Calculating Position Based On Array Geometry

The calculation of position is based firstly on the shape of the array. Obviously, different shaped arrays will lead to a difference in the placement as well as the number of sensors. Altering the shape of the array to accommodate different parameters can lead to very specific characteristics as well is varying performance. In our case the array is linear in shape, possessing two pairs of microphones. For the purpose of determining location of a source, we consider the array itself to lie along the x-axis, centered at the origin. This placement allows position equations to be derived in a more straightforward manner. It is obvious that the placement of each microphone will have a profound impact on calculation of position. The image shows a diagram of the array geometry relative to the acoustic source.

The parameters of the array consist of the distance between each microphone in the pairs, 'd' as well as the distance between the midpoints of each pair , 'D'. We derive the x* and y* position in detail for the source in question here. Shown below are the two equations for x and y that are finally obtained from this derivation.

Based on the two formulas derived for the x and y coordinates, we find two equations based on the known parameter of 'D', as well as the unknown variables for angle. The propagation of the wave form is directly related to the arrays ability to estimate the degree of arrival for each microphone pair. The concept and approach is detailed here. The derivation of the angle is performed here.