Sufficient condition for invertibility of square FIR MIMO systems

We derive a sufficient condition for a square FIR MIMO system to have a causal and stable IIR inverse. The condition requires that the spectral norm of the normalized channel impulse response (i.e., the first tap is the identity matrix) is below a certain bound. Intuitively, this means that the system has a strong first tap. This condition often is easier to check than the usual minimum-phaseness, where the roots of the systems determinant have to be computed. Simple approximations of the bound are found. Furthermore, we also give a negative result: the Wiener Filter, which approximates the inverse under low noise conditions, nevertheless always is non-causal. We apply our results to two inversion problems with causality and stability constraint. These problems arise in oversampled noise-shaping subband coding and residual interference cancellation in precoded systems, respectively.