Accuracy limits and Aliasing
To determine the direction of the position of the sound source, the signal from the microphones is digitized at a given sampling rate. For the implementation of the acoustic locator the sampling frequency (Fs) is set at 44,100 Hz. Therefore, the sampling time (ts) is equal to 1/Fs (1/44,100 = 22.68 μs).
The velocity of sound (Vs) at 20°C is 343m/s. The minimum distance change (dm) that can be detected is the distance travelled by a sound wave in 22.68 μs. Therefore the minimum distance change detected is equal to 7.78mm, since dm = Vs x ts = 343m/s x 22.68μs = 7.78mm
In the developed acoustic locator the distance between the microphones (Lm) is set to 10cm. The distance difference between the distances from the sound source to each microphone is dx = |RL – RRI. From figure 5, it can be observed that the maximum distance difference occurs when the source is in line with the two microphones, thus it is 10 cm.Therefore the number of distinct sound source directions detected is 12.9 for positive values of dx and 12.9 for negative values of dx, since 10cm/7.78mm = 12.9. This implies that for a 180 degree angle there are 25 distinct directions detected. As shown in figure 6, these directions are denser if the sound source is closer to the line formed by the two microphones, and less dense when the sound closed to perpendicular with the line formed by the microphones.
Figure 5
Figure 6
A way to determine the time difference between the arrivals of a sound signal to the microphones is by detecting either the peak values of two adjacent signal cycles, or by detecting the zero crossings of the two signals. If the number of signal samples between the two adjacent zero crossings is ‘n’ then the time difference is ‘n x ts’. This time difference corresponds to a distance difference equal to ‘n x 7.78mm ‘. An effect than can result in wrong detection results is aliasing. Aliasing occurs in cases where it is clear which microphone received the sound signal first. If we assume that the sound signals are pure sine waves, then as shown in figure 7(a) it is possible to have more than one zero crossings for each microphone in the samples examined. From the waveforms in figure 7(a) it can be seen that during the time frame (Δt) for which the signals are examined, there are two zero crossings for the red waveform and one for the blue. Therefore, there are two possible results: one by considering the first zero crossing of the red waveform, and one by considering the second. From the diagram in figure 7(a) it is clear that the correct result is achieved by considering the second zero crossing, however it is difficult to decide on this if the only information available is a number of samples.
From the waveforms in figure 7(b) it can be seen that if the frequency of the signal is reduced, then there is no possibility to get more than one zero crossings within a given time limit. This time limit is determined by the distance between the microphones. More specifically, if the maximum signal frequency corresponds to a period that has a duration less than the time needed by the sound to travel a distance equal to the distance between the microphones, then it is ensured that aliasing will not happen. In the diagram in figure 7(b) the time (Δt) corresponds the time needed by the sound to travel a distance equal to the distance between the microphones. Thus if the frequency is low enough, there is no possibility of having two zero crossings within this time frame. The maximum sound frequency that ensures no aliasing Fmax is the frequency with period equal to the time (Δt). Therefore (Fmax = Vs/dm = 343(m/s) / 10cm/343m/s = 3.43KHz). To ensure aliasing-free operation, the signals from the microphones should be low-passed with a low pass filter with a cut-off frequency at 3.43KHz. From the above, it can be set that increasing the distance between the microphones will improve the accuracy of the acoustic locator, but it will also reduce the maximum frequency for aliasing-free operation.
Figure 7