A hybrid global minimization scheme for accurate source localization in sensor networks
- Hamidreza Aghasi^{1}Email author,
- Hamidreza Amindavar^{2} and
- Alireza Aghasi^{3}
https://doi.org/10.1186/1687-6180-2011-81
© Aghasi et al; licensee Springer. 2011
Received: 8 March 2011
Accepted: 7 October 2011
Published: 7 October 2011
Abstract
We consider the localization problem of multiple wideband sources in a multi-path environment by coherently taking into account the attenuation characteristics and the time delays in the reception of the signal. Our proposed method leaves the space for unavailability of an accurate signal attenuation model in the environment by considering the model as an unknown function with reasonable prior assumptions about its functional space. Such approach is capable of enhancing the localization performance compared with only utilizing the signal attenuation information or the time delays. In this article, the localization problem is modeled as a cost function in terms of the source locations, attenuation model parameters, and the multi-path parameters. To globally perform the minimization, we propose a hybrid algorithm combining the differential evolution algorithm with the Levenberg-Marquardt algorithm. Besides the proposed combination of optimization schemes, supporting the technical details such as closed forms of cost function sensitivity matrices are provided. Finally, the validity of the proposed method is examined in several localization scenarios, taking into account the noise in the environment, the multi-path phenomenon and considering the sensors being not synchronized.
1 Introduction
A challenging and highly demanding signal-processing application is the localization of signal sources using the physical measurements at some sensors in the environment. Source localization has become an important task in various applications such as mobile communications, global positioning system (GPS), radar, sonar, navigation, seismology, and geophysics [1–5].
During the recent decades various algorithms have been proposed to estimate the location of the signal sources. These methods utilize different signal characteristics at different sensors and generally can be classified in three main categories: using the time difference of arrival (TDOA); analyzing the signal direction of arrival (DOA) at distinct arrays; and using the differences in the signal amplitude or received energy level. For a constant propagation speed, the TDOA among different sensors is proportional to the source-sensor range differences and may be estimated through methods such as cross-correlation (CC) [6] or its generalized version (GCC) [7]. The source locations can then be estimated using geometric methods such as linear, spherical or hyperbolic intersections [8–10]. To estimate the DOA, for narrowband signals, high resolution algorithms such as multiple signal classification (MUSIC) [11] and maximum likelihood (ML) [12] are proposed. In [13], those authors propose an approximate maximum likelihood method (AML) for wideband signals using spectral properties of the signal when rather long sample streams are available. In this method, the corresponding cost function can be directly expressed in terms of the source locations or in a far-field case, may be expressed in terms of the relative time delays followed by a post-processing step to find the source locations from the corresponding DOAs. The post-processing step may be carried out through geometric methods such as cross bearing or a machine learning approach such as the support vector machine (SVM) method [14]. Using the differences in the signal intensity or energy level for the purpose of localization is a more recent technique [15, 16]. Theoretically, this class of localization can be considered for both narrowband and wideband signals by only taking into account the attenuation information and usually neglecting the time delay information. For these methods, a precise attenuation model in the environment is inevitable for an accurate localization. Moreover, from an optimization perspective the resulting cost functions in these kind of approaches usually undergo many local optima and saddle points which require considering specific optimization schemes [17].
In this article, we manage the problem of localization of multiple wideband sources by coherently taking into account the TDOA and the amplitude attenuation pattern. Our method generalizes the AML approach to utilize the signal attenuation characteristics. We provide a more robust algorithm in which the targets should simultaneously satisfy the correct time delays among the sensors and provide sensible level of attenuation at each sensor. Unlike the aforementioned energy and intensity based methods which not only ignore the time delay stamps but also require knowing the signal attenuation model, we benefit using the delay information and as a generalization to our recent study in [18], leave the space for not knowing an exact signal attenuation model in the environment by suggesting an appropriate functional space for it. We minimize a cost function which is obtained through maximum likelihood approach from which the locations, attenuation model parameters, and the multi-path parameters are obtained. To apply the minimization, we propose a hybrid approach combining the differential evolution algorithm [19] with the Levenberg-Marquardt algorithm [20]. This combination provides a minimization scheme which is likely to globally search for the optima and rather quickly converges to the accurate results. Through simulations and Cramér-Rao bound, we verify the effectiveness of the novel method introduced in this paper.
This article is organized as follows. In Section 2, we propose a general form for the received signal at every sensor and later provide an adaptive model for the signal attenuation based on Laurent polynomials. In Section 3, a maximum likelihood estimation of the source location and attenuation parameters is proposed. We also provide the Cramér-Rao bound for this estimation problem. For the purpose of minimization in Section 4, a hybrid approach combining the differential evolution algorithm with the Levenberg-Marquardt is proposed for which the combination algorithm and closed form equations for calculation of the Jacobian are provided. In Section 5, we examine the efficiency of proposed method through some examples, and finally there are some concluding remarks in Section 6.
2 Problem definition
2.1 Signal model
where, X_{ m }(f), S_{ n }(f) and ξ_{ m }(f) are the data, signal, and noise spectra respectively. As stated in [13], we emphasize on the fact that, for (3) to be a valid equivalent form of (2), we need n_{ t } to be large enough to avoid edge effects and accordingly n_{ f } > n_{ t }. In general, having more samples from the signal better poses the problem.
2.2 A low-order representation of signal attenuation model
In many applications, the low-order representation of α(ρ) in (4) is acceptable enough to model the attenuation, and usually considering only few terms in the series (i.e., L rather small), would suffice for the localization problem.
3 A maximum likelihood estimation of the unknowns
3.1 Derivation
for ℓ = 1, ..., L. The matrix H(f) is related to the multi-path parameters and its elements
Similar to the idea in [13], for a real valued signal, we can only consider up to n_{ f } /2 frequency bins and form Q with blocks of Q(f) for f = 0, 1, ..., n_{ f } /2. We would like to highlight the fact that in [13], the zero frequency bin is ignored due to producing a constant term in the likelihood function; however, in our approach, the matrices K(0) and H(0) are still dependent on ρ_{ m }_{,}_{ n } and the multi-path parameters γ_{ m }_{,}_{ n }_{,}_{ p } and hence are worth being considered.
where ${x}_{{S}_{n}}$ and ${y}_{{S}_{n}}$ are, respectively, the x and y components of the position vector ${r}_{{S}_{n}}$. In case of multi-path, the parameters γ_{ m }_{,}_{ n }_{,}_{ p } and ${\widehat{\tau}}_{m,n,p}$ are also included in θ. The approach is clearly not only limited to 2D Cartesian systems and 3D Scenarios, and other coordinate systems may also be considered.
3.2 Cramér-Rao lower bounds for the estimated parameters
An analogous technique is used to derive $\partial \stackrel{\u0303}{K}\left(f\right)\u2215\partial {y}_{{S}_{n}}$.
Specifying the elements of the Fisher matrix F yields the CRLB values for all the estimations.
4 Minimization strategy
The minimization in (11) may be performed through various optimization schemes, most generally categorized as global and local optimizations. For a global optimization different approaches such as deterministic, stochastic, or evolutionary and metaheuristic methods may be considered [22–24]. Clearly for an accurate localization, global minimizers of (11) are required. However, in general, using global methods to optimize an arbitrary function may be iteratively or computationally expensive. As an alternative to this and specifically for a least squares cost function as (11), local search methods such as gradient descent and quasi-Newton methods may be considered [20]. Although these methods can be relatively faster than the global ones, there is always a chance of getting trapped into a local minima. In the context of localization, although for good initial estimates of the source relatively fast methods such as the gradient descent and alternating projection are proposed, to increase the chances of finding a global minima, the process usually involves exhaustive search methods such as the grid search and multiresolution search [13, 16].
For the purpose of this article, we consider a hybrid approach combining a global search method with a fast local search method [25, 26]. Hybrid methods have received considerable interests in different areas in the recent years [26–29]. More specifically, we consider a hybrid combination of the differential evolution (DE) algorithm [19] as successful evolutionary search with the Levenberg-Marquardt algorithm (LMA) [20, 30] as a rather fast and robust local search method. Before getting to the combination scheme, we provide a brief description of each method highlighting the main technical issues specifically in the context of our localization problem.
4.1 Differential evolution algorithm
DE is among the metaheuristic and evolutionary global optimization schemes. Simplicity and successful performance are the main advantages of this algorithm. Considering θ= [θ_{1}, θ_{2}, ..., θ_{ D }] to be the vector of problem unknowns with size D, at every generation G of the algorithm, N_{ P } parameter vectors θ_{ i }_{,}_{ G } = [θ_{1,i,G}, θ_{2,i,G}, ..., θ_{ D }_{,}_{ i }_{,}_{ G }], i = 1, 2, ..., N_{ P }, are generated. The initial population is randomly chosen with a uniform distribution in the search region. For this study, we consider the DE/rand/1/bin, which is a general and widely used strategy of this algorithm [19, 31]. For every generation, three main operations are performed as follows:
4.1.1 Mutation
where r_{1}, r_{2}, and r_{3} are randomly selected indices among 1, 2, ..., N_{ P }, and F ε 0[2] is a constant real scalar controlling the difference vector amplification.
4.1.2 Crossover
with d = 1, 2, ..., D. Here, C_{ R } ε [0,1] is the crossover constant, r(d)_{[0,1]} is the d^{th} evaluation of a uniform random number generator in [0,1] and k(i) ε {1, 2, ..., D} is a randomly chosen index ensuring that υ_{ i }_{,}_{ G } takes at least one of the elements of μ_{ i }_{,}_{ G }.
4.1.3 Selection
At this step a next generation population member θ_{ i }_{,}_{ G }_{+1} is produced by a selection among θ_{ i }_{,}_{ G } and υ_{ i }_{,}_{ G }. This selection is based on the fitness, and basically, the vector with the lower cost proceeds to the next generation.
4.2 A Levenberg-Marquardt algorithm for the local minimization
As the local minimization scheme, we suggest using the LMA. Our attention toward this algorithm is based on several advantages. LMA is basically considered as a Newton type method and provides a rather quadratic convergence. Meanwhile, this algorithm benefits from stability and uses a trust region approach [30]. The other feature of this method, considered as an advantage over other methods such as the gradient descent, is its suitability for cases where there are different variables of different types as the cost function arguments. In fact, LMA is almost independent of variable scaling, while for methods such as the gradient descent, minimizing a cost function dependent on a set of variables with different natures and scales requires appropriate parameter scaling to guarantee a proper convergence [30]. This is a demanding feature for our problem where the θ vector in general consists of the source locations, attenuations coefficients, and the multi-path parameters.
where Q is the vertical vector of length Mn_{ f } /2 shown in (12) and obtained for values θ^{(i)}at that iteration. The parameter λ^{(i)}is the damping factor, obtained at every iteration based on the trust region approach [20, 30]. The Jacobian matrix J_{ θ } contains the sensitivities of Q to every element of θ. In order to run the algorithm, we need to know the Jacobian matrix at every iteration, obtaining which is discussed in the Appendix.
4.3 The hybrid combination scheme
5 Simulation results
To examine the method developed in the previous section, we consider some localization examples in this section. In the first example, we consider a reverberation-free environment to show the efficiency of the method for such cases and provide a comparative study for this scenario. The second example brings more realistic issues such as the multi-path, and sensor synchronization error into the problem, and examines the performance of the proposed method for such cases.
which somehow normalizes α(·) and unifies the representation. Clearly, since the desired unknowns of the problem are the acoustic source coordinates, obtaining a multiple of the attenuation and multi-path coefficients is non-problematic. The true attenuation model to be used in this article is α(ρ) = ρ^{-1.25} [15].
5.1 Example 1
To provide a better understanding of the problem, in Figure 2, the cost function behavior for a known attenuation model is shown. In Figure 2a, the cost is shown when the source is located at point (4, 3) within the sensors convex hull. All positions are in meters. Figure 2b shows the cost when the source is located at (12, 10) outside the sensors region. In both cases the cost functions are rather well-behaved functions away from the sensors. Intuitively, for two neighboring points in the domain, sudden variation of the cost function with respect to both time delay criteria and attenuation model constraints is unlikely, and hence the resulting cost functions are usually expected to be rather slow varying and well behaved, away from the sensors.
5.2 Example 2
Localization Results for the Sensor Network Configuration in Figure 7.
Type of Problem | Synchronization Error Variance (mS) | G _{max} | Number of LMA Iterations | Estimated Target Coordinates | Localization Error (meters) | |
---|---|---|---|---|---|---|
Array Synchronization | Reverberation | |||||
■ | □ | 0.5 | 20 | 21 | (34.62, 24.91) | 0.386 |
■ | □ | 1.0 | 20 | 28 | (33.99, 24.71) | 1.053 |
■ | □ | 2.0 | 20 | 31 | (33.79, 24.66) | 1.254 |
□ | ■ | 0 | 20 | 24 | (34.89, 24.97) | 0.113 |
■ | ■ | 0.5 | 20 | 25 | (34.60, 24.89) | 0.418 |
Furthermore, a more challenging problem is when the reverberation is also taken into account. In theory, for the emitted signal to arrive at every measuring sensor, an individual multi-path filter should be considered. Although the formulation in this article is general, for the purpose of this example, we have made a reasonable and practical assumption that for all the sensors within each array, the filter representing the multi-path is identical. In general, the sensor network may be represented as a collection of several clusters with each being composed of sensors closely placed and each cluster treated as a single receiving node. This assumption prevents dealing with a large collection of unknowns (γ_{ m }_{,}_{ n }_{,}_{ p } and ${\widehat{\tau}}_{m,n,p}$) for every source-sensor pair and aggregates them into fewer parameters each assigned to the clusters.
6 Conclusion
In this article, we proposed an efficient method for localization of multiple wideband sources based on both signal attenuation and time delay information. The method developed in this article models the localization problem as a minimization problem and provides an additional flexibility of not being exactly aware of the signal attenuation model. We propose a certain function space for the unknown model, and tune it iteratively along to estimate the signal source locations. The minimization scheme used here is a hybrid algorithm, combining the differential evolution with the LMA. This combination increases the chances of finding a global minima while benefits from the speed and computational advantages of Newton methods. The accuracy and performance of the method is examined through several simulations depicting a noisy environment, a multi-path environment, and lack of synchronization among sensors. In the simulations, we compared our approach with the approximate maximum likelihood method which show the superiority of the proposed method.
Appendix
As mentioned earlier, in order to find columns of the Jacobian, we are required to find ∂ Q/∂θ, where θ is one of the unknown parameters ${x}_{{S}_{n}}$, ${y}_{{S}_{n}}$, β_{ ℓ }, γ_{ m }_{,}_{ n }_{,}_{ p }, or ${\widehat{\tau}}_{m,n,p}$. Since Q is a vector containing sub-vectors Q(f) for f = 0, 1, ..., n_{ f } /2, we will only find ∂ Q(f)/∂θ, and clearly forming ∂ Q/∂θ would be aligning the corresponding sub-vectors.
To complete the derivation we only need to have $\partial \stackrel{\u0303}{K}\left(f\right)\u2215\partial \theta $, which is already discussed in Section 3.2.
Declarations
Authors’ Affiliations
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