by Gold and Radar (1969) , The first step in the development of this 

 filter function is to assume an ideal filter response function. 



F,Cf) 





1 , < f < f 



f < f < f 

 c ' ' — n 



(12) 



F,(f) = 



, < |f| < f^ 



1 , f < Ifl < f 



' c — ' ' — n 



where f^ is the cutoff frequency, f,^ the Nyquist frequency, and F^(f) 



and F, (f) are the ideal low-pass and high-pass filter response functions. 



The ideal filter response function is then Fourier-transformed to 

 the time domain, giving the impulse response function, which is trun- 

 cated by using an appropriate window function and transformed back to 

 the frequency domain giving a complex frequency response function. The 

 number of points used in representing the filter function is allowed to 

 vary and the resulting convolution with the original time series is 

 accomplished by using the overlap-add method of convolving smaller ser- 

 ies with larger ones. This allows for more economical filtering proce- 

 dures . 



This gives three variables to choose from in the final filter func- 

 tion design: the length or number of points used in the filter, the 

 type of window used to truncate the impulse response function, and the 

 niomber of points to be truncated. 



This procedure is analogous to spectral estimation techniques except 

 for the truncation of the impulse response function. The larger the 

 niimber of points used in the filter function, the better the estimate. 

 The smoother the window function, the broader the transition band. In 

 addition, the ripple or Gibb's phenomena is reduced. Generally speaking, 

 the more points that are truncated (set to zero) the better the result- 

 ing approximation. In practice, the actual number is determined experi- 

 mentally by comparing results for different truncation values. This 

 results in setting approximately 20 percent of the inpulse response func- 

 tion to zero. The banning window function was used with 128 points in 

 the filter response function and 38 points being set to zero in the im- 

 pulse response function. That is: 



h(nAt) = w(nAt) h(nAt) 

 and 



ric 



1 + cos IT I^) , 1 <_ n ^ 45 



45 

 w(nAt) = <* , 45 < n < 83 (13) 



i (1 - cos u ^H^), 83 < n < 128, 



2 " 45 



56 



