Place of Development: University of Buenos Aires/École Nationale Supérieure d’Ingénieurs de Caen
Author: Pablo Manuel Delgado
Supervisors: C. Galarza and M. Frikel
Anechoic Chamber for acoustic source localization experiments in Fraunhofer IDMT.
This project was carried out as a part of my Masters Thesis in conjunction with the University of Buenos Aires and the École Nationale Supérieure d’Ingénieurs de Caen. Below is a very concise and simplified explanation of the main topics, and by no means complete.
Abstract: This work investigates most of the aspects related to locating and tracking moving targets via acoustic sensing using wireless sensor networks (WSNs). The majority of these localization methods has historically been based on position estimation via measuring time differences of arrival (TDOA) of the sound wavefront to the sensing points. Considering efficiency of the different localization algorithms on systems with limited storage, transmission and processing capabilities, a newer type of algorithms based on sound energy measurements (Energy Based – EB) and a corresponding propagation-attenuation model have taken prominence in later years. A comparison of the most relevant algorithms with an emphasis on energy based methods is carried out. In addition, an adaptation of an algorithm originally used in TDOA methods is proposed in a collaborative framework (the network is divided into independent processing clusters with local hierarchy and no centralized calculation) for the energy measurement model. Due to new considerations in approaching the disturbances affecting the energy model, this adaptive algorithm presents better performance in terms of communication cost and localization error compared to previously used energy based (EB) methods when the number of sensors in the network increases. Lastly, some considerations on implementation are taken into account under a specific hardware WSN solution, available in the consumer market.
A wireless sensor network for monitoring buildings to assess earthquake damage and other natural disasters. Source: IEEE 2013 Transactions on Sensors Journal
The task of locating a sound source in space (speaker, vehicle, event..) using acoustic measurements (usually microphones of all kinds) of the emitted signal propagated over a certain medium has been of interest for a long time. This localization can be useful for a variety of applications ranging from enhancing speech processing algorithms in noisy, enclosed environments (Cocktail Party Problem) or passive vehicle or target localization/tracking in the open field. For the latter type of tasks, a specially suited technology comprising of relatively cheap, lightweight, battery-powered microcontroller nodes with sensing and transmitting capabilities is usually employed.
Sound source localization
These nodes comprise what is known as a Wireless Sensor Network (WSN, see figure below). These WSNs usually show better performance than the classical localization systems in terms of localization accuracy vs equipment cost, since the nodes are usually cheaper than the costly precision acoustic sensors usually dedicated to these efforts, and can conform an arbitrary topology (sensor distribution) for enhanced resolution in the areas of interest.
An example of a Wireless Sensor Network mote of node, comprising of a microcontroller, an RF device and a physical sensor.
WSN motes can vary significantly in cost, size and features.
The signal processing theory associated with wireless sensor networks often differs in some cases from the classical approach in the sense that the signal acquisition is multiple (bears some similarity to array processing) and also that a special care must be put in optimizing the algorithms for efficient data fusion and calculation in resource constrained, battery powered computers. In general, the following subjects are usually taken into account:
- Transmission cost: the amount of data transmitted must be kept to a minimum within the performance specifications in order to extend the sensors’ (nodes) battery life.
- Computational complexity: the algorithm blocks, methods or functions implemented must usually run in resource constrained computers (micro controllers) which do not have much processing power.
- Underlying protocols: some of the localization methods require tight synchronization among the sensors’ time base, especially the ones based on signal phase or time of arrival, these synchronization efforts over a wireless network can be major resource consuming tasks.
Given N intervening sensors and K emitting sound sources, the received signal, a random variable (usually Wide Sense Stationary) on each sensor will be given by the following expression:
- is the i-th sensor index
- is the attenuation factor of the medium (estimated in a calibration phase)
- is the i-th sensor gain, supposed unitary without loss of generality
- is the signal emitted by the source k
- is the Cartesian position vector of sensor i
- position vector of the k-th source
- time delay of the wavefront between the sound source k and the sensor i
- is the associated noise corresponding to each measurement in the sensors, supposed AWGN with zero mean and variance , and identically, independently distributed (i.i.d.).
In simple terms, according to this propagation model, the signal measured on each sensor will be the sum of the delayed, attenuated versions of the signals emitted by each of the K sound sources. We also assume that the noise power that perturbs the measurement on each sensor is of equal magnitude across the network. Among other considerations regarding the model are:
- The signal is emitted omnidirectionally, the localization process will take place in a 2D scenario,, without loss of generality for the 3D, spatial case.
- The energy of the signal is approximately constant during the measurement interval .
- As evidenced by the preceding equation, the system is such in that linear superposition of the signals takes place.
- The influence of reflections, multipath influence and obstacles are negligible for the measurements.
Time Difference of Arrival (TDOA) based localization
The TDOA based localization is based on the estimation of the time delay of arrival of the wavefront to the different sensing points that participate. Given two different sensors i and j, one can establish the following equation for estimating this quantity:
where is the cross correlation function between the signals received at two different sensors i and j. Having estimated this delay, many geometrical considerations can be used in order to estimate the position of one acoustic source with respect to known positions of the sensors. In the 2D case, with at least 3 sensors and knowing the propagation velocity of the sound on that given medium, one possibility is to estimate the sound source position as stated by the scheme shown below:
Sound source localization estimating Time Difference of Arrival (TDOA)
this resulting system of equations can be linearized  as nonlinear systems can be difficult to solve in computationally efficient ways. Nevertheless, the determination of the TDOA can be a very resource consuming task due to the tight synchronization on the time base of each sensor. This synchronization will be a determinant factor on the accuracy of the location estimation. In addition, the transmission of the full set of samples corresponding to the variation of the signal through time during the measurement time window can lead to a considerable data overhead, resulting in a costly transmission. The energy method described below only needs to transmit one single energy value per sensor, greatly reducing transmission burden over the network.
Energy Measurement based localization
Considering the previously introduced signal model, the acoustic energy of the signals at each sensor on a time window can be calculated as:
where the TDOA was considered much shorter than the time window for the measurement and therefore neglected. The calculation is carried out at a sampling frequency for L samples. The multipath effects and other reflections and perturbations described on the model assumptions have usually shorter time constants and are consequently smoothed out by the sample sum.
With similar geometric considerations using the proposed model and the measurements, many different systems of equations can be derived for estimating the source position. The deviations from the model, modeled as noise in the previous equation, turn the associated localization problem in an optimization problem, for which a Maximum Likelihood (ML) estimator can be found ..
The squared sum of random variables from the energy calculation corresponds to a new random variable which has a distribution. Nevertheless, the ML estimator derived from optimizing a feasibility problem under these circumstances can lead to complicated calculations. One way to tackle this problem is to assume that, given the sufficient amount of samples L, can be approximated to respond to a normal distribution (as granted by the Central Limit Theorem). This normal distribution can be completely described by its moments of first and second order, derived from the energy equation for :
The complexity of these expressions comes from the fact that the cross products between the random variable representing the signal and the one representing noise were not neglected, as is common practice with the independence hypothesis . This simplification is valid when the number of samples on the energy calculation approaches to infinity, i.e. only on the limit. Nevertheless, for finite samples ignoring these cross product terms can sometimes lead to great errors on estimating the true values of the measurements. As it will be seen below, keeping these terms gives way to a better model for the estimation process.
This set of equations put in evidence that the incertitude on energy calculations not only will depend on the measurement noise power, but also on the distance from the source to the measuring nodes. This means that the error affecting the optimization problem is not identically distributed along the equations given by each measurement, even if the measurement noise is equal for all the sensors. There is also a cross correlation (albeit small) between measurements from different nodes.
With the previous considerations, the optimization problem can be associated to a cost function:
where estimates the source position, its energy and a parameter depending on its autocorrelation function. is the error covariance matrix of the problem.
A particular realization of the cost function for a 20×20 sq. meter field. The negative cones correspond to sensor positions. The global maximum corresponds to the source location estimated by the likelihood function derived from energy measurements ,.
As it can be seen from the above figure the cost function is nonconvex. This nonconvexity can lead to complex computational optimization methods for finding the true maximum, and therefore the source location. In addition, even if very precise when properly initialized, these complex iterative methods can converge to local extrema which do not reflect the true source location. These are some of the reasons why the maximum likelihood methods can be prohibitive for implementation on distributed algorithms for WSNs, given the fact that more often than not, the sensor nodes are implemented with very basic hardware and limited computational power.
Linearization: Weighted Least Squares
The optimization problem can be linearized in various ways. Thus, the nonconvex ML problem can be solved through a (linear) least squares approach. By rewriting the energy calculation expression as:
where represents the LS equation error, normally distributed with the moments presented on the previous section. We have then, rearranging and expanding:
The LS error in this equation system is not identically distributed along the set of equations. As it can be seen, its magnitude not only depends on noise power, but also on the relative distance from the source to the i-th intervening measuring node. This automatically leads to a weighted scheme in the LS problem. The corresponding solution will be then, as usual, of the form:
where is the normalized weighted matrix for the problem, and can be estimated locally only with the energy measurements and the estimated noise power . This matrix can also be considered diagonal if we neglect the cross-correlation terms, which is usually an acceptable simplification. Given these conditions, the weighted least squares problem can be solved iteratively , .
Adaptive algorithms in Wireless Sensor Networks
There are many methods for solving linear systems in a recursive way. In particular, the regular expressions for various versions of adaptive filters can be interpreted as iterative algorithms for accomplishing such a task, where each new measurement corresponds to a new equation or row in the system.
In the context of Wireless Sensor Networks, a given estimator of the final solution is propagated through the network and updated with the local measurements of each node. In addition, this propagation scheme can respond to a certain hierarchy on the nodes for granting (or accelerating) convergence and optimizing transmission power.
Solving iteratively on a WSN. The estimator, and possibly its momentary variance, are propagated through the network (or a cluster within radio reach) and updated on every node that has a measurement available.
Performances of different Energy Based localization methods and one TDOA linearized method , are compared in terms of the RMS error over Monte Carlo simulations.
The sensor positions will be randomly placed on a 20×20 squared meter surface, conforming an arbitrary topology. The source will perform a random trajectory on this area as well. The network size is varied from 6 to 10 nodes. The emitted acoustic signal is white noise generated via a pre-established power spectrum density with 1kHz bandwidth. A sample frequency of 5 kHz is used for the measuring nodes over a period of one second (5000 samples) with a Signal-to-Noise Ratio of 20 dB.
Experiment description of the acoustic source localization method comparison 
Root Mean Square Error
The RMS error in meters is calculated according to the following formula for the Monte Carlo simulations:
where is the amount of different random network topologies (sensor placements) used, the number of simulations carried out for each network topology. is the location estimate given by each of the tried methods over each simulation and the true source position.
The centralized methods consist on receiving all the raw measurements from each sensor node to a fusion center and solving the system of equations in a batch procedure (non iterative way). This might be not possible in situations where computational power is restricted and transmission time must be saved in the sensors in order to save energy. However, they provide the performance limit for distributed (iterative) algorithms that propagate the current location estimate across the network.
The centralized acoustic source localization methods compared here are:
- A linear nonweighted Least Squares optimization problem (LS_OS), similar to that implemented in .
- A linear weighted direct method (LS_WD)  following the extended propagation model described above.
- A correction method based on the direct method for further improving the estimation in some cases (LS_WDC)
- A linearization of the TDOA localization problem as described in 
This comparison will be the basis for determining the theoretical performance limit of some distributed versions  of these methods, more suitable for implementation on low cost WSN’s with no fusion center.
Root mean squared localization error (RMSE) with varying SNR
RMSE (meters) with varying SNR (db) for 1000 realizations over a 20×20 sq. meter surface, N= 7 sensors, and 5000 samples over 100 different sensor topologies for minimizing the biasing effects .
The parameters that present the most influence on the energy based localization methods are the signal-to-noise ratio and the number of samples used for the energy measurement. It can be seen clearly that under the 10 dB mark the LS-WDC (corrected, weighted method) deteriorates even more than the nonweighted schemes, so it can be seen that bad energy readings due to noise directly affect the estimation of weights for the WLS problem. Nonetheless, the weighted schemes remain vastly superior for higher SNR in all cases. The TDOA estimation (green) is expectedly better than all the EB methods for SNR > 15 dB, given that its an intrinsically more accurate, time-based method, but requires at the same time more resources in terms of synchronization and data transmission, as discussed previously. Under the 10 dB mark the TDOA method fails considerably due to the fact that the maximum value of the cross correlation function can no longer correspond to the true time delay between the two signals received at different sensors, therefore resulting in a bogus estimation of the true position.
Root mean squared localization error (RMSE) with varying network size
RMSE (meters) with varying network size for 1000 realizations over a 20×20 sq, meter surface, N= 7 sensors, and 5000 samples over 100 different sensor topologies for minimizing the biasing effects .
The improvement of the direct, weighted methods over nonweighted schemes is noticeable with the growing number of sensors in the network. Given the geometrical considerations for the nonweighted localization method, based on energy reading ratios over pairs of sensors , , the variance and covariance of the measurements can be considerably high when two different energy ratios from different sensors are similar in magnitude, therefore degrading the performance. On the other hand, the weighted schemes -as deducted on the previous section- do not rely on these energy ratios between a pair of sensors, and the direct weighted method is not as heavily affected by this phenomenon. The chance of having two different sensors with similar energy readings increases with the number of sensors, so the performance of non weighted, ratio-based schemes reach a limit very quickly. This is not the case with the weighted direct methods.
Root mean squared localization error (RMSE) with varying sample amount
RMSE (meters) with varying amount of samples used for the energy calculation or cross correlation function calculation. The equations represent the method of calculation of the weights for the associated WLS problem: Blue: nonweighted (identity weight matrix), Cyan: estimated weights with energy measurements, Red: theoretical weights based on sensor to source distance .
The amount of samples greatly influences the performance of the weighted schemes as seen above. The weighted schemes outperform the nonweighted methods with the sufficient amount of samples used for the weight estimation. However, an insufficient amount of samples can considerably reduce performance. This is evidenced by the difference between the theoretical weighted method curve (red) and the estimated weight method (cyan) that significantly differ for fewer samples. The case where fewer than 10 samples are used for the energy calculation is nonetheless unlikely in any application. On the other hand the TDOA has a threshold in accuracy corresponding to the case where the amount of samples used (at a given sampling rate) for the cross correlation function is not enough for reflecting the propagation time of the sound wave from the source to each of the sensors.
Distributed (adaptive) Methods
As it was mentioned on the previous paragraph, the performance analysis of the centralized methods provide a limit to which the distributed methods converge. The importance of implementing localization algorithms in an adaptive form comes from the fact that the hardware used might not be powerful enough to carry out all the computations needed at once. Among the prohibitive mathematical operations are the large matrix inversions required for solving the equation systems coming from the measurements. These iterative methods are often suitable for being implemented in a collaborative, distributed manner on a WSN, where each node only performs a relatively simple mathematical operation based on previous data calculated elsewhere on the network and its own measurements. The resulting updated estimator is then transmitted to other nodes and the process repeats itself until a certain threshold (usually set upon convergence criteria) is reached.
There are several adaptive, sequential methods for solving localization problems in a distributed manner. These methods usually focus on optimizing design parameters related to data communication overhead among the network, mathematical operations involved on local node hardware, precision and speed (number of iterations required) of convergence. One of the most popular adaptive methods successfully implemented on WSN’s is the method of Projection Onto Convex Sets or POCS . This method proved to be one of the most efficient in terms of achieving good precision of the location estimates given the design constraints recently mentioned, in particular requiring low communication overhead for transmission among nodes and a relatively low computational complexity for estimator updates on each node. There are, nevertheless, a few problems associated with the localization process based on POCS.
- Speed of convergence can easily deteriorate if the relaxation parameters of the algorithm are not properly initialized, resulting in unnecessary transmissions across the network for reaching a solution within the restrictions set by the convergence criterion.
- The localization method on which the POCS is implemented is nonlinear.
- The statistic or attribute that provides the basis for the localization method over which the POCS is applied is the computation of an energy ratio between two different energy measurements in two different sensors, if the source emission power is unknown ,. Therefore, for a network or cluster (subgroup of nodes participating in the localization process) comprised of N nodes, the total amount of energy pairs (ratios) would be then N(N-1)/2. This means that, in order to the algorithm to be properly initialized and the measurements be available on each one of the nodes (for a subsequent estimator update), the same (minimum) amount of initial transmissions need to take place, thus enlarging the communication overhead greatly. The number of transmissions can be prohibitively large when the number of sensors N increases. Accordingly, a localization method which utilizes direct energy measurements (as opposed to energy ratios) is preferred.
- As explained before, for the case in which an energy ratio approach is used for source localization, a great variance in the estimation can be present if two energy readings are similar in different sensors (i.e. energy ratio close to 1), thus greatly deteriorating the estimation and/or convergence. The chance of having similar energy readings on two difference sensors increases with the number of sensors in the network, and the convergence of the POCS algorithm gets worse.
- As implemented usually, the POCS algorithm for acoustic source passive localization based on energy readings does not incorporate the extended model (i.e. does not consider weights on the associated measurements for the optimization problem), therefore degrading the precision of the localization.
- In order for the POCS method to converge, the acoustic source must be inside the convex hull of the sensor node positions.
On the other hand, an iterative method for solving the linear WLS system of equations described in the previous section can be constructed . This method is direct, in the sense that it only requires local measurements for the basic statistic used for the estimation (it is not necessary to transmit the values across the network for producing the energy ratios), thus the convergence and precision will not be affected by similar energy measurements on two different sensors, as is the case in the POCS method. The method is a well known iterative algorithm for solving weighted, least squares problems. For each iteration of the algorithm, an estimation of the location is issued along with a corresponding variance of the estimator. These two values are subsequently updated across the network with new measurements until a convergence criterion is met. The weights are, as described, estimated with the measurements and can heavily influence speed of convergence if they are not accurate enough on a low SNR environment. The total amount of iterations possible is, in this case, equal to the number of sensors present in the network which participate in the localization process. This comes from the fact that, in the direct method, each measurement contributes with one equation per node, totaling N equations, as opposed to the energy ratio localization approach in which a total of N(N-1)/2 equations -and accordingly the same number of iterations- can be used.
RMS error vs signal-to-noise ratio for adaptive algorithms
For the RMSE vs SNR scenario, a network of 10 sensors randomly distributed over a 20×20 sq. meter surface along with a random deployment of the sound source over Monte Carlo simulations is carried out. The number of samples for calculating the energy is 5000.
RMSE vs SNR for adaptive algorithms.
The inclusion of weights in the LS problem due to an extended signal model is an advantage for high SNR, as is expected since the background noise terms are no longer dominant in the energy measurement variance, giving room to the terms that depend on the source-sensor distance that are compensated by the weights. It is also pointed out that a bad SNR leads to erroneous estimation of the weights, which can lead to a poorer performance than in the unweighted method.
RMS error vs number of sensors for adaptive algorithms
Considering the number of sensors in the network, the POCS method, the proposed adaptive WLS method and a centralized WLS centralized batch method (for control) are analyzed in terms of RMS error, in meters, in a 20×20 sq. meter surface with a signal-to-noise ratio of 30 dB and 5000 samples used for each energy calculation.
RMSE vs number of sensors used in adapptive distributed algorithms. The POCS algorithm does not improve with an increasing sensor number due to higher chance of large variance measurements (according to the energy ratio localization method).
It can be clearly seen that the POCS method reaches a RMSE limit for the given experiment parameters once the network has more than 11 participating sensors. This is due to the previously explained phenomenon involving a higher chance of high variances due to similar energy readings from different sensors, and thus inducing energy ratios near 1. This is not the case for the WLS direct method adaptive method (cyan), where equal energy readings do not interfere the accuracy due to the fact that each measurement updates the estimator independently as opposed to using energy ratios. The centralized method provides the limit of performance for the WLS method, and is well below the adaptive estimations.
Convergence vs estimator path
One of the determinant factors to economize when dealing with collaborative algorithms in Wireless Sensor Networks is total transmission time and data overhead. The amount of data and the distance over which it is transmitted is a key factor in power consumption. Given that these networks are usually composed of autonomous, battery-powered nodes, it is of great interest to design algorithms that can extend the autonomy to a maximum. In order to maximize battery power the data (i.e. the estimation’s current state) need to be transmitted through an optimum path. Since usually the transmission cost is proportional to the euclidean distance over which the data is transmitted, this optimization can be carried out using techniques for solving what is called a Travelling Salesman Problem, which consists in finding the shortest possible route for visiting each member, only once, of a given set of points in space. This way, the estimator is propagated (and updated) across the network in the most efficient way in terms of distance, and accordingly transmission cost.
On the other hand, the optimal propagation path for the estimator in terms of transmission cost might not be optimal for convergence speed or smoothness. Given the structure of the signal model on which the adaptive WLS algorithm is based ,, the process will converge smoothly only if the estimator is propagated in such a way that the LS error on each measurement is decreasing in magnitude . In other words, given that the LS error depends on the distance from the source, the estimator needs to be propagated from the closest measurement (i.e. highest energy reading) to the weaker ones, in which the variance is larger. This way, the adaptive gain of the algorithm can properly predict the noise present in the next measurement, thus making the algorithm converge smoothly.
RMSE vs number of iterations are shown below for the estimator propagation path above. For N=30 active sensor nodes in the network, the POCS method has a maximum number of N(N-1)/2=435 possible iterations while the direct adaptive WLS algorithm has only 35. The convergence rate seems to be smoother and faster on the WLS method. However, the sub-optimal propagation route chosen makes this unfeasible and/or inefficient for implementation on a battery powered wireless sensor network.
RMSE vs number of iterations for a decreasing energy reading path. The convergence is smooth but the transmission cost may not be minimum.
The cyan and purple lines correspond to each theoretical limit -calculated in a one-shot, centralized way- for each adaptive algorithm (i.e. the global solution). The theoretical, nonlinear solution for POCS is solved using a local maximum search routine based on the Nelder Mead method and is usually more accurate than the linear methods, albeit requiring more computational power.
If an optimization process is carried out for choosing the propagation route beforehand, the transmission energy for the localization process will be minimized, but the convergence (especially on the WLS method) can fail or be heavily perturbed. In this case, the simulation system tries to solve the Traveling Salesman Problem using a convex hull insertion algorithm.
Optimized propagation route of the localization data. Convergence of the WLS algorithm is heavily affected
This is because, as explained before, the adaptation gain will not predict correctly the noise levels of the next sample. The variance magnitudes are not defined by a smooth variation between measurements, as is the case with a decreasing energy reading propagation path. The result will be minimum transmission cost but harsh convergence and a tendency to huge instant deviations from the true location on the estimator (figure below).
RMSE vs iterations for the POCS and WLS methods. The WLS method shows huge error peaks when using the optimized data propagation path on the WSN, due to the uneven measurement variances across the nodes.
Improving convergence by estimating weights
In summary so far, there are two cases possible concerning the balance between an efficient transmission of the data and the smoothness of convergence of the distributed WLS algorithm. In the first case, the estimator es propagated in order to approach the nodes with an ascending variance in measurements, and the estimator converges smoothly. On the other hand, the data of the estimation is propagated to optimize energy (transmission) consumption. Through this route, the variances over the measurements can be dissimilar, thus the adaptation gain may not act as a proper buffer for these sudden changes in variance (or sample noise) and the algorithm can not properly adapt or predict when being updated through the sensors. The problem with the second case can nevertheless be relieved by properly estimating the associated weights.
As it can bee seen from the associated LS error equations, the normalized theoretical weight matrix to counteract the uneven distribution of the variance in measurements will contain elements which are mostly dependent on the distance between source and sensors. Given that the actual distance between source and sensor is not available in the localization process, these weights need to be estimated from measurements within each node . The extent to which these weights are accurately estimated determines the quality of the final location estimation, but also the adaptive ability of the algorithm facing abrupt changes in variance, as is the case when using an optimized route.
As it was shown on the centralized case for the influence of the number of samples involved in weight estimation on location accuracy, the same procedure can be used to show the performance limit of the WLS algorithm using theoretical weights, based on the distance between source and sensor nodes.
RMSE vs. iterations for the optimized data propagation path. Theoretical weights are used. The adaptation gain can properly estimate the variances across sensors and the convergence is smooth for the WLS method.
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