A unified class of inverse filter criteria using two cumulants for blind deconvolution and equalization

Cumulant (higher-order statistics) based inverse filter criteria maximizing J_r,m=|C_m|^r/|C_r| ^m, where m≠r and C_m (C_r) denotes the mth-order (rth-order) cumulant of the inverse filter output, have been proposed for blind deconvolution and equalization with only non-Gaussian output measurements of an unknown linear time-invariant (LTI) system. This paper shows that the maximum of J_r,m associated with the true inverse filter of the unknown LTI system, exists only for r to be even and m>r, otherwise, J_r,m is unbounded. The admissible values for (r,m)=(2s,l+s) where l>s⩾1 include (2,3), (2,4) and (4,6) proposed by Tugnait (see IEEE Trans. Signal Processing, vol.41, no.11, p.3196-3199, 1993), Wiggins (1978), and Shalvi and Weinstein (see IEEE Trans. Information Theory, vol.36, p.312-321, 1990) in addition to more new ones such as (2,5), (2,6) and (4,5). Some simulation results for the inverse filter criteria J_r,m with the proposed admissible values of (r,m) are then provided. Finally, we draw some conclusions