i_-r«a!B'«K«'« ! 5Saa. 



function (Vjf) of which there are (2/V+l) . These weights are 

 sequentially arranged as follows for the discrete weighting 

 function. 



W-n, W-n+1, . . . .W , W , V . . . .W 



-101 n § J- 



? 'v . 



The smoothed points are obtained by equation (12). For the "v 



weighting functions discussed in this section the weights 



may be positive or negative except the central weight W \ 



which is positive. However, any particular W% is identical . 



with the corresponding W_ k > in magnitude and sign. | .-•,' 



Table 1 gives in addition to r c , h, and N, r ac which is the 



actual cutoff frequency for our approximate filter and, 



as such, is the frequency for the first zero crossing of the 



frequency-response-curve with the frequency axis; and E the I 



maximum absolute error which is the departure of the fre- s 



quency-response ordinate from unity between zero frequency 



and r c , or the departure of the ordinate from zero, between 



the first zero crossing frequency and the freauency limit 



r = 0.5- 



Figure 6 is designed to portray the essential characteristics 



of the frequency-response curve in greater details. The t 



magnified scales at the upper left and the lower right 

 indicate the oscillatory nature of the frequency-response 

 function. The graph in fip-ure 6 shows the desired ideal 

 cutoff frequency r c and how it is related to the actual 

 cutoff frequency r ac which is the frequency of the first 

 zero crossing of the frequency-response curve. The value 

 E gives the maximum error amplitude in the gain between 

 zero frequency and the ideal cutoff frequency r c , and E 

 gives the maximum error amplitude between the actual cutoff 

 frequency r ac and the frequency limit (r = 0-5) for this 

 type of filter. In most cases E is approximately equal to 

 E . However, in any case the maximum absolute error is 

 taken as E. If E = E 2 , then E = E^ = E z - 



The two frequency-response curves A and B in figure 7 show the 

 general trend of the function. Both curves have the same ideal 

 cutoff frequency ( r c = O.Ol) and both have the same maximum 

 absolute error in the gain (l'= 0.002). It is seen in curve A 

 that with h = 0.08, which is relatively large, N is only 

 20 but the penalty is that the actual cutoff frequency, is 

 relatively large namely 0.165- In curve B, with h and r ac 

 relatively small, namely h - 0.01, r ac = 0.0275, it wa 

 necessary to increase N to 100 in order to retain the same 

 maximum absolute error as small as E = 0.002. The above 



19 



