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Sound Propagation Models for Beamforming

The design of beamforming algorithms is affected by the two different sound propagation models  – near-field and far-field. There are two main differences between these models. The first being that the signal level of a sound source between two sensors in the array are equivalent even though the distance between the source and sensor is different. The second difference is that the phase shift of the source signal between two microphones depends only on the direction of the source signal for the far-field model, while the near-field model depends on distance and direction. The far-field model is used more often because it is a simplified version of the near-field model which makes it easier to analytically derive solutions.

From their names it becomes apparent that distance plays the lead role in determining when it is appropriate to switch to using the simplified far-field model. A source signal of frequency, ƒ, is considered in the near-field when < 2cl⁄ λ, where l is the total size of the microphone array, d is the distance of the signal from the center of the microphone array and c is the velocity of sound.

Although the far-field model simplifies the propagation of sound to a plane-wave, increased distance between the microphone and speaker results in a decrease in signal-to-reverberation ratio. Reverberation causes the capture sound to have an increased temporal correlation. This temporal correlation can have negative effects on the convergence speed on many interference nulling beamforming algorithms. Therefore, design of a beamformer algorithm must be cognizant of the role distance has on sound capture.

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