Resolving RIPS Measurement Ambiguity in Maximum Likelihood Estimation

Transcription

1 14th International Conferene on Information Fusion Chiago, Illinois, USA, July 5-8, 011 Resolving RIPS Measurement Ambiguity in Maximum Likelihood Estimation Wenhao Li, Xuezhi Wang, and Bill Moran Shool of Automation, Northwestern Polytehnial University, Xian, P.R.China, * Melbourne Systems Laboratory, Dept. Eletrial & Eletroni Engineering, University of Melbourne, Australia, Abstrat A Radio Interferometri Positioning System RIPS) is able to measure the sum of distane differenes between four sensor nodes. If the loations of the three out of four sensor nodes are known, that of the fourth node may be determined using RIPS measurements via the tehniques of hyperboli positioning. One of the key issues with RIPS is the measurement ambiguity beause of the wrapping of distane by phase measurements. Solutions to this RIPS measurement ambiguity problem often require extensive omputational power whih is vital to some wireless network of small sensors. In this paper, we desribe an effiient stohasti method to pratially resolve RIPS measurement ambiguity under the riterion of Maximum likelihood estimation. Our simulation results demonstrate the effetiveness of the proposed algorithm. Keywords: Maximum Likelihood Estimation, Radio Interferometri Positioning System, Phase ambiguity removal, Mote loalization, Wireless sensor. 1 Introdution Loalization of wireless sensor network has beome an important issue reently. The most widely disussed loalization tehniques mainly fous on obtaining the distane between the nodes, suh as time differene of arrival TDOA), time of arrival TOA), or reeived signal strength indiator RSSI). These tehniques are very effetive but often require failities of higher ost. For small sensors of low ost and power, above tehniques may fae numerous hallenges. The RIPS tehnique was introdued to mote loalization in [1] and later disussed in [, 3, 4]. The main idea of RIPS is to generate frequeny interferene signal by a pair of nodes transmitting at lose frequenies, and then obtain the relative phase offset whih an be measured by the low-ost sensor with high auray. The RIPS tehnique has generated great opportunities for network node loalizations for small sensors like motes. A number of loalization algorithms using RIPS measurements were investigated in the literature [5, 6], where the impat of RIPS measurement noise has been onsidered. Beause RIPS measures the relative phase offset of two nodes, whih involves the sum of distane differenes between four nodes, the phase ambiguity will inevitably our due to the wrapping of distane in phase. To remove the ambiguity, a method desribed in [1] formulates the Diophantine equations by a set of phase measurements for many arrier frequenies, and then finding the potential value in a bin sorting manner. However, in the presene of measurement noise, this method has a lower auray. A least square phase unwrapping estimator algorithm was presented in [7] to estimate the frequeny of a signal based on lattie. In [8], a method whih is based on the Chinese Remainder Theorem has been desribed. Apart from its omputational omplexity, the approah does not address measurement noise properly. An alternative approah whih solve the ambiguity issues of phase measurements whih are observed by three phase-only sensors for both motion and miro-motion target traking has reently presented in [9]. In this paper, we propose a maximum likelihood estimation MLE) method to remove the RIPS measurement ambiguity by estimating its ground truth and this work is motivated by [10]. The joint likelihood funtion of RIPS measurements under multiple wavelengthes is approximated by a mixture of trunated Gaussian distributions, the MLE is applied to estimate the true value of the measurements. The proposed method an be seen as a diret appliation of Chinese Reminder theorem to a stohasti Diophantine problem in a finite distane interval. The effetiveness, effiieny and robustness of the proposed algorithm are demonstrated via simulation. In the next setion, RIPS measurement generation ISIF 414

2 and the issue of RIPS measurement ambiguity are desribed. The maximum likelihood algorithm for estimating the real value of RIPS measurement using signals of different frequenies is presented in Setion 3. Algorithm performane evaluation via simulation is given in Setion 4, followed by the Conlusions. Ambiguity Issue of RIPS Measurements A RIPS requires four nodes to generate the measurement whih is shown in Fig. 1, where the nodes A and B transmit pure sine waves at losed frequenies f A and f B whih satisfy the ondition f A f B ξ f A or f B. 1) In pratie, ξ 1 KHz. The reeived signal at time t is of the form [1] st) =a A os πf A t + ϕ A )+a B os πf B t + ϕ B )+νt) ) where νt) is the noise term of Gaussian distribution, a A and a B are amplitudes, and the ϕ A and ϕ B are phase offset of the two sine waves respetively. d AC d BC d AD d BD of the attenuation signal transmitted by X and reeived by Y. It an be shown that, when the noise is negleted temporarily, the relative phase offset of these two signals is of the form φ = πf A t A + d ) AC +πf B t B + d ) BC +πf A t A + d AD ) πf B t B + d BD = f π d AD d AC + d BC d BD ) ) mod π) + f π d AD d AC + d BC d BD ) mod π) 5) where f = fa+fb, f = fa fb. In view of 1), f/ 0, and the seond term in Eq.5) vanishes approximately. Therefore, the phase offset of the two signals reeived at C and D an be expressed as φ =π d AD d BD + d BC d AC λ mod π) 6) where the λ is the wavelength of the signal, and φ is the measurement of RIPS. Note that the φ is the phase offset of the low frequeny signal, f A f B, while λ is the wavelength of high-frequeny signal, i.e. λ = /f A + f B )/. In mote appliations, the range of the wavelength is around meters. An important remark is that synhronization is not required between the four sensors whih have a ommon lok rate. The distane differene in Eq.6) an be defined as d ABCD = d AD d BD + d BC d AC 7) Figure 1: Measurement priniple of RIPS. The reeived signal at node C is s C t) =a AC os πf A t t A d )) AC + a BC os πf B t t B d )) BC + ν 1 t) 3) Similarly, the reeived signal at node D is s D t) =a AD os πf A t t A d )) AD + a BD os πf B t t B d )) BD + ν t) 4) where is light speed, d XY is the Eulidean distane between the nodes X and Y and a XY is the amplitude Similarly, let the nodes A and C be the transmitter pair and the nodes B and D be the reeiver pair, we have d ACBD = d AD d CD + d BC d AB 8) If the loation of nodes A, B and C are known, the loation of node D an be alulated by finding the intersetion point of the two hyperboles defined by the d ABCD and d ACBD [5]. Unfortunately, either Eq.7) or Eq.8) annot be obtained diretly from the RIPS measurement system with a single transmission due to the modulo π in Eq.6). By the Chinese Remainder Theorem, it is possible to redue the phase ambiguity problem using multiple arrier frequenies f 1,,f m, m>0whih generate multiple RIPS measurements φ i } m i : φ i =π d ABCD λ i mod π). 9) 415

3 Equivalently, 9) an be rearranged as y i = d ABCD mod λ i ) 10) where y i = λ i φ i /π is deemed to be a RIPS measurement. The value of y i is bounded by the maximum value that λ i an take. So, we assume that y i λ. Note that the value of the real distane d ABCD and the measured distane y i is symmetri about 0 and d ABCD [ Λ, Λ]. However, we an easily show that in general d ABCD = d BACD 11) Suppose that y i < 0andd ABCD < 0, 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 ) 1) This result means that we may onsider a negative d ABCD as a positive d BACD by swapping the order of transmitters. Therefore, in this paper, we will only onsider the ase where the real distane d ABCD and d ACBD are in [0, Λ]. In the absene of measurement noise, the d ABCD may be obtained by solving the following Diophantine equations d ABCD = y 1 + n 1 λ 1 = y + n λ.. = y m + n m λ m 13) where n 1,n,,n m } are a set of unknown integers. In theory, three arrier frequenies are enough to determine the d ABCD. In the presene of measurement noise, the problem beomes more ompliated sine Eq.13) an no longer hold. In [1], a simple approah was presented to alulate the d ABCD by searhing the set of integers n 1,n,,n m } for m frequenies suh that the following inequality holds. n i λ i + y i ) n j λ j + y j ) <ε 14) where ε is a fration of the wavelengths and is determined by the phase measurement auray. The unambiguous RIPS measurement d ACBD is then given by d ABCD = 1 y i + n i λ i ) 15) m The auray of this method is very low beause multiple integers n i may satisfy the ondition 14) and it is also influened by the prior knowledge of the d ABCD. In addition, the value of ε is hard to determine. 3 Maximum likelihood formulation and solution Inspired by [10], we present a method to solve the ambiguity of RIPS measurement. The basi idea to formulate and thus find a maximum likelihood solution to the underlying problem is skethed as follows. 1. The onditional distribution py i X) is approximated as a wrapped and trunated Gaussian distribution, where X is defined as the ground truth state of the unambiguous RIPS measurement, e.g., X = d ABCD.. As stated in the setion, m remeasurements are obtained for resolving the ambiguity, i.e. y 1:m = y 1,,y m }. Therefore, we need to onsider the joint probability distribution py 1:m X). 3. How to find the maximum likelihood estimator whih gives the following solution ˆX =arg max py 1:m X) X [0, Λ] is desribed in this setion. Let i,j be the jth possible value of d ABCD with unwrapped phase aording to ith wavelength, where the j an be determined when the Λ is given, namely, for ith wavelength, we have Λ j = 16) where rounds the element to the nearest integer. Thus λ i i,j = y i + n i,j λ i, i =1,,,m 17) In the presene of measurement noise ω i,themeasurement orresponding to the wrapped phase 13) may be expressed as y i = y i,0 + ω i 18) where y i,0 signifies the noiseless wrapped measurement orresponding to Eq.10). In this work, the distribution of measurement noise ω i is assumed to be a trunated Gaussian, i.e., N 0, σ pω i )= i ), ω i <δλ i 0, Otherwise 19) where 0 <δ 0.5, i.e., the error of phase measurement will be restrited within one wave length. It is reported in [] that the noise of the relative phase offset may be as high as 0.1λ i, whih indiates that this assumption is reasonable. Therefore, Eq. 17) may be written as i,j = y i,0 + n i,j λ i + ω i = X + ω i 0) 416

4 wherewenotethat X = y i,0 + n i,j λ i 1) Thus the true value of X may fall in the following intervals X [ i,j δλ i, i,j + δλ i ]=Ξ i,j ) From the relationship y i + n i,j λ i = X + ω i,wehave y i X)modλ i )=ω i 3) where the modulo operation has been removed. To understand the proess better, onsidering a simple example involving two frequenies, f 1 and f with wavelengths λ 1 and λ respetively. From two noiseless measurements, y 1,0 and y,0,wehave whih is shown in Fig.. 1,j = y 1,0 + n 1,j λ 1 + ω 1,j = y,0 + n,j λ + ω 30) In view of 19), the following likelihood funtion an be obtained } exp [yi X)mod λi)], X Ξ py i X) = σi i,j 0, Otherwise. 4) f 1 y 1 C 11 C 1 C 13 C 14 v 1 v v 3 v 4 v 5 v 6 As we mentioned, to estimate X, the joint likelihood funtion py 1:m X) needs to be onsidered. For m measurements, the value of X will fall in one of the intersetions of Ξ i,j, i.e., m X v l = Ξ i,j, l =1,,,k 5) where the j is defined as Eq.16). Then the joint likelihood funtion an be written as m py 1:m X) = p y i X) 6) } 1 = m ) exp [y i X)modλ i ] /σi πσi Due to the modulo operation i.e, mod λ i ), the py 1:m X) is a multimodal funtion whih may have many loal maximum values in the interval of [0, Λ]. Therefore, diretly finding the global maxima is nontrivial. Considering the following manipulations: i,j X = y i + n i,j λ X y i + n i,j λ X) modλ i = i,j X) modλ i y i X) modλ i = i,j X) modλ i y i X) modλ i = i,j X l ) 7) where X l =: X v l and the subsript j of the i,j an be determined by j =argmin j X l i,j,. 8) Thus the joint likelihood funtion 7) in interval v l is written as p y 1:m X l ) 1 = m ) exp πσi } i,j X l ) /σi 9) f 0 y C 1 C C 3 Figure : Illustration of phase ambiguity removal via two frequenies, where i = and δ =0.5. The intervals labeled as v l,l =1,, 6 are possible regions where the true phase measurement may fall in. It indiates that, for the first measurement, i =1, when j = 1 and δ =0.5, the true value of X may our in the range [ 1,1 0.5λ 1, 1,1 +0.5λ 1 ] whih indiated by the dash line in the Fig.. Similarly, for the seond measurement, the true value of X may our in the range [,1 0.5λ 1,,1 +0.5λ 1 ]whenj =1. Thus,the true value of X may fall in the intersetion of these two ranges. So six intervals denoted by v l,l=1,, 6are found in the range [0, Λ] aording to Eq.5) as shown in Fig.. The i,j an be determined using Eq.7), for example, the i,j orresponding to the interval v 3 are 1, and,. The joint likelihood funtion p y 1:m X l ) is only nonzeroineahoftheintervalsv l,l=1,,. The funtion py 1:m X l ) attains its maximum value at the point X l if the funtion F = i,j X l ) /σi 31) attains the minimum value. The idea is to firstly estimate a loal parameter ˆX l at whih the maximum value of p y 1:m X l ) in eah interval v l is ahieved. The state of X an then be found at the global maximum of p y 1:m X), whih is one the finite set of estimated parameters ˆX l } at whih the joint likelihood p y 1:m X) is maximised. 417

5 Take the derivative of the Eq.31) with respet to X l and set it to zero. We obtain m ) ˆX l = i,j /σi /α 3) f 1 y 1 C 11 C 1 C 13 C 14 where α = m 1/σ i. That is, the joint likelihood funtion p y 1:m X) attains a loal maximum value at ˆX l in v l. Substituting ˆX l into Eq.31), we have B l = m 1 k=i+1 [ i,j k,j ) /σ i σ k ] /α 33) The ˆX l orresponds to the smallest B l is the estimate of X under the riterion of maximum likelihood estimation. This implies that m 1 ĉ i,j =arg min i,j v l k=i+1 [ i,j k,j ) /σ i σ k ] 34) Therefore, the maximum likelihood estimator ˆX MLE is given by m ˆX MLE = [ĉi,j /σi ] ) /α 35) The algorithm is summarised in the following steps: Step 1: Obtain the measurements y i and determine the set of integers A = n i,j } within [0, Λ] ; Step : Find those points i,j } via: i,j = y i + n i,j λ i, n i,j A 36) Step 3: Find the finite set of intervals v l } over the interval [0, Λ] aording to Eq.5); Step 4: Compute the B l in eah interval aording to Eq.33); Step 5: Find the minimum B l, and ompute the orresponding ĉ i,j ; Step 6: Compute the ˆX MLE aording to Eq.35). In this method, the values of ĉ i,j and B l have to be omputed and stored for eah interval v l and the ˆX MLE is then found over all of the intervals v l }. As δ approahes to 0.5, the number of intervals inreases dramatially. f 0 y C 1 C C 3 Figure 3: Illustration of phase ambiguity removal via two frequenies, where i = and δ =0.1. The shadowed intervals are the possible regions v l,l =1,, 3, where the true measurements may fall in. The ondition p ω i 0.5λ i ) = 0 guarantees that the error of measurement is less than a single wavelength, thus the value an be estimated orretly. Fig. 3 shows an example in whih three intervals v l,l=1,, 3 for the joint likelihood of non-zeros values) are found in the range [0, Λ] when p ω i 0.1λ i )=0. Inthis ase, only three intervals are formed. 4 Algorithm performane analysis and disussions The performane of the proposed algorithm is examined via a omputerised simulation, where the set of wavelengths of the transmitted signals are λ i } = [0.55, 0.56, 0.61, 0.63, 0.65, 0.67] meters and the threshold δ =0.5. The standard deviation of RIPS measurement noise is σ i =0.1λ i. We assume that the unknown true RIPS measurement d ABCD is uniformly distributed in [0, 300] meters, i.e., the maximum of d ABCD is Λ=300 meters. All results are averaged over 100 Monte Carlo runs. The simulation results using signals of five wavelengths λ i,i = 1,, 3, 4, 5) are shown in Fig. 4. The histogram of estimation error defined as ˆX MLE d ABCD from 100 runs) is presented in Fig. 4a) in terms of probability and the joint likelihood py 1:m X) in a single realisation omputed via the proposed algorithm is plotted in Fig. 4b), where the ground truth is at its maximum value. In another similar example shown in Fig. 5, total six wavelengths were used. A better performane is observed in the sense that the maximum value of the omputed joint likelihood is more separate from other loal maxima. The robustness of the proposed method is evaluated in terms of the probability of suessful estimating the 418

6 a) a) b) Figure 4: Simulation results when i=5. a) Normalised estimation error histogram from 100 runs). b) The value of py 1:m X l ) from a single run where the ground truth d ABCD = m. ground truth of a RIPS measurement within a speified estimation error. As shown in Fig. 6, suh a probability for estimation error ˆX MLE d ABCD 5mversusthe total number of different wavelengths used is given. In this simulation, all wavelengths are seleted randomly in the range of meter. Statistial results for the omparison of omputational omplexity and number of interseted intervals v l involved of the proposed algorithm in various parameter values are given in Table 1, where the results are grouped in two ases, i.e., Case 1: p ω i 0.5λ i )=0 and Case : p ω i 0.1λ i )=0. Table 1: Comparison of two ases Number of λ i Case 1 p ω i 0.5λ i )=0 Number of v l CPU time s s s Case p ω i 0.1λ i )=0 Number of the v l CPU time s 3.93 s 3.37 s Disussions: As demonstrated in Fig. 6, MLE performane of the proposed algorithm beomes robust as the number of different wavelengths used inreased. In b) Figure 5: Simulation results when i=6. a) Normalised estimation error histogram from 100 runs). b) The value of py 1:m X l ) from a single run with ground truth d ABCD =47.48 m. Probability Number of wavelength Figure 6: The probability of orret detetion with error ˆX MLE d ABCD 5 m versus number of different wavelengths used. the simulation examples, when using up to seven different signal frequenies, a reliable result an be ahieved. Computational omplexity will inrease as the standard deviation of RIPS measurement noise inreased. The separation between the arrier frequenies, f i f k, will influene the estimation auray and omputation load. Normally, the great separation will lead to the high auray and low omputation load than the small separation. The former 419

7 one an generate less intersetion intervals whih have high py 1:m X l ), so the true value always fall in the interval of maximum py 1:m X l ) with high probability and therefore, the estimation auray an be improved. 5 Conlusions In this paper, a MLE method is proposed to estimate true hyperboli distane from ambiguous RIPS observations whih wrap distane by phase measurements. Based on Chinese Reminder theorem, a joint likelihood of a mixture of trunated Gaussian distributions is onstruted by using multiple wavelengthes within a finite distane interval. This method an effetively handle RIPS measurement noise without a large number of realisations in different transmitting wavelengths being used. Simulation results show that a robust estimation performane an be ahieved by using only six different wavelengths in a pratial mote ommuniation transmitter onfiguration. Nevertheless, the effiieny and effetiveness of the proposed algorithm are yet to be onfirmed in real wireless mote network environment, whih is urrently investigated in our researh work. [6] M.R. Morelande, B. Moran and M. Brazil. Bayesian node loalisation in wireless sensor networks, In Pro. IEEE International Conferene on Aoustis, Speeh and Signal Proessing, Las Vegas, USA, 008. [7] I.C. MKilliam, B.G. Quin, I.V.L Clarkson and B. Moran. Frequeny estimation by phase unwrapping, IEEE Trans. on Signal Proessing, Vol. 58, No. 6, pp , 010. [8] C. Wang, Q. Yin, W. Wang. An effiient ranging method for wireless sensor networks, 010 IEEE International Conferene on Aoustis Speeh and Signal Proessing, Dallas, TX, USA, Marh 010. [9] Y. Cheng, X. Wang, T. Caelli, and B. Moran. Target Traking and Loalization with Ambiguous Phase Measurements of Sensor Networks, Pro. 36th International Conferene on Aoustis, Speeh and Signal Proessing ICASSP 011), Prague, Czeh Republi, -7, May 011. [10] I. Vrana. Optimum statistial estimates in onditions of ambiguity, IEEE Trans. on Information Theory, Vol. 39, No. 3, pp , Referenes [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 , Nov [] 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 IPSN06), Nashville, Tennessee, USA, April 19 1, 006. [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 007. [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, 010. [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 ICON007), Adelaide, Australia, pp , 19-1 Nov