Microphone array sound capture consists of a desired source signal accompanied with additive noise. It is therefore advantageous to approximate noise fields during testing of a beamformer. The coherence function,
![](https://vocal.com/wp-content/uploads/2021/07/Matrix.png)
is one representation utilized to estimate the spatial correlation of noise in a microphone array of M sensors. Here, we will discuss two noise field estimations: spherically isotropic and cylindrically isotropic. Both approximations assume the noise field is an infinite sum of uncorrelated plane waves. We also assume omnidirectional elements in the microphone array.
![](https://vocal.com/wp-content/uploads/2021/07/spat_corr_coeff-1-1014x1024.png)
Figure 1. Comparison of spherical and cylindrical spatial correlation
coefficients as a function of inter-sensor spacing over wavelength.
Consider the spatial coherence function of a plane wave,
where is the wavenumber,
is the incidence angle of the plane wave, and
is the distance between the nth and mth sensor. For a spherically isotropic field, the coherence function can be calculated using the unit surface area of a sphere, giving
Applying u-substitution, , the spherically isotropic coherence function is
The spherically isotropic coherence function best serves as a model for a three-dimensional diffuse field, where noise is travelling in all directions with equal power. The same computation utilizing the unit surface area of a cylinder yields
where J is the zero-order Bessel function of the first kind. The cylindrically isotropic noise field, or two-dimensional diffuse field, models noise within a room with an absorptive ceiling and floor causing the soundfield to propagates longitudinally within the enclosed space. As seen in Figure 1, correlation between two sensors decreases at a higher rate in a spherically isotropic field. This results in a higher directivity index (gain) in the three-dimensional diffuse field.