M * = z * + 2i7T7 W 



S(r) = W + 2 \ W k cos(2rthr) (2^) 



The normalized frequency r is defined by the relation 

 r = f/f a of which the range of r is from r = to r = 0.5- 

 Here / is in cycles per unit time, and f g is the sampling 

 rate. Consequently r , the normalized cutoff for the ideal 

 filter, is defined as follows: 



-Li. 



where f c is the ideal cutoff frequency in cycles per unit 

 time and f g is the sampling rate. The quantity N is defined 

 in terms of the total number of weights in the weighting 

 function, namely (2AT+l) . The normalized frequency for the 

 first zero crossing of the frequency-response curve is given 



h * r aC 



Equation (2l) gives the approximate weights, with sine 

 termination, corrected for high frequency oscillations 

 beyor." the first zero crossing of the frequency-response 

 curve. The quantity A 7(2^+1) in equation (24) is the correc- 

 tion to insure unity gain at zero frequency. Equation (24) 

 gives the best approximation of the weights for the filter 

 under the conditions specified. Equation (2?) gives the 

 best approximation of ordlnates for the frequency-response 

 function. The numerical results were obtained with the 

 Datatron 220 digital computer, and the coded programs 

 utilized the basic working formulas (2l) through (25) 

 inclusive. 



Numerical results were obtained for various combinations of 

 r c , h, and N. The normalized cutoff frequency for the ideal 

 filter is designated by r c and h is a parameter which permits 

 variation in the slope of the sine terminations . The use of 

 sine termination avoids sharp discontinuity at r c , thus 

 greatly reducing the high-frequency oscillations or ripples 

 associated with the Gibbs phenomenon. The number N is 

 associated with the total number of weights in the weighting 



18 



