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Unified performance measures in network localization
EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 48 (2018)
Abstract
With the evolving Internet of Things, locationbased services have recently become very popular. For modern wireless sensor networks (WSNs), ubiquitous positioning is elementary. Hence, the demand of everlasting and lowcost sensor nodes is rapidly increasing. In terms of energyefficiency, received signal strength (RSS)based direction finding is a prospective approach providing location information in lowpower sensor networks. Unfortunately, RSSbased direction finding is, as radiobased localization is in general, prone to multipath propagation of the wireless channel. Therefore, the impact of multipath fading as well as all other error source have to be modeled correctly and have to be considered in the design of a locating WSN.
In this paper, we derive the classical CramérRao Lower Bound (CRLB) for RSSbased directionofarrival (DOA). The drawbacks of the classical CRLB and its influence on the optimal network topology are discussed. The CRLB indicates that the minimum variance unbiased estimator (MVUE) does not exist for the problem of RSSbased DOA due to the nature of its measurement function. Hence, beyond the CRLB, we derive performance metrics for the maximum likelihood estimator (MLE) and compare position estimation errors for the MVUE and the MLE for different network topologies. Since both approaches, the CRLB and the maximum likelihood (ML) limits, are not capable of handling ambiguities, we introduce another measure for the variance of a measurement and its corresponding position estimate based on information theory. This way, the amount of information for a set of RSS measurements can be quantified exactly, even in the case of ambiguous probability densities. Thus, the proposed technique gives a holistic view on the information obtained from sensor measurements which can be utilized for network topology optimization.
1 Introduction
The use of wireless sensor networks (WSNs) is rapidly increasing these days. Also, new aspects of WSNs, like energyefficiency and locationawareness, are gaining more and more attention. There is a vast number of applications for locationaware WSNs, e.g., smart metering [1], collision avoidance [2], or animal tracking [3], just to name a few of them. In all of these examples, the sensor information is almost meaningless without any position information. Hence, localization is a core feature of today’s sensor networks [4].
Radiobased localization is one of the most popular techniques to provide location information in [5]. There is one thing all radiobased localization systems have in common: the wireless radio channel. Precise localization would be an easy challenge without its impairments of the wireless propagation channel. To design locating WSNs in an optimal way, these impairments have to be modeled in order to analyze the performance of the network localization. A straightforward approach is to model the errors of each sensor and combine the error distributions of all sensors with regard to the geometry of the WSN [6]. Estimation theory allows to compute the expected variance for the sensors evaluating the CramérRao Lower Bound (CRLB). In a similar manner, the position CRLB can be computed for noisy sensor estimates, e.g., ranging and direction finding, for a location estimate within a sensor network [7].
The effect of multipath and nonlineofsight propagation on the precision of the position estimates has been extensively analyzed in [8]. Shen et al. consider the use of prior knowledge of the user’s position [8]. Prior knowledge may be integrated with the informationtheoretic performance metric presented in this paper. As shown by [8], the use of prior information is reasonable, as it may be provided by e.g. maps. Hence, performance metrics for WSNs should be capable of incorporating prior information. In [9], the impairments due to multipath propagation are characterized. Multipath fading results from destructive and constructive interference of different propagation paths at the receiver. When ranging is considered with narrowband systems, multipath components can not be resolved. This inherently leads to biased estimates. The same is true for the spatial domain in received signal strength (RSS)based direction finding, where multipath propagation affects the accuracy of directionofarrival (DOA) estimates. In [10], anchored and anchorfree localization is compared. According to [10], accurate position estimates require for a good local geometry. Furthermore, in their conclusion, the authors state that a tool for the identification of the bottleneck in localization would be useful. The informationtheoretic approach, presented in this paper, provides exactly that type of information, i.e., quantifying the information gain of additional sensor nodes. In many fields, there has been a revival of information theory. Informationtheoretic measures are used to optimize MIMO radar waveforms [11]. The loss of subNyquist sampling has been characterized applying information theory [12, 13]. Within the context compressive sensing, fundamental limits [14] and bounds for kernelbased time delay estimation [15] have been derived based on information theory.
Although there are many manifestations of radio localization systems, the technique addressed in this paper is based on RSS measurements, more specifically RSSbased direction finding [16]. Although we focus on RSSbased DOA estimation, the presented framework is applicable in general. Hence, it may be applied to cooperative ranging or timedifferenceofarrival (TDOA)based localization systems. RSSbased DOA is a promising approach for several reasons. Field strength measurements do not demand for complex signal processing. In contrast to timeofflight measurements, RSS measurements do not require exhaustive synchronization of the sensors. RSSbased DOA is a rangefree approach. Hence, it does not depend on prior knowledge of the emitted power of the transmitter or the path loss exponent of the wireless channel [17]. A representative of such a RSSbased localization system is the BATS^{Footnote 1} system [18]. The BATS system is an energyefficient sensor network designed for wildlife monitoring, more precisely the tracking of bats. Within the WSN, position information is obtained from RSS measurements [19] that are gathered in the sensor network and stored on a central computing unit, cf. Fig. 1.
For optimal deployment of the wildlife monitoring network in the woods, a performance analysis of the network localization for a given topology is essential. Therefore, we provide a position CRLB for RSSbased DOA for the antenna array utilized in the BATS project [20]. Due to the nonlinearity, more specifically the missing curvature [21] of the measurement function for the problem of RSSbased direction finding, applying the classic CRLB results in an unbounded variance for some signal directions. This unbounded variance indicates the minimum variance unbiased estimator (MVUE) does not exist. Besides that, RSSbased DOA, as many other fundamental estimation problems, such as DOA estimation, frequency and phase estimation, involve parameters that are of a cyclic nature. There exist modifications of the CRLB that address estimation of periodic parameters [22, 23]. However, periodicity is still on the downside of the CRLB.
As a consequence, a performance metric based on the error distribution of the maximum likelihood estimator (MLE) is derived in [24]. The Bayesian estimation approach allows to integrate noninformative priors [21] in order to cope with cyclic parameters. This approach reveals more consistent results in terms of the position’s mean squared error (MSE). In [20, 24], bounds for RSSbased DOA have been derived for an MVUE and a biased estimator, the MLE, showing completely different results. Results from estimation bounds significantly affect the optimization of network topology in local positioning [7]. As shown in [24], choosing another estimation bound entirely changes the constraints of the sensor arrangement.
However, the maximum likelihood (ML) approach still lacks the capability to handle ambiguities that inherently arise in RSSbased DOA. Thus, we introduce a novel approach to characterize the information content for multimodal measurement error distributions. Based on information theory, the entropy of sensor measurements is quantified. Finally, the entropy of a set of measurements retrieved by a WSN is related to a corresponding variance of unimodal measurement distribution. Information theoretic measures, i.e., entropy and mutual information, allow for performance assessment of locating WSNs in a holistic manner incorporating local precision and the impact of ambiguities. Utilizing the proposed informationtheoretic performance metric allows for network topology optimization in heterogeneous sensor networks with arbitrary measurement error probability density functions (PDFs).
The main contributions of this paper are summarized as follows.

A review of RSSbased direction finding utilizing coupled dipole antennas is given.

We present the classical CRLB for RSSbased DOA. Furthermore, the impact of network topology, i.e., the arrangement of the sensor nodes, on the position resolution is assessed for the MVUE by applying the 2D position CRLB.

Estimation errors for the MLE are derived for RSSbased direction finding. Again, the influence of the topology of the WSN is analyzed. The error bound for the MLE is more consistent compared to the MVUE and features a significantly smaller mean squared position error.

Both findings above give indications of position resolution for the given networks. However, inherently arising ambiguities in RSSbased DOA are not considered. Therefore, we introduce informationtheoretic measures for the analysis of direction finding in WSNs. These informationtheoretic measures allow for joint analysis of resolution and ambiguities. Hence, they provide a holistic view on the performance of the locating WSNs.

We provide a design tool, based on mutual information, for the comparison and optimization of different network topologies considering different sensors types (e.g., unambiguous and ambiguous DOA sensors). The use of mutual information allows to exactly quantify the loss of information due to the presence of ambiguities, whereas the CRLB gives no useful information in that comparison.
The remainder of this paper is organized as follows. In Section 2, a brief review of RSSbased direction finding is given. Section 3 elaborates on the classic CRLB deriving the CRLB for RSSbased direction finding and corresponding position estimates with concluding remarks on network topology as well as its drawbacks. Performance limits of MLEs for RSSbased DOA are proposed in Section 3.2. These are compared to the result from classical CRLB analysis. In Section 4, a unified approach to quantify measurement uncertainty based on information theory is introduced and illustrated for multimodal measurement PDFs. A network design tool for optimal node arrangement is presented in Section 5. Furthermore, the impact of ambiguous sensors in different network localization scenarios is discussed. Section 6 concludes this paper.
2 A primer on RSSbased direction finding
For this paper, we consider RSSbased DOA estimation applying coupled dipole antennas [24]. Furthermore, it is assumed that localization takes place in the horizontal plane orthogonal to the two dipoles. Presuming a perfect linear dipole array, the radiation pattern of the dipoles in the horizontal plane (i.e., θ=90°) is constant over all impinging signal angles in azimuth ϕ∈ [ 0,2π]. Hence, the radiation pattern for N dipoles at a distance of
is given by the array factor [25]
where d_{i} are the corresponding distances of the dipole elements, λ is the wavelength, and c_{i} is the coupling factor. Considering only two dipoles at distances d_{0}=0 and d_{1}=d and θ=90°the array factor reduces to
For the considered antenna array, the dipoles are coupled in phase. Hence, the radiation pattern is given by
We define the radiation power patterns G(ϕ) (in dB) by
With two identical antennas rotated by 90° towards each other, the gain difference function is expressed by
The patterns described above are sketched in Fig. 2. Gain patterns for the rotation angles of 0° and 90°, respectively, are depicted. DOA estimation yields minimum variance at a high gradient of the gain difference function, as shown in the next section.
The RSS at a receiver a for a transmitted signal with power P_{TX} can be computed as follows
with L denoting the bulk path loss. G_{TX} and G_{a}(ϕ) are transmit and receive antenna gain, respectively. When considering a single signal source, i.e., no multipath propagation, the received signal strength difference is given by
due to the fact that both channels are stimulated by the same transmit power and exhibit equal path loss. Thus, the gain difference function does not depend on transmit power and path loss. Hence, it maybe estimated without prior knowledge of the the path loss exponent and the power emitted by the transmitter. This fact is, in contrast to rangebased localization based on RSS, a major benefit of RSSbased DOA estimation.
3 Conventional methods for performance assessment
Estimation theory is a core essential of many modern signal processing system. Those include, but are not limited to, radar, image analysis, communications, and localization. In estimation theory, a parameter θ is inferred from a set of measurement z. In parameter estimation, two basic concepts have to be distinguished [21]: Deterministic parameter estimation Parameters are assumed to be deterministic but unknown in classical estimation theory, whereas in Bayesian estimation theory, the parameter, that is to be estimated, is assumed an to be a random variable (RV). Thus, the data, in classical estimation, is described by a PDF of the form p(zθ). Bayesian estimation In contrast to that, in Bayesian estimation theory, is described by joint PDF p(x;θ)=p(zθ)p(θ) that is composed of the measurement likelihood p(zθ) and the prior PDF p(θ). Hence, Bayesian estimation theory allows to incorporate prior knowledge on the parameter θ.
In the sequel, the CRLB for classic parameter estimation is discussed. For the Bayesian approach, we derive an ML error bound for powerbased direction finding.
3.1 CramérRao Lower Bound
In parameter estimation, not only the estimate itself is of valuable interest, but also the distribution of its errors. Hence, it is desirable to quantify the deviation of the estimate from the true parameter value. Basically, estimators should be unbiased on the one hand. On the other hand, they should provide a minimum variance. For the MVUE, the CramérRao Lower Bound gives a lower bound on the variance of an estimator with a zeromean error [26].
When recalling the alternative form of the CRLB, [21] its respective limit on the variance is defined by
for a parameter estimate \(\hat {\theta }\) observed by noisy measurements r. In case of an unknown parameter, θ of a deterministic signal observed in additive white Gaussian noise (AWGN) by a series of observations
the general CRLB above can be strapped down to
where w∼N(0,σ^{2}). Equation 10 simplifies to
for a single observation in presence of a AWGN.
3.1.1 DOA CramérRao Lower Bound
The RSS measurements retrieved by the WSN are impaired by noise resulting from fading effects of the wireless propagation channel. Results from a channel measurement campaign [27] show that noise on RSS difference measurements is lognormal distributed with a constant variance over different ranges between transmitter and sensor node. When observing RSS difference measured in dB, the CRLB estimating DOA by RSS difference measurements is expressed by [20]
The resulting CRLB for RSSbased direction finding for the BATS sensor node is depicted in Fig. 3. It can be easily seen that the variance of the DOA estimate strongly depends on the direction of the impinging signal. Moreover, the variance is unbounded for signal directions near \(\phi \approx k\frac {\pi }{2}\). This disadvantageous behavior can be explained by the gradient of the gain difference function ΔG(ϕ) which approaches zero for the corresponding locations. Equivalently, it can be stated that the measurement function has no curvature [21] at the considered positions. Summarizing the results so far, the variance of unbiased DOA estimates inherently depends on the direction of the signal source. Hence, position estimation errors will also be dependent on the location of the tracked object.
3.1.2 Position CramérRao Lower Bound
The CRLB for DOA estimates has been derived in the section above. With the results for the variance of the DOA sensors, the variance of the corresponding position estimate can be computed applying the position CRLB [20, 28, 29] for a set of sensor nodes. In the scope of the paper, DOA estimates are considered in the horizontal plane. Hence, position state space is 2D. However, the concepts presented here may be easily extended to 3D position space. The covariance of a position estimate
is considered. As position estimation in general is vector parameter estimation, the Fisher information matrix (FIM) [21] can be stated as
ϕ denotes a vector of DOA measurements and p(ϕx) is the measurement likelihood for a given location x=[x,y]^{T} of the tracked object.
The joint likelihood for all sensors is given by the product of the likelihoods for the DOA observations ϕ_{k} at all nodes of the network. The loglikelihood is expressed by
where the subscript distinguishes the sensor nodes contributing to the estimate. For DOA measurements in presence of noise, the measured angles ϕ for a given position x are given by
with the measurement function
where Δx_{k}=x−x_{k} and Δy_{k}=y−y_{k}. The noise processes w are assumed to be mutually independent Gaussian random variables. The likelihood function for a single DOA measurement p(ϕ_{k}x) neglecting some scaling factors can be written as
It has to be noted that the variance \(\sigma _{\text {DOA}}^{2}(\mathbf {x})\) of the DOA estimates is dependent on the angle of the impinging signal. Thus, it is a function of the location of the transmitter. We are now able to compute the FIM merging Eqs. (14), (15), and (18) and resulting in
with d_{k}=∥x−x_{k}∥_{2} for position estimation from DOA obtained from RSS measurements. The position estimation error bound for DOAbased localization in a WSNs is retrieved by evaluating the trace of the inverted FIM:
With help of the derived CRLB for network localization utilizing RSSbased direction finding, different network topologies can be assessed with respect to their localization performance Fig. 4.
3.1.3 Optimal sensor node arrangement
The variance of direction estimates significantly changes with the angle of the impinging signal with respect to the orientation of the antenna array. Hence, not only the node positions influence the localization performance, but also the orientation of the sensor nodes is of importance. To elaborate the impact of node orientation on the location errors, two network configurations with identical node positions but different orientation of receive antennas have been defined. For the analysis, the node orientation is defined as the rotation of the receive antenna array in the horizontal plane with respect to the xaxis of the Cartesian coordinate system. The optimal rotation angles can be found by a parameter search. Optimal node orientation is found by minimization of the mean CRLB over the considered area of interest. The mean CRLB for the two examined areas is depicted for different orientation angles in Fig. 5. Orientation of the sensors is 0° and 35° for the tested networks configurations 1 and 2, respectively. In both cases, the nodes have a spacing of 50 m. In total, four nodes are used being arranged in a quadratic shape. A noise variance of σ_{ΔRSS}=5 dB is assumed for the cross fading of the two antennas of the sensor nodes.
The position CRLB for both networks is visualized in Fig. 4a, b. For the first network, the position error bound is very inhomogeneous. This network yields promising performance for the center of the area of interest. But on the other hand, the position resolution is poor in the outer areas. Having a look at network 2, the errors are distributed in a much more uniform way compared to network 1. Therefore, network 2 features good average results for the whole area of interest. Simulation results are shown in Table 1 for the two network configurations. Average position errors are computed for networks 1 and 2 considering two different regions x,y∈ [ 10,40] and x,y∈ [ −10,60]. These numbers are in line with visual representation of 2D error distribution presented in Fig. 4a, b.
In sum, DOA dependent noise variance causes the position errors to be position dependent. This, in consequence, makes the network localization sensible to node orientation. In addition to the fact that node orientation significantly affects position errors, also the area of interest has to be considered thoroughly for the assessment of localization performance and the design of locating sensor networks.
3.1.4 Results and discussion
Estimation variance significantly depends on the direction of the signal source when considering the MVUE for DOA estimation. Reflecting the results from above, topology optimization of the WSN is not only subject to the antenna gain patterns and its orientation. It also significantly depends on the area of interest for the object that is to be tracked. Prior knowledge of the spatial probability density of the tracked object allows for an optimal design of the locating network which results in a decrease in position estimation errors.
However, due to missing curvature of the RSSbased DOA measurement function, the MVUE yields an unbounded variance for angle estimates at multiples of 90°. This indicates that the MVUE may not even exist. In the next section, we will have a look on the maximum likelihood estimator. Furthermore, the classic CRLB can not handle ambiguities, which inherently exist in RSSbased DOA.
3.2 Bayesian estimation error bound
In general, the MLE \(\hat {\theta }_{\text {ML}}\) maximizes the likelihood function for a parameter of interest θ observed by noise measurements r [30]
We further assume that the parameter of interest is estimated for a particular realization of a random variable θ. Taking the logarithm yields to
For AWGN, we can write the following condition
where, in order to equate to zero, at least one of terms of the product has to be zero. As in general \({\frac {\partial }{\partial \theta }}g(\theta) \neq 0\), the inverse function g^{−1} maximizes the likelihood function. Therefore, in the considered case, the MLE is simply given by the inverse of the measurement function.
3.2.1 Computation of posterior density
In this section, the posterior density for a RSSbased angle estimate is computed. Again, we assume field strength difference measurements to be impaired by AWGN
where ϕ is the azimuth angle of the impinging signal and w is a AWGN process with \(\mathcal {N}(0, \sigma _{r}^{2})\). The likelihood function for the DOA estimation problem is given by [24]
Recalling (12) (cf. [20]), the CRLB for a DOA estimate from a single observation of two field strength measurements states as
Noting again, the MVUE has an unbounded variance for ϕ approaching multiples of 90°as shown as dashed line in Fig. 3. The variance to be unbounded makes sense due to the property of the gain difference function ΔG. The function ΔG has no curvature [21] at multiples of 90°. From an applications point of view, it is not reasonable to make use of an estimator with unbounded variance. Moreover, the unboundedness of the variance for the MVUE indicates that the estimator does exist. Allowing the estimator to be biased allows for less variant estimates.
In order to assess the performance of the MLE, we compute the full posterior PDF that can be derived by applying Bayes’ theorem:
Due to point and axial symmetry of the gain difference function, ΔGP(ϕ) may be limited to ϕ∈[0,90°] without loss of generality. As there is no prior knowledge on the distribution of the directions of a signal source, the prior density is given by a uniform distribution
This PDF is a noninformative prior [21]. The marginal likelihood P(r) for the RSS measurement results from
For the antenna considered in the scope of this paper (cf. Fig. 2), the marginal density is shown in Fig. 6a. Referring to (28), the posterior is computed by normalizing the likelihood with the marginal density for the field strength measurements. Figure 6b depicts the posterior PDF for RSSbased DOA. Exemplary posterior PDFs are sketched for some specific field strength measurements r∈ [ 0 dB,12 dB,18 dB,24 dB].
3.2.2 DOA estimation errors
With the posterior density of an angle estimate, the MSE for RSSbased direction finding is derived in this section. Considering the MLE introduced in (24), the inverse function of the stated problem is
where ΔG(r)^{−1} is the inverted gain difference function for the considered antenna array. However, there are some remarks on the measurement function f=ΔG(ϕ). Measurement values r are limited to r∈ [ min(ΔG(ϕ)), max(ΔG(ϕ))]. To transform arbitrary measurement values r∈ [−∞,∞] into parameter space, the definition of the inverted function ΔG^{−1}(r) function needs to be redefined:
With the posterior PDF, the MSE is computed by is defined by
The results for the DOA root mean squared error (RMSE) are shown in Fig. 7. Remarkably, the variance of the MLE is bounded for all azimuth angles ϕ. Furthermore, the MLE features a quite homogeneous distribution of DOA errors for all signal directions. In conclusion, there is no significant impact of the direction of the signal source on the variance of DOA estimates, though the MLE is biased.
3.2.3 Position estimation errors
Having derived the MSE for direction estimate, we can now evaluate the 2D position errors for a WSN featuring RSSbased DOA. As positioning errors are considered, the MLE position error bound describes an expectation on the MSE of the position estimator [21]. The expected MSE for a position estimate is denoted by
Recalling Eqs. (16) and (17), similar to the position CRLB, given by the error propagation law, the covariance matrix is expressed by [31]
where I denotes the identity matrix. Finally leading to
for location information obtained from RSSbased DOA measurements, with x_{2} denoting the euclidean norm.
In its compressed form, the position estimation error is expressed by
for DOAbased network localization when applying ML estimation.
3.2.4 Comparison of network topologies
The derivation of the position error bound for RSSbased network localization is now utilized to compare the performance of different network topologies for the ML approach. The DOA estimation error mse_{θ} is computed according to Eq. (33). Mean square position errors are evaluated for three different WSN topologies. The locating networks consist of four nodes and have a node spacing of 50 m. Receive antenna are being rotated by 0°, 30°, and 45° for the examined network topologies 1, 2, and 3, respectively. Sensor nodes have been arranged in quadratic shape. All networks have been examined for an area of interest of 50×50 m.
As elaborated in Section 3, the network topology, more specifically the rotation of the receive antenna arrays, is a crucial parameter in terms of position estimation errors. This holds true for the MVUE. However, results for the ML approach are different. In Fig. 8, the resulting position estimation errors for all the three networks are given. For the given rotation angles of the antenna arrays, there is no substantial difference in average RMSE. Results are shown in Table 2. The average RMSE for the MLE is constant for different node orientation angles, whereas the average RMSE significantly depends on the position of the tracked object for the MVUE. In conclusion, the rotation of the nodes is negligible for the design of the WSN when considering the MLE. These results are in contrast to those for the MVUE, where the RMSE is highly dependent of the node orientation.
3.2.5 Results and discussion
With the MLE, we have shown that the topology of the locating WSN does not have a significant impact on the positioning. These findings are in contrast to the results of the CRLB for the MVUE. For snapshot localization, the MLE should be preferred since it features smaller RMSEs in any case. Yet, unbiasedness comes at the cost of an increasing variance resulting in a larger MSE for positioning (cf. Fig. 9). Considering recursive filtering, one might prefer the MVUE over the MLE as the higher variance can be averaged out applying motion models. But still, the analysis of the MLE lacks the capability of considering ambiguous measurements. In the next section, we introduce informationtheoretic measures to provide an insight into the gain of information retrieved from a single measurement obtained from a WSN.
4 Informationtheoretic localization performance metrics
Despite the success of information theory in communications, information theory has not caught much attention in localization and navigation in the past, except for [32, 33]. Recently, there has been a revival of information theory in many fields. Informationtheoretic measures, like the mutual information, have been utilized to optimize MIMO radar waveforms [11]. The loss of information due to subNyquist sampling has been determined applying information theory [12, 13]. Currently, there are big advances in the field of compressive sensing. Lately, fundamental limits in compressed sensing [14] and bounds for kernelbased time delay estimation [15] are derived.
However, classical estimation theory is still in the focus of performance analysis today. Nevertheless, the CRLB is a local measure for the variance of an estimator. Considering a uniform linear antenna array, the CRLB [34] states that
where d is the distance between the antenna elements. In consequence, distance between elements should be maximized. However, ambiguities arise for a spacing of more than λ/2. These need to be considered in the design of sensor nodes. As the CRLB is a local measure, it is blind for ambiguities. Hence, the CRLB is only applicable when further constraints are introduced. Information theory, in contrast to the CRLB, has a unified view on the information gained from a set of sensor measurements considering precision and ambiguities.
The entropy H(X) of a discrete random variable X is defined by the expectation value of its information given by [35, 36]
Maximum entropy is achieved by a uniform probability distribution for a given number of discrete values. Physical quantities, as in network localization, generally have no discrete values. Measurement and state variables are continuous. Therefore, the limit of the entropy from discrete values to a continuous range would diverge. To circumvent this, the differential entropy is defined [36]
In contrast to entropy, the differential entropy can be negative and the differential entropy of a constant diverges. For an infinite interval (i.e., x∈]−∞,+∞[) and fixed variance, the normal distribution maximizes the differential entropy, whereas for a fixed interval (i.e., x∈ [ a,b]), the uniform distribution is maximum.
For illustration, we consider some wellknown distributions: uniform, normal, and von Mises distribution (Table 3). The von Mises distribution [37] is defined over ϕ∈ [ −π;+π]
where μ is the mean value and κ defines the concentration of the distribution. I_{0} is the modified Bessel function of order 0. m denotes the number of modes of the distribution and is understood as some scaling factor for the von Mises distribution. The concentration is reciprocal to dispersion. Hence, \(\frac {1}{\kappa }\) is comparable to the variance. Large concentration implies low variance and viceversa. For κ≫0, the von Mises distribution approaches the normal distribution with . Therefore, it is a good approximation of the wrapped normal distribution. Its differential entropy is given by [37]
where I_{1} is the firstorder modified Bessel function.
In Fig. 10, von Mises distributions (blue and orange) with different κ are depicted. It can be seen that with rising κ, i.e., higher concentration around the mean value, entropy, denoted by h, declines. Actually, the standard deviation of the blue density is three times larger than the deviation of orange density. κ=0 (uniform) yields maximum entropy. Furthermore, a repeated and scaled version (green line) of the von Mises distribution is sketched in Fig. 10. The green distribution has locally the same standard deviation as the orange one. Thus, in terms of the CRLB, both, the orange and the green, yield the same precision. However, when considering the total information gained from the measurement, the blue and the green distribution are comparable as they feature the same entropy. Actually, the multimodal PDF can be seen as ambiguous measurements. Even though a single lobe of a multimodal distribution features a small variance on a local scale, the entropy is the same compared to a broader lobe of a unimodal distribution.
To conclude, entropy does not give a hint if ambiguities are apparent. Neither does the CRLB. The variance inferred from the CRLB is a local measure. Hence, it does not give any information on the modality of the distribution. The mutual information only states how much information was gained by a measurement regardless if the resulting PDF is a broad unimodal Gaussian or a set of multiple sharp Gaussian distributions. Therefore, the CRLB is a local measure and the entropy quantifies the overall information gain. In fact, both approaches lead to the same result if the underlying distribution is a unimodal Gaussian.
5 Network topology optimization
In this section, we utilize the mutual information as a criterion for network topology optimization. For the illustration, we use a simple measurement model for the DOA estimation. The bearing measurements are assumed to be impaired by AWGN and ambiguous. Furthermore, we define the variance of the DOA measurements to be independent of the direction of the impinging signal. As the estimated angle \(\hat {\phi }\) is limited to the interval ϕ∈ [ −π;+π], its PDF is to be described by a wrapped normal distribution. Commonly, the wrapped normal distribution is approximated by the von Mises distribution.
For the rest of the section, we consider two different measurement likelihoods:

1
The ambiguous DOA sensor:
Provides multimodal DOA measurements with constant variance. Ambiguities arise due to point and axial symmetry exactly the same way as they arise for the antenna considered in the BATS project.

2
The unambiguous DOA sensor:
Provides unimodal DOA measurements with constant variance over angles. This unambiguous sensor, that features locally the same variance as the ambiguous one, is used as a reference. The ambiguous sensor is benchmarked against the unambiguous sensor.
Considering the BATS DOA sensor, the likelihood is state as
The likelihood of the unambiguous DOA sensor is given by
The PDFs for both of the estimators are depicted in Fig. 11. For reference, the entropy of a uniform U(−π,π) is h=2.65. It can be easily seen that, considering the entropy of both sensor types, the unambiguous sensor has a significantly lower entropy. Thus, the unambiguous sensor provides more information. This is quite obvious, as the unambiguous sensor perfectly resolves the ambiguities arising in the BATS DOA sensor. On the other hand, with respect to the CRLB, both yield the same performance.
5.1 A comparison of network topologies
The two considered DOA sensors are examined in two network localization scenarios. The impact of ambiguities in BATS DOA sensor on the localization performance is examined in the following. Therefore, the BATS DOA sensor is compared to an unambiguous sensor with same local precision as described above. It can be shown that even for a small number of sensor nodes, the ambiguities arising on sensor level can be almost completely resolved on position level.
Network localization scenario 1 consists of two sensor nodes placed at [ 0,25] and [ 30,25]. The object to be tracked resides at [ 25,35]. For network scenario 1, the likelihoods for the unambiguous DOA sensor and the BATS DOA sensor are depicted in Fig. 12. The depicted likelihoods involve all sensor nodes, i.e., the shown likelihood is the product of the likelihoods of the individual sensors. The likelihood is centered around the true position for the unambiguous bearing measurement, whereas in case of the ambiguous sensor, the likelihood PDF features four modes. In consequence, a tracking filter would have to be capable of propagating multiple hypotheses. Hence, in such a case, particle filters would have to be used instead of basic Kalman filters.
For the evaluation, a single snapshot measurement is considered. The prior information available is compared to entropy after the measurement. If there is no prior information available, we assume a noninformative prior, which relates to a 2D uniform distribution. Over an area of interest of 70×70 m, the entropy of a uniform is h=12.26. Entropy is computed for the networks for the ambiguous DOA sensor and the unambiguous one. The information gain of such a set of DOA measurements is given by the mutual information. Mutual information is computed from entropy and conditional entropy
For the network localization scenario 1, the corresponding entropy and mutual information is given in Table 4. As expected, ambiguous DOA has a larger entropy than unambiguous DOA.
Now, two additional nodes are placed at [ 0,50] and [50,0] for the second network localization scenario. The true position of the object is, as before, at [ 25,35]. Likelihoods for the ambiguous and unambiguous DOA sensor are illustrated in Fig. 13. As stated before, all sensors depicted in the figure are involved in the computation of the likelihood shown Fig. 13. It can be clearly seen that the true position becomes the dominant hypothesis for the ambiguous bearing sensor (cf. Fig. 12). On the other hand, for the unambiguous sensor, the position PDF becomes more diffuse. When more sensors are used, the impact of the ambiguities gets less important. Furthermore, the gain in information for additional sensors is not very significant when unambiguous DOA sensors are utilized. Effectively, additional sensors do not provide as much information as the first two sensors (cf. Table 4). In conclusion, for overdetermined sensor networks, ambiguous sensors are an option as the impact of ambiguities becomes negligible when a larger number of sensors is deployed. Additionally, accepting ambiguities commonly results in higher precision for the individual modes (cf. DOA estimation applying linear antenna arrays).
5.2 From entropy to localization precision
One disadvantage of the mutual information is that it is not as intuitive as a standard deviation. Thus, we are interested in computing variance from the entropy. The relation for variance and entropy assuming a normal distribution is expressed by
The variance of a symmetric 2D normal distribution (i.e., Σ=σ^{2k}, with Σ being of size k × k) is computed as follows
for a given entropy h. Nevertheless, it has to be noted that this expression is only a good approximation when standard deviation is small compared to the size of the area of interest. Aside from that, the complete mutual information is assumed to arise from a single 2D Gaussian. For a multimodal distribution, the entropy inferred variance is larger than the variance of the individual modes as the above stated transformation is only valid for unimodal Gaussian distributions.
In Table 4, standard deviation for the two example networks is given for the unambiguous and ambiguous DOA sensor, respectively. The standard deviation computed under the assumption of a unimodal 2D Gaussian proofs the conclusions drawn from Figs. 12 and 13. The impact of ambiguities is less important when using more sensor nodes. The total information gain is negligible for additional sensors in the case of unambiguous DOA.
5.3 Mutual information as a network design tool
In the section above, we have shown that the use of additional sensor nodes may mitigate the impact of ambiguities. Hence, it is desirable to compare different network topologies considering different sensor types. For the analysis, we consider the unambiguous and ambiguous DOA sensors introduced above. The design question to be answered is: How many nodes are required to realize a certain gain of information about the position of an object? For the sake of simplicity, we assume that all nodes are place equidistantly on a circle with fixed radius. Hence, the only parameters are the number of nodes and the sensor type.
Examples of network configurations for 2 and 4 nodes are given in Fig. 14a and b, respectively. For an area of interest of 70×70 m, the conditional entropy is computed for the PDF after the network localization measurement. We assume no prior information is available before the measurement. Mutual information is computed by (45) for all network topologies and sensor types under test. A comparison of the unambiguous and ambiguous DOA sensor is depicted in Fig. 15. It can be easily seen that utilizing an additional number of two sensor nodes allows to mitigate the negative impact of the ambiguities for the ambiguous DOA sensor.
Mutual information provides a tool to compare different network topologies including multimodal measurement PDFs, whereas the classic CRLB fails in this case. CRLB analysis would provide exactly the same results for the unambiguous and ambiguous DOA sensor, which is quite misleading. In contrast, information theory allows for exact quantification of the impact of ambiguities on the information that is gain on an object’s position. With the use of this design tool, based on mutual information, engineers are able to design the most effective networks for localization problems.
6 Conclusion
In this paper, different performance metrics have been analyzed and applied in the context of RSSbased direction finding. The first of those metrics is the CRLB. It has been shown that the CRLB strongly depends on the direction of the impinging signal for powerbased DOA estimation. Due to the nature of the DOA measurement function, the CRLB diverges for some angles. Hence, the MVUE is not a reasonable estimator in the case of RSSbased DOA.
Secondly, a performance metric has been derived for the MLE evaluating the posterior density for RSSbased DOA estimates. In this case, DOA estimation variance is nearly constant for all signal directions. Hence, the variance in position estimation is less dependent on the actual position of the tracked object. Furthermore, the MSE for the MLE is significantly smaller than for the MVUE. However, both the CRLB and the error bound for the MLE lack the capability of handling ambiguities.
In order to account for ambiguities, we have proposed a novel approach to assess the performance of localization systems that feature sensors with multimodal measurement likelihoods. Informationtheoretic measures are utilized to quantify the information gain of a set of measurements from multiple sensor nodes. With this unified approach, all aspects, i.e., local precision and multimodality, can be captured with a single measure: the mutual information. The presented informationtheoretic approach allows for unified optimization of location sensors and localization networks in effective way maximizing the mutual information. Moreover, mutual information could be used to quantify the loss when utilizing suboptimal estimators and for performance assessment of recursive filters. In total, informationtheoretic metrics provide a holistic view on the performance of tracking systems.
Notes
Dynamic Adaptable Applications for Bats Tracking by Embedded Communicating Systems, http://www.forbats.org/
Abbreviations
 AWGN:

Additive white Gaussian noise
 CRLB:

CramérRao lower bound
 DOA:

Directionofarrival
 FIM:

Fisher information matrix
 ML:

Maximum likelihood
 MLE:

Maximum likelihood estimator
 MSE:

Mean squared error
 MVUE:

Minimum variance unbiased estimator
 PDF:

Probability density function
 RMSE:

Root mean squared error
 RSS:

Received signal strength
 RV:

Random variable
 TDOA:

Timedifferenceofarrival
 WSN:

Wireless sensor network
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Funding
This work is funded by the German Science Foundation DFG grant FOR 1508, Research Unit BATS.
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All data are based on simulations. Scripts for data simulation and evaluation will be made available on demand for the review. Simulation data will be made available on the institutional website when the manuscript is published.
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The main contribution to this manuscript is by TN. MH contributed to the antenna design and development as well as the CRLB in RSSbased DOA. JT developed the initial ideas and the proposal of the BATS project. He continuously supported the project work with his expertise in navigation and localization. Furthermore, he did a thorough revision of the content published. All authors read and approved the final manuscript.
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Thorsten Nowak received his diploma in engineering from the University of Ulm, Germany, in 2009, and is currently working toward the Ph.D. degree at FriedrichAlexanderUniversität ErlangenNürnberg (FAU). In 2008, he joined Fraunhofer Institute for Integrated Circuits (IIS), where he was a research assistant at the Locating & Communication Systems Department within the Sensor Fusion & Event Processing Group. His focus was on multipath mitigation techniques, multisensor data fusion, localization systems, and RFID. Since 2013, he is with FriedrichAlexanderUniversität ErlangenNürnberg (FAU), working for the Department of Electrical, Electronic and Communication Engineering. He is a research assistant at the Insitute of Information Technology (Communication Electronics) and is involved in the Research Unit BATS—Dynamic Adaptable Applications for Bats Tracking by Embedded Communicating Systems funded by the German Research Foundation (DFG).
Multipath: He regularly serves in the program committee of the IEEE ICC and is a reviewer for the IEEE Systems Magazine, the International Journal of Microwave and Wireless Technologies, and the Elsevier Journals of Adhoc Networks and Computer Communications.
Markus Hartmann received his diploma in engineering, in 2009, from the University of Applied Sciences AmbergWeiden, Germany, and his master of engineering, in 2011, from the University of Applied Sciences Coburg, Germany. In 2010, he joined Fraunhofer Institute for Integrated Circuits (IIS), where he was a research assistant at the Locating and Communication Systems Department within the RFID and Radio System group. His research focus was on inductive lowfrequency localization technologies for a goal line decision system. He is currently working toward the Ph.D. degree at FriedrichAlexanderUniversity ErlangenNürnberg (FAU) in the area of localization based on phase and signal strength informations. His main focus is now on the performance evaluation and error analysis of field strength based localization approaches.
Jörn Thielecke received a PhD (Dr.Ing.) from the FriedrichAlexanderUniversität ErlangenNürnberg (FAU), Germany. In 1991, he joined Philips Kommunikations Industrie AG and in 1997 Ericsson leading a research group on radio access for cellular mobile radio communications. He was responsible for the Department of Communications at the Research Establishment for Applied Science (FGAN) from 2003 until 2004. Now, he is professor at FAU focusing on localization and navigation systems.
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Nowak, T., Hartmann, M. & Thielecke, J. Unified performance measures in network localization. EURASIP J. Adv. Signal Process. 2018, 48 (2018). https://doi.org/10.1186/s1363401805708
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DOI: https://doi.org/10.1186/s1363401805708