A Survey on Wireless Position Estimation <s> CRLBs for Position Estimation <s> The time-of-arrival (TOA), time-difference-of-arrival (TDOA) and signal strength (SS) methods have been widely accepted as three principal techniques for positioning a mobile station (MS) in a wireless communication system. To the best of our knowledge, previous studies tend to treat these methods separately, and less analytical results on their relationship have been reported. We first examine the link between the TOA and TDOA methods. We provide an analytical explanation for the claim that, given a set of BS locations and an MS position, the TOA method should achieve a higher positioning precision than its TDOA counterpart. However, the two positioning methods may attain the same level of accuracy under certain conditions. We then investigate the tradeoff between the accuracy limits of the TOA and SS based methods, which leads to our proposal of a hybrid distance estimation scheme that combines both TOA and SS data. <s> BIB001 </s> A Survey on Wireless Position Estimation <s> CRLBs for Position Estimation <s> The Cramer-Rao bound (CRB) on location estimation accuracy of two different hybrid schemes, time of arrival/received signal strength (TOA/RSS) and time difference of arrival/received signal strength (TDOA/RSS), is computed. For short-range networks, the hybrid schemes offer improved accuracy with respect to conventional TOA and TDOA schemes, particularly in the proximity of the reference devices. <s> BIB002 </s> A Survey on Wireless Position Estimation <s> CRLBs for Position Estimation <s> A process for drying organic waste, such as sewage sludge, by contacting the organic waste with hot vapors wherein the dried material is plasticized and extruded to form compacted granules having good flow characteristics and suitable for application by commercial fertilizer spreaders. A plasticizer and an extrusion aid are added to the dried material prior to extrusion. <s> BIB003
In this section, CRLBs for various position estimation algorithms are investigated. The CRLB for an unbiased estimateθ of θ can be expressed as where I θ is the FIM given by with p(z | θ) representing the conditional probability density function of the position related parameter estimate given θ, and I 1 ≥ I 2 meaning that I 1 − I 2 is positive semi-definite. The CRLB provides also a lower bound on the mean square error (MSE) of a position estimate as follows: where MMSE refers to minimum MSE. In the following, the theoretical limits are investigated in terms of MMSE expressions for TOA, TDOA, RSS and AOA based positioning systems. It can be shown that the MMSE for TOA based positioning is given by BIB003  where c is the speed of light and σ 2 i , for i = 1, . . . , N L , is the variance of the zero mean Gaussian noise in the LOS case. Note that the theoretical lower bound in  is independent of the parameter estimates related to the NLOS nodes, which means that the NLOS nodes do not contribute to the positioning accuracy in the absence of prior statistical information related to NLOS errors. In addition, the geometric configuration of the LOS nodes can affect the theoretical limit significantly through the last term in the denominator. For sufficiently large SNR and/or effective bandwidth β, σ is approximately equal to 8π 2 β 2 SNR i , where SNR i is the SNR for the ith signal  . In that case,  can be expressed as which shows the impact of the effective bandwidth on the MMSE. In BIB003 , it is shown that an ML estimator based on LOS delay estimates obtained via matched filtering (or correlation) can attain the MMSE expression in  for sufficiently large SNR and/or effective bandwidth. In other words, the two-step positioning receiver in Figure 10 is asymptotically optimal. As an example, consider the positioning scenario in Figure 11 , where the target node is surrounded by 6 reference nodes that are uniformly located on a circle. Assuming that all the signals have the same SNR, the theoretical lower bounds can be obtained as in Figure 12 , where the square root of the MMSE expression in (39) is plotted against the effective bandwidth for various numbers of NLOS nodes. Namely, for N L = 3, nodes 4, 5 and 6; for N L = 4, nodes 5 and 6; and for N L = 5, node 6 are the NLOS nodes. From the figure, it is observed that as the number of LOS nodes, the effective bandwidth and/or the SNR increases, the accuracy of the positioning system increases. For TDOA based positioning, the MMSE can be expressed as BIB003  where K i,j is given by As can be observed from (40), the MMSE is independent of the estimates related to NLOS nodes as in the TOA case. Also, it can be shown that the MMSE for TDOA based positioning is always larger than or equal to that for TOA based positioning BIB001 , which is expected due to the presence of an extra unknown parameter, timing offset, in TDOA systems. For RSS based positioning systems, the MMSE is expressed as BIB003 MMSE RSS = ln 10 10n where n is the path loss exponent, σ 2 sh,i is the variance of the log-normal shadowing for the ith signal, and d i = (x − x i ) 2 + (y − y i ) 2 is the distance between the target node and the ith reference node. Note that the accuracy of RSS based positioning depends heavily on the channel parameters, namely the path loss exponent and the shadowing variances. Also, the accuracy depends on estimates at all nodes, LOS and NLOS, since the effects of NLOS propagation are implicitly included in the RSS signal model, as studied in Section 2.1. For an AOA based positioning system employing ULAs, the MMSE can be expressed as BIB003 MMSE AOA = 3 4π 2 ∆ 2 N a (N a + 1)(2N a + 1) where N a is the number of antenna elements and ∆ is the inter-element spacing. Similar to the time based systems, the AOA based positioning utilizes the estimates from LOS nodes only. For hybrid positioning systems, two different categories can be considered depending on whether estimation errors for various types of position related parameters are correlated or not. Some lower bound expressions for such systems can be found in BIB003 , BIB002 . The analysis in this section assumes that there is no prior statistical information related to NLOS errors. In the presence of such prior information, the accuracy can be evaluated by means of generalized CRLB (G-CRLB), as investigated in BIB003 . In that case, an asymptotically optimal positioning receiver can be implemented as shown in Figure 13 . Note that estimates from both LOS and NLOS reference nodes are utilized in the presence of NLOS error statistics.