0.1 Beamforming theory

We used delay-and-sum beamforming in order to determine the direction of origin for our 500 Hz test signal. Beamformingtakes advantage of the fact that the distance from the source to each microphone in the array is different, which means that thesignal recorded by each microphone will be phase-shifted replicas of each other. The amount of phase-shift at eachmicrophone in the array can be calculated by thinking about the geometry of the situation, shown in . In our case, we are assuming that the source is in the far-field, whichmeans that the source is far enough away that its spherical wavefront appears planar at our array. The geometry is muchsimplier with that assumption, and shows the calculation for the extra time it takes to reach each microphonein the array relative to the array center. shows an example of the out of phase signals that might be recorded by a three microphone array.

In order to determine the direction of origin of a signal, we have to add a time delay to the recorded signal from microphonethat is equal and opposite of the delay caused by the extra travel time. That will result in signals that are perfectlyin-phase with each other. Summing these in-phase signals will result in constructive interference that will amplify the resultby the number of microphones in the array. The question is how to know what time delay to add that will produce the desiredconstructive interference. The only solution is to iteritively test time delays for all possible directions. If the guess iswrong, the signal will destructively interfere resulting in an diminished output signal, but the correct guess will result inthe signal amplification described above.

We can plot the resulting output amplitudes as a function of test angles to produce a beampattern for the array. A typicalbeampattern for a signal arriving from the $\frac{\pi }{2}$ direction is shown in for a two microphone array. Naturally, the peak is located at $\frac{\pi }{2}$ because time delays from that region produced the most constructive interference. Test values further from the trueangle resulted in diminished output signals. If the source originates from a different direction, such as $\frac{\pi }{3}$ as shown in , the peak moves to the new location.

The peak width is partially determined by the spacing of the microphones in the array. shows that as the spacing is increased, the peak width decreases. That trend willcontinue until the array length reaches the optimal length for the source frequency used. This length is half the wavelengthof the source signal as shown in the Design Decisions section.

shows the affect of adding more microphones to the array. The most interesting feature is the appearance ofside lobes in the beampattern. However, the global peak value is still located at the true origination angle.