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The distance between microphones in a two-microphone array for direction of arrival estimation, should be at least less than half of the wavelength of the signal that is being detected by the microphone, when the signal is periodic.

I am having trouble understanding this constraint on the distance between microphones. As far as I remember, the path difference between the sound arriving at one microphone and the other shouldn't be a multiple of half the wavelength, because if $x = \frac{\lambda}{2}$, the phase difference between the two signals for a continuous periodic signal would be zero, and thus we wouldn't be able to measure the time difference between the sound arriving at the two micropones.

However, I am unable to understand how a less than constraint appeared, and on the distance between the microphones and not on the path difference of the sound arriving at the two microphones.

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Say the microphones $A$ and $B$ are omnidirectional and exactly half a wavelength apart. If the sound field propagates in the direction of the vector $A\to B$, then a chosen part of the waveform will arrive at $...,$ $A,$ $B,$ $A,$ $B,$ $...$ at intervals that equal half the period of the waveform. If the sound field propagates in the direction of the vector $B\to A$, the chosen part of the waveform will arrive at $...,$ $B,$ $A,$ $B,$ $A,$ $...$ at identical intervals. You won't be able to tell the difference between the two cases from the arrival times because for a periodic waveform there is no "first arrival". This ambiguity is resolved by the constraint you mention, as you then know that you can safely choose for comparison arrival times at $A$ and $B$ that are separated by at most some known time less than half the period of the waveform.

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few points 1)speech signals are quasi periodic.
2)Speech/Sound is a broadband signal, comprising of many frequencies.
3)The case where TDOA seems to be half wavelength will come only when source is placed in line with microphone array and only for particular frequencies, hence we will get phase difference for all other cases.
4)The half wavelength criteria is for avoiding spatial aliasing. for understanding spatial aliasing you can refer to literature.

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  • $\begingroup$ Could you elaborate more on this? There are explanations for this requirement but I am yet to understand it fully. $\endgroup$ – Samyukta Ramnath Sep 29 '16 at 13:54
  • $\begingroup$ What I am trying to say is that if microphone distance is half wavelength, phase difference will be zero only in case when sound source is inline to a microphones. One more important thing is sound/speech is broadband signal hence you will still get phase difference for many frequencies even in above case. $\endgroup$ – arpit jain Sep 30 '16 at 5:24

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