smoothing function, on the contrary, decreases smoothly with 

 increasing frequency and asymptotically approaches zero, 

 thus avoiding the generation of undesirable high frequency 

 components in the smoothed output. 



(b) If, at pleasure, one defines for practical purposes 



the cutoff frequency of the Gaussian filter at the point 



where the frequency response is 1 per cent, it is seen that i 



for equal filtering intervals the equally weighted running ? 



mean filter has a lower cutoff frequency than the Gaussian 



filter. For the smoothing filters shown in figure 5 the 



equally weighted running mean filter has a cutoff frequency 



at 0.02W3 cycle per data interval, whereas the 1 per cent 



cutoff point for the corresponding Gaussian filter occurs 



at O.OjV cycle per data interval. I 



J 



For some types of smoothing the equally weighted running 



mean or the Gaussian filter may be satisfactory. Frequently, 



in other cases of smoothing or filtering the disadvantages 



of the filters discussed above may render them inadequate. 



A more satisfactory filter in which the above deficiencies i| 



are significantly reduced will be discussed below. 



FILTERS WHICH APPROXIMATE IDEAL FILTERS 

 WITH UNITY GAIN AND ZERO PHASE SHIFT 



In the preceeding sections we discussed frequency-response 

 functions having specified weighting functions. The proce- 

 dure may be reversed and filtering or weighting functions 

 having specified frequency-response functions -S(/) may be 

 obtained using equation (7) : 



V(t) = f P.(f) exp{-J2nft) df 



W(t) = I B{f)[cos 2xft-isin 2rcft\ df (17) 



\J -09 



For an even response -fl(/) the Imaginary part vanishes, and 



15 



