primarily concerned with the low-pass case since the high- 

 pass case is easily derived. 



LINEAR FILTERS 



A system is said to be linear if for all inputs f(t), g(t) , 

 and constants a, b 



S[af(t)-rbg(t) ]=aS[f (t)}+bS[g(t) } 



By superposition, these properties extend to any finite num- 

 ber of input functions . 



The input and output of a filter can be related by a differ- 

 ential equation, the solution of which gives the output for 

 any input. Notwithstanding, the differential equation 

 description in many cases is not the most convenient for 

 design purposes. More convenient modes of describing a 

 filter, which make use of outputs produced by special types 

 of inputs , employ the following functions : 



1. The weighting function. 



2. The frequency-response function. 

 3- The transfer function. 



The response of a linear filter to general types of inputs 

 may be described by its weighting function which is defined 

 as the response of the filter to a unit impulse function 

 after a time t has elapsed. The weighting function U{j), 

 frequently called the "impulsive response" of the filter 

 provides a complete characterization of the filter for >/(t) 

 vanishing when t<Q. 



The frequency-response function i?(/) relates a sinusoidal 

 input to the output that produces it and, for a stable filter, 

 is the Fourier transform of its weighting function. 



The transfer function, a generalization of the frequency- 

 response function, is defined as the Laplace transform of 

 the weighting function. 



Theory shows 1 ' 2 that, if St (i) is the input, the output 

 £ (t) iJ given by the following relation in which W^ (t) 



r 



\ 



