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. [ABSTRACT FROM AUTHOR]