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# Beamforming Theory

Module by: Elizabeth Gregory, Joseph Cole. E-mail the authors

Summary: In this section, we will summarize the aspects of beamforming theory we use for our project.

We used delay-and-sum beamforming in order to determine the direction of origin for our 500 Hz test signal. Beamforming takes advantage of the fact that the distance from the source to each microphone in the array is different, which means that the signal recorded by each microphone will be phase-shifted replicas of each other. The amount of phase-shift at each microphone in the array can be calculated by thinking about the geometry of the situation, shown in Figure 1. In our case, we are assuming that the source is in the far-field, which means that the source is far enough away that its spherical wavefront appears planar at our array. The geometry is much simplier with that assumption, and Equation 1 shows the calculation for the extra time it takes to reach each microphone in the array relative to the array center. Figure 2 shows an example of the out of phase signals that might be recorded by a three microphone array.

Δ m = x m cosθc Δ m x m θ c
(1)

In order to determine the direction of origin of a signal, we have to add a time delay to the recorded signal from microphone that is equal and opposite of the delay caused by the extra travel time. That 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. The question is how to know what time delay to add that will produce the desired constructive interference. The only solution is to iteritively test time delays for all possible directions. If the guess is wrong, the signal will destructively interfere resulting in an diminished output signal, but the correct guess will result in the 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 typical beampattern for a signal arriving from the π2 2 direction is shown in Figure 3 for a two microphone array. Naturally, the peak is located at π2 2 because time delays from that region produced the most constructive interference. Test values further from the true angle resulted in diminished output signals. If the source originates from a different direction, such as π3 3 as shown in Figure 4, the peak moves to the new location.

The peak width is partially determined by the spacing of the microphones in the array. Figure 5 shows that as the spacing is increased, the peak width decreases. That trend will continue until the array length reaches the optimal length for the source frequency used. This length is half the wavelength of the source signal as shown in the Design Decisions section.

Figure 6 shows the affect of adding more microphones to the array. The most interesting feature is the appearance of side lobes in the beampattern. However, the global peak value is still located at the true origination angle.

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