Cramér-Rao Lower Bounds for Hybrid Localization of Mobile Terminals

Transcription

1 Workshop on Positioning, avigation Communication. Mar 28, Hannover, Germany Cramér-Rao Lower Bounds for Hybrid Localization of Mobile Terminals Carsten Fritsche Anja Klein Technische Universität Darmstadt, Communications Engineering Lab, Merckstr. 25, Darmstadt, Germany Abstract While in outdoor scenarios the Global Positioning System (GPS) provides accurate mobile station (MS) location estimates in the majority of cases, in dense urban indoor scenarios GPS often cannot provide reliable MS location estimates, due to the attenuation or complete shadowing of the satellite signals. The existing cellular radio network (CR)-based localization methods, however, provide MS location estimates in almost every scenario, but they do not reach the accuracy of the MS location estimates provided by GPS. Hybrid localization methods combine MS location information available from measured values of the CR with MS location information provided by the measured values of GPS. In this paper, a hybrid localization method is proposed that combines received signal level timing advance measured values from the Global System for Mobile Communication (GSM) time of arrival measured values from GPS. The best achievable localization accuracy of the proposed hybrid localization method is evaluated in terms of the Cramér- Rao Lower Bound. It is shown that the hybrid localization method significantly improves the localization accuracy compared to existing CR-based localization methods. I. ITRODUCTIO The dem for localization methods offering precise mobile station (MS) location estimates has increased in the last few years. On the one h, this is due to the increasing interest in location based services (LBS) such as, e.g., fleet management, location sensitive billing, navigation other promising applications that are based on accurate knowledge of the MS location that will play a key role in future wireless systems []. On the other h, the requirement of the United States Federal Communications Commission (FCC) that all wireless service providers have to report the location of all enhanced 9 (E-9) callers with specified accuracy, has pushed research stardization activities in the field of MS localization [2]. Several localization methods have been proposed to solve the problem of locating a MS [3], [4]. Global navigation satellite system (GSS)-based localization methods, e.g., such as the Global Positioning System (GPS), utilize the time of arrival (ToA) principle, i.e., the MS location is determined from the propagation time of the satellite (SAT) signals. If the MS receives SAT signals from at least four SATs, a three dimensional (3-D) MS location estimate can be found, where the fourth SAT signal is needed to resolve the unknown clock bias between the SAT the MS clock [5]. GSS-based localization methods have the disadvantage that especially in indoor dense urban scenarios the number of SATs in view is often insufficient to determine a 3-D or even two dimensional (2-D) MS location estimate, due to the attenuation or complete shadowing of the SAT signals. Cellular radio network (CR)-based localization methods are based on, e.g., the received signal strength (), round trip delay time (), time difference of arrival (TDoA), angle of arrival (AoA) or ToA principle (an overview is given in [3], [4]). Although CR-based localization methods have the advantage that the MS location estimates are almost everywhere available, they do not reach the accuracy of MS location estimates of the GSS-based localization methods. In scenarios, where MS location estimates from the GSS are not available the MS location estimates from the CR do not reach the desired localization accuracy, it is a promising approach to combine the information provided by the measured values of the CR the GSS [6], [7]. The resulting hybrid localization methods are expected to enhance the accuracy availability of MS location estimates. The best achievable localization accuracy of hybrid localization methods can be evaluated in terms of the Cramér-Rao lower bound (CRLB). In [8], the CRLB for hybrid /ToA /TDoA for wireless sensor networks is presented. The impact of quantizing measured values on the CRLB for wireless sensor networks is presented in [9]. The localization accuracy of hybrid ToA/TDoA for cellular radio networks is investigated in []. In this paper, the CRLB of a hybrid localization method is evaluated that combines measured values from the Global System for Mobile Communication (GSM) ToA measured values from GPS. In existing GSM networks, only quantized (RXLEV) quantized (TA) measured values are available. Thus, the corresponding CRLB that combines quantized measured values is additionally investigated compared to CRLB of unquantized measured values. Simulation results show that the hybrid localization method will significantly improve the localization accuracy compared to existing CR-based localization methods. The remainder of this paper is organized as follows: In Section II, the well known general expressions for statistical models for the measured values, the Fisher information matrix their relationship to the CRLB are introduced. In Section III, the statistical models for the RXLEV TA are introduced the well known statistical model for the ToA measured value is presented. In Section IV, the CRLB for the hybrid localization method is determined. In Section V, the achievable localization accuracy of hybrid localization method is evalu /8/$ IEEE 57

2 Workshop on Positioning, avigation Communication. Mar 28, Hannover, Germany ated by means of simulations. Section VI concludes the work. II. CRAMÉR-RAO LOWER BOUD The Cramér-Rao Lower Bound (CRLB) gives a lower bound for the covariance matrix of any unbiased estimate of the unknown parameters []. Let ˆθ be an estimate of the n MS location vector θ, where n 2 for MS localization in 2-D scenarios n 3 in 3-D scenarios. Let further m [m,m 2,..., m ] T denote the vector of measured values that are available from the different localization methods, where [ ] T denotes the transpose of a vector. The covariance matrix of any unbiased estimator ˆθ satisfies the following inequality Cov(ˆθ) I(θ) () [], where Cov(ˆθ) E θ (ˆθ θ)(ˆθ θ) T }, E θ } denotes the expectation conditioned on θ, I(θ) denotes the Fisher information matrix (FIM), [ ] denotes the inverse of a matrix. The matrix inequality A B should be interpreted as the matrix (A B) is positive semidefinite. The CRLB matrix is defined as the inverse of the FIM I(θ). The(i, j)-th element of the FIM I(θ) is given by 2 } log p(m θ) [I(θ)] i,j E θ i, j, 2,..., n (2) θ i θ j [], where p(m θ) denotes the probability density function (pdf) of the measured values m conditioned on θ. In the following, the measured values m k m l, k,...,, l,...,, k l are assumed to be statistically independent of each other. This assumption is considered to be a good approximation if the measured values result from different localization methods, each suffering from different errors. The conditional pdf p(m θ) can thus be determined from p(m θ) p(m k θ), (3) where p(m k θ) denotes the likelihood function of each measured value m k. In order to evaluate the CRLB, p(m k θ) has to be known. The likelihood function p(m k θ) can be determined from a statistical model of the measured value that is introduced in the following. Let us assume that each measured value m k provides information about the MS location θ. Each measured value is additionally affected by errors that are assumed to be additive that can be statistically described by a rom variable n k with known probability density function p nk (n k ). The statistical model for the measured value is, thus, given by m k f k (θ)n k (4) [7], where f k (θ) models the functional relationship between the MS s location θ the error-free measured value m k, i.e., m k f k (θ). The likelihood function p(m k θ) of the measured value m k can be determined from (4), yielding p(m k θ) p ni (m k f k (θ)). (5) III. EXAMPLES FOR STATISTICAL MODELS OF MEASURED VALUES A. Introduction In the following, the statistical models of the measured values that can be obtained from the CR, namely, from the GSS, namely ToA, are presented. Due to the fact that in GSM the measured values are quantized to finite precision, the corresponding relationships between quantized unquantized measured values are additionally determined. The SAT location is given by x SAT [x SAT,y SAT,z SAT ] T. The base station (BS) location x BS [x BS,y BS ] T, as well as the MS s location θ [x, y] T to be estimated, are assumed to lie in the xy-plane. For the case of 3-D BS MS locations thus an estimation of the 3-D MS location vector, the statistical models can be obtained in a similar way. B. Round Trip Time In CRs, the is the time the radio signal requires to travel from the BS to the MS back. It is a parameter that is used to synchronize the transmitted bursts of the MSs to the frame of the receiving BS [2]. Let d BS (θ) denote the BS to MS distance given by d BS (θ) (x BS x) 2 (y BS y) 2, (6) c 3 8 m/s the speed of light. Then, the model for the functional relationship between the error-free measured value m the MS location θ is given by m f (θ) 2 d BS(θ) c. (7) The measured values are affected by errors resulting form the propagation conditions - line-of-sight (LOS) or non-line-of-sight (LOS) situation - inaccuracies of the measurement equipment. It is assumed that the errors can be modelled as a Gaussian rom variable n with mean μ accounting for the error due to LOS situations stard deviation σ [3]. Thus, the statistical model of the measured value m is given by m f (θ)n. (8) In GSM, the measured values m are rounded to the nearest integer bit period T b 48/3 μs, also known as Timing Advance (TA) measured values m TA [2]. The statistical model for the TA measured value m TA is given by m T m TA b m 2, T b < 62 m 63, (9) T b 62, where denotes the nearest integer smaller than or equal to the argument. 58

3 Workshop on Positioning, avigation Communication. Mar 28, Hannover, Germany C. Received Signal Strength In CRs, the value is an averaged value of the strength of a radio signal received by the MS from the BS. It is well known that the attenuation of the signal strength through a mobile radio channel is caused by three factors, namely path loss, fast fading slow fading [3]. Let A denote the reference path loss at a BS to MS distance of km B the path loss exponent. Then, the path loss PL in db is given by PL A B log ( dbs (θ) km ) () [3], where both parameters A B strongly depend on the propagation conditions the antenna settings of the transmitter. In a real system, the BS may be equipped with directional antennas in order to enhance the cell s capacity. For a fixed MS to BS distance this means that the MS located in the direction of maximum antenna gain receives a larger value than a MS having the same propagation conditions but which is located in the direction of minimum antenna gain. In the following, it is assumed that normalized antenna gain models are a-priori available. As long as the 2-D MS location is determined, it is a reasonable assumption to assume 2-D normalized antenna gain models. Let ϕ BS (θ) denote the MS to BS angle given by ( ) ybs y ϕ BS (θ) arctan. () x BS x Let further A m denote the maximum attenuation, ϕ the BS antenna s boresight direction ϕ 3dB the half-power beamwidth of the BS antenna in degrees. Then, the model for the normalized antenna gain in db scale is given by ( ) 2 ϕbs(θ) ϕ 2, ϕ BS(θ) ϕ g(ϕ BS (θ)) ϕ 3dB ϕ 3dB A m 2 A m, else (2) [4]. The model for the functional relationship between the error-free measured value m the MS location θ is given by m f (θ) P t PL g(ϕ BS (θ))}, (3) where P t denotes the BS s equivalent isotropic radiated power. The measured value m is further affected by errors due to fast fading slow fading. As the measured value is averaged over several time-consecutive measurements, the error due to fast fading can be removed, thus, has not to be taken into account in the statistical model for the errors. It is well known that the error in db due to slow fading can be modelled as an additive zero-mean Gaussian rom variable n with stard deviation σ [3]. Thus, the statistical model of the mean measured value m in db scale is given by m f (θ)n. (4) In GSM, the measured values m, measured in dbm, are quantized to finite precision, also known as received signal level (RXLEV) measured values m RXLEV [2]. The statistical model for the RXLEV measured value m RXLEV is given by m, dbm m RXLEV m dbm, < m dbm < 62 m 63, dbm 62, (5) where denotes the nearest integer larger than or equal to the argument. D. Time of Arrival The GSS-based localization methods utilize the concept of ToA, i.e., the MS is measuring the time the SAT signal requires to travel from the SAT to the MS [5]. In the following, the statistical model of the ToA measured value m ToA is determined. Let d SAT (θ) denote the SAT to MS distance given by d SAT (θ) (x SAT x) 2 (y SAT y) 2 zsat 2. (6) Then, the model for the functional relationship f ToA (θ) between the error-free measured value m ToA the MS s location θ is given by m ToA f ToA (θ) d SAT(θ). (7) c The measured value m ToA is affected by errors resulting from delays as the signal propagates through the atmosphere, propagation conditions - LOS or LOS situation - as well as resolution errors receiver noise. These errors, given by n ToA, are assumed to be Gaussian distributed with mean μ ToA accounting for the error due to LOS situations stard deviation σ ToA [3]. In general, the MS s clock is not synchronized to the clocks of the GPS SATs, resulting in an unknown receiver clock error [5]. The unknown receiver clock error can be compensated, e.g., by using timing information provided by the radio ressource protocol [5]. In the following, it is assumed that the MS s clock is synchronized to the clocks of the GPS SATs. Consequently, the statistical model of the measured value m ToA is given by m ToA f ToA (θ)n ToA. (8) IV. EVALUATIO OF THE CRAMÉR-RAO LOWER BOUD FOR HYBRID LOCALIZATIO A. Introduction In the following, the CRLB for the hybrid localization method is evaluated. Since the best achievable localization accuracy of the estimated MS location ˆθ [ˆx, ŷ] T is of main interest, it is reasonable to introduce the minimum mean square error (MMSE) of the MS location estimate ˆθ [3]. Let min( ) denote the minimum of the argument, v 2 the L 2 norm of the vector v tra} the trace of a matrix A. The MMSE of the MS location estimate is then given by MMSE min(e θ ˆθ θ 2 }) min(trcov(ˆθ)}) tri(θ) } [I(θ)], [I(θ)] 2,2, (9) Det I(θ)} 59

4 Workshop on Positioning, avigation Communication. Mar 28, Hannover, Germany where Det I(θ)} denotes the determinant of the symmetric 2 2 FIM I(θ). Further, let I (θ) I Q (θ) denote the FIMs of the unquantized quantized measured value, I (θ) I Q (θ) the FIMs of the unquantized quantized measured value I ToA (θ) the FIM of the unquantized ToA measured value. As long as the measured values are assumed to be statistically independent, the corresponding FIMs can be added up [3]. Thus, the FIM of the hybrid localization method for unquantized, ToA measured values can be determined from I(θ) I (θ)i (θ)i ToA (θ), (2) for quantized unquantized ToA measured values from I(θ) I Q (θ)i Q (θ)i ToA (θ). (2) In the following, the FIM of unquantized, ToA measured values quantized measured values are derived, from which the corresponding CRLBs for the hybrid localization method the MMSE of the MS location estimate can be determined. B. Cramér-Rao Lower Bound for unquantized measured values As long as the measured values are unquantized corrupted by additive errors that are zero-mean Gaussian distributed, the (i, j)-th element of I(θ) can be found from the well known expression [I(θ)] i,j σ 2 k f k(θ) θ i f k(θ) θ j (22) []. The FIM of measured values can be found from inserting (7) into (22). Let denote the number of available measured values u BS,k the unit vector originating at the MS location directed towards the k-th BS given by u BS,k u BSx,k u x u BSy,k u y (23) [], where u x, u y are the unit vectors in the x y directions. Let further a k be defined as ( ) 2 2 a k. (24) c σ,k Then, the elements of the FIM of measured values are given by: [I (θ)], [I (θ)],2 [I (θ)] 2,2 a k u 2 BS x,k, (25) a k u BSx,k u BSy,k, (26) a k u 2 BS y,k. (27) The FIM of measured values can be found from inserting (3) into (22). Let denote the number of available measured values. Let further b k w k (θ) be defined as b k B k σ,k log e () (28) w k (θ) g k(ϕ BS,k (θ)). (29) σ,k ϕ BS,k (θ) Then, the elements of the FIM of measured values are given by [ ] 2 bk u BSx,k w k (θ) u BSy,k [I (θ)],,(3) [( ) bk u BSx,k w k (θ) u BSy,k [I (θ)],2 ( )] bk u BSy,k w k (θ) u BSx,k, (3) [ ] 2 bk u BSy,k w k (θ) u BSx,k [I (θ)] 2,2.(32) The FIM of ToA measured values can be found from inserting (7) into (22). Let ToA denote the number of available ToA measured values u SAT,k the unit vector originating at the MS location directed towards the k-th SAT, given by u SAT,k u SATx,k u x u SATy,k u y u SATz,k u z, (33) where u z is the unit vector in the z direction. The projection of the unit vector u SAT,k into the xy-plane is given by p SAT,k u SATx,k u x u SATy,k u y. (34) Let further c k be defined as ( ) 2 c k. (35) c σ ToA,k The elements of the FIM of ToA measured values are given by [I ToA (θ)], [I ToA (θ)],2 [I ToA (θ)] 2,2 ToA ToA c k u 2 SAT x,k, (36) c k u SATx,k u SATy,k, (37) ToA c k u 2 SAT y,k. (38) [8]. The CRLB for the hybrid localization method of unquantized measured values can then be found from (2) (). In the following, a closed form expression of the MMSE of the MS location estimate is determined. For the sake of simplicity, it is assumed that the BS antennas have a uniform normalized antenna gain g k (ϕ BS,k (θ)). From this it follows 6

5 Workshop on Positioning, avigation Communication. Mar 28, Hannover, Germany that w k (θ).leta k,i denote the area of the parallelogram determined by u BS,k u BS,i [], given by A k,i u BS,k u BS,i 2, (39) B k,i denote the area of the parallelogram determined by u BS,k p SAT,i, given by B k,i u BS,k p SAT,i 2 (4) C k,i denote the area of the parallelogram determined by p SAT,k p SAT,i, given by C k,i p SAT,k p SAT,i 2, (4) where a b denotes the cross product between two vectors a b. Then, the MMSE of the MS location estimate for unquantized measured values is given by MMSE a k b 2 k d 2 BS,k where the determinant is given by Det I(θ)} i i>k i ToA i ToA i ToA ToA (θ) Det I (θ)} a k a i A k,i a k b 2 i d 2 BS,i (θ) A k,i a k c i B k,i i i>k ToA i i>k b 2 k c i d 2 BS,k (θ) B k,i c k p SAT,k 2, (42) b 2 k b2 i d 2 BS,k (θ) d2 BS,i (θ) A k,i c k c i C k,i. (43) C. Cramér-Rao Bound for quantized measured values In GSM, the measured values are quantized to K 64 levels, i,..., 63, i Z. Each level i is composed of a lower bound Q,lb (i), denoting the minimum measured in level i an upper bound Q,ub (i), representing the maximum measured in level i. According to (9), the lower upper bounds of the corresponding TA measured values m TA i are given by, i Q,lb (i) (44) (2i ) Tb 2, i,..., 63 Q,ub (i) (2i ) T b 2, i,..., 62, i 63. (45) The FIM of quantized measured values can be found, following the derivation given in [9]. Let Q denote the number of available TA measured values Φ( ) the stard normal cumulative distribution function of the argument. The conditional pdf p k ( θ) is given by p k ( θ) σ,k 2π exp [f,k(θ)] 2 2 σ 2,k }, (46) h k,i is defined as h k,i [ 63 exp 2 (r ) 2} exp 2 (r 2) 2}] 2, (47) 2π Φ(r 2 ) Φ(r ) i with the following functions r Q,lb(i) f,k (θ) σ,k (48) r 2 Q,ub(i) f,k (θ) σ,k. (49) Then, the elements of the FIM of quantized measured values are given by [I Q (θ)], [I Q (θ)],2 [I Q (θ)] 2,2 Q [( f,k (θ) p k ( θ) a k Q Q ) ] d 2 BS,k (θ) u 2 BS x,k d 2 BS,k (θ) a k u 2 BS x,k h k,i, (5) f,k (θ) p k ( θ) ) d 2 BS,k (θ) Q Q [( [( u BSx,ku BSy,k a k ] d 2 BS,k (θ) a k u BSx,k u BSy,k h k,i, (5) f,k (θ) p k ( θ) a k Q ) ] d 2 BS,k (θ) u 2 BS y,k d 2 BS,k (θ) a k u 2 BS y,k h k,i, (52) In GSM, the measured values are quantized to K 64 levels, i,..., 63, i Z. Again, each level i is composed of a lower bound Q,lb (i), denoting the minimum measuredinleveli an upper bound Q,ub (i), representing the maximum measured in level i. According to (5), 6

6 Workshop on Positioning, avigation Communication. Mar 28, Hannover, Germany the lower upper bounds of the corresponding RXLEV measured values m RXLEV i are given by, i Q,lb (i) (53) (i ), i,..., 63 Q,ub (i) i, i,..., 62, i 63. (54) The elements of the FIM of quantized measured values are given by Q [ ] bk u BSx,k w k (θ) u BSy,k [I Q (θ)], 2 h k,i, (55) Q [( ) bk u BSx,k w k (θ) u BSy,k [I Q (θ)],2 ( )] bk u BSy,k w k (θ) u BSx,k h k,i, (56) Q [ ] bk u BSy,k w k (θ) u BSx,k [I Q (θ)] 2,2 2 h k,i, (57) where Q is the number of available RXLEV measured values h k,i is defined in (47) with the exception that r r 2 have to be replaced by r Q,lb(i) f,k (θ) σ,k (58) r 2 Q,ub(i) f,k (θ) σ,k. (59) The CRLB for the hybrid localization method of quantized measured values unquantized ToA measured values can be then found from (2) (). The MMSE of the MS location estimate can be found from (9) (2). For the sake of clarity, the closed form expression of the MMSE of the MS location estimate is omitted. V. SIMULATIO SCEARIO AD RESULTS In the following, the simulation scenario is explained simulation results are presented. It is assumed that the BSs are organized in an ideal 2-D CR with hexagonal cell structure cell radius R cell. The BS locations are assumed to be fixed known, each BS is equipped with an omnidirectional BS antenna. In order to evaluate the CRLB of the hybrid localization method, only the BS nearest BSs are considered for MS localization as shown in Fig., where the nearest BS is assumed to be the serving BS. The GPS SAT locations are assumed to be known since they can be calculated from the navigation message that is transmitted by each SAT. The simulation parameters are given in Table I are assumed to be equal for all BSs all SATs for the sake of y 2R cell 2R cell 2R cell x 2R cell Fig.. CR with the MS ( ) BS 3BSs ( ) that are involved in the MS localization process. The BSs that are not involved in the MS localization process are marked as squares ( ). simplification. The following combinations of measured values are investigated: One TA RXLEV measured value from the serving BS one RXLEV measured value from BS neighbouring BSs (GSM method) Measured values of GSM method, in addition, one ToA measured value from one GPS SAT (Hybrid method) Measured values of GSM method, in addition, one ToA measured value from each of a total of two GPS SATs (Hybrid 2 method) Two ToA measured values provided by two different GPS SATs (2 Satellite method) The hybrid localization method is evaluated in terms of the average root mean square error (RMSE), which gives the localization accuracy for all possible MS locations in the CR is given by } RMSE E θ tr I(θ) }. (6) In Fig. 2, the simulation results for the RMSE in dependence of the cell radius for the GSM, Hybrid Hybrid 2 method for (dense) urban scenarios are shown. For comparison, the results for the RMSE assuming unquantized are given. The GSM method provides the worst results in terms TABLE I SIMULATIO PARAMETERS Parameter Value Parameter Value A in db 32.8 P t in dbm 33 B in db 3.8 σ in μs σ in db 8 σ ToA in ns of localization accuracy, which can be explained by the fact that the investigated GSM measured values cannot provide the same level of accuracy as the GPS measured values. The 62

7 Workshop on Positioning, avigation Communication. Mar 28, Hannover, Germany 5 4 quantized unquantized GSM 5 4 BS 3 BS 4 BS 5 GSM RMSE in m 3 2 Hybrid RMSE in m 3 2 Hybrid 2 Hybrid 2 Hybrid R cell in m Fig. 2. RMSE vs. R cell for the GSM, Hybrid Hybrid 2 method for quantized unquantized measured values for BS R cell in m Fig. 3. RMSE vs. R cell for the GSM, Hybrid Hybrid 2 method for quantized measured values for different BS 3, 4, 5}. localization accuracy can be significantly improved with the Hybrid method, by additionally taking into account one ToA measured value from one GPS SAT. In contrast to the Hybrid method, the Hybrid 2 method takes into account two ToA measured values from two different GPS SATs, leading to a further improvement of the localization accuracy. The impact of the quantization on the achievable localization accuracy can also be seen from Fig. 2. For the GSM Hybrid method the RMSE degrades only for large cell radii, whereas for the Hybrid 2 method, the quantization does not significantly affect the RMSE for almost the entire range of investigated cell radii. In Fig. 3, the RMSE in dependence of the cell radius for the GSM, Hybrid Hybrid 2 method for different values of BS is shown. If BS, thus, the number of available RXLEV measured values is increased, the RMSE can be improved. For the GSM method the improvement is significant, for the Hybrid method moderate for the Hybrid 2 method marginal. The marginal accuracy improvement of the Hybrid 2 method can be explained by the fact that the two ToA measured values from the GPS SATs have a higher level of accuracy than the RXLEV measured values from the CR, so that the improvement of the localization accuracy due to additionally taking into account RXLEV measured values from the CR is marginal. Moreover, the achievable localization accuracy of the Hybrid 2 method strongly depends on the geometric constellation of the SATs BSs relative to the MS. For the 2 Satellite method the geometric constellation of the SATs relative to the MS can be expressed by the geometric dilution of precision (GDOP) value for 2-D scenarios [6]. If the GPS satellites are close together in the sky, the geometry between the SATs the MS is said to be weak the corresponding GDOP value is large. If the GPS SATs are far away from each other, the geometry between the SATs the MS is said to be strong the corresponding GDOP value is small. In Fig. 4, the localization accuracy for the Hybrid 2 method the 2 Satellite method for different average GDOP values BS 3is presented. It can be clearly seen that the larger the GDOP value, the better is the performance of the Hybrid 2 method compared to the 2 Satellite method. Thus, in scenarios where the GDOP value tends to be large as, e.g., in urban street canyons, the localization accuracy of the Hybrid 2 method is expected to be significantly better than the localization accuracy of the 2 Satellite method. RMSE in m Hybrid 2, GDOP5 2 Satellite, GDOP5 Hybrid 2, GDOP 2 Satellite, GDOP Hybrid 2, GDOP2 2 Satellite, GDOP2 Hybrid 2, GDOP5 2 Satellite, GDOP R cell in m Fig. 4. RMSE vs. R cell for the Hybrid 2 2 Satellite method for different average GDOP values for BS 3. VI. COCLUSIO In this paper, the CRLB for a hybrid localization method is determined that combines RXLEV TA measured values from GSM ToA measured values from GPS. It has been shown by simulations that compared to combining only RXLEV TA measured values from GSM the localization accuracy can be significantly improved by additionally taking 63

8 Workshop on Positioning, avigation Communication. Mar 28, Hannover, Germany into account one or two ToA measured values from GPS. In scenarios with low GDOP values it is sufficient to combine two ToA measured values from GPS, whereas in scenarios with high GDOP values the corresponding hybrid localization method should be used. REFERECES [] A. Küpper, Location-based services, st ed. John Wiley & Sons, 25. [2] FCC : Third report order, Federal Communications Commission, Tech. Rep., Oct [3] F. Gustafsson F. Gunnarsson, Mobile positioning using wireless networks, IEEE Signal Processing Mag., vol. 22, no. 4, pp. 4 53, Jul. 25. [4] G. Sun, J. Chen, W. Guo, K. J. R. Liu, Signal processing techniques in network-aided positioning: a survey of state-of-the-art positioning designs, IEEE Signal Processing Mag., vol. 22, no. 4, pp. 2 23, Jul. 25. [5] E. D. Kaplan, Ed., Understing GPS: principles applications, st ed. Artech-House, 996. [6] S. Soliman, P. Agashe, I. Fernez, A. Vayanos, P. Gaal, M. Oljaca, gpsone tm : a hybrid position location system, in Proc. International Symposium on Spread Spectrum Techniques Applications, vol., Sept. 2, pp [7] C. Fritsche, A. Klein, H. Schmitz, Mobile station localization by combining measurements from different sources including reliability information, in Proc. ITG Fachtagung Innovations for Europe - Mobility, Aachen, Germany, Oct. 26, pp [8] A. Catovic Z. Sahinoglu, The Cramer-Rao bounds of hybrid TOA/ TDOA/ location estimation schemes, IEEE Commun. Lett., vol. 8, no., pp , Oct. 24. [9]. Patwari A. O. Hero, Using proximity quantized for sensor localization in wireless networks, in Proceedings 2nd International ACM Workshop on Wireless Sensor etworks Applications (WSA 3), Sept. 23, pp [] M. A. Spirito, On the accuracy of cellular mobile station location estimation, IEEE Trans. Veh. Technol., vol. 5, no. 3, pp , May 2. [] S. M. Kay, Fundamentals of statistical signal processing: estimation theory, st ed. Prentice-Hall, 993. [2] 3GPP TS 5.8, Technical specification group GSM/EDGE radio access network; Radio subsystem link control, Version 8.23., ov. 25. [3] H. K. Y. Qi H. Suda, Analysis of wireless geolocation in a nonline-of-sight environment, IEEE Trans. Wireless Commun., vol. 5, no. 3, pp , Mar. 26. [4] 3GPP TR , Technical specification group radio access network; Spatial channel model for MIMO simulations, Version 6.., Sept. 27. [5] 3GPP TS 4.3, Location services LCS; Mobile Station (MS) - Serving Mobile Location Centre (SMLC) - Radio Resource LCS Protocol (RRLP), Version 8.8., Jun. 27. [6]. Levanon, Lowest GDOP in 2-D scenarios, in IEE Proc. Radar, Sonar avigation., vol. 47, no. 3, Jun. 2, pp