Multi-channel multi-variate equalizer design (Q1810944)

Distortion equalization can occur when operating Synthetic Aperture Radar (SAR). The modelling of SAR leads to a Multiple-Input Multiple-Output (MIMO) system whose describing matrix contains the reflectivity coefficients and its delayed versions. In a general setting, a SAR is described (in the operational domain, i.e. \(Z\)-transform in more dimensions) as follows: \(y=Hx\), where \(y\) is the output, an \(M\times 1\) matrix, \(x\) is the input, an \(N\times 1\) matrix, and \(H\) is an \(M\times N\) matrix whose elements are polynomials in \(z_1,z_2,\dots, z_n\), with real or complex coefficients, and \(n\) is the dimension of the space. In terms of system theory the MIMO under consideration is a linear, finite impulse response, time invariant filter. It must be pointed out that the matrix \(H\) is not always square but in most cases of interest the following inequality is valid: \(M\geq N\). To achieve the equalization, one needs to find the left inverse of \(H\), i.e. an \(N\times N\) matrix denoted by \(G\) so that \(GH= I_N\), where \(I_N\) is the unit matrix of rank \(N\). Before presenting the proper contribution, the authors give a constructive proof of a basic theorem, already known in the literature and ensuring necessary and sufficient conditions for the existence of \(G\). After pointing out the difficulties when applying linear algebra for finding \(G\) (such as high volume of calculations, difficulties in evaluating the dimensions of the polynomials entering the left inverse), the authors propose a new procedure using Gröbner bases of polynomials. Also, approximate solutions are given for the case when the exact inverse does not exist. Some examples are given. We think the paper represents a very valuable contribution in the area of MIMO identification.