Optimization of linear filters under power-spectral-density stabilization

Geometric-mean filters compose a family of filters indexed by a parameter k varying between 0 and 1. They have been used to provide frequency-based filtering that mitigates the noise suppression of the optimal-linear Wiener filter in the blurred-signal-plus-noise model. For k = 0 and k = 1, the geometric-mean filter gives the inverse filter and the Wiener filter for the model, respectively. The geometric mean for k = 1/2 has previously been derived as the optimal linear filter for the model under power-spectral-density (PSD) equalization. This constraint requires the PSD of the filtered signal to be equal to the PSD of the uncorrupted signal that it estimates. This paper defines the notion of PSD stabilization, in which the PSD of the restored signal is equal to a predetermined function times the PSD of the uncorrupted signal. A particular parameterized stabilization function yields the geometric-mean family as the optimal linear filter for the model under PSD stabilization. Relative to unconstrained optimization, geometric means are suboptimal; however, we consider a parameterized model for which the noise is such that the geometric-mean filters provide optimal linear filtering. In the altered signal-plus-noise model for which the geometric mean is optimal, the blur is the same as the original model in which the geometric mean is defined, but the noise PSD is a function of the Fourier transform of the blur and the PSD of the original noise. Since the altered model depends on k, we consider a robustness question: what kind of suboptimality results from applying the geometric mean for k(1) to the model for which the geometric mean for k(2) is optimal?