本論文基於前人所提出的抵達時間差之機率模型來做出改善，使其對於有室內殘響的結果更加準確，並且發展任意多對麥克風的定位方法。為了處理室內殘響的問題，我們將殘響視為另外一種噪音，並重新加入到我們的模型當中，可以得知改善後的模型，對於抵達時間差的預測更為穩健。我們可以直觀地利用此機率模型做出2對麥克風的定位結果，但如果聲源擺在麥克風對之間的連線附近，其定位結果會下降很多，因此我們發展出可以延伸到任意多對麥克風的定位方法。從最大定位誤差結果得知，利用3對麥克風來做定位，比起用2對麥克風，下降超過20%的誤差。

A probabilistic time difference of arrival (TDOA) estimation method is extended for multiple pairs of microphones to collaboratively perform source localization. To deal with reverberation, we treat the reverberant signal as a separate type of noise, and the TDOA estimation result outperforms existing approaches such as an interaural phase difference (IPD) method [22], the GCC-PHAT [12], and the adopted method based on probabilistic modeling [21]. Then, the task of sound source localization using arbitrarily many pairs of microphones is formulated as a maximum likelihood problem, and the x-y coordinate of the sound source is estimated by gradient descent minimization of the cost function. Simulation in a reverberant room shows that, when using two pairs of microphones for source localization, large errors tend to occur when the source and the microphones are on the same line. Nevertheless, by adding a third pair of microphones, the proposed algorithm is able to reduce the maximum mean square localization error by more than 20%. And the relationship with different γ and 〖RT〗_60 is tested. The simulation shows that, no matter what 〖RT〗_60 is, γ=0.5 has the better performance than γ=0.1 and 0.3 in our system. However, γ=0.7 has the lower position error when 〖RT〗_60 is 0.45 s, and γ=0.9 has the lower position error when 〖RT〗_60 is 0.6 s.

摘要…………………………………………………………………………………….i
Abstract……………………………………………………………………………..…ii
Contents……………………………………………………………………………….I
List of Figures……………………………………………….………………………..III
List of Tables……………………………………………………………………….IV
Introduction 1
Research motivation……………………………………………………….1
Literature review…………………………………………………………….1
Thesis organization………………………………………………………….3
Probabilistic modeling review [21] 4
The modified system model 8
3.1 Two pairs of microphones for source localization……………………..…8
3.2 Dealing with reverberation………………………………………………..13
3.3 Extending to arbitrary many pairs of microphones………………………..15
3.4 Simulating reverberant rooms…………………….………………………..19
3.5 Reverberation time……………………………….………………………..22
Results and discussion 24
4.1 TDOA estimation under reverberation……………………………………..24
4.2 Parameter γ v.s. RT_60…………………………………………………..27
4.3 Localization results…………………………………………………..29
Conclusion 33

Appendix
Cramer-Rao Lower Bound (CRLB) and Fisher Information 35
1.0 Cramer-Rao Lower Bound (CRLB) Theorem……………………………..35
1.1 Finding MVUE when only |X[k]| is unknown…………..………………36
1.1 Finding MVUE when |X[k]| and σ^2 are unknown…………..…………37
Taylor series expansion and unbiased estimator 39
2.1 Finding r ̂ by Taylor series expansion……………………………..39
2.2 r ̂ is an unbiased estimator………………………………….……………..40
References 41

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