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/sub n/)/sub n/spl isin/z/ having a strictly negative Kurtosis /spl gamma//sub 2/= E[|s/sub n/|/sup 4/]-2(E[|s/sub n/|/sup 2/])/sup 2/-|E[s/sub n//sup 2/|/sup 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[s/sup 2/]=0. When this constraint is relaxed, (s/sub n/)/sub n/spl isin/z/ is said to be noncircular symmetric: It is shown that the minimizers achieve deconvolution if and only if 2|E[s/sub n//sup 2/]|/sup 2/<-/spl gamma//sub 2/(s). If this condition is not met, the global minimizers are found to be finite-impulse-response filters with two taps.