"1 1 



I i 



In designing a filter whicn approximates the ideal filter 

 specified above one should select, h and N as small as 

 possible for a chosen r c such that the resulting gain of 

 the filter does not depart from the ideal gain by more than 

 some predetermined tolerance. 



The use of the filter is illustrated by the following 

 example. In figure 10 a spectral density curve is plotted 

 with variance per unit band as the ordinate and frequency 

 in cycles per week as the abscissa. This graph (unfiltered 

 data) shows principal peaks at frequencies of 7 and 1^ 

 cycles p^.r week. A numerical filter was designed to elim- 

 inate the- 1^- cycle component. 



For the original data f s the sampling rate was 168 per week. 

 The nontL-lized frequency r ir related to the frequency / 

 in cycles per week by the relation/ = rf s . 



For this problem a filter is needed which will retain the 

 7-cycle-per-week component and eliminate higher components. 

 Note that filter number kj, table 3> in the Appendix, with 

 r c = 0.05, h = 0.01, r ac =- 0.08, N = 70, and E = 0.002 can 

 be used. 



Then 



f c = (0.05) (168) = 8 A cycles per week 



f ac = (0.08)(l68) = 13.4 cycles per week 



Frequency components below 8.k cycles per week are retained 

 but those above 13-^ cycles pei* week are eliminated. The 

 effectiveness of this filtering operation is seen in figure 11. 

 Observe that the lk- cycle-per-week component has been prac- 

 tically eliminated in the graph (figure ll), but the 7-cycle- 

 per-week component is left unchanged in shape and magnitude. 



2k 



