SOURCE LOCALIZATION IN THE PRESENCE OF SENSOR MANIFOLD UNCERTAINTIES AND SYNCHRONIZATION ERROR

Transcription

1 SOURCE LOCALIZATION IN THE PRESENCE OF SENSOR MANIFOLD UNCERTAINTIES AND SYNCHRONIZATION ERROR A Thesis Presented to the Faculty of Graduate School University of Missouri Columbia In Partial Fulfillment Of the Requirements for the Degree Master of Science by Yue Wang Dr Dominic KC Ho, Thesis Supervisor May 2011

2 The undersigned, appointed by the dean of the Graduate School, have examined the thesis entitled SOURCE LOCALIZATION IN THE PRESENCE OF SENSOR MANIFOLD UNCERTAINTIES AND SYNCHRONIZATION ERROR presented by Yue Wang a candidate for the degree of Master of Science and hereby certify that, in their opinion, it is worthy of acceptance Dr Dominic KC Ho Dr Tony Han Dr Yi Shang

3 ACKNOWLEDGEMENTS First, I would like to sincerely thank my advisor, Dr Dominic Ho, for his intellectual and patient guidance Without his help and guidance, I would not be able to complete my research I would also like to express my gratitude to DrTony Han, DrYi Shang for their time and consideration in reading my dissertation, and or their suggestions in completing my research I would like to thank Ming Sun, Le Yang, Zhenhua Ma, Liyang Rui, Shanjie Chen, my colleagues in the Communications Lab I would like to express my sincere thankfulness to my father, Weimin Wang, my mother, Yirong Shi, and all my other family members Without your love, support and encouragement, I would not be able to accomplish my goal I also want to say thank you to all my friends Thanks for the help from all of you ii

4 Abstract Passive source localization is a commonly used technology which can be applied to many areas, such as radar, sonar, microphone array, sensor network and wireless communication system If an unknown source radiates some signals, the signal will be received by some receivers The source location can be estimated based on the received signals using passive source localization technology A lot of positioning methods have been derived on this subject, such as time of arrivals (TOAs), time differences of arrival (TDOAs), angle of arrivals (AOAs) This thesis is mainly based on Chan and Ho s two stage closed form TDOAs source localization method However, Chan and Ho s method assume the sensor positions are known and all of the sensors are perfectly synchronized Three topics that affect the accuracy of the source localization are discussed in this thesis in the presence of sensor position manifold uncertainties, in the presence of clock-bias error and in the presence of both sensor position manifold uncertainties and clock-bias error At first, we develop an estimator for source localization when measurement noise and sensor position manifold uncertainties are present We modify a weighting matrix that accounts for the sensor position manifold error to improve the source location estimation Then, we use simulation to analyse the performance of the proposed estimator The simulation result shows that the proposed method reaches the CRLB performance for both the near-field and distant sources in the small error region Furthermore, the proposed method has been provn that its performance reaches CRLB theoretically Secondly, we develop an algorithm for source localization in the presence of measurement noise and unknown but fixed clock offsets The main idea of this estimator is to group the sensors with the same synchronization clocks together to form m N sub-arrays We transformed the original TDOA values to the new TDOA values of which the reference sensors are different for different sub-arrays so that the clock offsets are eliminated within a sub-array The simulation results show that the proposed method reaches the CRLB performance for both the near-field and distant sources in the small error region The performance of the proposed method in reaching CRLB has been provn theoretically under the small noise condition Finally, we develop an estimator for source localization in the presence of measurement noise, iii

5 sensor position manifold uncertainties and clock-bias error The weighting matrix takes into account of the measurement noise and the sensor position manifold uncertainties In addition, we group the sensors with the same synchronization clocks together to form m N sub-array and use the new transformed TDOA values which allow us to eliminate the clock offsets within a subarray The simulation result shows that the proposed method reaches the CRLB performance for both the near-field and distant sources in the small error region The performance of reaching the CRLB has been proved theoretically under the small noise condition iv

6 Contents Abstract iii 1 Introduction 1 11 Background 1 12 Localization Accuracy Manifold Uncertainty Clock Bias Uncertainty Manifold error and Clock Bias uncertainty 5 13 Motivation Manifold Uncertainty Clock Bias Uncertainty Combination of Manifold and Clock Bias Uncertainty 8 14 Contribution 8 2 Localization Basics Cramer-Rao lower Bound(CRLB) TDOA localization Taylor-series Method Closed-form Two Stage Method Summary 17 3 Cramer-Rao lower Bound(CRLB) CRLB Due to Measurement noise only 18 v

7 32 CRLB for TDOA Source Localization in the Presence of Measurement noise and Manifold Sensor Uncertainties CRLB for TDOA Source Localization in the Presence of Measurement Error and Clock-bias Uncertainties CRLB for TDOA Source Localization in the Presence of Combination of manifold and Clock-bias Uncertainties Summary 27 4 A New Estimator and Performance Analysis of Source Localization in the Presence of Manifold Sensor Position Uncertainties Mathematic Derivation Simulation Mathematic Proof of Optimum Performance of the Proposed Estimator for Source Localization in the Presence of Sensor Position Manifold Uncertainties Summary 41 5 A New Estimator and Performance Analysis of Source Localization in the Presence of Clock-bias Error Mathematic Derivation Simulation Mathematic Proof of Optimum Performance of the Proposed Estimator for Source Localization in the Presence of Clock-bias Error Summary 57 6 Source Location Estimator and Performance Analysis in the Presence of Measurement Noise, Sensor Position Manifold Uncertainties and Clock-bias Error Mathematic Derivation Simulation Mathematic Proof of the Optimum Performance of the Proposed Estimator for Source Localization in the Presence of Sensor Position Manifold Uncertainties and Clock-bias Error 71 vi

8 64 Summary 76 7 Conclusion and Future Work Conclusion Future Work 78 vii

9 Chapter 1 Introduction 11 Background Passive source localization is a commonly used technology which can be apply in many areas, such as radar [1,2], sonar [3-6], microphone array [7-9], sensor network [10-14] and wireless communication [15-18] system If an unknown source radiate some signal, and the signal will be received by some receivers The source location can be estimate base on the received signals using the passive source localization technology After many years study on this subject, a lot of methods have been derived using time of arrivals(toas), time differences of arrival(tdoas) [19-25], angle of arrivals(aoas) [26-28] TOA refers to the travel time of a radio signal from a single transmitter to a remote single receiver TOA after multiplied by signal propagation speed is the distance from the transmitter to the receiver TOA defines a circle at the receiver position in which the emitting source will lie In absence of noise, the source location is at the intersection of the circles from three receivers Because TOA method need the absolute time of arrival,time stamping the signal and synchronization between transmitter and receiver is needed Figure 11 shows the TOA localization approach, where s 1, s 2, s 1 are receivers the intersection of the three circles is the source location 1

10 Figure 11: Source localization using TOAs TDOA is the difference of arrival time of a source signal arrived at two different receivers TDOAs can be calculated by subtraction of two TOA measurements, or it can be obtained by cross-correlating the two received signal A constant TDOA locus is a hyperbola Source location can be estimated by solving a nonlinear hyperbolic equations Both TOA and TDOA techniques need synchronization However, no time-stamping the signal is needed for TDOA Figure 12 will illustrates the TDOA localization methods, where s 1, s 2, s 1 are receivers The intersection of hyperbolic curves give the location AOA refers to angel of arrival AOA uses direction information instead of distance information to estimate the source location AOA defines a bearing line that passes through the source Figure 13 demonstrates the AOA localization method, where s 1, s 2, s 1 are receivers, The intersection of bearing lines is the source location This thesis is based on TDOAs for source localization The proposed estimators, however, can be easily extended to TOA and AOA position localization algorithm in a straight forward manner 2

11 Figure 12: Source localization using TDOAs Figure 13: Source localization using AOAs 3

12 12 Localization Accuracy Localization accuracy depend on different factors, for example, measurement noise, geometric distribution of the sensor and source, nonlinear of sight, sensor position error and clock-bias error In this thesis, we will discuss sensor position manifold uncertainties and clock-bias error and combination of the two error 121 Manifold Uncertainty Previous study on TDOA localization usually assumes sensor positions are known and accurate However, sensor positions may have some error Some previous work has been done on this subject[29,30] However, they paid more attention on handling independent sensor uncertainties In practice, sensor uncertainty may have some relations between each other If these relations are considered, better localization accuracy can be achieved than considering all sensor uncertainty as independent Gaussian noise For example, in Figure 14, the entire sensor array contains two sub-arrays The sensors in each sub-array are mounted at fixed position In this case, each sub-array can be drifted and all its sensors have the same amount of position error If all the sensors have independent error, it will be sixteen dimensions of uncertainty However, since all sensors in one sub-array move together, it only has four dimensions of uncertainty So, there is some hope that the estimation accuracy can be improved 122 Clock Bias Uncertainty The TDOA localization method needs synchronization among the sensors However, a sensor array may have more than one clocks This thesis provide a method which does not need to synchronize different clocks within an array Some work has been done in sensor self localization without synchronization [31,32] It uses differential TDOA (dtdoas) to cancel out unknown clock offsets However, it requires all sensors to transmit some signals In this thesis, sensors don t need to transmit signals Source location can be estimated using a passive sensor array For example, consider Figure 15 where, 4

13 Figure 14: Illustration of manifold sensor uncertainty s 1, s 2, s 1, are receivers and r 1, r 2, r 3 are TOAs we have two clocks and the receivers have the same clock are grouped together Each sub-array has one own clock, clock 1 and clock 2 we choose sensor 1 as reference sensor, the TDOAs are r 21 = r 2 r 1 and r 31 = r 3 r 1 In absence of TDOAs noise, because s 1 and s 3 have different clocks and hence r 31 contains clock bias As result, using r 21 and r 31 to localize the source will give very large localization error 123 Manifold error and Clock Bias uncertainty The two kinds of error may happen at the same time, and this thesis will consider this situation as well Figure 16 illustrate this situation, which both manifold sensor uncertainty and clock bias error exist 5

14 Figure 15: Illustration of clock bias uncertainty Figure 16: Illustration of in the presence of sensor position manifold uncertainties and clock-bias uncertainties 6

15 13 Motivation 131 Manifold Uncertainty In a sensor array, quite often sensors are fix on a structure and the structure may be moved or rotated because of wind, vibration In this situation, the position uncertainties of al the sensors in the structure are the same If the dependency can be explored properly, a more accurate source location estimate can be achieved than considering all sensor uncertainties are independent For example, consider a sensor array with a number of sensor mounted, which is placed along the highway to detect wild animals that might cross the road The sensing system is need to inform the driver to avoid the accident But, because of the wind, the rain and the vibration caused by vehicles, the position of the sensor array may be moved or rotated But the relative position of the sensors remain unchanged We shall call this sensor position uncertainties as sensor manifold uncertainties 132 Clock Bias Uncertainty If a sensor array has several sub-arrays, and each sub-array has its own clock, different sub-arrays will have clock bias with one anotherwhen their clock are not synchronized for example, this could be due to the large geometric distances among the sub-arrays This thesis will provide a method to solve this problem For example, consider the situation that one sensor array has several sub-arrays The sub-arrays are separated and the sensors in one sub-array is close to each other Sensors can be synchronized using Bluetooth technology Because of the small range coverage of Bluetooth, only sensors in the same sub-array can be synchronized Each sub-array has its own clock and the clock of different sub-arrays are not synchronized In this case, method proposed in this thesis can be applied 7

16 133 Combination of Manifold and Clock Bias Uncertainty If one sensor array has several sub-arrays that are far from each other, each sub-array has its own clock When the sub-arrays are fixed on several objects, the sensors will have manifold uncertainty In this case, it is reasonable to propose a algorithm which can handle both manifold uncertainty and clock bias uncertainty together 14 Contribution This thesis addresses the problem of TDOA localization in the presence of sensor manifold uncertainties and /or synchronization clock bias errors The main contributions include: (1) The study of localization accuracy in the presence of these errors through the CRLB analysis based on Gaussian noise model (2) The development of estimators to obtain the optimum source location estimate (3) The analysis of the proposed estimators For the first contribution, the study through the CRLB analysis based on Gaussian noise model For the second contribution, three algorithms are proposed in the thesis The first one is to solve source localization in the presence manifold uncertainty of the sensors Since the position uncertainties in different sensors are related dimension of uncertainties can be reduced Estimation accuracy can be improved by exploring manifold sensor error uncertainties than assuming all sensor uncertainties are independent If there is no relationship among the sensor positions, the method proposed in this thesis also can handle this situation The new method can be viewed as an generalization for the previous study of independent position errors as well The second algorithm is the estimation of the source position in the presence of clock bias error among different sub-arrays in a sensor array It can be applied to the situation where a sensor array has more than one clock and only the sensors within a sub-array can have perfect synchronization The third estimator is to address the situation when both manifold sensor uncertainties and clock bias uncertainties are presence at the same time 8

17 For the third contribution, we have conducted the analysis of the performance of the proposed estimators in both simulations study and theoretical proofs The three proposed estimators achieve optimal performance both in simulation and in theory over the small error region 9

18 Chapter 2 Localization Basics In this Chapter, we will introduce some basic topics in localization, including basic idea about the Cramr-Rao lower Bound(CRLB), the TDOA localization algorithm, the Taylor-series technique and the closed-form two-stage algorithm 21 Cramer-Rao lower Bound(CRLB) Cramer-Rao lower Bound(CRLB) provide a lower bound on the variance of any unbiased estimator[33,34] It gives optimal variance and alerts us if it is impossible to find an unbiased estimator whose variance is less than the bound CRLB is obtained from probability density function(pdf) of the collection of data measurements x that is parameterized with respect to the unknown parameter θ It is assumed that if PDF satisfies [ ] E lnp(x;θ) = 0 (21) θ Then the variance of any unbiased estimator ˆθ must satisfies ] [ cov(ˆθ) E 1 2 lnp(x;θ) (22) θθ T In this thesis, we will compare compare CRLB with simulation result to see if mean-square error reach CRLB which is the optimal permance Then, we will prove variance of the estimated source location is equals to CRLB theoretically 10

19 Figure 21: Source localization using TDOAs 22 TDOA localization As mentioned in Chapter 1, TDOA is the difference of arrival time of a source signal arrived at two different receivers A constant TDOA locus is a hyperbola Source location can be estimated by solving a set of nonlinear hyperbolic equations Figure 21 shows a example of TDOA localization s 1, s 2, s 3 are receivers The intersection of hyperbolic curves is the source location We shall denote the true distance between the source and i th receiver as r o i r o i = u o s i (23) where represents the 2-norm TDOA measurement, after multiplied by the signal propagation speed, between sensor i and sensor 1 is r i1 = r i r 1 + n i1, i = 2, 3,, M (24) n i1 is the TDOA noise, that is assumed to be zero-mean Gaussian noise In TDOA localization, we use the measurements r i1, i = 2, 3,, M to estimate source location u o 11

20 23 Taylor-series Method The basic idea of Taylor-series method [19] is an iterative algorithm to locate the source It starts with a initial guess and improving the estimation at each step by adding the local linear least-sum square error correction to the previous estimation Let u o = [x o, y o, z o ] T to be the source location Sensor positions are represented by S = [s T 1,, s T M ] where s i = [x i, y i, z i ] T for i=1,, M is the sensor position of the i th sensor The range from source to the i th sensor is r o i = u o s i = (x o x i ) 2 + (y o y i ) 2 + (z o z i ) 2 (25) Then the true TDOA is related to u o by r o i1 = r o i r o 1, i = 2, 3,, M (26) Let n i1 be TDOA noise for the i th sensor We represent r i1 as r i1 = r o i1 + n i1 (27) In matrix form f(u o ) = T = M E (28) where T [ T = r21 o rm1] o (29) [ ] T M = r 21 r M1 (210) [ ] T E = n 21 n M1 (211) The error term E has a covariance matrix Q ] Q = E [EE T (212) If u g is the guess value then we can express the true source location u o as u = u g + u (213) We expand f(u) in Taylor-series f(u) u=ug +A u = M E (214) 12

21 where [ T u = (u o s 2) T r2 o A = f(u) u (uo s 1) T r o 1 = T u,, (uo s M ) T r o M (uo s 1) T r o 1 ] (215) (216) (214) can be rewritten as A u = W E (217) where W = M f(u) u=ug (218) Choosing u that gives least weighted squared error u = [A T Q 1 A] 1 A T Q 1 W (219) We update u g by replace it with u g = u g + u g (220) Then, we repeat (219) and (220) The final estimate is obtained when u g converges to a stable value and u g goes to zero Although the Taylor-series method can provide the least weighted squared error estimator for the TDOA localization problem, it needs a initial guess which is close enough to the source location Otherwise it can only provide local minimum solution rather than the globe minimum source location 24 Closed-form Two Stage Method Another approach to solve this localization problem is the closed-form two stage solution proposed by Chan and Ho [20] The advantage of this method is that no initial guess is needed and it is not iterative This closed form method reaches CRLB in small noise condition We will introduce the idea of Chan and Ho s two stage method in this section Stage 1: Let u o = [x o, y o, z o ] T to be the source location The sensor positions are represented by S = [s T 1,, s T M ]T where s i = [x i, y i, z i ] T for i=1,, M is the sensor position of the i th 13

22 sensor The range from source to the i th sensor is r o i = u o s i = (x o x i ) 2 + (y o y i ) 2 + (z o z i ) 2 (221) Then the TDOA is r i1 = r i r 1, i = 2, 3,, M (222) Let n i1 be TDOA noise for the i th sensor n i1 = c t i1, where t i1 is TDOA noise and c is the signal propagation speed we represent r i1 as r i1 = r o i1 + n i1 (223) In absence of noise, TDOA for the i th sensor is r o i1 = r o i r o 1, i = 2, 3,, M (224) rewrite (224) as r o i1 + r o 1 = r o i, i = 2, 3,, M (225) Squaring both sides, gives r o2 i1 + 2r o i1r o 1 + r o2 1 = r o2 i (226) Substituting (221) into (226) and simplifying r o2 i1 + 2r o i1r o 1 = s T i s i s T 1 s 1 2(s i s 1 ) T u o (227) Expressing ri1 o = r i1 n i1 and ignoring n 2 i1we have r 2 i1 2r i1 n i1 + 2r i1 r o 1 2r o 1n i1 = s T i s i s T 1 s 1 2(s i s 1 ) T u o (228) (228) can be rewritten as 2r o i n i1 = r 2 i1 s T i s i + s T 1 s 1 + 2(s i s 1 ) T u o + 2r i1 r o 1 (229) In matrix form, it can be expressed as ɛ 1 = B 1 n = h 1 G 1 ϕ o 1 (230) 14

23 where B 1 = 2 r o 2 r o 3 r o M r21 2 s T 2 s 2 + s T 1 s 1 h 1 = rm1 2 st M s M + s T 1 s 1 (s 2 s 1 ) T r 21 G 1 = 2 (s M s 1 ) T r M1 (231) (232) (233) T n = [n 21,, n M1 ] T (234) ϕ o 1 = [ u o r1] o (235) In this case, weighted least-square (WLS) technique [33] can be applied and the result is ϕ 1 = (G T 1 W 1 G 1 ) 1 G T 1 W 1 h 1 (236) and the covariance matrix of ϕ 1 is cov(ϕ 1 ) = (G T 1 W 1 G 1 ) 1 (237) The weighting matrix W is calculated using W 1 = E[ɛ 1 ɛ T 1 ] 1 = (B T 1 QB 1 ) 1 (238) Stage 2: In Stage 2, we make use r 1 in ϕ 1 to refine the estimation of the source location Let ϕ o 2 = [(x o x 1 ) 2, (y o y 1 ) 2, (z o z 1 ) 2 ] T (u o s 1 ) (u o s 1 ) = ϕ 2 (239) Since ϕ 1 (1 : 3) is an estimator of u o, ie ϕ 1 (1 : 3) = u o + ϕ 1 (1 : 3), where ϕ 1 (1 : 3) is the estimation noise Thus replacing u o in (239) by ϕ 1 (1 : 3) ϕ 1 (1 : 3) (ϕ 1 (1 : 3) o s 1 ) (ϕ 1 (1 : 3) o s 1 ) 2(ϕ 1 (1 : 3) o s 1 ) ϕ 1 (1 : 3) o = (u o s 1 ) (u o s 1 ) (240) 15

24 or 2(ϕ 1 (1 : 3) ϕ 1 (1 : 3)) ϕ 1 (1 : 3) = (ϕ 1 (1 : 3) s 1 ) (ϕ 1 (1 : 3) s 1 ) (u o s 1 ) (u o s 1 ) (241) where the second order noise ϕ 1 (1 : 3) ϕ 1 (1 : 3) has been ignored Recall that ϕ 1 (4) = r1 o + r 1, where r i is the estimation noise (ϕ 1 (4) ϕ 1 (4)) 2 = (u o s 1 ) T (u o s 1 ) (242) Expanding the left side and ignoring ϕ 1 (4) 2 2ϕ 1 (4) ϕ 1 (4) = ϕ 1 (4) 2 (u o s 1 ) T (u o s 1 ) (243) We express (241) and (243) in matrix form ɛ 2 = B 2 ϕ 1 = h 2 G 2 ϕ 2 (244) where B 2 = 2 diag(uo s 1 ) r o 1 (245) h 2 = (ϕ (1 : 3) s 1 1 ) (ϕ (1 : 3) s 1 1 ) (246) ϕ 1 (4) G 2 = (247) The WLS solution for stage 2 is ϕ 2 = (G T 2 W 2 G 2 ) 1 G T 2 W 2 h 2 (248) The covariance matrix of ϕ 2 is cov(ϕ 2 ) = (G T 2 W 2 G 2 ) 1 (249) where the weighting matrix is W 2 = E[ɛ 2 ɛ T 2 ] 1 = [B 2 cov(ϕ 1 )B T 2 ] 1 The source location estimate u = [x, y, z] T can be obtained from ϕ 2 u = P[ ϕ 2 (1), ϕ 2 (2), ϕ 2 (3)] T + s 1 (250) 16

25 where P = diag[sgn(ϕ 1 (1) x 1 ), sgn(ϕ 1 (2) y 1 ), sgn(ϕ 1 (3) z 1 )] (251) According to (250), subtracting both sides by s 1, squaring and taking differential, u = B 1 3 ϕ 2 (252) where x o x 1 B 3 = 2 y o y 1 z o z 1 (253) The covariance matrix of the final source position estimator is cov(u) = B 1 3 cov(ϕ 2)B T 3 (254) 25 Summary In this chapter, we introduce the basic idea of CRLB, which is a benchmark of optimal variance for any unbiased estimator Taylor series[4] and Chan and Ho s method[3], the two source localization method, are also introduced in this chapter Taylor series needs initial guess and only converge to the local minimum solution and Chan and Ho s method needs small noise condition 17

26 Chapter 3 Cramer-Rao lower Bound(CRLB) Cramer-Rao lower Bound(CRLB) provides a lower bound on the variance of any unbiased estimator It gives optimal variance and alerts us it is impossible to find an unbiased estimator whose variance is less than the bound In this chapter, we will construct mathematic models for manifold sensor uncertainties, clockbias error and the presence of the two kinds of uncertainties Cramer-Rao lower Bound of the source location estimate under manifold sensor uncertainties, clock-bias error and the both kind of error will be derived according to the mathematic models 31 CRLB Due to Measurement noise only We will derive CRLB for TDOA source localization when the noise appear in TDOA measurements only Let u o = [x o, y o, z o ] T to be the source location Sensor positions are represented by S = [s T 1,, s T M ]T where s i = [x i, y i, z i ] T for i=1,, M is the sensor position of the i th sensor The ri o is the range from source to the ith sensor If we choose sensor 1 as reference and denote range difference of ri o and ro 1 as ri1 o = ro i ro 1, the TDOA measurement sensor i and sensor 1 is r i1 = r o i1 + n i1 (31) 18

27 where n i1 is the measurement noise that is assumed as zero mean Gaussian Let us collect(m-1) TDOA measurements as r = [r 21, r 31,, r M1 ] T The pdf of r parameterized on u o is p(r; u o 1 ) = ( ) M 1 1 2π Q r 1 2 exp{ 1 2 (r ro ) T Q 1 r (r r o )} (32) where Q r is the covariance matrix of measurement noise and r o = [r o 21, r o 31,, r o M1 ]T As mentioned in Chapter 2, if the pdf satisfies, Then the variance of any unbiased estimator u of u o must satisfy [ ] E lnp(r;u o ) = 0 (33) u o ( [ CRLB(u) E 2 lnp(r;u o ) u o u ot ]) 1 (34) To obtain CRLB, first we calculate lnp(r; u) lnp(r; u o ) = M 1 ln(2π) ln Q r 1 2 (r ro ) T Q 1 r (r r o ) (35) The expectation of first order derivative of lnp(r; u o ) is [ ] [ E lnp(r;u o ) = E u 1 o 2 ( ro u ) T Q 1 o r (r r o ) 1 2 (r ro ) T Q 1 r ( ro u ) o ] = 0 (36) So that (33) is satisfied ( [ cov(u) E 2 lnp(r;u) u o u ot ]) 1 (37) where and [ ] E 2 lnp(r;u) = ( ro u o u ot u o )T Q 1 r ( ro u o ) (38) r o [ u o = (u s 2) r2 o (uo s 1) r,, (uo s M ) 1 o r (uo s 1) M o r1 o ] T (39) 32 CRLB for TDOA Source Localization in the Presence of Measurement noise and Manifold Sensor Uncertainties In this section, we will discuss mathematic model of TDOA Source Localization in the presence of Manifold Sensor Uncertainties in addition to measurement noise CRLB for TDOA source 19

28 localization will be obtained based on that model In the case of sensor manifold error, the position error of different sensors are related to each other Let p be a random vector of length L where L 3M Then we model the sensor position vector as s = s o + Vp (310) s o 1 = + s o M V 1 V M p L 1 (311) 3M L where V = [V T 1,, V T M ] T is a known matrix v i is transformation matrix which shows the relation between random vector p and the position uncertainties of the i th sensor If V is a 3M 3M identity matrix and p is a 3M 1 random vector, the manifold sensor position uncertainties become independent uncertainties f is assumed to be zero mean Gaussian We take both measurement noise and sensor position manifold uncertainties into account in deriving the CRLB The unknown source position is u o = [x o, y o, z o ] T, the measurement vector is x = [r T, p T ] T and the unknown parameter vector is θ o = [u ot, p T ] T The pdf of x can be written as Thus lnp(x; θ) = M p(x; θ) = ( ) M 1 1 2π Q r ( ) L 1 2π Q p 1 2 ln(2π) 1 2 ln Q r 1 2 (r ro ) T Q 1 r exp{ 1 2 (r ro ) T Q 1 r (r r o )} exp{ 1 2 pt Q 1 r p} (312) The expectation of first order derivative of nature logarithm of the pdf is [ ] lnp(x;θ) u E lnp(x;θ) = E o θ = E lnp(x;θ) p ( ro (r r o ) L 2 ln(2π) 1 2 ln Q p 1 2 pt Q 1 p r (313) u ) T Q 1 o r (r r o ) = 0 (314) Q 1 p p 20

29 and (33) is satisfied Thus the CRLB of θ is where ( [ cov(θ) E [ ] E 2 lnp(x;θ) θ θ T The partial derivatives in (316) are = E [ E 2 lnp(x;θ) u o u ot ] [ E 2 lnp(x;θ) u o p T ] [ E 2 lnp(x;θ) p u ot ] [ ] E 2 lnp(x;θ) = ( ro p p T 2 lnp(x;θ) θ θ T 2 lnp(x;θ) u o u ot 2 lnp(x;θ) p u ot ]) 1 (315) 2 lnp(x;θ) u o p T 2 lnp(x;θ) p p T (316) = ( ro u )T Q 1 r ( ro u ) (317) = ( ro u o )T Q 1 r ( ro p ) (318) = ( ro p )T Q 1 r ( ro u o ) (319) p )T Q 1 r ( ro p ) + Q p (320) where Q r is covariance matrix of measurement noise vector n and Q p is the covariance of random vector p r o [ u = (u o s 2) r2 o r o [( p = (u o s 2) T V 2 r (uo s 1) T V 1 2 o r1 o (uo s 1),, (uo s M ) (uo s 1) r1 o rm o r1 o ) T (,, (u o s M ) T V M r o M (uo s 1) T V 1 r o 1 ] T (321) ) T ] T (322) In this study, we are interested only in CRLB of the source position u It is the upper 3 3 block of CRLB(θ) Denote the four block matrix in (316) as [ ] [ [ ] E 2 lnp(x;θ) E E 2 u lnp(x;θ) o u ot θ θ T [ ] [ E 2 lnp(x;θ) E p u ot X = Y T According to the inverse of block matrix, X 1 is ] 2 lnp(x;θ) u o p T ] 2 lnp(x;θ) p p T (323) Y (324) Z CRLB(u) = (X YZ 1 X T ) 1 (325) = X 1 + X 1 Y(Z Y T X 1 Y) 1 Y T X 1 (326) In this section, CRLB in presence of manifold sensor position uncertainties in TDOA localization system is derived It provide a benchmark for the performance of a estimator for the source location when the measurement noise and manifold sensor position uncertainties are presented 21

30 33 CRLB for TDOA Source Localization in the Presence of Measurement Error and Clock-bias Uncertainties In this section, we first derive mathematic model of TDOA source localization in the presence of clock-bias error The CRLB of the source localization in the presence of clock-bias error will be obtained using that model The unknown source position is u o = [x o, y o, z o ] T and the known sensor position is s i = [x i, y i, z i ] T Let us further assume that the sensor array can be decomposed into N sub-arrays Within each sub-array the sensors are synchronized with the same clocks However, the clocks among different sub-arrays are not synchronized and have clock bias δ j, j = 2, 3,, N with respect to the first sub-array Let us choose the first sensor as reference for TDOA measurements The TDOA measurements r i1 = u o s i u o s 1 +n i1, i = 2,, m 1 r i1 = u o s i u o s 1 +δ o 2 + n i1, i = m 1 + 1,, m 2 (327) r i1 = u o s i u o s 1 +δ o N + n i1, i = m N 1 + 1,, m N where δ = [δ 2,, δ N ] T is the clock-bias vector we shall assume in this study δ is deterministic and not random {s 1,, s m1 }, {s m1+1,, s m2 },, {s mn 1+1,, s mn } are different subarrays We can write the TDOA measurements in vector form as r = r o + n + Fδ o (328) where r = [r 21,, r MN,1] T, n = [n 21,, n MN,1] T and m 1 1 m 2 m 1 {}}{{}}{ 0,, 0, 1,, 1,0,, 0 0,, 0, 1,, 1, 0,, 0 F = m N 1 +1 m N m N 1 {}}{{}}{ 0,,,, 0, 1,, 1 T (329) 22

31 Denote the unknown parameter vector be θ o = [u o, δ o ] T and the measurement vector be x = r, the pdf in the presence of measurement noise and clock-bias error is p(x; θ o 1 ) = ( ) M 1 1 2π Q r 1 2 exp{ 1 2 (r ro Fδ o ) T Q 1 r (r r o Fδ o )} (330) The natural logarithm of the pdf is lnp(x; θ) = M 1 ln(2π) ln Q r 1 2 (r ro Fδ o ) T Q 1 r (r r o Fδ o ) (331) Taking expectation of the nature logarithm of the PDF, [ ] lnp(x;θ) u E lnp(x;θ) = E o θ = E lnp(x;θ) δ o ( ro u ) T Q 1 o r (r r o Fδ o ) (332) F T Q 1 r (r r o Fδ o ) = 0 (333) and (33) is satisfied ( [ CRLB(θ o ) E 2 lnp(x;θ) θ θ T ]) 1 (334) where [ ] E 2 lnp(x;θ) θ θ T The partial derivatives in (335) are = E 2 lnp(x;θ) u o u ot 2 lnp(x;θ) δ o u ot 2 lnp(x;θ) u δ ot 2 lnp(x;θ) δ o δ ot (335) [ ] E 2 lnp(x;θ) = ( ro u o u ot u o )T Q 1 r ( ro u o ) (336) [ ] E 2 lnp(x;θ) = ( ro u o δ ot u o )T Q 1 r F (337) [ ] E 2 lnp(x;θ) = F T Q 1 δ o u ot r ( ro u o ) (338) [ ] E 2 lnp(x;θ) = F T Q 1 δ o δ ot r F (339) where Q r is covariance matrix of measurement noise vector n, and r o [ u o = (u s 2) r2 o (u s1) r,, (u s 1 o M ) r (u s1) M o r1 o ] T (340) 23

32 We are interested in the source position u o The CRLB of the source position u is the upper left 3 3 block of CRLB(θ o ) Denote [ ] E 2 lnp(x;θ) θ θ T [ E = [ E X = Y T ] 2 lnp(x;θ) u o u ot 2 lnp(x;θ) δ o u ot According to the block matrix inversion formular, CRLB(u o ) ] [ ] E 2 lnp(x;θ) u [ o δ ot ] E 2 lnp(x;θ) δ o δ ot (341) Y (342) Z CRLB(u o ) = (X YZ 1 X T ) 1 (343) = X 1 + X 1 Y(Z Y T X 1 Y) 1 Y T X 1 (344) In this section, the CRLB for the TDOA source localization problem which contains clock-bias error is derived It provides the minimum achievable variance of an estimator for source location in the presence of measurement noise and clock-bias error 34 CRLB for TDOA Source Localization in the Presence of Combination of manifold and Clock-bias Uncertainties In this section, we shall consider the presence of all three kinds of errors in estimating the source position We shall started by construct a mathematic model for the measurement and sensor positions Then,we will derive the CRLB using the models The measurement vector is x = [r T, p T ] T and the unknown parameter vector is θ o = [u ot, p T, δ ot ] T Following Section 32, the sensor position vector is modeled as s = s o + Vp (345) s o 1 = + s o M 24 V 1 V M p L 1 (346) 3M L

33 where p is a zero-mean Gaussian random vector, V = [V T 1,, V T M ] T and V i is transformation matrix which shows the relation between random vector p and the position uncertainties of i th sensor From Section 33, TDOA measurements are r i1 = u o s i u o s 1 +n i1, i = 1,, m 1 r i1 = u o s i u o s 1 +δ o 2 + n i1, i = m 1 + 1,, m 2 (347) r i1 = u o s i u o s 1 +δ o N + n i1, i = m N 1 + 1,, m N Or in vector form, r = r o + n + Fδ o (348) where r = [r 21,, r MN,1] and m 1 1 m 2 m 1 {}}{{}}{ 0,, 0, 1,, 1,0,, 0 0,, 0, 1,, 1, 0,, 0 F = m N 1 +1 m N m N 1 {}}{{}}{ 0,,,, 0, 1,, 1 T (349) where δ o = [δ o 2,, δ o N ]T is the clock-bias The pdf of x is p(x; θ o 1 ) = ( ) M 1 1 2π Q r ( ) L 1 2π Q p 1 2 Thus, the nature logarithm of p(x; θ) exp{ 1 2 (r ro Fδ o ) T Q 1 r (r r o Fδ o )} exp{ 1 2 pt Q 1 p p} (350) lnp(x; θ) = M 1 ln(2π) ln Q r 1 2 (r ro Fδ o ) T Q 1 r (r r o Fδ o ) L 2 ln(2π) 1 2 ln Q p 1 2 pt Q 1 p p (351) 25

34 Expectation of first order derivative of nature logarithm of the PDF is [ ] E lnp(x;θ o ) θ o = E = E = 0 lnp(x;θ o ) u o lnp(x;θ o ) p lnp(x;θ o ) δ o ( ro u ) T Q 1 o r (r r o Fδ o ) Q 1 p p F T Q 1 r (r r o Fδ o ) (352) and (33) is satisfied Thus where [ ] E 2 lnp(x;θ o ) θ o θ ot The partial derivatives in (354) are ]) 1 ( [ CRLB(θ o ) E 2 lnp(x;θ o ) θ o θ ot (353) = E 2 lnp(x;θ o ) u o u ot 2 lnp(x;θ o ) 2 lnp(x;θ o ) p u ot 2 lnp(x;θ o ) 2 lnp(x;θ o ) u o p T u o δ ot 2 lnp(x;θ o ) p p T p δ ot 2 lnp(x;θ o ) 2 lnp(x;θ o ) 2 lnp(x;θ o ) δ o u ot δ o p T δ o δ ot (354) [ ] E 2 lnp(x;θ) = ( ro u o u ot u o )T Q 1 r ( ro u o ) (355) [ ] E 2 lnp(x;θ) = ( ro u o p T u o )T Q 1 r ( ro p ) (356) [ ] E 2 lnp(x;θ) = ( ro p u ot p )T Q 1 r ( ro u o ) (357) [ ] E 2 lnp(x;θ) = ( ro p p T p )T Q 1 r ( ro p ) + Q p (358) [ ] E 2 lnp(x;θ) = ( ro u o δ ot u )T Q 1 r F (359) [ ] E 2 lnp(x;θ) = F T Q 1 δ o u ot r ( ro u o ) (360) [ ] E 2 lnp(x;θ) = F T Q 1 δ o δ ot r F (361) where Q r is covariance matrix of measurement noise vector n, Q p is the covariance of random vector p r o [ u = (u s 2) r2 o r o [( p = (u s 2) T V 2 r (u s1)t V 1 2 o r o 1 (u s1) r,, (u s 1 o M ) r (u s1) M o r1 o ) T,, ( (u s M ) T V M r o M (u s1)t V 1 r o 1 ] T (362) ) T ] T (363) 26

35 The CRLB of u o is the upper left 3 3 block of CRLB(θ o ) Denote [ ] [ ] E 2 lnp(x;θ o ) E 2 lnp(x;θ o ) [ ] u o u ot u o p [ ] [ T ] [ E 2 lnp(x;θ) = E 2 lnp(x;θ o ) E 2 lnp(x;θ o ) E θ θ T p u ot p p [ ] [ T ] [ E E 2 lnp(x;θ o ) E 2 lnp(x;θ o ) δ o u ot X 11 X 12 X 13 }3 X T 12 X 22 X 23 }L = X T 13 X T 23 X 33 }N }{{}}{{}}{{} 3 L N δ o p T [ E 2 lnp(x;θ o ) u o δ ot 2 lnp(x;θ o ) p δ ot 2 lnp(x;θ o ) δ o δ ot Using the inversion formula, the upper left (L + 3) (L + 3) block of the CRLB(θ o ) is X 11 X 12 X T 12 X 22 X 13 X 1 33 X 23 [ ] X T 13 X23 T = X 11 X 13 X 1 33 XT 13 X 12 X 13 X 1 33 XT 23 X T 12 X 23 X 1 33 XT 13 X 22 X 23 X33 1 X T ] ] ] (364) (365) (366) (367) Thus, the upper left 3 3 block of (366) is CRLB(u o ) 1 = (X 11 X 13 X 1 33 XT 13) (X 12 X 13 X 1 33 XT 23)(X 22 X 23 X 1 33 XT 23) 1 (X T 12 X 23 X 1 33 XT 13)(368) Substituting (355)-(361) into (368), give the CRLB of the source position when the observation noise, manifold sensor position uncertainties and clock-bias error are present 35 Summary In this chapter, we derive four CRLB The first one is the CRLB only in presence of measurement noise Then we derive the CRLB in the presence of measurement noise and sensor position manifold uncertainties, the CRLB in the presence of measurement noise and clock-bias error, and the CRLB when all the three kind of noise exist 27

36 Chapter 4 A New Estimator and Performance Analysis of Source Localization in the Presence of Manifold Sensor Position Uncertainties In this chapter, we will develop an estimator for source localization when measurement noise and sensor position manifold uncertainties are present and analysis its performance This estimator has closed-form solution and is obtained based on the two stage approach from Chan and Ho s closed-form algorithm The simulation result of the proposed algorithm will be given In the last part of this chapter, we will prove the theoretically the performance of the proposed algorithm reaches the CRLB 28

37 41 Mathematic Derivation Denote u o = [x o, y o, z o ] T as unknown source location and s o i as the unknown sensor position of i th sensor, i = 1,, M s i is the known noisy sensor position of the i th sensor, where s i = s o i + s i and s i is the sensor position error We shall model s i = V i p, where p is a random vector with length L and V i is the transformation matrix relates the random vector p and the position uncertainties of the i th sensor Thus s i = s o i + V i p (41) Denote s = [s T 1,, s T M ]T and V = [V T 1,, V T M ] T, then S = S o + Vp (42) From the emitted source signal in reaching the sensors, we have M-1 TDOA r i1 measurements r i1 = r o i1 + n i1, i = 2, 3,, M (43) where r i1 is the true TDOA and n i1 is the noise that is zero mean Gaussian Sensor s 1 is the reference sensor in computing TDOAs The true range from source to the i th sensor is r o i = u o s o i = (x o x o i )2 + (y o y o i )2 + (z o z o i )2 (44) Stage 1: Without loss of generality, true TDOA measurement is equal to r o i1 = r o i r o 1, i = 2, 3,, M, (45) or r o i1 + r o 1 = r o i, i = 2, 3,, M (46) For the i th sensor, we substituting (44) into (46) and after squaring r o2 i1 + 2r o i1r o 1 + 2(s o i s o 1) T u o + s ot 1 s o 1 s ot i s o i = 0 (47) Expressing r o i1 = r i1 n i1 and s o i = s i + V i p, we have r 2 i1 2r i1 n i1 + n 2 i1 + 2r i1 r o 1 2r o 1n i1 + 2(s i V i p s 1 + V 1 p) T u o +(s 1 V 1 p) T (s 1 V 1 p) (s i V i p) T (s i V i p) = 0 (48) 29

38 Ignoring the 2 nd order noise terms and rearranging 2r o i n i1 + 2(u o s i ) T V i p 2(u o s 1 ) T V 1 p = r 2 i1 s T i s i + s T 1 s 1 + 2(s i s 1 ) T u o + 2r i1 r o 1 (49) Because r o 1 is the true value, it depends on the true position of s 1 We can express it as r o 1 = u o s o 1 = u o s 1 + s 1 r o 1 + ρ T u o,s 1 V 1 p (410) where r o 1 = u o s 1 and ρ uo,s 1 (uo s 1) u o s 1 represents the unit vector from s 1 to u o (49) now becomes 2ri o n i1 + 2(u o s i ) T V i p 2(u o s 1 + r i1 ρ u o,s 1 ) T V 1 p = ri1 2 s T i s i + s T 1 s 1 + 2(s i s 1 ) T u o + 2r i1 r 1 o (411) [ T The unknown vector is ϕ o 1 = u ot r 1] o In matrix form, we can express (411) as ɛ 1 = B 1 n + Dp = h 1 G 1 ϕ o 1 (412) where B 1 = 2 r o 2 r o 3 (u o s 2 ) T V 2 (u o s 1 + r 21 ρ uo,s 1 ) T V 1 D = 2 (u o s M ) T V 2 (u o s 1 + r M1 ρ u o,s 1 ) T V M r21 2 s T 2 s 2 + s T 1 s 1 h 1 = rm1 2 st M s M + s T 1 s 1 (s 2 s 1 ) T r 21 G 1 = 2 (s M s 1 ) T r M1 r o M (413) (414) (415) (416) 30

39 and n = [n 21,, n M1 ] T is the noise vector In this case, weighted least-square (WLS) technique can be applied to estimate ϕ 1 and the result is ϕ 1 = (G T 1 W 1 G 1 ) 1 G T 1 W 1 h 1 (417) The covariance matrix of ϕ 1 is cov(ϕ 1 ) = (G T 1 W 1 G 1 ) 1 (418) The weighting matrix W 1 is calculated using W 1 = E[ɛ 1 ɛ T 1 ] 1 = (B 1 QB T 1 + DQ p D T ) 1 (419) where Q is the covariance matrix of n and Q p is the covariance matrix of p Stage 2: Stage 2 follows stage 2 in Chan and Ho s method[3], we refine the estimation of the source location using r 1 in ϕ 1 Let ϕ o 2 = [(x o x 1 ) 2, (y o y 1 ) 2, (z o z 1 ) 2 ] T (u o s 1 ) (u o s 1 ) = ϕ o 2 (420) Since ϕ 1 (1 : 3) is an estimator of u o, ie ϕ 1 (1 : 3) = u o + ϕ 1 (1 : 3), where ϕ 1 (1 : 3) is the estimation noise Thus replacing u o in (420) by ϕ 1 (1 : 3) ϕ 1 (1 : 3) (ϕ 1 (1 : 3) o s 1 ) (ϕ 1 (1 : 3) o s 1 ) 2(ϕ 1 (1 : 3) o s 1 ) ϕ 1 (1 : 3) o = (u o s 1 ) (u o s 1 ) (421) or 2(ϕ 1 (1 : 3) s 1 ) ϕ 1 (1 : 3) = (ϕ 1 (1 : 3) s 1 ) (ϕ 1 (1 : 3) s 1 ) (u o s 1 ) (u o s 1 ) (422) where the second order noise ϕ 1 (1 : 3) ϕ 1 (1 : 3) has been ignored Recall that ϕ 1 (4) = r o 1 + r 1, where r i is the estimation noise (ϕ 1 (4) ϕ 1 (4)) 2 = (u o s 1 ) T (u o s 1 ) (423) Expanding the left side and ignoring ϕ 1 (4) 2 2ϕ 1 (4) ϕ 1 (4) = ϕ 1 (4) 2 (u o s 1 ) T (u o s 1 ) (424) We express (422) and (424) in matrix form ɛ 2 = B 2 ϕ 1 = h 2 G 2 ϕ 2 (425) 31

40 where B 2 = 2 diag(uo s 1 ) r o 1 (426) h 2 = (ϕ (1 : 3) s 1 1 ) (ϕ (1 : 3) s 1 1 ) (427) ϕ 1 (4) G 2 = (428) The WLS solution for stage 2 is ϕ 2 = (G T 2 W 2 G 2 ) 1 G T 2 W 2 h 2 (429) The covariance matrix of ϕ 2 is cov(ϕ 2 ) = (G T 2 W 2 G 2 ) 1 (430) where the weighting matrix is W 2 = E[ɛ 2 ɛ T 2 ] 1 = [B 2 cov(ϕ 1 )B T 2 ] 1 The source location estimate u = [x, y, z] T can be obtained from ϕ 2 u = P[ ϕ 2 (1), ϕ 2 (2), ϕ 2 (3)] T + s 1 (431) where P = diag[sgn(ϕ 1 (1) x 1 ), sgn(ϕ 1 (2) y 1 ), sgn(ϕ 1 (3) z 1 )] (432) According to (431), subtracting both sides by s 1, squaring and taking differential, u = B 1 3 ϕ 2 (433) where x o x 1 B 3 = 2 y o y 1 z o z 1 (434) The covariance matrix of the final source position estimator is cov(u) = B 1 3 cov(ϕ 2)B T 3 (435) 32

41 x i y i z i Table 41: True Sensor Position 42 Simulation The weighting matrix W 1 in first stage is depend on the unknown true source and sensor positions In practice, We set W 1 to Q at first After we have an initial estimate of the source location, we use it to obtain W 1 Then, we make use of the updated W 1 to obtain more accurate estimation of ϕ 1 We repeat the stage 1 computation several times, to obtain more accurate result In our implementation, the number of times to repeat is set to 3 The TDOA measurements are obtained according to r = r o + n where r is the TDOA measurements with noise, r o is the true TDOA values and n is the noise vector In simulation, r o is calculated by ri1 o = uo s i u o s 1 and the covariance matrix of n is Q Besides measurement noise, sensor position manifold uncertainties in the simulation are given by s = s o + Vp The covariance matrix of p is Q p V = [V T 1,, V T M ] T where v i is 3 3 identity matrix In the simulation, we compare mse(mean-square error) with CRLB It is obtained according to mse = K k=1 (uo u (k) ) T (u o u (k) ) K, where K is the number of ensemble runs of the proposed solution We set K to 5000 Table 41 is the sensor positions used in the simulations Figure 41 is the distribution of the sensors Figure 42 is the simulation result for a near-field source at u o = [600, 550, 650] T Measurement noise matrix Q is set to c 2 σ 2 in the diagonal elements and 05c 2 σ 2 otherwise Q p, which is the covariance matrix of p, is set to a L L identity matrix with the noise power of c 2 σ 2 The 33

42 Figure 41: Sensor Array Distribution CRLB of source localization in the presence of manifold sensor position error is higher than the CRLB of source localization which has accurate sensor positions According to this simulation, mean-square error of the source location estimate reaches CRLB when σ is small Figure 43 is the simulation result for a near-field source u o = [600, 550, 650] T The measurement noise matrix Q is set to c 2 σ 2 in the diagonal elements and 05c 2 σ 2 otherwise Q p is set to 01 times L L identity matrix At the beginning when σ 2 is small, the performance of source localization in the presence of manifold sensor position error is mainly effected by the sensor manifold uncertainties because sensor manifold uncertainties is much larger than measurement noise at first As the measurement noise power increases, the measurement noise power becomes much larger than the noise power of manifold sensor uncertainties and it dominates the performance In addition, the CRLB in the presence manifold sensor uncertainties becomes close to the CRLB in absence of sensor manifold error From this simulation, mean-square error of source localization reaches CRLB when the measurement noise power is small Figure 44 is the simulation result for a near-field source at u o = [600, 550, 650] T The Measurement noise matrix Q is fixed to 1 in the diagonal elements and 05 otherwise times 01 Q p is set to a L L identity matrix times the noise power of c 2 σ 2 Because the measurement noise power is fixed, The CRLB of the source location estimate in presence of measurement noise only is a horizontal line When the power of the sensor position manifold error is small, the two CRLBs 34

43 Figure 42: Near-field source localization in the presence of manifold sensor position uncertainties Figure 43: Near-field source localization in the presence of manifold sensor position uncertainties -fix Q p 35

44 Figure 44: Near-field source localization in the presence of manifold sensor position uncertainties -fix Q with and without sensor position manifold error are close to each other When the power of sensor position manifold error become much larger than the fixed measurement noise power, the CRLB of source localization in presence of manifold uncertainties is higher than the one without manifold uncertainties The mean-square error of the source location estimate reaches the CRLB when the sensor position manifold error is small Figure 45 is the simulation result for the distant source at u o = [2000, 2500, 3000] T The measurement noise matrix Q is set to c 2 σ 2 in the diagonal elements and 05c 2 σ 2 otherwise Q p, which is the covariance matrix of p, is set to a L L identity matrix with the noise power of c 2 σ 2 Comparing to the near-field source localization in Figure 42, the mean-square error diverges from CRLB earlier for the distant source However, the mean-square error remains to reach the corresponding CRLB if the noise is small enough Figure 43 is the simulation result for a far-field source u o = [2000, 2500, 3000] T The measurement noise matrix Q is set to c 2 σ 2 in the diagonal elements and 05c 2 σ 2 otherwise Q p is set to a 01 times an L L identity matrix At the beginning when σ 2 is small, the performance of source localization in the presence of manifold sensor position error is dominated by the sensor manifold uncertainties because it is much larger than the measurement noise As the measurement noise power increases, the measurement noise power dominates the localization 36

45 Figure 45: Distant source localization in the presence of manifold sensor position uncertainties performance In addition, the CRLB in the presence manifold sensor uncertainties becomes close to the CRLB in absence of sensor manifold error From this simulation, the mean-square error of source localization reaches the CRLB when the measurement noise power is small, but the performance diverges earlier from the bound than that of the near-field case Figure 47 is the simulation result for the distant source u o = [2000, 2500, 3000] T The measurement noise power is fixed Q is fixed to 01 times 1 in the diagonal elements and 05 otherwise Q p is set to a L L identity matrix times the noise power of c 2 σ 2 The CRLB of the source location estimate in presence of measurement noise only is a horizontal line as σ 2 increase When the power of sensor position manifold error is small, the two CRLBs of with and without sensor position manifold error are close to each other When the power of sensor position manifold error becomes much larger than the fixed measurement noise power, CRLB of source localization in presence of manifold uncertainties is higher than the one without manifold uncertainties The mean-square error of source localization reaches CRLB when the sensor position manifold error is small, but the performance diverges from the bound earlier than the near-field case 37

46 Figure 46: Distant source localization in the presence of manifold sensor position uncertainties -fix Q p Figure 47: Distant source localization in the presence of manifold sensor position uncertainties -fix Q 38

47 43 Mathematic Proof of Optimum Performance of the Proposed Estimator for Source Localization in the P- resence of Sensor Position Manifold Uncertainties In this section, we will prove that the Mean-square Error of the proposed estimator achieves the CRLB performance The CRLB of source localization in the presence of sensor position manifold error has been given in Chapter 3 Now, we evaluate the covariance of the source location estimate from the proposed estimator cov(u) 1 = B T 3 cov(ϕ 2 ) 1 B 3 = B T 3 G T 2 B 1 2 GT 1 B T 1 (Q + B 1 1 DQ pd T B T 1 ) 1 B 1 1 G 1B 1 2 G 2B 3 (436) Let G 3 = B 1 1 G 1B 1 2 G 2B 3 G 4 = B 1 1 D, (437) After using matrix inversion Lemma, we have cov(u) 1 = G T 3 (Q + G 4 Q p G T 4 ) 1 G 3 = G T 3 Q 1 G 3 G T 3 Q 1 G 4 (Q 1 p + G T 4 Q 1 G 4 )G T 4 Q 1 G 3 (438) Substituting (413), (416), (426), (428) and (434), G 3 becomes G 3 = B 1 1 G 1B 1 2 G 2B 3 = = x 2 x 1 r o 2 x M x 1 r o M (s 2 s 1) T r o j (s M s 1) T r o M + r21(xo x 1) r o 2 ro 1 + r M1(x o x 1) r o M ro 1 + r21(uo s 1) T r o 2 ro 1 + r M1(u o s 1) T r o M ro 1 y 2 y 1 r o 2 y M y 1 rm o + r21(yo y 1) r o 2 ro 1 + r M1(y o y 1) r o M ro 1 z 2 z 1 r o 2 z M z 1 r o M + r21(zo z 1) r o 2 ro 1 + r M1(z o z 1) r o M ro 1 (439) 39