### 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.

The position *p* *=* [*x, y*]^{T}of a mobile station (MS) is determined relative to *B* reference objects (ROs) whose positions *p*_{
b
}*=* [*x*_{
b
}*,y*_{b}]^{T}(1 ≤ *b* ≤ *B*) are known. For each RO *b*, the TOA {\widehat{\tau}}_{1,b} is estimated. The TOA corresponds to the distance between this RO and the MS {r}_{b}={\widehat{\tau}}_{1,b}c, where *c* is the speed of light. The estimated distances *r*= [*r*_{1},..., *r*_{
B
}]^{T}are called *pseudo-ranges* since they consist of the true distances *d*(*p*) *= [d*_{1}*(* *p* *),..., d*_{
B
}(*p*)]^{T}and estimation errors *η*= [*η*_{1},..., *η*_{
B
}]^{T} with covariance matrix {\mathit{C}}_{\eta}=\mathsf{\text{diag}}\left({\sigma}_{\eta 1}^{2},\dots ,{\sigma}_{\eta B}^{2}\right):

\mathit{r}=\mathit{d}\left(\mathit{p}\right)+\mathit{\eta}.

(42)

The true distance between the *b* th RO and the MS is a nonlinear function of the position *p* given by

{d}_{b}\left(\mathit{p}\right)=\sqrt{{\left(x-{x}_{b}\right)}^{2}+{\left(y-{y}_{b}\right)}^{2}}.

(43)

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) [22] and the weighted least-squares (WLS) method [23, 24].

The TSA is based on a linearization of the nonlinear function (43). Given a starting position {\widehat{\mathit{p}}}_{0} (initial guess), the pseudo-ranges can be approximated by a first-order Taylor series

\mathit{r}\cong \mathit{d}\left({\widehat{\mathit{p}}}_{0}\right)+\mathit{J}\left({\widehat{\mathit{p}}}_{0}\right)\left(\mathit{p}-{\widehat{\mathit{p}}}_{0}\right)+\mathit{\eta},

(44)

in which *J*(*p*) is the Jacobian matrix of (43) with entries

{\left[\mathit{J}\left(\mathit{p}\right)\right]}_{b1}=\frac{\delta {d}_{b}\left(\mathit{p}\right)}{\delta x},\phantom{\rule{1em}{0ex}}{\left[\mathit{J}\left(\mathit{p}\right)\right]}_{b2}=\frac{\delta {d}_{b}\left(\mathit{p}\right)}{\delta y}.

(45)

Defining \Delta {\mathit{r}}_{0}=\mathit{r}-\mathit{d}\left({\widehat{\mathit{p}}}_{0}\right) and \Delta {\mathit{p}}_{0}=\mathit{p}-{\widehat{\mathit{p}}}_{0} results in the following linear relationship

\Delta {\mathit{r}}_{0}\cong \mathit{J}\left({\widehat{\mathit{p}}}_{0}\right)\Delta {\mathit{p}}_{0}+\mathit{\eta},

(46)

that can be solved according to the least-squares approach:

\Delta {\widehat{\mathit{p}}}_{0}={\left(\mathit{J}{\left({\widehat{\mathit{p}}}_{0}\right)}^{T}\mathit{W}\mathit{J}\left({\widehat{\mathit{p}}}_{0}\right)\right)}^{-1}\mathit{J}{\left({\widehat{\mathit{p}}}_{0}\right)}^{T}\mathit{W}\Delta {\mathit{r}}_{0}.

(47)

The weighting matrix *W* is given by the inverse of the covariance matrix {\mathit{C}}_{\eta}:\mathit{W}=\mathsf{\text{diag}}\left(\frac{1}{{\sigma}_{\eta 1}^{2}},\dots ,\frac{1}{{\sigma}_{\eta B}^{2}}\right). A new position estimate {\widehat{\mathit{p}}}_{1} is obtained by adding the correction factor \Delta {\widehat{\mathit{p}}}_{0} to the starting position {\widehat{\mathit{p}}}_{0}. This procedure is performed iteratively,

{\widehat{\mathit{p}}}_{i+1}={\widehat{\mathit{p}}}_{i}+\Delta {\widehat{\mathit{p}}}_{i},

(48)

until the correction factor \Delta {\widehat{\mathit{p}}}_{i} 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.

The WLS method [23, 24] solves the set of nonlinear equations described by (42) and (43) in closed form. Hence, this method is non-iterative and less costly than the TSA. The basic idea is to transform the original set of *nonlinear* equations into a set of *linear* equations. For this purpose, one RO is selected as reference. Without loss of generality, the first RO is chosen here. By subtracting the squared distance of the first RO from the squared distances of the remaining ROs, a linear least-squares problem with solution

\widehat{\mathit{p}}={\left({\mathit{S}}^{T}\mathit{W}\prime \mathit{S}\right)}^{-1}{\mathit{S}}^{T}\mathit{W}\prime \mathit{b}

(49)

is obtained, in which

\mathit{S}=\left[\begin{array}{cc}\hfill {x}_{2}-{x}_{1}\hfill & \hfill {y}_{2}-{y}_{1}\hfill \\ \hfill {x}_{3}-{x}_{1}\hfill & \hfill {y}_{3}-{y}_{1}\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill \\ \hfill {x}_{B}-{x}_{1}\hfill & \hfill {y}_{B}-{y}_{1}\hfill \end{array}\right]

(50)

and

\mathit{b}=-\frac{1}{2}\left[\begin{array}{c}\hfill {r}_{2}^{2}-{r}_{1}^{2}-{R}_{2}^{2}+{R}_{1}^{2}\hfill \\ \hfill {r}_{3}^{2}-{r}_{1}^{2}-{R}_{3}^{2}+{R}_{1}^{2}\hfill \\ \hfill \vdots \hfill \\ \hfill {r}_{B}^{2}-{r}_{1}^{2}-{R}_{B}^{2}+{R}_{1}^{2}\hfill \end{array}\right]

(51)

with {R}_{b}^{2}={x}_{b}^{2}+{y}_{b}^{2}. The weighting matrix *W'* is given by:

\mathit{W}\prime =\mathsf{\text{diag}}\left(\frac{1}{{\sigma}_{\eta 2}^{4}},\dots ,\frac{1}{{\sigma}_{\eta B}^{4}}\right).

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 {\sigma}_{{\widehat{\tau}}_{1,b}}^{2} is determined via the linearization^{g} or the likelihood method. This TOA variance is transformed into a pseudo-range variance {\sigma}_{\eta b}^{2} by a multiplication with *c*^{2}. If no information about the estimation error *η* is available, the weighting matrices correspond to the identity matrix *I*(no weighting at all).

The Cramer-Rao lower bound (CRLB) provides a benchmark to assess the performance of the estimators [14]:

\mathsf{\text{CRLB}}\left(\mathit{p}\right)=\sum _{d=1}^{2}{\left[\mathit{I}{\left(\mathit{p}\right)}^{-1}\right]}_{dd},

(52)

where

\mathit{I}\left(\mathit{p}\right)=\mathit{J}{\left(\mathit{p}\right)}^{T}\mathit{W}\mathit{J}\left(\mathit{p}\right)

(53)

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.

The positioning accuracy depends on the geometry between the ROs and the MS and, thus, varies with the position *p*. This effect is called geometric dilution of precision (GDOP) [22, 25]. In order to separate the influence of the geometry from the influence of the estimation errors *η* on the positioning accuracy, it is assumed that all pseudo-ranges are affected by the same error variance {\sigma}_{\eta}^{2}=1, i.e., *W*= *I*. Given this assumption, the GDOP is the square root of the CRLB:

\mathsf{\text{GDOP}}\left(p\right)=\sqrt{\mathsf{\text{CRLB}}\left(\mathit{p}\right){|}_{\mathit{W}=\mathit{I}}}.

(54)

### 5.2 Numerical results

In the following, the overall performance of the proposed system concept using soft information is evaluated. For this purpose, two scenarios with different GDOP as shown in Figure 5 are considered. The ROs are denoted by black circles and the GDOP is illustrated by contour lines. For both scenarios, *B =* 4 ROs are located inside a quadratic region with side length \sqrt{2}R, where *R = 2T*_{
s
}*c* is the distance from every RO to the middle point of the region. For the first scenario, the ROs are placed in the lower left part of the region, which results in a large GDOP on average. The second scenario has a small GDOP on average since the ROs are placed in the corners of the region. For the communication links between the MS and the ROs, the same setup as described in Section 4.2 is applied. Furthermore, power control is assumed, i.e., the SNR for all links is the same. All results reported throughout this paper are for one-shot measurements.

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 (Δ*τ*_{2} ∈ [*T*_{
s
},*2T*_{
s
}]) and a two-path channel (*M* = 2) with small excess delay \left(\Delta {\tau}_{2}\in \left[\frac{{T}_{s}}{10},{T}_{s}\right]\right). For all channel models, the LOS delay *τ*_{1},_{
b
}for each link *b* is calculated from the true distance *d*_{
b
}(*p*). The excess delay of the multipath component Δ*τ*_{2} 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 {\widehat{\tau}}_{1,b} 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.

Different channel realizations are generated during the Monte Carlo simulations. Since different channel realizations result in different weighting matrices *W*, a mean CRLB is introduced,

\overline{\mathsf{\text{CRLB}}}\left(\mathit{p}\right)=E\left\{\sum _{d=1}^{2}{\left[\mathit{I}{\left(\mathit{p}\right)}^{-1}\right]}_{dd}\right\},

(55)

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*.

The simulation results are shown in Figure 6. There are eight different graphs (6a, b, c, d, e, f, g, h) arranged in an array with two columns and four rows. In the first column, the results for the simulations over SNR are shown. The second column contains the results for the simulations over space at 30 dB. In each row, the results for a fixed simulation setup are illustrated. All graphs show the root mean squared error (RMSE) of \widehat{\mathit{p}} normalized with respect to *d*_{
s
}*= cT*_{
s
}for positioning without soft information ("wo"), with soft information from the likelihood method ("like"), and with soft information from the linearization method ("lin"). The square root of the mean CRLB (normalized with respect to *d*_{
s
}), which is denoted simply as CRLB in the following, is plotted for comparison ("crlb"). Curves labeled with "L" were obtained for the first scenario with large average GDOP, and curves labeled with "S" were obtained for the second scenario with small average GDOP.

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., [24]).

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.

As mentioned in Section 2.2, PSO does not assure global convergence. For both two-path channels, PSO sometimes converges prematurely. In most of these cases, it converges to a boundary of the search space, such that the premature convergence can be detected (outage). In Figure 7, the outage rates are shown for both two-path channels: The dashed lines (i) and (iii) denote the probability that the delay estimation fails for one RO and the solid lines (ii) and (iv) denote the probability that two or more ROs fail. If the delay estimation fails for one RO, the position of the MS can be determined nevertheless since only three ROs are necessary for positioning in two dimensions. Only if two or more ROs fail, the position estimation fails, too. By adding more ROs, the outage rate for positioning can be decreased to an arbitrary small amount. The outage rates for the two-channel models differ significantly. For the two-path channel with large excess delay (Δ*τ*_{2} ∈ [*T*_{
s
}, 2*T*_{
s
}]), the outage rates (i) and (ii) are negligible. In contrast, the outage rates (iii) and (iv) for the two-path channel with small excess delay \left(\Delta {\tau}_{2}\in \left[\frac{{T}_{s}}{10},{T}_{s}\right]\right) are quite high at low SNR but decrease significantly with increasing SNR. The smaller the excess delay is, the higher is the probability that PSO converges prematurely.