Delay-and-Sum Beamforming
If a time delay is added to the recorded signal from each microphone that is equal and opposite of the delay caused by the extra travel time, it will result in signals that are perfectly in-phase with each other. Summing these in-phase signals will result in constructive interference that will amplify the result by the number of microphones in the array. This is known as delay-and-sum beamforming. For DOA (direction of arrival) estimation, one can iteratively test time delays for all possible directions. If the guess is wrong, the signal will destructively interfere, resulting in a diminished output signal, but the correct guess will result in the signal amplification described above.
The problem is, before the incident angle is estimated, how could it be possible to a time delay that is 'equal' and opposite of the delay caused by the extra travel time? It is impossible. The solution is to try a series of angles at sufficiently high resolution, and calculate the resulted mean output signal of the array using Eq. (3). The trial angle that maximizes the mean output is an estimation of DOA given by the delay-and-sum beamformer. Adding an opposite delays to the input signals is equivalent to physically turning the sensor array. Therefore, it is also known as beam steering.
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