## In which way does the array contrast depend on the number of channels?

There is only an indirect relationship between the number of microphones and the contrast of a microphone array. The array contrast is defined as the level difference between the main lobe and maximum side lobe of the beam pattern which primarily depends on the shape of the array. It is important that the average distance between two adjacent microphones is less than half the wavelength

$$\ d_{min}<(\lambda/2)$$ $$\ d_{min}=(\lambda/2)$$

Otherwise, aliasing effects occur. Aliasing causes so-called "Grating Lobes". These are side lobes with a particularly high amplitude which reduce the contrast. The figure shows the array patterns of ring arrays with different numbers of microphones which are given by a sine source directly $$\ d_{min}=(\lambda/2)$$ in front of the arrays. For a general conclusion, all values have been normalized with respect to the wavelength. As long as the distance of two adjacent microphones is higher than half the wavelength, the beam patterns vary considerably. If this distance is lower than half the wavelength, the array patterns will not change if the number of channels increases.

The above examples present harmonic situations. All distances have been normalized with respect to the wavelength. The simulated patterns can consequently be applied to a real ring array for one frequency value only depending on the number of microphones and the array diameter. Some $$\ d_{min}<(\lambda/2)$$ array patterns may have a better array contrast than the simulations with a microphone distance of 0.6 $$\lambda$$. At the same time, however, the array pattern changes at another frequency value which may lead to a bad array contrast.

To ensure that the array delivers the same results throughout the frequency range used, the wavelength of the highest frequency should be smaller than the double distance between two adjacent microphones.

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