Characterization of the undesirable global minima of the Godard cost function: case of noncircular symmetric signals

The deconvolution of a filtered version of a zero-mean normalized independent and identically distributed (i.i.d.) signal (s_n)_n∈z having a strictly negative Kurtosis γ₂= E[|s_n|⁴]-2(E[|s_n|²])²-|E[s_n²|²] is addressed. This correspondence focuses on the global minimizers of the Godard function. A well-known result states that these minimizers achieve deconvolution at least if the input signal shows the symmetry E[s²]=0. When this constraint is relaxed, (s_n)_n∈z is said to be noncircular symmetric: It is shown that the minimizers achieve deconvolution if and only if 2|E[s_n²]|²<-γ₂(s). If this condition is not met, the global minimizers are found to be finite-impulse-response filters with two taps.