D τ which corresponds to the TDOA between microphone 1 and 2 The TDOA estimate

D τ which corresponds to the tdoa between microphone

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D τ ), which corresponds to the TDOA between microphone 1 and 2. The TDOA estimate is the time lag that maximizes ) ( 12 τ R : ) ( max arg ˆ 12 12 τ τ τ R D = Note that finding 12 ˆ τ requires a simple, one-dimensional search. In general, Equation 4.9 has multiple local maxima. The amplitudes and corresponding time lags of these maxima depend on a number of factors. These factors include the separation distance of the microphones, the nature of the source signal and noise signals, and the choice of the weighting function ) ( 12 ω Ψ . 4.1.1 Maximum Likelihood (ML) Weighting Function When there is no multipath propagation (no reverberation), and the source and noise terms are uncorrelated Gaussian signals, the maximum likelihood (ML) weighting yields an estimator that is asymptotically unbiased and efficient. The ML weighting function is given in terms of the power spectral densities of the source signal, ) ( ω S , and noise signals, ) ( 1 ω V and ) ( 2 ω V , [64]: 3 This finite range is determined by the distance between the microphones divided by the speed of sound. 44
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1 2 1 2 1 12 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( + + Ψ ω ω ω ω ω ω ω ω V S V S V V S ML The idealized conditions for which this weighting is optimal are rarely encountered in practice. It has been shown that even mild reverberation greatly degrades performance of the ML estimator [19]. The coherence function, on which the ML estimator is based, is estimated in practice using a temporal averaging technique, such as the one described in [18]. However, this process can be problematic for non-stationary source signals, such as long segments of speech. An approximation to this weighting, which has been shown to work well with speech signals, operates on a single, short segment of speech, and can be given in terms of the magnitude spectra of the microphone signals and noise signals [11]: 2 1 2 2 2 2 2 1 2 1 12 ) ( ) ( ) ( ) ( ) ( ) ( ˆ ω ω ω ω ω ω X V X V X X ML + Ψ Here ) ( 1 ω X and ) ( 2 ω X are the received microphone spectra, and ) ( 1 ω V and ) ( 2 ω V represent the additive noise components that are assumed to be estimated over silence regions. When reverberation is present in the noise terms, the basic assumption that the noise and source signals are uncorrelated is violated. While ML-type weightings are widely used in speech-array applications, they are inadequate in reverberant environments and will not be used in the experiments of this thesis. 4.1.2 The Phase Transform (PHAT) Weighting Function Another weighting function, known as the phase transform (PHAT) [64], is sub-optimal under reverberation-free conditions, yet performs considerably better than ML in realistic environments. It is a popular form of GCC because of its robustness to reverberation. GCC-PHAT whitens the microphone signals, which in turn whitens the cross-spectrum, ) ( ) ( 2 1 ω ω X X . It is defined as follows: ) ( ) ( 1 ) ( 2 1 12 ω ω ω X X Ψ (4.10) GCC-PHAT has been shown to be effective in real environments [72][93]. It will be studied in more detail later in this thesis.
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