Passive Source Localization using Acoustic Sensor Arrays

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Passive Source Localization using Acoustic Sensor Arrays

The class of passive source localization estimators based on the measurements of an acoustic sensor array, can be roughly subcategorized into direct and indirect methods. The indirect methods first estimate the time differences of arrival between multiple sensor pairs and then use these delays to estimate the source position in a second step. Direct methods, on the other hand, generally carry out the position estimation procedure in one step, by transferring the received microphone signals into the frequency domain. Out of the class of direct methods probably the MUltiple SIgnal Classification (MUSIC) algorithm due to Schmidt [Sch86], and a variety of beamforming techniques, such as the steered response power beamforming approach [DBS01] and the steered filter-and-sum beamformer [SMR99] are currently the most widely applied.
The choice of which subclass is best suited for a given estimation scenario is strongly dependent on the physical characteristics and features of the source(s) and the propagation properties between the source(s) and the sensor array, Chen et al. [CYH02] characterize the basic features and properties as follows
• Narrow-band versus wide-band signal: signals can be divided into the class of narrow-band or wide-band sources by the ratio of their highest to their lowest frequency component. The sound produced from wheeled and tracked vehicles may range from 20Hz-2kHz, resulting in a ratio of about 100, and consequently are referred to as wide-band signals. On the contrary, transmitted radio signals are usually narrowband, since their ratio is typically close to unity.
• Far-field versus near-field source: the wavefront of an emitted signal is curved and the curvature depends on the distance to the source. If the sensors are assumed to be close to the source, the signal is said to be in the near-field. If the distance becomes large, the wavefront becomes planar, and the source is said to be in the far-field. For far-field sources, only the direction of arrival (DOA) can be estimated, meaning that the direction towards the source can be estimated. For near-field sources its position can be estimated.
• Free-space versus reverberant space propagation: in free-space the emitted signal is not reflected by obstacles, such as walls, and the emitted signal arrives at the microphones only by the direct path (straight line from source to sensor). Most indoor applications are fairly reverberant, meaning that the received signal is the sum of the direct path signal and signals reflected from the walls and other obstacles. Strongly reverberant environments make the position estimation rather difficult, since the reflected signals might be interpreted as being an independent source.
• Single versus multiple source: the choice of estimator is made upon to the number of present sources. The direct techniques were originally derived for narrow-band sources in the far-field, estimating the direction of arrival (DOA) only. A variety of extensions for broad-band signals in the near-, as well as in the far-field are proposed in literature. These extensions do generally divide the entire frequency range of the broad-band signal into frequency bins and then carry out the estimation procedure for each frequency bin, using the standard algorithms for narrow-band signals. This procedure results in position estimates for each frequency bin, which must then be combined to obtain a single source position estimate. Obviously, this results in a considerable computational burden.
On the other hand, the computational burden of the TDOA-based position estimators is quasi independent of the bandwidth of the emitted source signal, and can easily be utilized in near-field as well as far-field scenarios.

Aim and Objectives of this Thesis

This thesis investigates the class of time difference of arrival (TDOA) based position estimators and tries to give a comprehensive overview of the existing TDOA-based estimators and discusses their strengths and weaknesses. While a large number of articles has been published on how to best implement the TDOA-based estimators in all kind of situations, relatively little information can be found on how to best place the spatially separated microphones. Intuitively it can be argued that if the microphones are all placed in the near neighborhood of each other, that the received signals will be very similar. Consequently, only little information can be obtained from the individual microphones and the estimation procedure is likely to be of poor quality. Contrary, if the microphones are spread, the information obtained from an individual microphone becomes more important and the estimation quality will possibly increase. The question of how to best place the sensors in order to obtain an optimal performance builds the second major part of this thesis. It will be discussed how the quality of an estimator can be measured, and how this quality measure can then be used to optimize the sensor configuration.
It will be seen that a sensor configuration being optimal for one source position might be insufficient for other locations of the object. This can lead to severe limitations of the estimation procedure, if a moving source is considered. One way of addressing this problem is to predict the future source position and then to adapt the sensor network. This procedure is studied using recursive Bayesian estimation techniques.
Finally, a low-cost system is presented, which uses a low-cost digital signal processor (DSP) to analog-digital convert the data of multiple microphones and then transmit these to a standard PC, which carries out the actual estimation procedure.

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Optimal Sensor Configuration

The performance of a position estimator based on a, not necessarily acoustic, sensor array is strongly dependent on the geometry of the microphone network and the relative position of the source w.r.t. this array. The problem of maximizing the estimator’s accuracy by using an optimal sensor configuration has been addressed in literature for a variety of different position estimator, minimizing a variety of cost functions, describing the estimator’s performance.
One of the most often utilized cost functions is the trace of the Cramer Rao lower bound (CRLB). This performance measure was used by e.g. Abel [Abe90], Aranda et al. [AMB05], and Yang and Scheuing [YS05], who all considered the problem of passive acoustic source localization. While Abel [Abe90] derived an analytic solution, constraining the sensors to lie on a line segment, Yang and Scheuing derived an analytic solution without any constraints on the sensor positions: the optimal configuration is obtained if the sensors are evenly spread around the source position. Consequently, the configuration will usually be optimal for a single source position. Aranda et al. found the same configuration and relaxed the problem of a single source position, by assuming a moving sensor network.
An extension of the Cramer Rao lower bound is used by Jourdan and Roy [JR06] for ultra-wideband ranging sensors. Instead of using a moving sensor network for achieving an optimal performance over an entire area or trajectory, they minimize the average cost function over the considered area or path.

Table of contents :

0 Introduction Fran¸caise 
1 Introduction 
1.1 Passive Source Localization using Acoustic Sensor Arrays
1.2 Aim and Objectives of this Thesis
1.3 State of the Art
1.3.1 TDOA-based Position Estimation
1.3.2 Optimal Sensor Configuration
1.3.3 Passive Acoustic Source Tracking
1.4 Structure of the Thesis
2 Estimator Optimization 
2.1 Statistical Estimation Theory
2.1.1 Unbiased and Minimum Variance Estimators
2.1.2 Cramer Rao Lower Bound
2.1.3 Linear Model Estimators
2.1.4 Maximum Likelihood Estimator
2.1.5 Dilution of Precision
2.2 Least-Squares Estimation
2.2.1 Linear Least-Squares Estimator
2.2.2 Nonlinear Least-Squares Estimator
2.2.3 Linearized Estimator
2.3 Condition Number
2.4 Optimal Internal Parameter Selection
2.4.1 Comparison of Cost Functions
2.4.2 Quasi-static Source
2.4.3 Moving Source
2.5 Chapter Summary
3 TDOA-Based Passive Source Localization 
3.1 Problem Statement
3.2 Time Delay Estimation
3.2.1 Generalized Cross-Correlation
3.2.2 Least Mean Square Adaptive Estimator
3.2.3 Adaptive Eigenvalue Decomposition
3.2.4 Time Delay Estimator Evaluation
3.2.5 Time Delay Estimation Summary
3.3 TDOA-based Measurement Model
3.3.1 Cramer Rao Lower Bound
3.4 Iterative TDOA-based Estimators
3.4.1 Maximum Likelihood Estimator
3.4.2 Linearized Estimator
3.4.3 Iterative Estimator Evaluation
3.5 Linear Approximation Estimators
3.5.1 Spherical Intersection Estimator
3.5.2 Weighted Least-Squares Estimator
3.5.3 Weighted Linear Least-Squares Estimator
3.5.4 Linear Correction Least-Squares Estimator
3.5.5 Hyperbolic Interpolation Estimator
3.6 Linear Intersection Estimator
3.7 Optimal Sensor Configuration
3.7.1 CRLB Optimal Sensor Configuration
3.7.2 Optimal Sensor Configuration for the Linearized Estimator
3.7.3 Optimal Sensor Configuration for the LLS Estimator .
3.7.4 Optimal Sensor Configuration for the Combined LLS, Linearized Estimator
3.7.5 Optimal Sensor Configuration of the LI estimator
3.8 Analytic Linear Correction Least-Squares Estimator
3.8.1 Evaluation
3.9 Chapter Summary
4 Source Tracking 
4.1 Recursive Bayesian Estimation
4.1.1 Linear System, Gaussian Noise: Kalman Filter
4.1.2 Nonlinear System, Gaussian Noise
4.1.3 Extended Kalman Filter
4.1.4 Unscented Kalman Filter
4.1.5 Nonlinear System, Non-Gaussian Noise
4.2 TDOA-based Passive Source Tracking
4.2.1 System Equation
4.2.2 TDOA Measure Tracking
4.2.3 Position Measure Tracking
4.3 Optimal Sensor Configuration
4.4 Time Delay Estimations
4.4.1 Most Likely Local Maximum
4.4.2 Weighted Probability Density Function
4.5 Chapter Summary
5 DSP-based Passive Source Localization 
5.1 System Hardware
5.1.1 PC-DSP communication
5.1.2 Multi-DSP System
5.1.3 Single-DSP System
5.2 Algorithm Implementation
5.2.1 Data Acquisition
5.2.2 Microphone Circuit
5.2.3 System Realization
5.3 Time Delay Estimation
5.3.1 Interpolation
5.3.2 Evaluation
5.4 System Evaluation
5.5 Chapter Summary
6 Conclusion and Outlook 
6.1 Conclusion
6.2 Outlook
A Matrix Algebra 
A.1 Induced Matrix Norms
A.2 Condition Number
B Geometric Dilution of Precision for GPS

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