the expression reduces to 



W{t) 



■i: 



B(f)cus2nft df 



(15) 



which is observed to be the Fourier cosine transforms of 

 -#(/)• Here the frequency response can be specified and the 

 weighting or smoothing function computed. If one specifies 

 B(f) with 



R(f) = 1 0§/s/ c 

 = / > f 



(19) 



then 



f: 



W{t) = 2 / lcos (2 i( /0 df 



(20) 



V(t) = (irt) sin 2it f t 



/ 



The chief disadvantage of this smoothing function is its 

 slow damping which renders it impractical for many important 

 purposes. Furthermore if this function is truncated at some 

 convenient distance on each side of the origin (a necessary 

 procedure in most practical cases), the frequency response 

 of this smoothing function departs significantly from the 

 desired response at most frequencies. 



The design of a low -pass numerical filter which eliminates 

 many of the undesired properties of those filters or smooth- 

 ing functions discussed above has been achieved by determin- 

 ing a set of weights W^ such that the actual frequency 

 response of the filter defined by equation (19) will approx- 

 imate best in the least square sense the ideal frequency 

 response. 



In the design of this filter the weights W k are determined 

 subject to the following conditions: 



16 



