The simplest method of modifying the directivity pattern of a microphone array is delay-sum beamforming, visually depicted in Figure 2.
According to Fourier transform theory, a positive phase shift in the frequency domain is equivalent to a delay in the time domain. To implement delay-sum beamforming, manual time delays, which exactly cancel the time delays inherent to the array’s geometry, must be applied to each microphone in the array. By neutralizing these inherent delays, each microphone signal originating from the target direction is guaranteed to be in phase, resulting in constructive interference.
Choosing the angle in which the signals experience constructive interference is the same as steering the main lobe of the directivity pattern to that angle. In Figure 3, time delays were applied so that a plane wave incident at 40 degrees would constructively interfere.
Notice how the main lobes, originally located at 0 degrees and 180 degrees, are shifted to 40 degrees and 140 degrees.
Adaptive Beamforming: Generalized Sidelobe Cancellation
The primary flaw of delay-sum beamforming is its weak signal to noise ratio compared to adaptive beamforming techniques. Generalized Sidelobe Cancellation, henceforth referred to as GSC, is an adaptive beamforming technique which utilizes two signal processing paths in order to improve noise properties (Figure 4).
The first processing path implements the conventional delay-sum beamformer and generates a conventional signal. The second path is an adaptive noise canceller which generates a set of adaptive weights. These adaptive weights are used to eliminate the signal of interest while also minimizing the noise power.