For an array of M microphones there are a total of 2 1 M M possible pairwise

For an array of m microphones there are a total of 2

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For an array of M microphones, there are a total of 2 ) 1 ( M M possible pairwise combinations (i.e. there are " M choose 2 " 2-combinations of an M -set). Any subset of these pairings can be used for TDOA estimation, resulting in a multitude of TDOA estimates. By taking the root mean square of the individual TDOA errors, a single RMS error characterizes the accuracy of all TDOA estimates taken from the same array. It is convenient to define the RMS error in terms of TDOA vectors. At most, such a vector has 2 ) 1 ( M M elements, one for every possible pairwise TDOA. Hence, a complete TDOA vector is given by: [ ] ) 1 ( 2 4 2 3 2 1 3 1 2 1 M M M M τ τ τ τ τ τ τ K K K τ Denoting the vector of known TDOAs by and the vector of estimated TDOAs by , the RMS error can be defined as follows: 0 τ τ ˆ ( ) ( ) 0 0 ˆ ˆ ) ˆ ( τ τ τ τ τ sqrt E RMS (4.17) 4.4 Source Localization by Minimization of the RMS TDOA Error The source can be localized by minimizing the RMS TDOA error, as defined by Equation 4.17. By searching over a pre-defined set of spatial coordinates, the RMS error can be computed for each candidate point. Instead of using , which is the actual TDOA vector, the TDOAs corresponding to each candidate point are used to compute this error. Denoting the candidate point by 0 τ d r , the corresponding TDOA vector, , can be constructed from the following elements: τ { q l M q l c d d d d q l lq = , 1 , for K } r r r r τ 49
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For a fixed set of microphone locations, M d d r K r 1 , and a given TDOA-estimate vector, , the RMS error is a function of the candidate location, τ ˆ d r , and it can be defined as follows: ( ) ( ) ) , ( ˆ ) , ( ˆ ) ( 1 1 M M d d d d d d sqrt d E r K r r r K r r r τ τ τ τ (4.18) An estimate of the source’s location is given by the candidate location that minimizes : ) ( d E r ) ( min arg ˆ d E d d r r r = Since each candidate point is a 3-element vector with the Cartesian coordinates of the candidate location, Equation 4.18 is a function of three spatial variables. To minimize this error, a search must be performed over these variables, and it can be computationally intensive. This computational burden can be eased by using a simplex search [74], for example, which works well because ) ( d E r tends to be smooth and unimodal. As discussed in Section 2.6, far field conditions limit the ability of an array to estimate range. When the range of the source is ambiguous, the RMS TDOA error is a function of only two spatial dimensions, azimuth and elevation. Hence, by defining the candidate delays in terms of these spatial variables, the RMS TDOA error can be minimized over direction of arrival (DOA) instead of source- location. While this obviously eases the computational load, the source cannot be completely localized in 3-D space. Generally, the DOAs from multiple far-field arrays can be used, via triangulation, to yield an estimate of the source’s location.
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