In the process of replicating selective hearing using the 16-element microphone array, several mathematical concepts are fundamental.
At long distances (far-field assumption), sound propagates through air as a plane wave. It is easy to see that when the incoming sound signal has a non-zero azimuth angle, each microphone on the array receives the propagating plane wave at a different time. The exact time delay between microphone elements is a function of both the array geometry as well as the angle of incoming sound. This relation is pictured in Figure 1.
Although the 16 elements in the microphone array apply equal gain to incoming signals in all directions, the gain applied by the entire linear array has a directional dependency. This dependency, also known as directivity, compares the radiation intensity in a particular direction to the average radiation intensity in every direction. The expression for directivity is:
The directivity pattern of the microphone array essentially gives us the directions in which the array “listens” most closely. The function used to generate directivity patterns can be found in the Resources Section.