Revisiting the Kronecker Array Transform

It is known that the calculation of a matrix-vector product can be accelerated if this matrix can be recast (or approximated) by the Kronecker product of two smaller matrices. In array signal processing, the manifold matrix can be described as the Kronecker product of two other matrices if the sensor array displays a separable geometry. This forms the basis of the Kronecker Array Transform (KAT), which was previously introduced to speed up the calculations of acoustic images with microphone arrays. If, however, the array has a quasi-separable geometry, e.g., an otherwise separable array with a missing sensor, then the KAT acceleration can no longer be applied. In this letter, we review the definition of the KAT and provide a much simpler derivation that relies on an explicit new relation developed between Kronecker and Khatri-Rao matrix products. Additionally, we extend the KAT to deal with quasi-separable arrays, alleviating the restriction on the need of perfectly separable arrays.