Wiener filter smoothing

This is the noncausal Wiener filter for the clean signal with an adjustable noise level determined by the constraint a in Eq. (19.81). This filter is commonly implemented using estimates of the two power spectral densities. Let fy 9) andfy,(9) denote the estimates of fy 9) and /,y(0), respectively. These estimates could, for example, be obtained from the periodogram or the smoothed periodogram. In that case, the filter is implemented as... 

Faced with the problem of noise corruption alone [s(x) = <5(x)], Norbert Wiener (Goldman, 1953) devised his well-known smoothing filter... 

In deriving the Wiener smoothing filter, one asks what filter y(x) results in the smallest mean-square error between the object estimate... 

Noise is a serious problem where z(co) is small. We see that the estimate 6(cd) carries with it none of the sensible noise treatment that modified filters like those of the Wiener type provide. We may, however, supply the required noise suppression between iterations by smoothing. The polynomial filters described in Section III.C.5, for example, may be used. Very little has been done, however, to determine how suppression might be accomplished optimally. Usually it is treated in an ad hoc way. 

There are several ways of detecting peaks in such noisy signals. The Wiener-Hopf filter minimizes the expectation value of the noise power spectrum and may be used to optimally smooth the original noisy profile [19]. An alternative approach described by Hindeleh and Johnson employs knowledge of the peak shape. It synthesizes a simulated diffraction profile from peaks of known width and shape, for all possible peak amplitudes and positions, and selects that combination of peaks that minimizes the mean square error between the synthesized and measured profiles [20], This procedure is illustrated... 

---