a 



SUMMARY AND CONCLUSIONS 



Three types of statistical filters have been discussed 

 relative to their effectiveness in performing smoothing or 

 filtering of data in time series . 



1. The equally weighted running mean type of (.2^+1) consec- 

 utive equal weights, has a frequency-response function which 

 decreases smoothly down to its cutoff frequency f ac which is 

 determined by N. This filter is computationally simple and 

 has a relatively sharp cutoff, but its frequency response 



oscillates above the first zero crossing introducing un- | 



desirable ripples in the data. This filter is less critical 



than desired for many purposes. jj 



2. The Gaussian smoothing function, though devoid of the 

 oscillatory defect in the above type, has a frequency response 

 which drops smoothly approaching zero asymptotically. This 

 function does not have a cutoff but for practical purposes a 

 cutoff may be arbitrarily defined as the frequency for which 



its frequency response is 1 per cent. With weighting functions » 



of the same N , the equally weighted running mean type has a 



much sharper cutoff than the Gaussian type. Where the demand \ 



for sharp cutoff is not severe, the Gaussian smoothing filter 



may be used to advantage . 



3. In the equally weighted running mean and the Gaussian 

 types of filters the weighting function is specified and from 

 it the frequency response function is evaluated. It is 

 possible to reverse the procedure, by specifying the desired 

 characteristics of the frequency response and from these 

 conditions determining the corresponding weighting function. 

 The weighting function V{t) is the inverse Fourier transform 

 of its corresponding frequency-response funct.on (see equa- 

 tions (6) and (?))• This type of filter discussed earlier- 

 specified a gain of unity and zero phase shift for all fre- 

 quencies. To further improve the fidelity of this filter, 

 the frequency-response was terminated at its ideal cutoff 

 frequency, r c , by sine termination, and the weights further 

 corrected to insure unity gain at zero frequency. 



By a proper selection of the parameters r c , h, and // (see 

 list of definitions) filters with maximum absolute error 

 less than 1 per cent have been designed. Weights for 118 

 such filters are given in the Appendix. 



Computer programs have been written for the Burroughs Data- 



tron 220 computer and were used in obtaining the numerical | 



results for all filters discussed in this report. 



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