0.2 Developing the array model and processing techniques

We continue to develop the properties for the uniform linear array (ULA) that has been discussed previously . With the important relationship that we found to avoid spatial aliasing, $d\le \frac{{\lambda }_{\min }}{2}$ , we now consider the theoretical background of the ULA. Once we understand how the array will be used, we will look at a method called'beamforming'that directs the array's focus in specific directions.

Far-field signals

We saw previously that a very nice property of the ULA is the constant delay between the arrival of a signal at consecutive sensors. This is only true, however, if we assume that plane waves arrive at the array. Remember that sounds radiate spherically outward from their sources, so the assumption is generally not true! To get around that problem, we only consider signals in the far-field , in which case the signal sources are far enough away that the arriving sound waves are essentially planes over the length of the array.

Far-field source

A source is considered to be in the far-field if $r>\frac{2L^2}{\lambda }$ , where r is the distance from the source to the array, L is the length of the array, and $\lambda$ is the wavelength of the arriving wave.

Near-field sources are beyond the scope of our project, but they are not beyond the scope of array processing. For more information on just about everything related to array processing, take a look at Array Signal Processing: Concepts and Techniques , by Don H. Johnson and Dan E. Dudgeon, Englewood Cliffs, NJ: Prentice Hall, 1993.

Array properties

Depending on how the array will be used, it may be important (as it was in our project) that the microphones used be able to receive sound from all directions. We used omni-directional microphones, which are exactly what they sound like -- microphones that hear in all directions. If you don't need this ability, you can look into other options, but keep in mind that the theoretical development here assumes omni-directional capability, so you will need to do some research on array processing techniques that suit your needs. In fact, it would be a good idea no matter what! Our project used a simple array design, but it took a while to learn all of the theory and to figure out how to implement it.

Our array comprises six generic omni-directional microphones. We built an array frame out of PVC pipe to hold each microphone in place with equidistant spacing between the sensors. Physical limitations of the microphones and PVC connecting pieces prevented us from using a very small spacing; for our array, we had a sensor spacing of d=9.9 cm. Since we know that we need to have $d\le \frac{{\lambda }_{\min }}{2}$ to avoid spatial aliasing, we are able to calculate the highest frequency that the array is capable of processing: $f_{\max }=1600\mathrm{Hz}$ . (Actually, $f_{\max }$ is a little higher than 1600 Hz as you can verify, but to be on the safe side we kept it a bit lower.)