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CONTENTS 



Page 



3 INTRODUCTION 



k LINEAR FILTERS 



8 SMOOTHING OF TIME SERIES 



8 FREQUENCY - RESPONSE OF SOME SMOOTHING 



FUNCTIONS 

 8 Equally Weighted Running-Mean Smoothing 



Functions with (2M-l) Consecutive 

 Weights 

 Yd Gaussian Filter 



15 FILTERS WHICH APPROXIMATE IDEAL FILTERS 



WITH UNITY GAIN AND ZERO PHASE SHIFT 



27 SUMMARY AND CONCLUSIONS 



28 REFERENCES 



29 DEFINITIONS 



31 APPENDIX: FILTER DATA 



ILLUSTRATIONS 



Figure 



10 1 Frequency response of equally weighted 



running mean weight functions. 



11 2 Variations of cutoff frequency with N for 



the equally weighted running mean 



filtering function. 

 13 3 Frequency response of the Gaussian weighting 



function, 

 l^ k Variation of the cutoff frequency for the 



Gaussian filtering function. 

 1^ 5 Frequency response of forty-one term 



equally weighted running mean compared 



with an equivalent normal curve smoothing 



function. 



20 6 Frequency response function 



21 7 Frequency response versus the normalized 



frequency 



22 8 Variation of the maximum absolute gain 



error with h 



23 9 Variation of the maximum absolute gain 



error with N 

 25-26 10-11 Spectral density function versus, frequency. 



