Energy-Based Acoustic Source Localization Methods: A Survey <s> Weighted Least-Squares Method <s> The localization of an acoustic source can be based on the energy measurements at a number of spatially separated microphones. This is because the amount of source energy attenuation at a microphone is proportional to the square of the distance between the source and the microphone. This paper develops an algebraic closed-form solution for the acoustic source localization problem using energy measurements, under the condition of direct line-of-sight and free space propagation. First-order analysis is applied to the proposed solution to study its performance, where only the linear noise terms are kept in obtaining the mean-square localization error. The first-order analytical results show that the proposed solution reaches the Cramer-Rao lower bound (CRLB) accuracy for Gaussian noise as the signal-to-noise ratio tends to infinity. In addition, the proposed solution provides much better accuracy than other closed-form solutions available in literature. Improvement on the proposed solution that extends its operating range beyond the threshold noise level was made by imposing nonnegative constraints. Simulations are included to corroborate the performance of the proposed method. <s> BIB001 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Weighted Least-Squares Method <s> In this paper, energy-based localization methods for source localization in sensor networks are examined. The focus is on least-squares-based approaches due to a good tradeoff between performance and complexity. A suite of methods are developed and compared. First, two previously proposed methods (quadratic elimination and one step) are shown to yield the same location estimate for a source. Next, it is shown that, as the errors which perturb the least-squares equations are nonidentically distributed, it is more appropriate to consider weighted least-squares methods, which are observed to yield significant performance gains over the unweighted methods. Finally, a new weighted direct least-squares formulation is presented and shown to outperform the previous methods with much less computational complexity. Unlike the quadratic elimination method, the weighted direct least-squares method is amenable to a correction technique which incorporates the dependence of unknown parameters leading to further performance gains. For a sufficiently large number of samples, simulations show that the weighted direct solution with correction (WDC) can be more accurate with significantly less computational complexity than the maximum-likelihood estimator and approaches Cramer-Rao bound (CRB). Furthermore, it is shown that WDC attains CRB for the case of a white source. <s> BIB002
The unconstrained least-square mentioned above is closed-form and is computationally attractive. However, it is not able to reach the Cramer-Rao bound (CRB). In BIB001 , a two-stage closed-form least-squares method was proposed, which can achieve CRB when the signal-to-noise ratio (SNR) tends to infinity. From Equation (5), we can see that the measurement y i (t) is highly nonlinearly related to the unknown source location θ and source energy S. Generally, we need to take the energy ratio to remove the dependency of S. Without loss of generality, we choose Sensor 1 as the reference. Then, the ratios of the energy measurements with respect to the reference can be written as: where 1 is assumed to hold, which means the signal-to-noise ratio in the energy measurements should be large enough BIB001 . Expanding the right-hand side and ignoring the second-order noise term yield: The first term is the true value of q i1 , and the second term is noise. If we collect q i1 , ∀i = 2, . . . , N to form the vector q = q • + q = [q 21 , q 31 , . . . , q N1 ] T , then the covariance matrix of q, Q has elements: After obtaining the formulation in Equation (15), the source localization problem was solved by a two-stage closed-form solution proposed in BIB001 . The basic idea of the two-stage least-squares method is that in the first stage, θ and d 2 1 are considered as independent unknown variables. The source estimate can be obtained using the standard weighted least-squares method. Then, in the second stage, the relationship between θ and d 2 1 is explored to improve the result. The details of the algorithm can be found in BIB001 . In BIB002 , the authors claimed that the errors that perturb the least-squares equations (based on energy model) are not i.i.d, zero-mean Gaussian random variables. Hence, the noise term in Equation (5) is not white, but a colored one. In BIB002 , a weighted one-step least-squares (WOS) method and weighted direct least-squares method (WD) are proposed to solve the formulated energy-based source localization problem. The WD method yields the same the location estimate as WOS. However, WD offers lower computational complexity than WOS. Similar to the two-stage weighted least-squares method stated above, a correction technique can be used to improve the location estimation by incorporating the relationship of two dependent unknown variables, which is assumed independent in the first stage. The WD with correction (WDC) method can attain CRB for the case of a white source.