Though not implemented, nearfield calculations are both more computationally intensive and accurate. If it is assumed that the microphones have some sort of center for distance, then the center can be designated as the origin for the coordinate system. A point source at a point (xs, ys, zs) would then emit a signal s(t). A microphone at a point (xm, ym, zm) would then receive a signal m(t). Assuming that signal propogates uniformly with speed v and that signal strength is equal to the original signal strength divided by the square of the distance, we can conclude that the received signal is:

Figure 1

In this project, it was assumed that the array was always operating in farfield, an approximation in which the source is assumed to be far enough away that the spherical waves it emits can be approximated with plane waves. It is accurate in the limit where the distance between the microphones and the source is large enough so that the angle between the source and each microphone does not change significantly.

Since all of the processing is done in a digital environment, we must work with samples of the signals and not the signals themselves. Because of this, it is not possible to implement an arbitrary time shift as any shift must be done in increments of the sample period. To remedy this, the signals were interpolated digitally by upsampling them and then putting them through a lowpass filter with cutoff corresponding to the amount of upsampling. An equiripple filter was chosen for the lowpass filter as there appears to be no constraints as to the exact shape of the filter and because an equiripple filter would avoid the Gibb’s phenomena found in a direct approximation of an ideal lowpass filter. Using this interpolation, greater resolution can be achieved in the time shifts, though the drawback is the large amount of additional data that must now also be processed. In fact, even though the concept of delay and sum is incredibly simple, the amount of computation that must be done because of the upsampling is often prohibitively high. It is impossible for the amount of interpolation to be too high, but if it is too low, then it is entirely possible that the direction of the source will be inaccurate or entirely wrong as the algorithm will be unable to shift the signals to where they match enough.

Given a fixed sampling frequency, there is always the ”normal” aliasing associated the Nyquist Theorem, restricting the fully reconstructable signals to those that are bandlimited to half of the sampling frequency. Something similar occurs with array spacing, and if proper care is not taken, aliasing may occur in spatial dimensions. Using the spatial analogue of the Nyquist Theorem, the minimum spacing between microphones must be at most half the wavelength corresponding to highest frequency present. Thus, to achieve any resolution at all for higher frequency signals, smaller arrays must be used; however, with a smaller array, the precision with which a direction can be determined is diminished. It appears that there is an uncertainty principal at odds with beamforming in its spatial dimensions.

This is the main focus of the project: to try to locate a source using an array of microphones and then focus the array in the direction of the source, obtaining greater suppression of noise than would be possible using only one microphone. Since the direction of the source is unknown, we decided to scan for the source by sweeping all possibilities. This is where the far field approximation significantly reduces computational complexity. Using nearfield, any algorithm would be forced to evaluate all possible combinations of three coordinates. With farfield, there are only two angles to deal with as opposed to three coordinates so there is far less to compute.

Due to lack of computing power, we were forced to make a few, less-than-desirable assumptions in order to make the algorithm run at all without crashing. One of these simplifications was using only three of the microphones to perform the sweep of possible angles. A further simplification was to assume that the three microphones could be broken into two pairs in the calculations for determining the pair of angles from which the maximum was coming from. Further, hardware and computer limitations limited sampling to a rate of 8000 Hz from each of eight microphones and made the processing cost of upsampling prohibitive beyond a factor of around 10.