filtering interval, with w{t) = 1/r for | t \ % T/2, and 

 V(i) = when | i | > 7/2, then 



?(/) * 2J r cos(2«jt) eft 



-1 

 *(/) * (n/T) ain(«/T) (13A) 



Equation (13A) gives a very ^ood approximation of the fre- 

 quency response of the equally weighted running mean. 



In general, the iTequency -response function R( f) expresses 

 the relative amplitude and phase of the input and output as 

 a function of frequency, and is cefined only for stable 

 filters. 



Since the weighting functions for the equally weighted 

 running mean r.re even, phise shift is eliminated except for 

 l80° shifts after the first zero crossing of the frequency 

 axis. Because of this l80* shift, undesirable oscillations 

 occur which introduce high frequency ripples into the output. 

 Thir behavior of the frequency-response function beyond the 

 first zero crossing constitutes the significant disadvantage 

 in smoothing with equal weights. This can be seen by 

 inspection of the curves in figure 1. A brief discussion 

 of the coordinates used in plotting these curves may be 

 instructive. 



In requiring the weighting function w(t) to be even, the 

 frequency-response function if (/) becomes a pure, rtal quan- 

 tity. A further condition that the mean of the original 

 time function is preserved requires that the sum of the 

 weights of the weighting function W (t) be equal to unity. 

 It follows from this condition and equation (ll) that the 

 value of the ordinate of X{t) is unity at zero frequency. 

 The frequency / is plotted as the abscissa in cycles per 

 time interval between data values . This time interval is 

 called the data interval, and frequencies are expressed in 

 cycles per data interval in discussing equally weighted 

 running mean and Gaussian frequency-response functions. 



From a study of the curves for *he equally weighted running 

 mean type filter in figure 1, a good idea can be obtained of 

 its departure from the ideal low-pass filter with the same 



