes where the ambiguity of measurement is resolved in the measurement domain, ambiguous RIPS measurements are mapped into the loation spae to loalize a free node without expliitly resolving the measurement ambiguity. It is shown when the frequenies of transmitted sine waves satisfy the ondition desribed in Setion III, the number of transmissions required for a free node loalization an be redued ompared to a CRT based method. An algorithm based on this idea is therefore derived. Results from both derivation and simulation show that the proposed method is apable of saving power by using as little as three transmissions for loalizing a free node, whih is important to the loalization appliations of self ollaborative sensor networks. The impat of measurement noise is also onsidered. REFERENCES [1] M. Maroti, B. Kusy, G. Balogh, P. Volgyesi, K. Molnar, A. Nadas, S. Dora, and A. Ledezi. Radio interferometri positioning, Institute for Software Integrated Systems, Vanderbilt University, Teh. Rep. ISIS- 5-62, Nov. 25. [2] B. Kusy, A. Ledezi, M. Maroti and L. Meertens. Node-Density Independent Loalization, Proeeding of the 5th International Conferene on Information Proessing in Sensor Networks IPSN6), Nashville, Tennessee, USA, April 19 21, 26. [3] B. Kusy, J. Sallai, G. Balogh, A. Ledezi, V. Protopopesu, J. Tolliver, F. DeNap and M. Parang. Radio interferometri traking of mobile wireless nodes, 5th International Conferene on Mobile Systems, Appliation, and Servies, San Juan, Puerto Rio, June 27. [4] S. Szilvasi, J. Sallai, I. Amundson, P.Volgyesi and A. Ledezi. Configurable hardware-based radio interferometri node loalization, Pro. IEEE Aerospae Conferene, Big Sky, MT, 21. [5] X. Wang, B. Moran and M. Brazil. Hyperboli Positioning Using RIPS Measurements for Wireless Sensor Networks, Proeedings of the 15th IEEE International Conferene on Networks ICON27), Adelaide, Australia, pp , Nov. 27. [6] Y. Cheng, X. Wang, T. Caelli, X. Li and B. Moran, Optimal Nonlinear Estimation for Loalization of Wireless Sensor Networks, IEEE trans. Signal Proessing, vol. 59, no. 12, pp. 1 12, Deember, 211. [7] M.R. Morelande, B. Moran and M. Brazil, Bayesian node loalization in wireless sensor networks, In Pro. IEEE International Conferene on Aoustis, Speeh and Signal Proessing, Las Vegas, USA, 28. [8] X. G. Xia and G. Y. Wang. Phase Unwrapping and A Robust Chinese Remainder Theory, IEEE Signal Proessing Letter, Vol.14, no.4, pp , 27. [9] X. W. Li and X. G. Xia. A Fast Robust Chinese Remainder Theorem Based Phase Unwrapping Algorithm, IEEE Signal Proessing Letter, Vol.15, pp , Ot. 28. [1] W. C. Li, X. Wang and B. Moran. Resolving RIPS measurement ambiguity in maximum likelihood estimation, Proeedings of the 14th International Conferene on Information Fusion, pp. 1-7, 211. [11] W. J. Wang and X. G. Xia. A Closed-Form Robust Chinese Remainder Theorem and Its Performane Analysis, IEEE transations on Signal Proessing, Vol.58, no.1, pp , 21. [12] Y. Q. Cheng, X. Z. Wang, T. Caelli, X. Li and B. Moran. Traking and Loalizing Moving Targets in the Presene of Phase Measurement Ambiguities, IEEE Transations Signal Proessing, Vol.59, No.8, pp , Aug. 211 [13] R. G. MKilliam, B.G. Quin, I.V.L Clarkson and B. Moran. Frequeny Estimation by Phase Unwrapping, IEEE Transations on Signal Proessing, Vol. 58, No. 6, pp , June, 21. [14] I. V. L. Clarkson. Frequeny Estimation, Phase Unwrapping and the Nearest Lattie Point Problem, International Conferene on Aoustis, Speeh, and Signal Proessing ICASSP 99), Vol. 3, pp , Mar [15] E. Agrell, T. Eriksson, A. Vardy and K. Zeger. Closest Point Searh in Latties, IEEE Transations Information Theory, Vol.48, No.8, pp , Aug. 22. [16] I. V. L. Clarkson. An Algorithm to Compute a Nearest Point In the Lattie A N, Applied Algebra, Algebrai Algorithms and Error- Correting Codes, M. Fossorier, H. Imai, S. Lin, and A. Poli, Eds. New York: Springer, 1999, vol.1719, Leture Notes in Computer Sine, pp [17] R. G. MKilliam, I. V. L. Clarkson, and B. G. Quinn. An Algorithm to Compute the Nearest Point In the Lattie A N, IEEE Transations Information Theory, Vol. 54, No. 9, pp , Sep