Performance of cumulant based inverse filters for blind deconvolution

Chi and Wu (1963) proposed a class of inverse filter criteria J _r,m using rth-order and rth-order cumulants (where r is even and m>r⩾2) for blind deconvolution (equalization) of a (nonminimum phase) linear time-invariant (LTI) system with only non-Gaussian measurements. The inverse filter criteria J_r,m for r=2 are frequently used such as Wiggins´ (1978) criterion, Donoho´s (1981) criteria, and Tugnait´s (1993) inverse filter criteria for which the identifiability of the LTI system is based on infinite signal-to-noise ratio (SNR). We analyze the performance of the inverse filter criteria J_2,m (r=2) when the SNR is finite. The analysis shows that the inverse filter associated with J_2,m is related to the minimum mean square error (MMSE) equalizer in a nonlinear manner, with some common properties such as perfect phase (but not perfect amplitude) equalization. Furthermore, the former approaches the latter either for higher SNR, cumulant-order m, or for wider system bandwidth. Moreover, as the MMSE equalizer does, the inverse filter associated with J_2,m, also performs noise reduction besides equalization. Some simulation results, as well as some calculation results, are provided to support the proposed analytic results