Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 160', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Optimal Wiener Filtering with the FFT 547 Optimal Wiener Filtering with the FFT There are a number of other tasks in numerical processing that are routinely handled with Fourier techniques. One of these is filtering for the removal of noise from a corrupted signal. The particular situation we consider is this There is some underlying uncorrupted signal u t that we want to measure. The measurement process is imperfect however and what comes out of our measurement device is a corrupted signal c t . The signal c t may be less than perfect in either or both of two respects. First the apparatus may not have a perfect delta-function response so that the true signal u t is convolved with smeared out by some known response function r t to give a smeared signal s t s t p r t - t u r dr or S f R f U f where S R U are the Fourier transforms of s r u respectively. Second the measured signal c t may contain an additional component of noise n t c t s t n t We already know how to deconvolve the effects of the response function r in the absence of any noise we just divide C f by R f to get a deconvolved signal. We now want to treat the analogous problem when noise is present. Our task is to find the optimal filter t or f which when applied to the measured signal c t or C f and then deconvolved by r t or R f produces a signal u t or U f that is as close as possible to the uncorrupted signal u t or U f . In other words we will estimate the true signal U by U f C f W R f In what sense is U to be close to U We ask that they be close in the least-square sense I e t - u t 2 dt I I U f - U f I 2 df is minimized. Substituting equations and the right-hand side of becomes i 1 Sf Nf f _ S f 2 df J 1 R f R f d Zo - - - o 2l r 21 K -T 2 t . 2 t j. 2I R f Sf i- f Nf mf jdf Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 The signal S and the noise N are uncorrelated so their cross .
