WIRELESS localization using radio location systems has

Transcription

1 862 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004 Cramer Rao Bounds for the Estimation of Multipath Parameters and Mobiles Positions in Asynchronous DS-CDMA Systems Cyril Botteron, Member, IEEE, Anders Høst-Madsen, Senior Member, IEEE, and Michel Fattouche, Member, IEEE Abstract Commercial applications for the location of subscribers of wireless services continue to expand. Consequently, finding the Cramer Rao lower bound (), which serves as an optimality criterion for the location estimation problem, is of interest. In this paper, we derive the deterministic s for the estimation of the specular multipath parameters and the positions of the mobiles in an asynchronous direct sequence code division multiple access (DS-CDMA) system operating over specular multipath fading channels. We assume a multilateral radio location system the location estimates are obtained from some or all of the estimated signal parameters at different clusters of antennas of arbitrary geometry. Extension for unilateral and composite radio location techniques is also discussed. As an application example, we use numerical simulations to investigate the effects of specular multipath and multiple access interference (MAI) on the positioning accuracy for different radio location techniques. Index Terms Amplitude estimation, angle estimation, asynchronous DS-CDMA, Cramer Rao lower bound, location, position, specular multipath, time estimation. I. INTRODUCTION WIRELESS localization using radio location systems has become an important research area over the past few years. A major application is personal safety, such as in the location-based emergency service (E-911), which is a requirement for the wireless carriers in the United States [1]. Other applications include intelligent transportation systems, accident reporting, automatic billing, fraud detection, and other emerging services [2], [3]. Radio location systems attempt to locate a mobile station (MS) by measuring the radio signals travelling between the MS and a set of fixed stations (FSs) of known coordinates. They can be classified as unilateral (or handset-based), multilateral (or network-based), or composite [3]. In a unilateral system, an MS forms an estimate of its own position based on radio signals received from the FSs, and in a multilateral system, an estimate of the MS s location is based on a signal transmitted by the MS and received at multiple FSs. Manuscript received January 10, 2001; revised April 17, The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Randolph L. Moses. C. Botteron was with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada. He is now with the Institute of Microtechnology, University of Neuchâtel, CH-2000 Neuchâtel, Switzerland ( cyril.botteron@unine.ch). A. Høst-Madsen is with the Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, HI USA ( madsen@spectra.eng.hawaii.edu). M. Fattouche is with Cell-Loc Inc., Calgary, AB T2A 6T8 Canada ( michel.fattouche@cell-loc.com). Digital Object Identifier /TSP In order to quantify the impact of different signal parameters, environment parameters, and to understand the relative contribution to accuracy of different signal measurements, we derive in this paper the for multilateral or network-based radio location techniques. However, instead of assuming that the signal measurements are (zero-mean) Gaussian distributed 1 with known covariance matrix (see, e.g., [4] [7]), we assume for our derivation a specular multipath environment and Gaussian distributed noise at the antenna receivers so that we can express the directly in terms of the signal and environment parameters. Extension of the for unilateral or handset-based and composite techniques is also discussed. Using the for the location estimation, we investigate using numerical simulations the effects of the multiple access interference (MAI) and specular multipath for a K-user asynchronous DS-CDMA system for which we also present the deterministic and asymptotic for the joint estimation of the signal parameters. Three different multipath scenarios, the number and relative clustering of the main reflectors around the MS are varied, are also discussed. This paper has been organized as follows. Section II describes the notations and some general model assumptions. Section III presents the for the location estimation for the mostly used radio location techniques. In Section IV, the deterministic and asymptotic s for the joint estimation of the time, bearing, and amplitude parameters in a K-user asynchronous DS-CDMA system are derived. In Section V, the effects of MAI and specular multipath on the relative contribution to accuracy of different radio location techniques are investigated. Finally, the conclusions are discussed in Section VI. II. RADIO LOCATION MODEL AND ASSUMPTIONS We consider a specular multipath environment and a multilateral positioning system, the location of a MS is estimated based on the measurements of the signal transmitted by the MS and received at FSs of known coordinates. Without loss of generality, we assume that. We consider the most commonly used signal parameter measurements, 2 i.e, the angle of 1 Because the signal measurements (such as time, bearings, etc.) cannot in general be expressed as a linear function of the received signal samples at the antenna receivers, their distribution should not be expected to be Gaussian, even if the signal samples are statistically Gaussian distributed. 2 For a more complete description of these different techniques, see, e.g. [8] [10], and the references therein X/04$ IEEE

2 BOTTERON et al.: CRAMER RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS 863 arrival (AOA), the time of arrival (TOA), the time difference of arrival (TDOA) between multiple FSs, and the strength of arrival (SOA) measurements. These measurements can be used individually or in combination (in which case, they are called mixed measurements) to produce a position location. Note that the model and theory developed in this paper can be easily extended to consider other measurements and positioning techniques (e.g., the extension of the model for unilateral and composite techniques is discussed in Section III-B). For clarity, the generic assumptions used in this paper are summarized and discussed below. A1) The environment between the MS and the FS(s) exhibits specular multipath due to a finite number of dominant reflectors located in the far field of the receiver(s). A2) Every dominant path results from the superposition of many component waves delay spreads are much smaller than the inverse bandwidth of the signal and the dominant path delays. A3) For short observation intervals, the signal parameters (e.g., the number of dominant paths, their attenuation, direction of arrival, and time-delay) can be assumed constant. A4) The transmitted signal is narrowband with respect to the reciprocal of the excess delay across the receiving antenna arrays (i.e., the signal bandwidth is assumed to be very small compared to the carrier frequency). A5) The noise present at the antenna receivers is zero-mean Gaussian distributed, spatially independent between the different FSs and the MS, and independent of the signal and position parameters. A6) In addition to the nonline of sight (NLOS) paths coming from the main reflectors, a direct path always exists between the MS and the FSs. Assumptions A1 A4 are the conventional assumptions used to model the direction and time estimation of the specular multipath arrivals from a narrowband source located in the far field and in the same 2-D plane as an array of sensors (see, e.g, [11] [13]). The Gaussian noise assumption A5 is a basic and mild assumption that is justified in practice, especially for the outdoor channels. Note that only the noise between the different FSs and the MS is assumed to be spatially independent. However, the noise processes are not assumed to be spatially independent between the different antenna receivers at one particular FS, and the noise covariance matrix may be different from one FS to another. Finally, assumption A6 is a little more restrictive but necessary to allow for an unbiased position estimator to exist. III. FOR THE LOCATION ESTIMATION We define the following parameter vectors for the developments to come. -dimensional complex vector, observation vector contains (in complex baseband representation) all the received samples from all the antenna receivers at the th FS. : source-location parameter vector : signal parameter vector -dimensional real vector containing the positions in Cartesian coordinates of all the main reflectors and the MS(s) (i.e., the locations of all the sources) plus any additional nuisance parameters (such as the noise parameters) necessary for to completely parameterize the probability density function (p.d.f.) of the observation vector. -dimensional real vector, contains the signal parameters (e.g., AOA, TOA, SOA, ) that characterize all of the specular multipath arrivals at the th FS plus any additional nuisance parameter (such as the noise parameters) that is necessary for the vector to completely parameterize the p.d.f. of the observation vector at the th FS. A. Derivation of the Location Since the source-location parameters contained in form a set of linearly independent parameters, the matrix for any unbiased estimate of could be calculated, under standard regularity conditions and based on the above definitions and assumptions, using the standard formula as the inverse of the Fisher information matrix (FIM), i.e., as (see, e.g., [14], [15]) (1) denotes the expectation operator taken with respect to, and the derivatives are evaluated at the true values of. Assuming an matrix of random samples that consists of independent, -dimensional random vectors normally distributed as (see assumption A5), another method to calculate the elements of the FIM would be to use the standard formula for the complex multivariate normal (MVN) model, i.e., (see, e.g., [15]) tr (2) However, unless a very simple system geometry [i.e., the position of the FSs and main reflectors with respect to the location of the MS(s)] is assumed, deriving the for the location of the sources using (1) or (2) is too difficult for the problem at hand. An alternative and simpler approach is to first calculate the for the signal parameters of the sources at the FSs and then use the following proposition

3 864 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004 Proposition 1: Based on the above definitions and assumptions, define... TABLE I CORRESPONDING PARAMETERS FOR MULTILATERAL AND UNILATERAL RADIO LOCATION TECHNIQUES diag (3) is the for the joint estimation of the signal parameters of the sources at the th FS. Express the signal parameters contained in as a function of the location parameters contained in as the functions for or with. We then have (4) the elements of the matrix are defined as provided that the functions do not affect the noise parameters. Proof: See Appendix A. 1) Discussion: We make the following observations. Proposition 1 looks like a standard transformation of parameters, as discussed in [14] and [15]. However, a transformation of parameters is defined for functions with, as here,. Because of the relationship between parameters, not all values of are valid is constrained to a submanifold of, and therefore is not a matrix (inverse of a Fisher matrix). The dimension of the vectors do not need to be the same for. In other worlds, the for the location of the sources can be obtained using Proposition 1 for a different number of samples, number of antennas, type of antennas, and noise covariance matrix at each of the FS. B. Extension for Unilateral and Composite Techniques For an unilateral technique the MS estimates its position based on the received signals coming from FSs, the observation vector at the MS will contain the superposition of the multipath arrivals from FSs. Thus, it can be expressed as the received samples in contains (in complex baseband representation) the contributions of the multipath signals coming from the th FS, and is assumed Gaussian distributed (see assumption A5) with covariance matrix. We observe that the above system model is similar to a -user system model for multilateral radio location techniques a single FS is used to estimate the location of the MSs (see Table I). Consequently, the for the signal parameters in a -user system model for multilateral techniques can be applied for a single-user unilateral technique by using the correspondence of parameters given in Table I. In this case, the signal (5) (6) parameter vector and the matrix in Proposition 1 will only contain the signal parameters of the sources estimated at the MS and the for those signal parameters, respectively. Finally, for a composite radio location technique the received signals at the MS and the FSs are jointly used to estimate the location of the MS, we deduce from Assumption A5 (i.e., assuming that the noise is spatially independent between the MS and the FSs, which is reasonable in practice) that Proposition 1 can still be used. In this case, the signal parameter vector will contain both the signal parameters estimated at the MS and at the FSs, and will contain in block diagonal form the matrix for the estimation of the signal parameters at the MS in addition to the matrices for the estimation of the signal parameters at the FSs. C. for the Location of a MS If the number of specular multipaths at the FSs is large and we are only interested in bounding the positioning accuracy for the location of a MS (and not for the location of other sources), using (3) (5), which involves the inversion of a large matrix, is cumbersome. Furthermore, most position estimators only use a subset of the signal parameters estimated at FSs to estimate the location of an MS. For example, if we assume a bearing-based radio location system, then only the estimated AOAs of the direct paths coming from the MS and received at FSs will be used. In addition, if a radio location system only uses the estimated TOAs of the direct paths coming from the MS and received at FSs and has no knowledge of the time of transmission (TOT), then it will use these TOAs to jointly estimate the unknown MS s position with the unknown MS s TOT. Thus, we now consider the case when some of the estimated signal parameters are used to estimate the location of a MS. Proposition 2: In addition to assumptions A1 A6, let us assume that the unbiased position estimator only uses the subset of the jointly estimated signal parameters contained in to estimate the parameters contained in, contains all the parameters necessary for a function that defines a continuous and bounded mapping from onto to exist. Then, the covariance matrix for any unbiased estimator is bounded as follows: cov (7) diag, and the elements of the matrix are defined as (8)

4 BOTTERON et al.: CRAMER RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS 865 provided that a) exists, and b) is positive definite (meaning exists). Proof: See Appendix B. 1) Discussion: Proposition 2 is very important since it allows the calculation of the for the location of a MS, assuming an unbiased estimator that only uses some of the estimated signal parameters without inverting the whole matrix. The matrices composing can easily be obtained from the matrices by removing the rows and columns that correspond to the nuisance parameters contained in,, i.e., the signal parameters that are not used to estimate the position parameter vector. Using the appropriate transformation matrix and matrices, Proposition 2 can be used for any positioning method (including multilateral, unilateral, and composite radio location methods using mixed measurements), signal waveform, or system geometry. It is worth noting that the for that we would obtain by removing the rows and columns corresponding to the subset from rather than from would not be the same but correspond to assume that the subset is known by the positioning system. Using the well-known fact that an augmentation of the nuisance parameters can only result in an increase of the corresponding (see, e.g. [14]), we can write (9) the path loss between the MS and the distance as [16] th FS separated by a db (10) is the path loss at the reference distance, and is the path loss exponent that indicates the rate at which the path loss increases with distance. If the MS transmits with a fixed power of dbm, we can thus express the power received at the th FS as dbm (11) represents the total system losses (in decibels). On the other hand, if the transmitted power is controlled by the th serving FS and assuming perfect power control (p.c.), the received power at the th FS can be expressed as dbm (12) SNR (in decibels) and (in decibel meters) are the desired SNR (without antenna gain) and the received noise at the th serving FS, respectively. By defining the known 2-D position of the th FS in Cartesian coordinates as and the MS s position as, we can thus write the th signal parameter as, D. Expressions for the Transformation Matrix We now give some analytical expressions for the transformation matrix in Proposition 2, assuming four different radio location estimators that we categorize based on the type of direct path measurements they use to estimate the position of the MS: SOA-based: using the signal parameter vector containing the estimated modulus of the direct paths fadings (we assume an estimator that has knowledge of the path loss model); AOA-based: using the signal parameter vector containing the estimated direct paths bearings; TD-based: using the signal parameter vector containing the estimated direct paths propagation-times obtained by subtracting the MS s TOT assumed to be known at the FSs from the estimated direct paths TOAs; TOA-based or TDOA-based: using the signal parameter vector containing the estimated direct paths TOAs. In this case, we assume that the MS s TOT is also unknown and jointly estimated with the location of the MS. In order to find an analytical expression for the transformation matrix for a SOA-based radio location estimator, we write and denotes the speed of light (in meters per second). By computing the derivatives of with respect to [see (8)], we obtain the transformation matrices,,, and corresponding to the parameter vectors,,, and, respectively, as (13) (14) (15) (16)

5 866 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004 is given assuming a fixed transmitted power or perfect power control at the th FS by or, respectively, and Note that for a mixed measurement vector containing any combination of the above measurements, the transformation matrix will simply be a combination of the columns of (13) (16) corresponding to the parameters in and rows corresponding to. IV. EXPRESSIONS FOR The derivation for direction and time estimation using antenna arrays in a specular multipath environment can be found in many publications. For example, assuming a single user and no MAI, the conditional (sometimes also called deterministic) for the estimation of the bearings and time delays was derived in [11], [13], and [17], assuming unknown deterministic path fading parameters. A compact derivation for parametric estimation of superimposed signals was also presented in [18]. For a K-user asynchronous DS-CDMA communication system the MAI is explicitly modeled, the for the time delays and path fadings treated as unknown deterministic parameters was derived without bearing estimation in [19] and [20]. However, the for the joint estimation of the path amplitudes, bearing, and time-delays only appeared (without proof) in [21]. Thus, we now briefly review the system model in [21] and present the deterministic and asymptotic for the joint estimation of the path amplitudes, bearing, and time-delays in asynchronous DS-CDMA systems. A. Asynchronous -User DS-CDMA System Model Besides the generic assumptions A1 A6 (see Section II), we assume that the complex path fading amplitudes can be considered constant over the duration of one symbol interval. The users code waveforms are assumed to be rectangular and periodic with period, is the chip period, and is the processing gain. The modulation is BPSK, i.e., the th user baseband signal is formed by pulse amplitude modulating the data stream with a period of the -long code waveform as. Using complex envelope representation and collecting the received samples at the th FS during the th symbol interval from the th element of a -element antenna array in a single vector, we can write 3 (for more details, see, e.g., [19], [21], [22]) and the complex noise vector is defined similarly to. denotes the number of dominant paths from the th user impinging on the antenna array. The elements of are and for, and, is the th element of the steering vector toward the AOA, and denotes the complex path amplitude during the th symbol interval. The columns of represent the shifted code sequences for the dominant paths coming from the th user and impinging on the antenna array. They are defined as (18) (19) is the path delay, such that is an integer, and, and (i.e., we assume no oversampling). The vector and the permutation matrix are defined as an- Finally, by stacking the received vectors from all the tennas in a single vector, we obtain (20) denotes the Kronecker product, and, are defined similarly to. B. Deterministic for the DS-CDMA Signal Parameters Based on the above signal model, we define the unknown signal parameter vector as Im Im (21) (17) 3 For ease of notation, we often drop the FS subscript c unless it is necessary to avoid confusion with the previous notations. (22)

6 BOTTERON et al.: CRAMER RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS 867 Assuming that the received noise at the antenna array is zero-mean complex circular Gaussian with variance, the log-likelihood function with respect to and conditioned on the transmitted bits (practically, the transmitted bits are readily available from the serving base stations) can be written as and is defined as diag const. By applying the formula (1) to the above log-likelihood function, it can be shown that the FIM for takes the form (23) crb (24) Im Im Im (25) (26) (27) and the matrices,,, and are defined as (28) (29) diag (30) (31) diag (32) The matrices and are defined similarly to. The elements of these matrices are (36) We observe that the for the signal parameters belonging to the th user is independent of the received powers of the other users (this fact can be easily verified by noting the diagonal form of and using the same procedure as in [19, Sec. IV]). C. Extension for the SOA Estimation We now extend the previous derivation to obtain the deterministic for the joint estimation of the bearing, time, and amplitude parameters using the following assumption: A7) The real parameter vector of estimated path amplitudes is obtained by averaging the estimated path amplitudes for the users over time intervals, i.e., as Im (37) is defined in (22). From (37), we can thus derive the for the joint estimation of the bearing, time, and amplitude parameters starting with the FIM expression in (23) and using a transformation of parameters as (38) the th element of the matrix is defined as and can be written as diag (39) otherwise (33) Im Im (34) diag otherwise otherwise (35) 4 Because the derivation of the deterministic assuming unknown deterministic path fading parameters for the above system model is somewhat lengthly and follows the same steps as in, e.g., [19] and [23], a complete derivation is omitted. and denotes the element-wise (Schur-Hadamard) matrix product. D. Asymptotic for the DS-CDMA Signal Parameters Proposition 3: Assuming that the transmitted data symbols are i.i.d. with and that the channel is static (with

7 868 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004 multipath), the asymptotic for the joint estimation of the complex path fading, bearing, and time-delay parameters contained in Im,, is given by As and Im Im Im Im Im Im diag (40) (41) (42) (43) (44) (45) (46) the elements of the matrices and are defined as otherwise (47) (48) otherwise. Proof: See Appendix C. 1) Discussion: We make the following observations. For a finite and sufficiently large observation interval and a static multipath channel, we can approximate the DS-CDMA expression with the asymptotic as As (49) The final expression for the asymptotic DS-CDMA is similar to the conditional (deterministic) expressions derived in both [11, App. A] and [13, App. ]. The expressions in [11] and [13], however, were derived under quite different assumptions. The asymptotic for the th user is independent of the number of asynchronous DS-CDMA users in the system. Because the asymptotic is not conditioned on the transmitted symbol sequence (as opposed to the deterministc ), it is much easier to calculate. If the MAI can be neglected or regarded as white Gaussian noise or a single user is assumed, the conditional (deterministic) derived in [11] and [13] can be used instead of the asymptotic (40) to provide further insights into the effects of a pulse shaping filter and slowly time-varying channel. V. NUMERICAL RESULTS A. Effect of MAI on the Estimation of the Signal Parameters for a K-User DS-CDMA System We use numerical simulations to compare the deterministic K-user (38), the MAI from the users is explicitly modeled with the asymptotic [see (40) and (49)] and the conditional single-user given in [11], [13], assuming a single-user transmitting a digital sequence modulated by a raised cosine pulse with excess bandwidth 0.35 truncated to. The sensor array is an uniform linear array (ULA) with half-wavelength element spacing composed of or three dipole sensors. Each user is assigned a different Gold code sequence of length. The number of multipath rays per user is or 2. Their relative time-delays and bearings are randomly generated for every Monte Carlo simulation as (in radians) and (in chip periods). The received path fading amplitudes have unit amplitude, and their phases are randomly generated for every Monte Carlo trial as (in radians). The SNR is calculated as the ratio of the received power of the first ray to the noise variance, and the noise variance is db. For each experiment, the s for the SOA, AOA, and TD parameters are averaged over the random phase of the amplitudes, random bearings, and random time delays using 100 Monte Carlo simulations. 1) sults: We considered two different experiments. In the first one (see Figs. 1 and 2), we simulated users and plotted the standard deviation of the bearing and time-delay estimates for the first user (i.e, ) as a function of the observation interval and assuming a ULA of sensors. In the second experiment (see Figs. 3 5), we simulated, 5, 10, 15, 20, and 25 users and plotted the quotient of the standard deviation obtained using the K-user deterministic formula with the conditional single-user formula as a function of the number of users. The observation interval was, and the number of antenna elements and multipaths per user was varied from to and to, respectively. We make the following observations. The conditional single-user gives similar results as the K-user deterministic (see Figs. 1 and 2). The smaller variance of the conditional single-user can be attributed to the different assumptions used for its derivation (e.g., the single user assumption, the pulse shaping filter, and the assumption that a channel estimate whose estimation noise is white and Gaussian with variance already exists (see [11] and [13]).

8 BOTTERON et al.: CRAMER RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS 869 Fig. 1. Standard deviation for ^. K =5users, P =3antennas, Q =1 path per user. Fig. 3. Quotient of the standard deviation obtained using the det. K-user formula with the cond. one-user formula. M = 50symbols, P = 1 antenna, Q =1path per user. Fig. 2. Standard deviation for ^. K =5users, P =3antennas, Q =1 path per user. Fig. 4. Quotient of the standard deviation obtained using the det. K-user formula with the cond. one-user formula. M = 50symbols, P = 3 antennas, Q =1path per user. The increase of the variance when users are assumed relative to the variance when a single user is assumed is about the same for the estimation of the bearing, time, and amplitude parameters (see Figs. 3 5). Thus, we can expect that an augmentation of the number of users will similarly affect a radio location technique using any of these measurements. A larger number of multipaths per user also results in an increase of the variance for the estimation of the signal parameters (see Figs. 4 and 5). The trends in performance previously observed in the literature as a function of the observation interval, numbers of users, number of antennas, spreading sequence length, etc. are also verified in our calculations of the s. B. Influence of Specular Multipath on the Radio Location Accuracy We use numerical simulations to investigate the influence of specular multipath on the achievable positioning accuracy for Fig. 5. Quotient of the standard deviation obtained using the det. K-user formula with the cond. one-user formula. M = 50symbols, P = 3 antennas, Q =2paths per user.

9 870 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004 different radio location methods and environments. Since the effects of the number of DS-CDMA users on the estimation variance have already been discussed in Section V-A, we only consider a single user and use the conditional for the signal parameters derived in [13]. Thus, the results we obtain are valid for both a TDMA or a DS-CDMA system the MAI can be modeled as white Gaussian noise. As in Section V-A, we thus assume a single user transmitting a digital sequence modulated by a raised cosine pulse with excess bandwidth 0.35 truncated to. No power control is assumed, the transmitted power is normalized to 30 dbm, and the received noise variance at the FSs is dbm. The number of symbol intervals and code sequence length are and, respectively. The received power at the th FS is calculated using (10) and (11) with db,, and db (corresponding to an urban area, an FS s antenna height of 30 m, and a carrier frequency of MHz [16, Tab. III]). We assume that each FS is equipped with a six-dipole uniform circular array with a interelement spacing. 1) System Geometry and Distance Normalization: Because different positioning methods may not depend on the system geometry the same way, numerical results obtained for a given system geometry may not be valid for another geometry. This is particularly a problem when we want to compare different positioning techniques since the comparison will only be valid for the specific system geometry considered for the comparison. In order to extend the comparison for any system geometry s size, we normalize all the distances in units. Thus, the results we obtain can be generalized for any scaling of the system geometry using Proposition 4. Proposition 4: If the system geometry s distances are expressed in units, the radio location techniques discussed previously will be proportional to no power control (p.c. perfect p.c. no p.c. perfect p.c. no p.c. perfect p.c. is the path loss exponent. Proof: see Appendix D. 2) sults: The system geometry we consider is a five-hexagonal cell geometry with FSs centered in each cell (see Fig. 6). For each experiment, the for the estimation of the MS s position was calculated for 500 Monte Carlo trials, in each trial, a new MS s position and new main reflectors positions were generated. Table II shows the threshold for which the probability that is assuming an uniform distribution of the MS s position within cell and a total of main reflectors uniformly distributed within the five hexagonal cells. Tables III and IV show the threshold for which the probability that is, assuming a uniform distribution of the MS within cell and a total of main reflectors uniformly distributed within a circle of radius and, respectively, and centered on the MS s position. Fig. 6. Five-hexagonal cells geometry. The Cartesian coordinates (b, b ) of the FSs are in D units. TABLE II THRESHOLD X FOR WHICH prob( (w) X) =p ASSUMING V MAIN REFLECTORS UNIFORMLY DISTRIBUTED WITHIN THE HEXAGONAL CELLS. (D IS THE SEPARATION DISTANCE BETWEEN THE FSS IN KILOMETERS, AND T ISTHEPULSE PERIOD IN MICROSECONDS) We make the following observations. From the normalization factors in Tables II IV (see also Proposition 4), we note that the relative contribution to accuracy of time measurements (i.e., TOA or TD) will not be reduced as much as the contribution to accuracy of SOA or AOA measurements when the cells size is increased. As a consequence, we can infer that time measurements will contribute more to the accuracy of radio location estimators than bearing measurements when the cells size becomes larger than a given threshold. Furthermore, we note that that this threshold will be function of the pulse s period and the number of antennas at the FSs (for more details on the influence of the number of antennas and pulse s shaping filter, see [24] and [25]). Comparing the results in Tables II IV, we note that for all the radio location methods considered, the augmentation of the number of main reflectors has a larger effect on the accuracy of radio location estimators when the main reflectors are more closely scattered around the MS, which is consistent with the observation that the delay spread

10 BOTTERON et al.: CRAMER RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS 871 TABLE III THRESHOLD X FOR WHICH prob( (w) X) =p ASSUMING V MAIN REFLECTORS UNIFORMLY DISTRIBUTED AROUND THE MS WITHIN A CIRCLE OF RADIUS D=2. (D ISTHESEPARATION DISTANCE BETWEEN THE FSSINKILOMETERS, AND T ISTHEPULSE PERIOD IN MICROSECONDS) TABLE IV THRESHOLD X FOR WHICH prob( (w) X) =p ASSUMING V MAIN REFLECTORS UNIFORMLY DISTRIBUTED AROUND THE MS WITHIN A CIRCLE OF RADIUS D=5. (D IS THE SEPARATION DISTANCE BETWEEN THE FSSINKILOMETERS, AND T IS THE PULSE PERIOD IN MICROSECONDS) and angle spread of the multipath arrivals at the FSs becomes smaller when the reflectors are more closely scattered around the MS, making the estimation of the signal parameters more difficult. If we only consider the relative contribution to accuracy of the time measurements with reference to the bearing measurements and as a function of the number of main reflectors for the three multipath scenarios we considered (and regardless of the cells size and pulse s period), we make the interesting observation that the contribution to accuracy of the time measurements relative to the bearing measurements increases with the number of main reflectors (see Fig. 7). This increase is also more pronounced when the main reflectors are more closely scattered around the MS. This fact is consistent with the observation that in the presence of closely time-spaced multipath arrivals, it will be easier for a detector to detect the direct path s time of arrival (which will simply correspond in detecting the first time of arrival) than detecting the direct path s AOA (which will be surrounded with the other multipath AOAs). From the results in Tables II IV, we note that the measurements that suffer the most from an augmentation of the number of main reflectors are the SOA measurements. This is despite the fact that for the derivation of the for the SOA estimation, we assumed an unbiased estimator that has perfect knowledge of the path loss model, which is a very strong assumption that will not hold well in practice. We also note that the achievable positioning accuracy of the TOA measurements is not as good as for the TD measurements. This can be easily explained since the MS s TOT is assumed to be known for the TD measurements and unknown for the TOA measurements. VI. CONCLUSION In this paper, we presented the for the estimation of the specular multipath parameters in asynchronous DS-CDMA systems and the for the estimation of the MS s position Fig. 7. Normalized quotient of the Threshold X for the AOA-based over the TD-based positioning method as a function of the number of main reflectors for a probability p =0:67 and p =0:90. valid for multilateral, unilateral, and even composite radio location techniques. Because the for the MS s position can be expressed as a function of the for the estimation of the signal parameters, it can be easily derived for many different system models and provide interesting insights into the physics of the localization problem. It is thus a valuable tool to compare different positioning methods, to assess the effects of different environment or system design parameters, or to evaluate if a given cellular system can fulfill positioning requirements, such as for the E-911 services. One main limitation of our approach comes from the assumption that the matrices for the estimation of the signal parameters at the FSs must exist. A second limitation comes from the unbiasedness assumption, i.e., the for the positions can be used as a benchmark or optimality criterion for any unbiased radio location estimator. However, if the estimator s bias in not negligible compared with its variance, then the will not provide a meaningful bound (note that

11 872 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004 in [24] and [25], we considered another approach to relieve these two limitations). For the numerical simulations we considered, the MAI in a K-user asynchronous DS-CDMA system was shown to affect similarly the estimation of the amplitude, time, and delay parameters, which means that the contribution to accuracy of radio location estimators using any of these measurements would be similarly affected with the augmentation of the number of users. We also investigated, using numerical results, the influence of specular multipath on the contribution to accuracy of different radio location estimators for a five-hexagonal cell geometry and three multipath scenarios. In the first scenario, we assumed an uniform distribution of the main reflectors within every cell, and in the second and third scenarios, we assumed that the main reflectors were scattered around the MS within a circle of radius and, respectively, denotes the distance between two FSs. As expected, the main reflectors had a greater nuisance effect on the achievable accuracy in the last two scenarios. However, we also noticed that the time measurements were not as much affected by an augmentation of the number of main reflectors as were the bearing measurements. Thus, for a radio location system using both time and delay measurements, the contribution to accuracy of the time measurements relative to the bearing measurements will augment with the number of reflectors. That contribution will also be augmented when the main reflectors are more closely scattered around the MS or when the cells size is increased. APPENDIX A PROOF OF PROPOSITION 1 Based on assumption A5, i.e., assuming that the position and noise parameters are independent, we rearrange the elements of the source-location parameter vector as, only contains the parameters for the noise covariance matrix and the parameters for the location parameters in. Thus, we can write, in block partition form, the FIM expression given in (2) as (50) of the matrix (4) as in (3) and using assumption A5, we rewrite (53) diag (54) diag (55) (56) (57) In order to prove Proposition 1, we must prove that (4) is correct. Note that this is equivalent to proving that (51) and (52) can be expressed as and, respectively. The first identity is easy to prove, since the noise parameters in are assumed independent of the signal parameters and statistically independent between different FSs. Without loss of generality, we assume that the parameters in are ordered as in, i.e.,. Thus, can be written as [see (54)], which is the same as, since for [see (56)]. To prove the second identity, we note that the vector in (52) can also be expressed as a function of the parameters in, i.e., as. Thus, by applying the chain rule of differentiation [26, Th. 9.15], we can write the derivatives contained in (52) as (58) tr (51) (52), and denotes the total number of elements in. Inserting the last equation in (52) and noting that the vectors and only contain real parameters, we obtain Since the signal parameter vector must also contain the same noise parameters as in (under the assumption that the functions do not affect the noise parameters), we also rearrange the elements of as, contains the noise s parameters, and contains the signal parameters (plus any other nuisance parameter, such as an MS s TOT if it is assumed unknown at the FSs). arranging in a similar order the elements Since, we also have. Thus, using the fact that the noise at the FSs is Gaussian distributed, the above expression can also be expressed in terms of the FIM

12 BOTTERON et al.: CRAMER RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS 873 expression for the estimation of the signal parameters contained in, i.e., as In order to obtain a simple expression for the for, we rearrange the elements in the vector as (63) (59), so that we can write the transforma- in block diagonal form as tion matrix Because the noise is also assumed statistically independent between different FSs, we can write as diag. Finally, by noting that [see (57)], we obtain diag (60) (64) the elements of are defined in (8). By also rearranging the rows and the columns of the matrix according to, we can express the FIM for [given as the inverse of (62)] as which completes the proof. APPENDIX B PROOF OF PROPOSITION 2 Since by assumption the position estimator only uses the subset of signal parameters to estimate the position parameters contained in, the for can be obtained by assuming that the remaining parameters in, which are not included in, are all nuisance parameters, i.e., by assuming that they provide no information to estimate and are thus independent of the parameters contained in. Let us denote this subset of nuisance parameters as,. Because these nuisance parameters are assumed to be independent of, no longer contains all the necessary parameters to completely parameterize the p.d.f. for the observation vector. However, if we define a vector as (61) then can be used to parameterize the p.d.f. for the observation vector. 5 Based on this new parametrization of the p.d.f for, we note that Proposition 1 still holds and can thus be used to find the matrix for (which also contains the for the parameters in )as (62) the elements of the matrix. are defined as 5 For example, if we consider a location system that only uses the bearing parameters in to estimate the source-location parameters contained in w, then will contain the remaining signal parameters in that are not used to estimate w (such as the time- and the strength-of-arrival parameters). Since these parameters are assumed independent of the location parameters in w, they cannot be expressed as a function of w and, therefore, must also be added to so that completely parameterizes the p.d.f. for the observation vector x. (65) The for can now be obtained by taking the inverse of the Schur complement corresponding to of (65), i.e., as (66) denotes the right lower block of a 2 2 block partition matrix, and the last equality was obtained by noting that is the matrix for the estimation of the signal parameters contained in. APPENDIX C PROOF OF PROPOSITION 3 For a static channel, we have. Thus, we can write the asymptotic FIM for Im as As Im Im Im Im Im Im (67)

13 874 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004 Im Im Im We start with the evaluation of. From the block vertical form of [see (32)] and the block diagonal form of, we rewrite as. Im..... Thus, the th block matrix of is. Because the transmitted bits are assumed i.i.d. with, it is easy to show that the th element of can be written as By noting that the matrices and have the same form as, and proceeding in the same manner, it is easy to verify that the th block matrices of the other elements of the asymptotic FIM can be written as in (42) (46). Finally, by noting that all the elements of the asymptotic FIM given by (67) are block diagonal, we conclude that by rearranging the elements in, we can write the asymptotic for the estimation of the signal parameters of the users as As diag As As, As is given in (40). APPENDIX D PROOF OF PROPOSITION 4 First, we note from the FIM expression (23) that the for the estimation of the bearing and time parameters is inversely proportional to the square of the received power, while it is independent of the received power for the estimation of the path amplitudes [this is easily verified by noting that in (23) is independent of, as and are proportional to. Thus, by applying the formula for the inverse of a partitioned matrix, we obtain that, as Im Im is independent of. Finally, using (38), we obtain that is also independent of the received powers]. Second, we note from the path loss formulas (10) (12) that the received power for any multipath at any FS will be proportional to only in the absence of power control, as it will be independent of, assuming perfect power control. Third, we note from (13) (16) that the transformation matrices for the time measurements are independent of, as the transformation matrix for the bearing measurements is inversely proportional to, and the one for the amplitude measurements is also inversely proportional to, assuming perfect power control, and inversely proportional to in the absence of power control (i.e., with fixed transmitted power). Thus, inserting the above proportionalities in the formula for the estimation of the MS s position [see (7)] concludes the proof. ACKNOWLEDGMENT The authors are grateful to the anonymous reviewers for their valuable comments. Consequently, is a -block diagonal matrix. By noting that the th element of is and the th element of is,we can rewrite the th block matrix of as. (68) REFERENCES [1] FCC acts to promote competition and public safety in enhanced wireless 911 services, FCC News: CC Docket , Sept. 15, [2] H. Koshima and J. Hoshen, Personal locator services emerge, IEEE Spectrum, vol. 37, pp , Feb [3] T. S. Rappaport, J. H. ed, and B. D. Woerner, Position location using wireless communications on highways of the future, IEEE Commun. Mag., vol. 34, pp , Oct [4] Y. T. Chan and K. C. Ho, A simple and efficient estimator for hyperbolic location, IEEE Trans. Signal Processing, vol. 42, pp , Aug [5] C. W. ed, R. Hudson, and K. Yao, Direct joint source localization and propagation speed estimation, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 3, 1999, pp

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Acoust., Speech, Signal Processing, vol. 37, pp , May [24] C. Botteron, A. Høst-Madsen, and M. Fattouche, Statistical theory of the effects of radio location system design parameters on the positioning performance, in Proc. IEEE 56th Veh. Technol. Conf., vol. 2, Sept. 2002, pp [25], Effects of system and environment parameters on the performance of network based mobile station position estimators, IEEE Trans. Veh. Technol., vol. 53, pp , Jan [26] W. Rudin, Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, Cyril Botteron (M 99) was born in Switzerland in He received the Dipl.-Ing. degree from the University of Applied Sciences, Le Locle, Switzerland, in 1991 and the Ph.D. degree in electrical engineering from the University of Calgary, Calgary, AB, Canada, in In 2000, he participated in the high-level design of a network-based positioning system with Cell-Loc, Inc., Calgary. Currently, he is a research scientist and group leader with the Institute of Microtechnology, University of Neuchâtel, Neuchâtel, Switzerland. His research interests include statistical and discrete-time signal processing techniques, RF systems design, and applications to wireless communications, including ultra-wideband radio technology and global navigation satellite systems. Anders Høst-Madsen (M 95 SM 02) was born in Denmark in He received the M.Sc. degree in electrical engineering in 1990 and the Ph.D. degree in mathematics in 1993, both from the Technical University of Denmark, Lyngby. From 1993 to 1996, he was with Dantec Measurement Technology A/S, Copenhagen, Denmark, from 1996 to 1998, he was an assistant professor at Kwangju Institute of Science and Technology, Kwangju, Korea, and from 1998 to 2000, he was an assistant professor at the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada, and a staff scientist at TRLabs, Calgary. In 2001, he joined the Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, as an assistant professor. He was also a visitor at the Department of Mathematics, University of California, Berkeley, in His research interests are in statistical signal processing, information theory, and wireless communications, including multiuser detection, equalization, and ad-hoc networks. Dr. Høst-Madsen currently serves as Editor for multiuser communications for the IEEE TRANSACTIONS ON COMMUNICATIONS. Michel Fattouche (M 82) received the M.Sc. and Ph.D. degrees in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 1982 and 1986, respectively. He is chief technical officer of Cell-Loc, Calgary, AB, Canada, and a tenured professor with the Department of Electrical and Computer Engineering, University of Calgary, he has taught and conducted research since He is currently on a leave of absence to allow him more time to dedicate to technology development at Cell-Loc. He has been affiliated with TR Labs, Calgary, since 1989, he is currently an adjunct professor, and he is on the board of directors of PsiNaptic Communications, Calgary. He currently holds nine U.S. patents and has ten additional patents pending. He has been published in a number of well-respected publications in the field of digital wireless communications and has spoken at several industry events. Dr. Fattouche is a member of the Association of Professional Engineers, Geologists, and Geophysicists of Alberta. He was named Prairies Entrepreneur of the Year 2000 for Communications and Technology as part of Ernst and Young s Entrepreneur of the Year (EOY) Program.