- Open Access
Joint communication and positioning based on soft channel parameter estimation
© Schmeink et al; licensee Springer. 2011
Received: 30 November 2010
Accepted: 23 November 2011
Published: 23 November 2011
A joint communication and positioning system based on maximum-likelihood channel parameter estimation is proposed. The parameters of the physical channel, needed for positioning, and the channel coefficients of the equivalent discrete-time channel model, needed for communication, are estimated jointly using a priori information about pulse shaping and receive filtering. The paper focusses on the positioning part of the system. It is investigated how soft information for the parameter estimates can be obtained. On the basis of confidence regions, two methods for obtaining soft information are proposed. The accuracy of these approximative methods depends on the nonlinearity of the parameter estimation problem, which is analyzed by so-called curvature measures. The performance of the two methods is investigated by means of Monte Carlo simulations. The results are compared with the Cramer-Rao lower bound. It is shown that soft information aids the positioning. Negative effects caused by multipath propagation can be mitigated significantly even without oversampling.
Interest in joint communication and positioning is steadily increasing . Synergetic effects like improved resource allocation and new applications like location-based services or a precise location determination of emergency calls are attractive features of joint communication and positioning. Since the system requirements of communication and positioning are quite different, it is a challenging task to combine them: Communication aims at high data rates with little training overhead. Only the channel coefficients of the equivalent discrete-time channel model, which includes pulse shaping and receive filtering in addition to the physical channel, need to be estimated for data detection. In contrast, positioning aims at precise position estimates. Therefore, parameters of the physical channel like the time of arrival (TOA) or the angle of arrival (AOA) need to be estimated as accurately as possible [2, 3]. Significant training is typically spent for this purpose.
In this paper, a joint communication and positioning system based on maximum-likelihood channel parameter estimation is suggested . The estimator exploits the fact that channel and parameter estimation are closely related. The parameters of the physical channel and the channel coefficients of the equivalent discrete-time channel model are estimated jointly by utilizing a priori information about pulse shaping and receive filtering. Hence, training symbols that are included in the data burst aid both communication and positioning.
On the one hand, in [5–7], it is proposed to use a priori information about pulse shaping and receive filtering in order to improve the estimates of the equivalent discrete-time channel model. However, the information about the physical channel is neglected in these publications. On the other hand, channel sounding is performed in order to estimate the parameters of the physical channel [8–10]. But, to the authors best knowledge, the proposed parameter estimation methods are not applied for estimation of the equivalent discrete-time channel model. The estimator proposed in this paper combines both approaches: Channel estimation is mandatory for communication purposes. By exploiting a priori information about pulse shaping and receive filtering, the channel coefficients can be estimated more precisely and positioning is enabled. Hence, synergy is created.
This paper focusses on the positioning part of the proposed joint communication and positioning system. Most positioning methods suffer from a bias introduced by multipath propagation. Multipath mitigation is, thus, an important issue. The proposed channel parameter estimator performs multipath mitigation in two ways: First, the maximum-likelihood estimator is able to take all relevant multipath components into account in order to minimize the modeling error. Second, soft information can be obtained for the parameter estimates. Soft information corresponds to the variance of an estimate and is a measure of reliability. This information can be exploited by a weighted positioning algorithm in order to improve the accuracy of the position estimate.
On the basis of confidence regions, two different methods for obtaining soft information are proposed: The first method is based on a linearization of the nonlinear parameter estimation problem and the second method is based on the likelihood concept. For linear estimation problems, an exact covariance matrix can be determined in closed form. For nonlinear estimation problems, as it is the case for channel parameter estimation, there are different approximations to the covariance matrix, which are based on a linearization. These approximate covariance matrices are generated by most nonlinear least-squares solvers (e.g., Levenberg-Marquardt method) anyway and can be used after further analysis . Confidence regions based on the likelihood method are more robust than those based on approximate covariance matrices since they do not rely on a linearization, but they are also more complex to calculate. Heuristic optimization methods like genetic algorithms or particle swarm optimization offer a comfortable procedure to determine the likelihood confidence region as demonstrated in . Both methods are only approximate, and their accuracy depends on the nonlinearity of the estimation problem. In , Bates and Watts introduce curvature measures that indicate the amount of nonlinearity. These measures can be used to diagnose the accuracy of the proposed methods.
The remainder of this paper is organized as follows: The system and channel model is described in Section 2. The relationship between channel and parameter estimation is explained and the nonlinear metric of the maximum-likelihood estimator is derived. General aspects concerning nonlinear optimization are discussed. In Section 3, the concept of soft information is introduced. Based on confidence regions, two methods for obtaining soft information concerning the parameter estimates are proposed. In order to further analyze the proposed methods, the curvature measures of Bates and Watts are introduced in Section 4. The curvature measures are calculated for the parameter estimation problem and a first analysis of the problem is given. Afterward, positioning based on the TOA is explained in Section 5, and the performance of the two soft information methods is investigated by means of Monte Carlo simulations. The results are compared with the Cramer-Rao lower bound. Finally, conclusions are drawn in Section 6.
2 System Concept
2.1 System and channel model
where M is the number of resolvable propagation paths. The parameters f μ (t) and τ μ (t) denote the complex amplitude and the propagation delay of the μ th path at time t, respectively. Without loss of generality, it is assumed that the multipath components are sorted according to ascending delay: τ1(t) < τ2(t) < ··· < τ M (t). The delay of the first arriving path is called TOA. Positioning is based on the assumption that the TOA corresponds to the distance between transmitter and receiver. This is only true if a line-of-sight (LOS) path exists. In urban or indoor environments, the LOS path is often blocked. In these so-called non-LOS (NLOS) scenarios, the modeling error reduces the positioning accuracy significantly. Additionally, positioning typically suffers from a bias introduced by multipath propagation even if a LOS path exists. In order to analyze the multipath mitigation ability of the proposed soft channel parameter estimator, this paper restricts itself to LOS scenarios. However, the influence of NLOS is discussed in Section 5.2.
In the following, it is assumed that the channel is quasi time-invariant over the training length (block fading). Thus, the time index k in (4) can be omitted.
For simulation of communication systems, it is sufficient to consider excess delays. Without loss of generality, τ1= 0 can be assumed then. The effective channel memory length L is, therefore, determined by the excess delay τ M -- τ1 plus the effective width T g of g(τ).
2.2 Channel parameter estimation
Using the assumptions above, the estimation error ε is zero mean and Gaussian with covariance matrix . For a pseudo-random training sequence, the matrix (X H X) becomes a scaled identity matrix with scaling factor K t - L, and the covariance matrix of the estimation error reduces to .
The parameters θ can be estimated by fitting the model function (8) to the least-squares channel estimates . Hence, the channel estimates are not only used for data detection, but they are also exploited for positioning. Furthermore, refined channel estimates ĥl are obtained by evaluating (8) for the parameter estimate .b On the one hand, positioning is enabled since the TOA τ1 is estimated. On the other hand, data detection can be improved because refined channel estimates are obtained.
with respect to θ. The second approach in (10) may seem more natural to some readers since the parameters are estimated directly from the received samples. But since both approaches are equivalent, as proven in the "Appendix", it seems more convenient to the authors to apply the first approach: Channel estimates are usually already available in communication systems and the metric derived from (9) is less complex than the metric derived from (10). Hence, only the first approach is considered in the following.
The minimization of the metric in (11) cannot be solved in closed form since is nonlinear. An optimization method has to be applied. In order to chose a suitable optimization method to find , different system aspects have to be taken into account, and a tradeoff depending on the requirements has to be found. The goal is to find the global minimum of . Unfortunately, has many local minima due to the superposition of random multipath components. Consequently, the optimization method of choice should be either a global optimization method or a local optimization method in combination with a good initial guess, i.e., an initial guess that is sufficiently close to the global optimum. Both choices involve different benefits and drawbacks. To find a good initial guess is difficult and, therefore, may be seen as a drawback itself. But in case a priori knowledge in form of a good initial guess is available, a search in the complete search space would be unnecessary.
For channel parameter estimation, it is suggested to divide the problem into an acquisition and a tracking phase. In the acquisition phase, a global optimization method is applied, and in the tracking phase, the parameter estimate of the last data burst may be used as an initial guess for a local optimization method. This is suitable for channels that do not change too rapidly from data burst to data burst. In this paper, particle swarm optimization (PSO) [15–17] is suggested for the acquisition phase, and the Levenberg-Marquardt method (LMM) [18, 19] is proposed for the tracking phase.
PSO is a heuristic optimization method that is able to find the global optimum without an initial guess and without gradient information. PSO is easy to implement because only function evaluations have to be performed. So-called particles move randomly through the search space and are attracted by good fitness values in their past and of their neighbors. In this way, the particles explore the search space and are able to find the global optimum. It is a drawback that PSO does not assure global convergence. There is a certain probability (depending on the signal-to-noise ratio) that PSO converges prematurely to a local optimum (outage). Furthermore, PSO is sometimes criticized because many iterations are performed in comparison to gradient-based optimization algorithms.
The LMM belongs to the standard nonlinear least-squares solvers and relies on a good initial guess. The gradient of the metric has to be supplied by the user. For the LMM, convergence to the optimum in the neighborhood of the initial guess is assured. Second derivative information is used to speed up convergence: The LMM varies smoothly between the inverse-Hessian method and the steepest decent method depending on the topology of the metric . Furthermore, an approximation to the covariance matrix of the parameter estimates is calculated inherently by the LMM. The LMM is designed for small residual problems. For large residual problems (at low signal-to-noise ratio), it may fail (outage).
3 Soft Information
3.1 Definition of soft information
The concept of soft information is already widely applied: In the area of communication, soft information is used for decoding, detection, and equalization. In the field of navigation, soft information is exploited for sensor fusion . This paper aims at obtaining soft information for the parameter estimates in order to improve the positioning accuracy before sensor fusion is applied.
In general, the true value of the parameters is not known. Therefore, the asymptotic covariance matrix cannot be determined and an approximation has to be found. Different approximate covariance matrices are given in the literature that should be used with caution since the approximation may be very poor [11, 21]. In the following section, a short description of confidence regions is included because they are closely related to soft information: Some of the confidence regions rely on the approximate covariance matrices mentioned above.
3.2 Confidence regions
In , Donaldson and Schnabel investigate different methods to construct confidence regions and confidence intervals. Confidence regions and intervals are closely related to soft information since they also indicate reliability: The estimated parameters do not coincide with the true parameters θ because of the measurement noise. A confidence region indicates the area around the estimated parameters in which the true parameters might be with a specific probability. This probability is called the confidence level and is often expressed as a percentage. A commonly used confidence level is 95%.
are included in the likelihood confidence region. This region does not have to be elliptical but can be of any form. The likelihood method is approximate for nonlinear problems as well but more precise and robust than the linearization method since it does not rely on linearization. There is an exact method, which is called lack-of-fit method, that is neglected in this paper due to its high computational complexity and because the likelihood method is already a good approximation according to . The accuracy of the linearization and the likelihood method strongly depends on the problem and on the parameters. Donaldson and Schnabel  suggest to use the curvature measures of Bates and Watts , which are introduced in Section 4, as a diagnostic tool. With these measures, it can be evaluated whether the corresponding method is applicable or not.
3.3 Proposed methods to obtain soft information
After this excursion to confidence regions, the way of employing this knowledge for obtaining soft information is now discussed. The first and straightforward idea is to use the variances of the approximate covariance matrix Capprox in (18). This method is simple, and many optimization algorithms like the LMM already compute and output Capprox or similar versions of it. But without further analysis (see Sections 4 and 5), it is questionable whether this method is precise enough.
are selected from the table and form the likelihood confidence region. It can be observed that the density of points near the parameter estimate is higher than at the border of the likelihood confidence region. The reason is that the particles are attracted by good fitness values near the optimum and oscillate in its neighborhood before convergence occurs. Hence, all points form a distribution with mean and variance, where the mean coincides with the parameter estimate . Therefore, the variance of this distribution can be used as soft information.
In Section 5, the performance of both methods is evaluated and compared. Prior to that the curvature measures of Bates and Watts  are introduced for further analysis and understanding.
4 Curvature Measures
4.1 Introduction to curvature measures
In , Bates and Watts describe nonlinear least-squares estimation from a geometric point of view and introduce measures of nonlinearity. These measures indicate the applicability of a linearization and its effects on inference. Hence, the accuracy of the confidence regions described in Section 3 can be evaluated using these measures. In the following, the most important aspects of the so-called curvature measures are presented.
Geometrically, (26) describes the distance between and in the (L + 1)-dimensional sample space. If the parameter vector is changed in the P-dimensional parameter space (search space), the vector traces a P-dimensional surface in the sample space, which is called solution locus. Hence, the function maps all feasible parameters in the P-dimensional parameter space to the P-dimensional solution locus in the (L+1)-dimensional sample space. Because of the measurement noise, the observations do not lie on the solution locus but anywhere in the sample space. The parameter estimate corresponds to the point on the solution locus with the smallest distance to the point of observations .
where is the Jacobian matrix as defined in (14) evaluated at . The informational value of inference concerning the parameter estimates highly depends on the closeness of the tangent plane to the solution locus. This closeness in turn depends on the curvature of the solution locus. Therefore, the measures of nonlinearity proposed by Bates and Watts indicate the maximum curvature of the solution locus at the specific point . It is important to note that there are two different kinds of curvatures since two different assumptions are made concerning the tangent plane. First, it is assumed that the solution locus is planar at and, hence, can be replaced by the tangent plane (planar assumption). Second, it is assumed that the coordinate system on the tangent plane is uniform (uniform coordinate assumption), i.e., the coordinate grid lines mapped from the parameter space remain equidistant and straight in the sample space. It might happen that the first assumption is fulfilled, but the second assumption is not. Then, the solution locus is planar at the specific point , but the coordinate grid lines are curved and not equidistant. If the planar assumption is not fulfilled, the uniform coordinate assumption is not fulfilled either.
and to compare them to in order to assess the accuracy of the confidence regions . If the confidence region based on the linearization method (19) with the approximate covariance matrix shall be applied, both the planar assumption and the uniform coordinate assumption have to be fulfilled. That means that the maximum relative curvatures ΓNand ΓT have to be small compared with . The confidence region based on the likelihood method (20) is more robust since only the planar assumption needs to be fulfilled and only ΓN needs to be small compared with .
4.2 Analysis of the parameter estimation problem
Parameters of the investigated channel models and the corresponding maximum rel. curvatures at different SNRs
M = 1
M = 2
θ1 = 0.4454
θ1 = 0.6401
θ4 = -0.3464
θ2 = -0.7715
θ2 = -1.1086
θ5 = 0.8363
θ3 = 3.81 T s
θ3 = 3.81 T s
θ6 = 4.62 T s
ΓN @ 10 dB
ΓT @ 10 dB
ΓN @ 30 dB
ΓT @ 30 dB
ΓN @ 50 dB
ΓT @ 50 dB
5.1 Positioning based on the time of arrival
There are many different approaches to determine the position, e.g., multiangulation, multilateration, fingerprinting, and motion sensors. This paper focusses on radiolocation based on the TOA, which is also called multilateration. Furthermore, two-dimensional positioning is considered in the following. An extension to three dimensions is straightforward.
Thus, positioning is again a nonlinear problem.f There are alternative ways to solve the set of nonlinear equations described by (42) and (43). In this paper, two different approaches are considered: The iterative Taylor series algorithm (TSA)  and the weighted least-squares (WLS) method [23, 24].
until the correction factor is smaller than a given threshold. If the initial guess is close to the true position, few iterations are needed. If the starting position is far from the true position, many iterations may be necessary. Additionally, the algorithm may diverge. Hence, finding a good initial guess is a crucial issue. For the numerical results shown in Section 5.2, the position estimate of the WLS method is used as initial guess for the TSA.
Both, the TSA and the WLS method, apply a weighting matrix that contains the variances of the pseudo-range errors. Reliable pseudo-ranges have higher weights than unreliable ones and, thus, have a stronger influence on the estimation results. Typically, the true variances are not known. They can only be estimated as described in Section 3: For each link b, the variance of the TOA is determined via the linearizationg or the likelihood method. This TOA variance is transformed into a pseudo-range variance by a multiplication with c2. If no information about the estimation error η is available, the weighting matrices correspond to the identity matrix I(no weighting at all).
is the Fisher information matrix. If the estimator is unbiased, its mean squared error (MSE) is larger than or equal to the CRLB. If the MSE approaches the CRLB, the estimator is a minimum variance unbiased (MVU) estimator.
5.2 Numerical results
Three different channel models with memory length L = 10 are investigated: a single-path channel (M = 1), a two-path channel (M = 2) with large excess delay (Δτ2 ∈ [T s ,2T s ]) and a two-path channel (M = 2) with small excess delay . For all channel models, the LOS delay τ1, b for each link b is calculated from the true distance d b (p). The excess delay of the multipath component Δτ2 for both two-path channels is determined randomly in the corresponding interval. The smaller the excess delay is, the more difficult it is to separate the different propagation paths. The power of the multipath component is half the power of the LOS component. The phase of each component is generated randomly between 0 and 2π. For each link, channel parameter estimation is performed and soft information based on the linearization method and on the likelihood method is obtained. For PSO, I = 50 particles and a maximum number of iterations T = 8,000 are applied.h The estimated LOS delays are converted to pseudo-ranges r b , and the position of the MS is estimated with the TSA and the WLS method applying the different soft information methods. For comparison, positioning without soft information is performed. The position estimate of the WLS method is used as initial guess for the TSA. Furthermore, in the WLS method, the RO with the best weighting factor is chosen as reference.
The performance of the estimators is evaluated by Monte Carlo simulations and the results are compared with the Cramer-Rao lower bound (CRLB). On the one hand, simulations are performed over SNR since the accuracy of the soft information methods depends on the SNR. In each run, a new MS position p is determined randomly inside the region of Figure 5. On the other hand, simulations are performed over space for a fixed SNR in order to assess the influence of the GDOP. A fixed 4 × 4 grid of MS positions is applied in this case.
where the expectation is taken with respect to the channel realizations. For the simulations over SNR, the expectation is additionally taken with respect to the random positions p.
At first, the results for the single-path channel are discussed because this scenario represents an optimal case: Both soft information methods are accurate (see Section 4.2) and due to power control, the pseudo-range errors for all ROs should be the same. Hence, positioning without and with weighting is supposed to perform equally well. The first row of Figure 6 contains the results for the WLS method, whereas the second row shows the results for the TSA. As supposed previously, the RMSE curves for positioning without soft information and with soft information from the likelihood and the linearization method coincide. The TSA is furthermore a MVU estimator since the RMSE approaches the CRLB for all SNRs and for all positions. The WLS method performs worse: There is a certain gap between the CRLB and the RMSE. In Figure 6b, it can be observed that this gap depends on the position and, thus, on the GDOP: The larger the GDOP is, the larger is the gap. Hence, the gap between RMSE and CRLB in Figure 6a is smaller for the second scenario ("S") since the GDOP is smaller on average. For the two-path channels, a similar behavior of the WLS method was observed. Therefore, only the results for the TSA are considered in the following due to its superior performance.
The third and fourth row of Figure 6 show the simulation results for the two-path channels with large and small excess delay, respectively. It was observed in Section 4.2 that the likelihood method is generally accurate even for multipath channels. In contrast, the accuracy of the linearization method depends on the excess delay and the SNR. The smaller the excess delay, the higher is the nonlinearity of the problem and the less accurate is the linearization method. The accuracy increases with SNR. Hence, it is supposed that the likelihood method outperforms the linearization method. Only at very high SNR, both methods are assumed to perform equally well. Surprisingly, the linearization and the likelihood method show approximately the same performance for all cases. The linearization method performs even slightly better in most cases. Only for very low SNR and a small excess delay the likelihood method outperforms the linearization method. The likelihood method seems to be more susceptible to the GDOP. Hence, the inaccuracy of the covariance matrices at low SNR barely influences the positioning accuracy. Actually, it seems that the absolute value of the weights in the weighting matrices W and W' is not crucial. Rather a correct ratio of the weights is relevant. Thus, rough soft information is sufficient as long as the ratio of the pseudo-range variances is accurate. This is fulfilled even for the inaccurate covariance matrices of the linearization method. Hence, it is suggested to apply the linearization method because of its lower computational complexity.
For the two-path channel with large excess delay (Figure 6e, f), the RMSE with or without soft information is almost the same since the multipath components can already be separated by the estimator quite well. For a small excess delay (Figure 6g, h), the RMSE with soft information is much closer to the CRLB than without soft information. With respect to SNR, a gain of approximately 7-10 dB is achieved (see Figure 6g). Furthermore, positioning with soft information is less susceptible to the GDOP (see Figure 6h). Thus, soft information is well suited to mitigate severe multipath propagation. The smaller the excess delay is, the more important it is to apply soft information for positioning.
The influence of the GDOP can be neglected for the scenario with small average GDOP. The curves labeled with "S" indicate that even for one-shot estimation without oversampling a positioning accuracy much smaller than the distance corresponding to the symbol duration, d s , is achieved for all channel models.
For all simulations, a LOS path has been assumed so far. Hence, the estimated TOA corresponds to distance between transmitter and receiver. However, in urban or indoor environments, the LOS path is often blocked as already mentioned in Section 2.1. Therefore, the influence of NLOS propagation is discussed here. In case of NLOS, a modeling error is introduced that reduces the positioning accuracy significantly. The proposed soft channel parameter estimator does not take a priori information about the physical channel (e.g., probability of NLOS) into account and, hence, is not able to detect such a modeling error. The obtained soft information can only be used to mitigate multipath propagation. In order to mitigate NLOS effects, further processing has to be done (e.g., ).
Nevertheless, multipath mitigation is an important issue. The multipath mitigation ability of the proposed soft channel parameter estimation has been presented for M = 2 paths due to clarity and simplicity reasons. The influence of the number of multipath components is as follows: The complexity of the soft channel parameter estimator increases with the number of multipath components. Furthermore, the reliability of the estimates decreases with M. Hence, the positioning accuracy deteriorates. If M is large and the scatterers are closely spaced (dense multipath), the estimator becomes biased and the positioning accuracy saturates. In general, it is suggested to consider only the dominant paths if M is large.
It was mentioned before that the TSA may diverge. Divergence occurred for large GDOP when the initial guess was far from the true position.i This happened only rarely. The initial guess is determined by the WLS method which is very susceptible to the GDOP. Hence, the starting position may be far away from the true position for large GDOP.
In this paper, a channel parameter estimator based on the maximum-likelihood approach is proposed for joint communication and positioning. The parameters of the physical channel (e.g., TOA) and the equivalent discrete-time channel model are estimated jointly. In order to mitigate multipath propagation effects and to improve the positioning accuracy, soft information concerning the parameter estimates is used. Two different methods to obtain soft information are proposed: The linearization and the likelihood method. The accuracy of the methods depends on the nonlinearity of the parameter estimation problem, which is evaluated by the curvature measures of Bates and Watts. It is shown that the likelihood method is always accurate for the parameter estimation problem. The linearization method is only accurate in a single-path channel or at high SNR for a multipath channel. Nevertheless, Monte Carlo simulations for a two-dimensional positioning problem show that this has only very little influence on the positioning. The positioning algorithms that exploit the soft information obtained by the linearization and the likelihood method perform equally well. For severe multipath propagation, the RMSEs for the weighted positioning algorithms are closer to the CRLB than the RMSE of positioning without weighting. A gain of approximately 7-10 dB can be achieved. Hence, multipath propagation effects can be mitigated significantly, even for one-shot estimation without oversampling. Based on these results, it is suggested to apply the linearization method because of its lower computational complexity.
aFor oversampling with factor J it follows: . bThe mean squared error of the channel estimates ĥ l is reduced in comparison to the mean squared error of the least-squares channel estimates , if the number of parameters, 3M, is less than the number of channel coefficients, L + 1, to be estimated. For simulation results please refer to . cCapprox corresponds to in  for a complex-valued problem instead of a real-valued problem. dThe superscript T, which denotes tangential, should not be mistaken for the superscript T , which denotes the transpose of a matrix. eIn  a simplified method to determine the maximum relative curvatures is introduced based on linear transformations of the coordinates in the parameter and the sample space. This method is neglected here because it is out of the scope of this paper. fIn a two-dimensional TOA scenario at least three ROs are required. For positioning in three dimensions a fourth RO is needed. gFor the linearization method the variance of the TOA corresponds to the 3rd diagonal entry of the approximate covariance matrix Capprox. h Furthermore, channel parameter estimation was performed for the LMM described in  with the true parameters θ as initial guess. Since PSO and the LMM provided approximately the same performance, only PSO is considered here for conciseness. iThe outliers due to divergence were not considered in the calculation of the RMSE.
In the following, the equivalence of the maximum-likelihood estimators based on (9) and (10) is shown. First, both metrics are stated in vector/matrix notation. Then, the equivalence of both metrics is proven given the assumptions of Section 2.2. For readability, the terms h = h(θ) and are introduced, where θ denotes the true parameter set and denotes the hypothetical parameter set.
Comparing (59) and (60) shows that (58) is valid with .
- Raulefs R, Plass S, Mensing C: The where project: combining wireless communications and navigation. In Proceedings of 20th Wireless World Research Forum (WWRF). Ottawa, Canada; 2008.Google Scholar
- Pahlavan K, Levesque AH: Wireless information networks. Wiley, Hoboken, New Jersey; 2005. ch. 13: RF location sensingView ArticleGoogle Scholar
- Cheung K, So H, Ma W-K, Chan Y: A constrained least squares approach to mobile positioning: algorithms and optimality. EURASIP J Appl Signal Process 2006, 23. p (2006) Article ID 20858Google Scholar
- Schmeink K, Block R, Hoeher PA: Joint channel and parameter estimation for combined communication and navigation using particle swarm optimization. In Proceedings of Workshop on Positioning, Navigation and Communication (WPNC). Dresden, Germany; 2010.Google Scholar
- Khayrallah A, Ramesh R, Bottomley G, Koilpillai D: Improved channel estimation with side information. In Proceedings of IEEE Vehicular Technology Conference (VTC Spring). Volume 2. Phoenix, Arizona; 1997:1049-1053.Google Scholar
- Liang JW, Ng B, Chen JT, Paulraj A: GMSK linearization and structured channel estimate for GSM signals. In Proceedings of IEEE Military Communications Conference (MILCOM). Volume 2. Monterey, California; 1997:817-821.Google Scholar
- Lee H-N, Pottie GJ: Fast adaptive equalization/diversity combining for time-varying dispersive channels. IEEE Trans Commun 1998, 46(9):1146-1162. 10.1109/26.718557View ArticleGoogle Scholar
- Feder M, Weinstein E: Parameter estimation of superimposed signals using the EM algorithm. IEEE Trans Acoust Speech Signal Process 1988, 36(4):477-489. 10.1109/29.1552MATHView ArticleGoogle Scholar
- Fleury BH, Tschudin M, Heddergott R, Dalhaus D, Pedersen KI: Channel parameter estimation in mobile radio environments using the SAGE algorithm. IEEE J Sel Areas Commun 1999, 17(3):434-450. 10.1109/49.753729View ArticleGoogle Scholar
- Richter A, Landmann M, Thomä RS: Maximum likelihood channel parameter estimation from multidimensional channel sounding measurements. In Proceedings of IEEE Vehicular Technology Conference (VTC Spring). Jeju, Korea; 2003:1056-1060.Google Scholar
- Donaldson JR, Schnabel RB: Computational experience with confidence regions and confidence intervals for nonlinear least squares. In Proceedings of 17th Symposium on the Interface of Computer Sciences and Statistics. Lexington, Kentucky; 1985:83-93.Google Scholar
- Schwaab M, Biscaia EC Jr, Monteiro JL, Pinto JC: Nonlinear parameter estimation through particle swarm optimization. Chem Eng Sci 2008, 63(6):1542-1552. 10.1016/j.ces.2007.11.024View ArticleGoogle Scholar
- Bates DM, Watts DG: Relative curvature measures of nonlinearity. J R Stat Soc Ser B (Methodological) 1980, 42(1):1-25.MATHMathSciNetGoogle Scholar
- Kay SM: Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice-Hall, Upper Saddle River, New Jersey; 1993.MATHGoogle Scholar
- Kennedy J, Eberhart R: Particle swarm optimization. In Proceedings of IEEE International Conference on Neural Networks. Volume 4. Perth, Australia; 1995:1942-1948.View ArticleGoogle Scholar
- Bratton D, Kennedy J: Defining a standard for particle swarm optimization. In Proceedings of IEEE Swarm Intelligence Symposium (SIS). Honolulu, Hawaii; 2007:120-127.Google Scholar
- Blum C, Li X: Swarm Intelligence in Optimization. In Swarm Intelligence: Introduction and Applications, ser. Natural Computing Series. Edited by: C Blum, D Merkle. Springer; 2008:43-85.View ArticleGoogle Scholar
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP: Numerical Recipes in C++: The Art of Scientific Computing. Cambridge University Press, Cambridge; 2002.Google Scholar
- Nocedal J, Wright SJ: Numerical Optimization. Springer, New York; 1983.Google Scholar
- Hoeher PA, Robertson P, Offer E, Woerz T: The soft-output principle: reminiscences and new developments. Eur Trans Telecommun 2007, 18(8):829-835. 10.1002/ett.1200View ArticleGoogle Scholar
- Dennis JE Jr, Schnabel RB: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, New Jersey; 1983.MATHGoogle Scholar
- Kaplan ED, (ed.): Understanding GPS: Principles and Applications. Artech House, Boston; 1996.Google Scholar
- Sayed AH, Tarighat A, Khajenouri N: Network-based wireless location: challenges faced in developing techniques for accurate wireless location information. IEEE Signal Process Mag 2005, 22(4):24-40.View ArticleGoogle Scholar
- Guevenc I, Chong CC, Watanabe F, Inamura H: NLOS identification and weighted least-squares localization for UWB systems using multi-path channel statistics. EURASIP J Appl Signal Process 2008, 1-14. (2008) Article ID 271984Google Scholar
- Langley RB: Dilution of precision. GPS World 1999, 10(5):52-59.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.