T I 



THE GAUSSIAN FILTER 



The undesirable oscillations associated with the frequency- 

 response curve, after the first zero crossing of the fre- 

 quency axis by the frequency-response curve, in the squally 

 weighted running mean filter can be suppressed if not 

 entirely eliminated by the use of a weighting function in 

 which the weigh ts decrease in magnitude outward from the 

 central or piincipal weight V Q . One useful set, of weights 

 may be made proportional to the ordinates of the normal 

 probability curve. In this case the contii:. ous analytical 

 expression for the weights is available: 



V(t) = ^jta 2 )" 1 ^ exp(-* 2 /2a 2 ) (A) 



It is known that the total area under the curve is unity, 

 and the mean of the original time function is preserved. 

 The frequency-response function can be calculated by sub- 

 stituting the expression (l4) in the basic equation (10). 

 The expression for the frequency response is given by the 

 integral 



R(f) = 2 f (2jta 2 )-V 2 exp(-t 2 /2a 2 ) cos(2it/t) dt (15) 

 J o 



Evaluating the integral (l^) gives the frequency-response 

 function for the Gaussian type filter. The result of this 

 integration is given by equation (l6) below: 



*(/) = exp(-2*V/ 2 ) (16) 



The general features of the function in (16) are shown 

 graphically in figure 3. The frequency response for the 

 normal curve smoothing function decreases smoothly with 

 frequency, approaching zero asymptotically. Theoretically, 

 zero response is never reached. However, the cutoff fre- 

 quency f c will be defined arbitrarily as the frequency for 

 which the frequency -response is 1 per cent. It follows 

 from equation (l6) that f . as defined above, is controlled 

 by the parameter a. The graph in figure k gives some indica- 



12 



