Contributed Review: Source-localization algorithms and applications using time of arrival and time difference of arrival measurements

The general 3D range equations for source localization using TOA and TDOA are

$${\displaystyle \begin{array}{l}TOA:\hfill \\ s{t}_i={\left[{(x-x_i)}^2+{(y-y_i)}^2+{(z-z_i)}^2\right]}^{1/2},\hfill \\ TDOA:\hfill \\ s\mathrm{\Delta }{t}_{ij}={\left[{(x-x_i)}^2+{(y-y_i)}^2+{(z-z_i)}^2\right]}^{1/2}\hfill \\ -{\left[{(x-x_j)}^2+{(y-y_j)}^2+{(z-z_j)}^2\right]}^{1/2},\hfill \\ i,j=1,\dots ,N,\hfill \end{array}}$$

(2)

where s is the signal propagation velocity, t_i is the signal traveling time from the source to sensor i, and Δt_ij indicates the time difference between travel times t_i and t_j. The terms x_i, y_i, z_i represent the position of sensor i, and N is the number of sensors (N ≥ 4). The terms x, y, z are the coordinates for the position of the source that are to be determined. Nonlinearity of the source localization problem is introduced into the algorithm by the square-root terms (distance formula) in Equations (2). These make estimation of a source’s location potentially complex and expensive.

To obtain the measurements of TOAs and TDOAs, a group of sensors first receives signals emitted from the source. The sources and receivers must be time-synchronized to ensure precision of the arrival times. In a passive system, the TDOA measurement can be made by cross-correlating the signals received at two different sensors. Alternatively, TOA measurements can be gathered at the sensors and converted into TDOA measurements by calculating the differences.

In geometric positioning, TOAs are then converted to range estimates by multiplying the propagation speed of light or sound in the appropriate medium. For each TOA, if accurately measured and assumed to be free of noise, the range estimate to the source results in a circular locus of possible source positions with the receiver at the center. This is commonly called the circular line of position (LOP). In a 2D space, at least three such LOPs are needed to estimate the source’s location. The position of the source is at a single point where all three LOPs intersect (Figure 1, solid line). In practical situations, TOAs have measurement errors and the circular LOPs may not have a unique intersection point. The trilateration will yield multiple intersections bounded by the errors in the TOA measurements (Figure 1, dashed line). The estimate of source position then is obtained by solving for an optimal solution. In a 3D space for TOAs measured without error, the source location locus for each TOA is the surface of a sphere. Three TOA sphere surfaces intersect at two points, so a fourth TOA is needed to determine the position of the source. TOAs can also be used to estimate TDOAs (Equation (2)), which are source range-difference estimates. In the 2D case, the possible location of a source for each TDOA is given by a hyperbolic LOP in which the focal points of the hyperbola are the positions of the two receivers used in the TDOA computation. At least two hyperbolas (Figure 2, solid line) formed using two TDOAs computed from TOAs for three receivers are needed to find the intersections of the two hyperbolas, the potential locations of the signal source. Because TOAs and therefore TDOAs have measurement errors, the location of a source may be estimated by propagating the errors through the computation and estimating the source location along with the errors in the estimates (Figure 2, dashed line). In the 3D case, a hyperboloid is defined by each TDOA, and at least three TDOAs need to intersect at a unique point to identify a source location. Intersection of LOPs, a geometric construct, is the most basic and intuitive method for source position estimation. Starting from this idea, numerous methods have been developed, validated, and published in the literature. The most used TOA- and TDOA-based source-localization algorithms when LOS between receivers and sources exist are summarized in Table III.

Direct solutions may be used when TOA measurement errors are not considered. The source position can be directly calculated from different sets of geometric equations (Schmidt, 1972Schmidt (1972) Schmidt, R. O., “A new approach to geometry of range difference location,” IEEE Trans. Aerosp. Electron. Syst. 8, 821–835 (1972). https://doi.org/10.1109/TAES.1972.309614; Fang, 1990Fang (1990) Fang, B. T., “Simple solutions for hyperbolic and related position fixes,” IEEE Trans. Aerosp. Electron. Syst. 26, 748–753 (1990). https://doi.org/10.1109/7.102710; and Caffery, 2000Caffery (2000) Caffery, Jr., J. J., “A new approach to the geometry of TOA location,” in IEEE Vehicular Technology Conference (IEEE, Boston, 2000).). The computational complexity of this approach to source localization is low, but frequently does not result in a source location solution when TOA measurement errors exist. Furthermore, when additional LOPs are available in an ODS, the source position needs to be estimated by averaging all the possible source locations (LOP intersections). A linear least-squares (LS) approach is an alternative source location estimation technique (Caffery, 2000Caffery (2000) Caffery, Jr., J. J., “A new approach to the geometry of TOA location,” in IEEE Vehicular Technology Conference (IEEE, Boston, 2000).) that can provide a global solution using simple computational steps for situations when only the minimum number of sensors (MRS) is available. It can also be used in the ODS case. However, it is a suboptimal technique because, by assuming sufficiently small measurement errors, the source location position estimate accuracy is not high. As a result, other types of approaches have been developed to improve source location estimate accuracy when there is error in TOA measurements. Measurement errors are usually assumed to be independent, zero-mean Gaussian variables with the same variance for all receivers in an LOS environment. Weighted LS estimators are computed by introducing a weight matrix into the LS cost function. A favorite LS weight matrix is the inverse of the covariance matrix of TOA measurement errors (Chan et al., 2006aChan et al. (2006a) Chan, Y. T., Hang, H. Y. C., and Ching, P. C., “Exact and approximate maximum likelihood localization algorithms,” IEEE Trans. Veh. Technol. 55, 10–16 (2006a). https://doi.org/10.1109/TVT.2005.861162).

The general LS cost function for TOA measurements and TDOA measurements are

$${\displaystyle J_{\text{TOA}}=\sum _{i=1}^N{(t_i\text{s}-||\tilde{\mathbf{x}}-{\mathbf{x}}_{\mathit{i}}|\left|\right)}^2,}$$

(3)

$${\displaystyle J_{\text{TDOA}}=\sum _{i=2}^N{((t_i-t_1)\text{s}-||\tilde{\mathbf{x}}-{\mathbf{x}}_{\mathit{i}}||+||\tilde{\mathbf{x}}-{\mathbf{x}}_1|\left|\right)}^2.}$$

(4)

The general maximum likelihood (ML) cost function for TOA measurements and TDOA measurements are

$${\displaystyle J_{\text{TOA}}=\sum _{i=1}^N\frac{{(t_i\text{s}-||\tilde{\mathbf{x}}-{\mathbf{x}}_{\mathit{i}}|\left|\right)}^2}{{\sigma }_i^2},}$$

(5)

$${\displaystyle J_{\text{TDOA}}=\sum _{i=2}^N\frac{{((t_i-t_1)\text{s}-||\tilde{\mathbf{x}}-{\mathbf{x}}_{\mathit{i}}||+||\tilde{\mathbf{x}}-{\mathbf{x}}_1|\left|\right)}^2}{{\sigma }_{d,\hspace{0.17em}i}^2},}$$

(6)

where t_i is the measured TOA at sensor i with or without measurement errors, x_i is the coordinate vector of sensor i, ${\sigma }_i^2$ is the variance of TOA measurements at sensor i, and ${\sigma }_{d,i}^2$ is the variance of TDOA measurement at sensor i. The objective of the algorithm is to estimate the source coordinate vector $\tilde{\mathbf{x}}$ that minimizes the cost function J. Iterative nonlinear minimization is required for an optimal solution. A common solution technique is to create an iterative algorithm based on an initial position estimate obtained using the Gauss-Newton method, the steepest descent method, or the combined Levenberg-Marquardt method, which typically have high computation requirements. A good initial source position estimate also is needed to find the global minimum. Therefore, a Taylor series (TS) is often used to linearize the nonlinear equations by updating a LS solution (Foy, 1976Foy (1976) Foy, W. H., “Position-location solutions by Taylor-series estimation,” IEEE Trans. Aerosp. Electron. Syst. 12, 187–194 (1976). https://doi.org/10.1109/TAES.1976.308294 and Torrieri, 1984Torrieri (1984) Torrieri, D. J., “Statistical theory of passive location systems,” IEEE Trans. Aerosp. Electron. Syst. 20, 183–198 (1984). https://doi.org/10.1109/TAES.1984.310439). The Taylor series methods refine the initial guess by iterating a procedure determining location estimation errors. Compared with LS methods that ignore measurement errors, TS methods can provide accurate location estimation at reasonable noise levels. However, they still require a fairly accurate initial estimate of source location, and convergence on a source location estimate is not guaranteed. The ML approach estimates source position by minimizing the cost function of the probability density function of measurements (Ziskind and Wax, 1988Ziskind and Wax (1988) Ziskind, I. and Wax, M., “Maximum likelihood localization of multiple sources by alternating projection,” IEEE Trans. Acoust. Speech 36, 1553–1560 (1988). https://doi.org/10.1109/29.7543). This algorithm is similar to a weighted nonlinear LS approach when measurement errors are zero-mean Gaussian distributed (Chan et al., 2006aChan et al. (2006a) Chan, Y. T., Hang, H. Y. C., and Ching, P. C., “Exact and approximate maximum likelihood localization algorithms,” IEEE Trans. Veh. Technol. 55, 10–16 (2006a). https://doi.org/10.1109/TVT.2005.861162). Li et al. (2014)Li et al. (2014) Li, X., Deng, Z. D., Sun, Y., Martinez, J. J., Fu, T., McMichael, G. A., and Carlson, T. J., “A 3D approximate maximum likelihood solver for localization of fish implanted with acoustic transmitters,” Sci. Rep. 4, 7125 (2014). https://doi.org/10.1038/srep07215 improved on an algorithm published by Chan et al. (2006b)Chan et al. (2006b) Chan, Y. T., Tsui, W. Y., So, H. C., and Ching, P. C., “Time-of-arrival based localization under NLOS conditions,” IEEE Trans. Veh. Technol. 55, 17–24 (2006b). https://doi.org/10.1109/TVT.2005.861207 using an efficient approximate ML algorithm that included coupling with bad-measurement filters for 3D source localization.

To avoid iterative algorithms, two-stage, closed-form LS estimators have been extensively developed for ML approximation (Friedlander, 1987Friedlander (1987) Friedlander, B., “A passive localization algorithm and its accuracy analysis,” IEEE J. Oceanic Eng. 12, 234–245 (1987). https://doi.org/10.1109/JOE.1987.1145216; Schau and Robinson, 1987Schau and Robinson (1987) Schau, H. C. and Robinson, A. Z., “Passive source localization employing intersecting spherical surfaces from time-of-arrival differences,” IEEE Trans. Acoust. Speech 35, 1223–1225 (1987). https://doi.org/10.1109/TASSP.1987.1165266; Smith and Abel, 1987aSmith and Abel (1987a) Smith, J. O. and Abel, J. S., “Closed-form least-squares source location estimation from range-difference measurements,” IEEE Trans. Acoust. Speech 35, 1661–1669 (1987a). https://doi.org/10.1109/TASSP.1987.1165089; Smith and Abel, 1987bSmith and Abel (1987b) Smith, J. O. and Abel, J. S., “The spherical interpolation method of source localization,” IEEE J. Oceanic Eng. 12, 246–252 (1987b). https://doi.org/10.1109/JOE.1987.1145217; Chan and Ho, 1994Chan and Ho (1994) Chan, Y. T. and Ho, K. C., “A simple and efficient estimator for hyperbolic location,” IEEE Trans. Signal Process. 42, 1905–1915 (1994). https://doi.org/10.1109/78.301830; Brandstein and Silverman, 1997Brandstein and Silverman (1997) Brandstein, M. S. and Silverman, H. F., “A practical methodology for speech source localization with microphone arrays,” Comput. Speech Lang. 11, 91–126 (1997). https://doi.org/10.1006/csla.1996.0024; Huang et al., 2001Huang et al. (2001) Huang, Y., Benesty, J., Elko, G. W., and Mersereau, R. M., “Real-time passive source localization: A practical linear-correction least-squares approach,” IEEE Trans. Speech Audio Process. 9, 943–956 (2001). https://doi.org/10.1109/89.966097; and Cheung et al., 2004Cheung et al. (2004) Cheung, K. W., So, H. C., Ma, W. K., and Chan, Y. T., “Least squares algorithms for time-of-arrival-based mobile location,” IEEE Trans. Signal Process. 52, 1121–1130 (2004). https://doi.org/10.1109/TSP.2004.823465). These LS solutions can provide good initialization for iterative estimators, which converge with less computational effort to a source position estimate with higher accuracy (Smith and Abel, 1987aSmith and Abel (1987a) Smith, J. O. and Abel, J. S., “Closed-form least-squares source location estimation from range-difference measurements,” IEEE Trans. Acoust. Speech 35, 1661–1669 (1987a). https://doi.org/10.1109/TASSP.1987.1165089 and Chan et al., 2006aChan et al. (2006a) Chan, Y. T., Hang, H. Y. C., and Ching, P. C., “Exact and approximate maximum likelihood localization algorithms,” IEEE Trans. Veh. Technol. 55, 10–16 (2006a). https://doi.org/10.1109/TVT.2005.861162). Some researchers have compared the performance of algorithms (Yu and Oppermann, 2004Yu and Oppermann (2004) Yu, K. and Oppermann, I., “Performance of UWB position estimation based on time-of-arrival measurements,” in Proceedings of the International Workshop on Ultra Wideband Systems. Joint with Conference on Ultra Wideband Systems and Technologies (IEEE, Kyoto, 2004), pp. 400–404.; Shen et al., 2008Shen et al. (2008) Shen, G., Zetik, R., and Thomä, R. S., “Performance comparison of TOA and TDOA based location estimation algorithms in LOS environment,” in Proceedings of the 5th Workshop on Positioning, Navigation and Communication (IEEE, Hannover, Germany, 2008), pp. 71–78. https://doi.org/10.1109/WPNC.2008.4510359; Gezici et al., 2008Gezici et al. (2008) Gezici, S., Güvenç, I., and Sahinoglu, Z., “On the performance of linear least-squares estimation in wireless positioning systems,” in Proceedings of the IEEE International Conference on Communications (IEEE, 2008), pp. 4203–4208.; and So, 2011So (2011) So, H. C., “Source localization: Algorithms and analysis,” in Handbook of Position Location: Theory, Practice, and Advances, edited byZekavat, S. and Buehrer, R. (John Wiley & Sons, 2011).). An LS methodology that used squared range or squared range-difference measurements (Beck et al., 2008Beck et al. (2008) Beck, A., Stoica, P., and Li, J., “Exact and approximate solutions of source localization problems,” IEEE Trans. Signal Process. 56, 1770–1778 (2008). https://doi.org/10.1109/TSP.2007.909342) is an attractive approach in that the method performs very well even at high noise levels.

Because real environments are complex and the geometries of sources and sensors can be quite variable, algorithms have been developed for special cases. For instance, estimation of the location of a source under NLOS conditions is one of the main challenges in localization. Güvenç and Chong (2009)Güvenç and Chong (2009) Güvenç, İ. and Chong, C. C., “A survey on TOA based wireless localization and NLOS mitigation techniques,” IEEE Commun. Sur. Tutorials 107–124 (2009). https://doi.org/10.1109/surv.2009.090308 presented an overview of different TOA NLOS localization algorithms with varying levels of computational complexity and prior information. Early researchers considered treatment methods that identify the NLOS sensors in an array using TOA measurements, then estimate the location of the source without using the NLOS sensors, or reduce the error in position estimates by weighting the NLOS less (Wylie and Holtzman, 1996Wylie and Holtzman (1996) Wylie, M. P. and Holtzman, J., “The non-line of sight problem in mobile location estimation,” in Proceedings of the 5th IEEE International Conference on Universal Personal Communications (IEEE, Cambridge, MA, USA, 1996), pp. 827–831. https://doi.org/10.1109/ICUPC.1996.562692; Chen, 1999Chen (1999) Chen, P. C., “A non-line-of-sight error mitigation algorithm in location estimation,” in Proceedings of the IEEE Wireless Communications and Networking Conference (IEEE, New Orleans, 1999), pp. 316–320.; and Chan et al., 2006bChan et al. (2006b) Chan, Y. T., Tsui, W. Y., So, H. C., and Ching, P. C., “Time-of-arrival based localization under NLOS conditions,” IEEE Trans. Veh. Technol. 55, 17–24 (2006b). https://doi.org/10.1109/TVT.2005.861207). This approach is effective when a small number of the distributed sensors are NLOS, but become computationally intensive when there are many distributed sensors and may not be possible in situations where the signal source is moving. Because NLOS-caused errors are always positive, this constraint can be imposed on the search for a position estimate (Wang et al., 2003Wang et al. (2003) Wang, X., Wang, Z., and O’Dea, B., “A TOA-based location algorithm reducing the errors due to non-line-of-sight (NLOS) propagation,” IEEE Trans. Veh. Technol. 52, 112–116 (2003). https://doi.org/10.1109/tvt.2002.807158; Venkatraman et al., 2004Venkatraman et al. (2004) Venkatraman, S., Caffery, Jr., J. J., and You, H. R., “A novel TOA location algorithm using LOS range estimation for NLOS environments,” IEEE Trans. Veh. Technol. 53, 1515–1524 (2004). https://doi.org/10.1109/TVT.2004.832384; and Cong and Zhuang, 2005Cong and Zhuang (2005) Cong, L. and Zhuang, W., “Nonline-of-sight error mitigation in mobile location,” IEEE Trans. Wireless Commun. 4, 560–573 (2005). https://doi.org/10.1109/twc.2004.843040), but normally, prior knowledge of NLOS error statistics is required to correct a LOS estimate. The procedure for estimating a source location when some distributed sensors are NLOS is more complicated using TDOAs because the NLOS error for a reference sensor will be transferred to all TDOAs computed using that sensor. Additional research is needed to learn how to mitigate NLOS errors effectively and efficiently under real world conditions with moving sources and large receiving networks.

Proficiency in estimating the locations of fixed sources is becoming less important as interest in detecting, locating, and tracking mobile sources has increased. Frequency difference of arrival (FDOA) measurements with independent sources of errors are combined with TDOAs to immediately estimate the location and velocity of a source (Ho and Chan, 1997Ho and Chan (1997) Ho, K. C. and Chan, Y. T., “Geolocation of a known altitude object from TDOA and FDOA measurements,” IEEE Trans. Aerosp. Electron. Syst. 33, 770–783 (1997). https://doi.org/10.1109/7.599239; Ho and Xu, 2004Ho and Xu (2004) Ho, K. C. and Xu, W., “An accurate algebraic solution for moving source location using TDOA and FDOA measurements,” IEEE Trans. Signal Process. 52, 2453–2463 (2004). https://doi.org/10.1109/TSP.2004.831921; and Anderson et al., 2005Anderson et al. (2005) Anderson, R. J., Rogers, A. E., and Stilp, L. A., “Method for estimating TDOA and FDOA in a wireless location system,” United States Patent 09/908998 (2005).). Kalman filter (Leonard and Durrant-Whyte, 1991Leonard and Durrant-Whyte (1991) Leonard, J. J. and Durrant-Whyte, H. F., “Mobile localization by tracking geometrical beacons,” IEEE Trans. Rob. Autom. 7, 376–382 (1991). https://doi.org/10.1109/70.88147) or Monte Carlo approaches (Sheng et al., 2005Sheng et al. (2005) Sheng, X., Hu, Y. H., and Ramanathan, P., “Distributed particle filter with GMM approximation for multiple targets localization and tracking in wireless sensor network,” in Proceedings of the 4th International Symposium on Information Processing in Sensor Networks (IEEE Press, Los Angeles, 2005), pp. 24–31.) can be integrated into tracking algorithms for mobile source localization to improve their performance. These additions can detect discontinuities from continuously estimated positions and smooth the source’s trajectory by replacing the position estimate outliers with predicted or refined estimates. Kalman filters, which are optimal recursive Bayesian estimators for linear Gaussian problems, are the best-known filters. Extended (Welch and Bishop, 2006Welch and Bishop (2006) Welch, G. and Bishop, G., “An introduction to the Kalman filter,” in Report TR 95–041 (Department of Computer Science, University of North Carolina, 2006).) and unscented Kalman filters (Julier and Uhlmann, 1997Julier and Uhlmann (1997) Julier, S. J. and Uhlmann, J. K., “New extension of the Kalman filter to nonlinear systems,” Proc. SPIE 3068, 182–193 (1997). https://doi.org/10.1117/12.280797) are examples of nonlinear filtering by linearization. Particle filters are the most general class of filters for nonlinear and non-Gaussian problems (i.e., sequential Monte Carlo) (Gordon et al., 1993Gordon et al. (1993) Gordon, N. J., Salmond, D. J., and Smith, A. F., “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” IEE Proc. F Radar Signal Process. 140, 107–113 (1993). https://doi.org/10.1049/ip-f-2.1993.0015; Kitagawa, 1996Kitagawa (1996) Kitagawa, G., “Monte Carlo filter and smoother for non-Gaussian nonlinear state space models,” J Comp Graph Stat 5, 1–25 (1996). https://doi.org/10.1080/10618600.1996.10474692; and Ristic et al., 2004Ristic et al. (2004) Ristic, B., Arulampalam, S., and Gordon, N. J., Beyond the Kalman Filter: Particle Filters for Tracking Applications (Artech House, 2004).). Computational cost is a major disadvantage of Monte Carlo estimators. The novel challenge of estimating the locations of mobile sources using mobile sensors is now being investigated (Ho and Xu, 2004Ho and Xu (2004) Ho, K. C. and Xu, W., “An accurate algebraic solution for moving source location using TDOA and FDOA measurements,” IEEE Trans. Signal Process. 52, 2453–2463 (2004). https://doi.org/10.1109/TSP.2004.831921 and Hu and Evans, 2004Hu and Evans (2004) Hu, L. and Evans, D., “Localization for mobile sensor networks,” in Proceedings of the 10th Annual International Conference on Mobile Computing and Networking (ACM, Philadelphia, 2004), pp. 45–57.). Optimal design of a sensor system (i.e., network, array) is a challenging task that can significantly affect tracking accuracy.

3D localization and tracking requires at least four different sensors to form the necessary nonlinear localization equations. From Equation (2), the precision of the estimate of a source’s location can be estimated as a function of the errors in the measurements of sensor locations, TOA/TDOA, and signal velocity. Under NLOS conditions, an additional term for the distance bias caused by the blockage of the direct path between source and sensor should be included in the localization equations.

In many cases, accurate estimates of sensor locations can be obtained from careful field surveys and highly accurate GPS location estimates. In homogeneous media, the speed of a signal can be accurately estimated by using known relationships between the characteristics of the transmitted signal (e.g., sound or radio) and of the medium through which the signal will propagate (e.g., air or water). For instance, a fifth-order polynomial can accurately describe the dependence of sound speed in water on temperature (Marczak, 1997Marczak (1997) Marczak, W., “Water as a standard in the measurements of speed of sound in liquids,” J. Acoust. Soc. Am. 102, 2776–2779 (1997). https://doi.org/10.1121/1.420332). In heterogeneous media, where the signal speed differs from path to path, “isodiachron” algorithms can improve the accuracy of location estimates (Spiesberger, 2004Spiesberger (2004) Spiesberger, J. L., “Geometry of locating sounds from differences in travel time: Isodiachrons,” J. Acoust. Soc. Am. 116, 3168–3177 (2004). https://doi.org/10.1121/1.1804625).

However, the errors in TOA and TDOA measurements can be quite complicated, depending on multiple factors (Mao et al., 2007Mao et al. (2007) Mao, G., Fidan, B., and Anderson, B. D. O., “Wireless sensor network localization techniques,” Comput. Networks 51, 2529–2553 (2007). https://doi.org/10.1016/j.comnet.2006.11.018), such as SNR, integration time, signal bandwidth, multipath propagation, and possible NLOS propagation. In practical implementations, hardware limitations and transmission channel degradation may become the dominant sources of error (Krizman et al., 1997Krizman et al. (1997) Krizman, K. J., Biedka, T. E., and Rappaport, S., “Wireless position location: Fundamentals, implementation strategies, and sources of error,” in Proceedings of the IEEE Vehicular Technology Conference (IEEE, Phoenix, AZ, 1997), pp. 919–923. and Caffery and Stuber, 1998Caffery and Stuber (1998) Caffery, Jr., J. J. and Stuber, G. L., “Overview of radiolocation in CDMA cellular systems,” IEEE Commun. Mag. 36, 38–45 (1998). https://doi.org/10.1109/35.667411). Considering hardware limitations, TDOA measurements require highly precise synchronization between sensors, whereas TOA measurements require accurate synchronization between a transmitter and sensor. Such hardware requirements can greatly increase the complexity and cost of localization systems. Minimizing the errors in time of arrival and other time-based measurements is the greatest challenge in many source localization studies because of the large uncertainties in such measurements. For example, ultra-wide-band signals are used for wireless positioning (Gezici et al., 2005Gezici et al. (2005) Gezici, S., Tian, Z., Giannakis, G. B., Kobayashi, H., Molisch, A. F., Poor, H. V., and Sahinoglu, Z., “Localization via ultra-wideband radios: A look at positioning aspects for future sensor networks,” IEEE Signal Process. Mag. 22, 70–84 (2005). https://doi.org/10.1109/MSP.2005.1458289) because of their time domain high-resolution capability.

Benchmarks are needed to assess the performance of localization estimators. The Cramér-Rao Lower Bound (CRLB), which gives the minimum variance of unbiased estimators, is widely used as a measure of the precision attainable for parameter estimates from a given set of observations (Van Trees et al., 1968Van Trees et al. (1968) Van Trees, H. L., Snyder, D., Baggeroer, A. B., Cahn, M., Cruise, T., Mohajeri, M., Collins, L., Eckberg, Jr., A., Wert, E., and Kurth, R., “Detection and estimation theory,” in Report QPR No. 90, Research Laboratory of Electronics (RLE) at the Massachusetts Institute of Technology (MIT) (1968). and Kay, 1993Kay (1993) Kay, S. M., Fundamentals of Statistical Signal Processing (PTR Prentice-Hall, 1993).). The CRLB was compared with the mean square errors of different localization algorithms under low SNR conditions (Gustafsson and Gunnarsson, 2005Gustafsson and Gunnarsson (2005) Gustafsson, F. and Gunnarsson, F., “Mobile positioning using wireless networks: Possibilities and fundamental limitations based on available wireless network measurements,” IEEE Signal Process. Mag. 22, 41–53 (2005). https://doi.org/10.1109/MSP.2005.1458284 and Macagnano et al., 2012Macagnano et al. (2012) Macagnano, D., Destino, G., and Abreu, G., “A comprehensive tutorial on localization: Algorithms and performance analysis tools,” Int. J. Wireless Inf. Networks 19, 290–314 (2012). https://doi.org/10.1007/s10776-012-0190-4). Using TOA and TDOA measurements, So (2011)So (2011) So, H. C., “Source localization: Algorithms and analysis,” in Handbook of Position Location: Theory, Practice, and Advances, edited byZekavat, S. and Buehrer, R. (John Wiley & Sons, 2011). used the corresponding Fisher information matrix with zero-mean Gaussian distributed measurement errors to compute CRLB in LOS scenarios. An example of 2D localization for TOA measurements is

$${\displaystyle \mathit{CRLB}\left(\mathbf{x}\right)={\left[I^{-1}\left(\mathbf{x}\right)\right]}_{1,1}+{\left[I^{-1}\left(\mathbf{x}\right)\right]}_{2,2},I\left(\mathbf{x}\right)=\left[\begin{array}{cc}\hfill \sum _{i=1}^N\frac{{\displaystyle {(x-x_i)}^2}}{{\displaystyle {\sigma }_i^2{d}_i^2}}\hfill & \hfill \sum _{i=1}^N\frac{(x-x_i)(y-y_i)}{{\sigma }_i^2{d}_i^2}\hfill \\ \hfill \sum _{i=1}^N\frac{(x-x_i)(y-y_i)}{{\sigma }_i^2{d}_i^2}\hfill & \hfill \sum _{i=1}^N\frac{{(y-y_i)}^2}{{\sigma }_i^2{d}_i^2}\hfill \end{array}\right],d_i=\left|\right|\mathbf{x}-{\mathbf{x}}_{\mathit{i}}\left|\right|,\mathbf{x}={[x,y]}^{\mathit{T}}.}$$

(7)

In NLOS scenarios, the CRLB depends on LOS signals, assuming that the NLOS sensors can be accurately identified (Güvenç and Chong, 2009Güvenç and Chong (2009) Güvenç, İ. and Chong, C. C., “A survey on TOA based wireless localization and NLOS mitigation techniques,” IEEE Commun. Sur. Tutorials 107–124 (2009). https://doi.org/10.1109/surv.2009.090308). Qi and Kobayashi (2002)Qi and Kobayashi (2002) Qi, Y. and Kobayashi, H., “Cramér-Rao lower bound for geolocation in non-line-of-sight environment,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, Orlando, 2002), pp. 2473–2476. derived an explicit formulation of the CRLB for NLOS geolocation. CRLBs for the TOA and TDOA based source localization estimates are determined by (1) the positions of the sensors, (2) the positions of the source, and (3) variances of measurement noise, σ². Independent of σ², the CRLB indicates that achievable localization accuracy is related to the geometry of the distributed sensors relative to the source locations.

Another approach to estimate source position accuracy is to directly compute position errors using distance equations, which does not require expensive Monte-Carlo simulations. This type of theoretical error analysis was performed on 2D MRS systems and compared with field measured positions (Smith et al., 1998Smith et al. (1998) Smith, G., Urquhart, G., Maclennan, D., and Sarno, B., “A comparison of theoretical estimates of the errors associated with ultrasonic tracking using a fixed hydrophone array and field measurements,” Hydrobiologia 371/372, 9–17 (1998). https://doi.org/10.1023/A:1017039325661). Wahlberg et al. (2001)Wahlberg et al. (2001) Wahlberg, M., Møhl, B., and Madsen, P. T., “Estimating source position accuracy of a large-aperture hydrophone array for bioacoustics,” J. Acoust. Soc. Am. 109, 397–406 (2001). https://doi.org/10.1121/1.1329619 extended the mathematical approach of this method (Watkins and Schevill, 1971Watkins and Schevill (1971) Watkins, W. A. and Schevill, W. E., “Four hydrophone array for acoustic three-dimensional location,” Report Reference No. 71–60, Woods Hole Oceanographic Institution, 1971.) to formulate a linear error propagation model. The position accuracy for 2D and 3D tracking for both MRS and ODS sensor systems was estimated as a function of TDOA errors, sound velocity errors, and sensor position errors, and they concluded sensor position errors had a large impact on source position estimate accuracy. Later, Ehrenberg and Steig (2002)Ehrenberg and Steig (2002) Ehrenberg, J. E. and Steig, T. W., “A method for estimating the ‘position accuracy’ of acoustic fish tags,” ICES J. Mar. Sci. 59, 140–149 (2002). https://doi.org/10.1006/jmsc.2001.1138 presented a derivation of an expression that showed the dependence of source position error on TOA and sound velocity estimate errors, which are independent of the algorithm used to determine the source position estimate. To minimize errors in source position estimates, in addition to optimization of signal processing, an optimal distributed sensor geometry is required to decrease the sensitivity of estimated source positions to TOA/TDOA measurement errors (Ehrenberg and Steig, 2003Ehrenberg and Steig (2003) Ehrenberg, J. E. and Steig, T. W., “Improved techniques for studying the temporal and spatial behavior of fish in a fixed location,” ICES J. Mar. Sci. 60, 700–706 (2003). https://doi.org/10.1016/S1054-3139(03)00087-0).

Radar and sonar source localization systems have been used to detect, localize, and track signal sources for decades (Altes, 1979Altes (1979) Altes, R., “Target position estimation in radar and sonar, and generalized ambiguity analysis for maximum likelihood parameter estimation,” Proc. IEEE 67, 920–930 (1979). https://doi.org/10.1109/PROC.1979.11355; Krim and Viberg, 1996Krim and Viberg (1996) Krim, H. and Viberg, M., “Two decades of array signal processing research: The parametric approach,” IEEE Signal Process. Mag. 13, 67–94 (1996). https://doi.org/10.1109/79.526899; and Le Chevalier, 2002Le Chevalier (2002) Le Chevalier, F., Principles of Radar and Sonar signal Processing (Artech House, 2002).). Radar and sonar have been widely used for civilian, military, and scientific purposes, and the range of applications is large (Minkoff, 1992Minkoff (1992) Minkoff, J., Signals, Noise, and Active Sensors: Radar, Sonar, Laser Radar (John Wiley & Sons, 1992).). The applications of sonar and radar systems are separated into active (Bekkerman and Tabrikian, 2006Bekkerman and Tabrikian (2006) Bekkerman, I. and Tabrikian, J., “Target detection and localization using MIMO radars and sonars,” IEEE Trans. Signal Process. 54, 3873–3883 (2006). https://doi.org/10.1109/TSP.2006.879267) and passive system (Carter, 1981Carter (1981) Carter, G. C., “Time delay estimation for passive sonar signal processing,” IEEE Trans. Acoust. Speech 29, 463–470 (1981). https://doi.org/10.1109/TASSP.1981.1163560) problems. Active sonars and radars transmit signals that are reflected back from targets (i.e., an echo), whereas passive sonars and radars do not transmit and only receive signals transmitted by sources (Stergiopoulos, 2000Stergiopoulos (2000) Stergiopoulos, S., Advanced Signal Processing Handbook: Theory and Implementation for Radar, Sonar, and Medical Imaging Real Time Systems (CRC press, 2000).). There are many similarities between radar (using radio waves) and sonar (using sound) signal processing, and developments for both types of systems have contributed significantly to the solutions to underwater localization problems (Vaccaro, 1998Vaccaro (1998) Vaccaro, R. J., “The past, present, and the future of underwater acoustic signal processing,” IEEE Signal Process. Mag. 15, 21–51 (1998). https://doi.org/10.1109/79.689583).

Determination of the location of nodes in wireless sensor networks accurately and at low cost is very important (Patwari et al., 2005Patwari et al. (2005) Patwari, N., Ash, J. N., Kyperountas, S., HeroIII, A. O., Moses, R. L., and Correal, N. S., “Locating the nodes: Cooperative localization in wireless sensor networks,” IEEE Signal Process. Mag. 22, 54–69 (2005). https://doi.org/10.1109/MSP.2005.1458287 and Mao et al., 2007Mao et al. (2007) Mao, G., Fidan, B., and Anderson, B. D. O., “Wireless sensor network localization techniques,” Comput. Networks 51, 2529–2553 (2007). https://doi.org/10.1016/j.comnet.2006.11.018). Knowledge of the precise location of wireless network sensors is fundamental for a wide range of applications (i.e., event detection and tracking), such as disaster relief operations (e.g., wildfire detection), environmental monitoring, biodiversity mapping, and precision agriculture (Karl and Willig, 2007Karl and Willig (2007) Karl, H. and Willig, A., Protocols and Architectures for Wireless Sensor Networks (John Wiley & Sons, 2007).). Accuracy and precision are the most important characteristics for a WSN localization system. Thus, TDOA methods are more suitable and commonly used for WSNs, compared with the low-cost but less accurate RSS method (Boukerche et al., 2007Boukerche et al. (2007) Boukerche, A., Oliveira, H. A., Nakamura, E. F., and Loureiro, A. A., “Localization systems for wireless sensor networks,” IEEE Wireless Commun. 14, 6–12 (2007). https://doi.org/10.1109/MWC.2007.4407221).

Numerous methods have been developed to reduce the location estimation errors for WSN nodes, thereby enhancing the event detection and tracking functions of WSN. Optimization of the design of sensor networks for specific applications is one of the most important components of source localization systems. The design of the sensor network determines the applicability of localization algorithms and thus has been investigated for years (Sohrabi et al., 2000Sohrabi et al. (2000) Sohrabi, K., Gao, J., Ailawadhi, V., and Pottie, G. J., “Protocols for self-organization of a wireless sensor network,” IEEE Pers. Commun. 7, 16–27 (2000). https://doi.org/10.1109/98.878532; Langendoen and Reijers, 2003Langendoen and Reijers (2003) Langendoen, K. and Reijers, N., “Distributed localization in wireless sensor networks: A quantitative comparison,” Comput. Networks 43, 499–518 (2003). https://doi.org/10.1016/S1389-1286(03)00356-6; Römer and Mattern, 2004Römer and Mattern (2004) Römer, K. and Mattern, F., “The design space of wireless sensor networks,” IEEE Wireless Commun. 11, 54–61 (2004). https://doi.org/10.1109/mwc.2004.1368897; Al-Karaki and Kamal, 2004Al-Karaki and Kamal (2004) Al-Karaki, J. N. and Kamal, A. E., “Routing techniques in wireless sensor networks: A survey,” IEEE Wireless Commun. 11, 6–28 (2004). https://doi.org/10.1109/MWC.2004.1368893; and Wang, 2008Wang (2008) Wang, Y., “Topology control for wireless sensor networks,” in Wireless sensor Networks and Applications, edited by Li, Y., Thai, M. T., and Wu, W. (Springer, 2008).). To improve source localization, different types of signals (e.g., acoustic sound, ultrasound, or radio) transmitted by sources may result in different estimation errors (Boukerche et al., 2007Boukerche et al. (2007) Boukerche, A., Oliveira, H. A., Nakamura, E. F., and Loureiro, A. A., “Localization systems for wireless sensor networks,” IEEE Wireless Commun. 14, 6–12 (2007). https://doi.org/10.1109/MWC.2007.4407221). Whitehouse (2002)Whitehouse (2002) Whitehouse, C., “The design of Calamari: An ad hoc localization system for sensor networks,” M.S. thesis, University of California at Berkeley, 2002 Electrical Engineering and Computer Science. tested an acoustic source tracking system and observed source location estimation errors of approximately 23 cm. In other experiments computing the distance of radio and ultrasound signals using various network design scenarios (Savvides et al., 2001Savvides et al. (2001) Savvides, A., Han, C. C., and Strivastava, M. B., “Dynamic fine-grained localization in ad-hoc networks of sensors,” in Proceedings of the 7th Annual International Conference on Mobile Computing and Networking (ACM, Rome, 2001), pp. 166–179.), the errors were as low as 2–3 cm. Novel computational techniques are also proving helpful in overcoming specific localization problems in WSNs (Kulkarni et al., 2011Kulkarni et al. (2011) Kulkarni, R. V., Förster, A., and Venayagamoorthy, G. K., “Computational intelligence in wireless sensor networks: A survey,” IEEE Commun. Sur. Tutorials 13, 68–96 (2011). https://doi.org/10.1109/SURV.2011.040310.00002). For example, stochastic particle swarm optimization was recognized as an efficient tool to use to solve local-minimum problems for localization and tracking in mobile WSN environments (Gopakumar and Jacob, 2008Gopakumar and Jacob (2008) Gopakumar, A. and Jacob, L., “Localization in wireless sensor networks using particle swarm optimization,” in Proceedings of the IET International Conference on Wireless, Mobile and Multimedia Networks (IET, Beijing, 2008), pp. 227–230. https://doi.org/10.1049/cp:20080185).

GPS provides coded satellite signals that can be processed to compute a receiver’s three dimensional position and velocity by using four or more GPS satellites (Kaplan and Hegarty, 2005Kaplan and Hegarty (2005) Kaplan, E. and Hegarty, C., Understanding GPS: Principles and Applications (Artech House, 2005).). The global GPS system has benefited civil, commercial, and military users worldwide (Hofmann-Wellenhof et al., 2013Hofmann-Wellenhof et al. (2013) Hofmann-Wellenhof, B., Lichtenegger, H., and Collins, J., Global Positioning System: Theory and Practice (Springer Science & Business Media, 2013).) in wide-ranging applications such as ground surveying, land vehicle navigation and tracking, marine navigation, air traffic control, aircraft landing, general aviation, and geodesy (Parkinson, 1996Parkinson (1996) Parkinson, B. W., Global Positioning System: Theory and Applications (AIAA, 1996).). A variety of error sources are encountered using the GPS system. Among these are selective availability, clock and ephemeris errors, ionospheric delays, tropospheric delays, multiple paths, signal noise, receiver errors, poor satellite coverage, and satellite geometry (Dana, 1997Dana (1997) Dana, P. H., “Global Positioning System (GPS) time dissemination for real-time applications,” Real-Time Syst. 12, 9–40 (1997). https://doi.org/10.1023/A:1007906014916; Kiefer and Lillesand, 2004Kiefer and Lillesand (2004) Kiefer, R. W. and Lillesand, T. M., Remote Sensing and Image Interpretation (John Wiley & Sons, 2004).; and Misra and Enge, 2006Misra and Enge (2006) Misra, P. and Enge, P., Global Positioning System: Signals, Measurements and Performance, 2nd ed. (Ganga-Jamuna Press, Lincoln, MA, 2006).). The accuracy of GPS depends on receiver location and the presence of obstructions that may block satellite signals. With technical improvements to GPS receivers, it is possible to measure a location on Earth at high frequency (e.g., 5 Hz) at a centimeter level of precision using the phase differential positioning method for timing measurements (Schutz and Herren, 2000Schutz and Herren (2000) Schutz, Y. and Herren, R., “Assessment of speed of human locomotion using a differential satellite global positioning system,” Med. Sci. Sports Exercise 32, 642–646 (2000). https://doi.org/10.1097/00005768-200003000-00014).

Electronically steerable arrays of microphones that apply TDOA technologies are used in a variety of ways in speech data acquisition systems (Brandstein and Silverman, 1997Brandstein and Silverman (1997) Brandstein, M. S. and Silverman, H. F., “A practical methodology for speech source localization with microphone arrays,” Comput. Speech Lang. 11, 91–126 (1997). https://doi.org/10.1006/csla.1996.0024). Microphone arrays are capable of automatic detection, localization, and tracking of active “talkers” in an enclosed environment in the presence of reverberation (DiBiase et al., 2001DiBiase et al. (2001) DiBiase, J. H., Silverman, H. F., and Brandstein, M. S., “Robust localization in reverberant rooms,” in Microphone Arrays, edited by Brandstein, M. S. and Ward, D. (Springer, Berlin, Heidelberg, 2001). and Ma et al., 2006Ma et al. (2006) Ma, W. K., Vo, B. N., Singh, S. S., and Baddeley, A., “Tracking an unknown time-varying number of speakers using TDOA measurements: A random finite set approach,” IEEE Trans. Signal Process. 54, 3291–3304 (2006). https://doi.org/10.1109/tsp.2006.877658).

The use of cellular mobile systems to position mobile stations will play a fundamental role in future wireless communication networks (Caffery, 2006Caffery (2006) Caffery, Jr., J. J., Wireless Location in CDMA Cellular Radio Systems (Springer Science & Business Media, 2006).). Applications of wireless localization using code division multiple access cellular networks include E-911, fraud detection, cellular system design and resource management, fleet management, and intelligent transportation systems (Caffery and Stuber, 1998Caffery and Stuber (1998) Caffery, Jr., J. J. and Stuber, G. L., “Overview of radiolocation in CDMA cellular systems,” IEEE Commun. Mag. 36, 38–45 (1998). https://doi.org/10.1109/35.667411). These positioning algorithms utilize TDOA approaches (Zhu and Zhu, 2001Zhu and Zhu (2001) Zhu, L. and Zhu, J., “A new model and its performance for TDOA estimation,” in Proceedings of the 53rd Vehicular Technology Conference (IEEE, Rhodes, Greece, 2001), pp. 2750–2753. and Mensing and Plass, 2006Mensing and Plass (2006) Mensing, C. and Plass, S., “Positioning algorithms for cellular networks using TDOA,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (IEEE, Toulouse, 2006), pp. 513–516.) or hybrid TDOA/AOA (Jami et al., 1999Jami et al. (1999) Jami, I., Ali, M., and Ormondroyd, R., “Comparison of methods of locating and tracking cellular mobiles,” in Proceedings of the IEE Colloquium on Novel Methods of Location and Tracking of Cellular Mobiles and Their System Applications (IEE, London, 1999), pp. 1–6. https://doi.org/10.1049/ic:19990238 and Cong and Zhuang, 2002Cong and Zhuang (2002) Cong, L. and Zhuang, W., “Hybrid TDOA/AOA mobile user location for wideband CDMA cellular systems,” IEEE Trans. Wireless Commun. 1, 439–447 (2002). https://doi.org/10.1109/twc.2002.800542).