We design the filters in such a way that the response of the system is the prescribed theoretical function vk for the k-th source (an unit-amplitude Dirac’s Delta function in the case of the example, as the 0th-order function is omnidirectional). So we set up a linear equation system of P equations, imposing that:

Lets call vk the right-hand vector of known results (they will be different for higher-order shapes). Once this matrix of N inverse filters are computed (for example, employing the Nelson/Kirkeby method), the output of the microphone array, synthesizing the prescribed 0th-order shape, will again be simply: