System reconstruction based on selected regions of discretized higher order spectra

We consider the problem of system reconstruction from arbitrarily selected slices of the nth-order output spectrum. We establish that unique identification of the impulse response of a system can be performed, up to a scalar and a circular shift, based on any two one-dimensional (1-D) slices of the discretized nth-order output spectrum, (n 3), as long as the distance between the slices and the grid size satisfy a simple condition. For the special case of real systems, one slice suffices for system reconstruction. The ability to choose the slices to be used for reconstruction enables us to avoid regions of the nth-order spectrum, where the estimation variance is high, or where the ideal polyspectrum is expected to be zero, as is the case for bandlimited systems. We show that the obtained system estimates are asymptotically unbiased and consistent. We propose a mechanism for selecting slices that result in improved system estimates. We also demonstrate via simulations the superiority, in terms of estimation bias and variance, of the proposed method over existing approaches in the case of bandlimited systems.