Angle of Arrival

The fundamental notion behind acoustic source localization is the concept that the angle of arrival of an incoming sound wave can be determined with a pair of microphones, spaced somewhat apart from each other. This concept rests on the constraint that the sound wave of interest originated at a far enough distance from the microphones that it can be approximated as a plane wave rather than a wave of circular shape. This simplification is valid under the circumstances given by the inequality,

Where R is the radial distance to the sound source, d is the distance between the microphones, and  is the wavelength of the incoming wave. Approximating the wave as planar is necessary to simplify the geometry involved in calculating of the angle of arrival.

Assuming this approximation holds, the angle of an incoming sound wave can be calculated as a function of the delay between the arrival of the sound wave between at microphones.

If this time delay between sensors can be accurately estimated by processing the microphones' data, all that remains in deriving the angle of arrival is some knowledge of the system's geometrical set up. Intuitively, this can be understood by noticing that if a sound source were to emit a wave directly in front of both microphones, there would be no delay whatsoever in the time period between each microphone's receiving of the wave. If, however, the source was shifted to, say, the left, the signal would reach the left microphone first, and some time delay between the left and right sensor's signals could be calculated. The extent of this delay would then be a function of the amount that the microphone was shifted left, and with this knowledge the angle of arrival could be calculated.

The full derivation of this is provided in Joshua York's project.  The expression for angle of arrival is provided by,

   

Where  is a function of the speed of sound, , the time delay, and d the length between the two microphones.

It is important to note that the final result should be in discrete time because the signal is processed in the digital domain. Thus the delay is ultimately given in terms of the difference in number of samples between the two microphones.  Converting from continuous time to discrete time,

  ,

Where n is the number of samples, and  is the sampling frequency of the data acquisition system. Substituting, the final expression is found,

  .