5-2] FOURIER ANALYSIS 241 



Continuing with the example adopted in Equation 5-3, the form of ^(r) 

 should be easy enough to find in this case. 



/• oo 



<p{t) = / (e-^'+^^)(e-')dt, T < 



^(r) = ^-m/ e-'-'dt= i^-M (5-9) 



^(0) = i 

 Also, the absolute value of the spectrum is easily obtained from Equation 

 5-4 in this case: 



By virtue of the relationship indicated in Equations 5-6 and 5-7, the 

 functions given by the two equations above must constitute a Fourier 

 transform pair, and the total energy in the signal is |. 



We consider next the effect of transmission through a linear network on 

 the time history and spectrum of a signal. Linear networks are conveniently 

 characterized in terms of either their impulse response or their transfer 

 function. The impulse response, sometimes called the network weighting 

 function, is simply the transient output of the network for a unit impulse^ 

 at the time / = 0. The transfer function is most commonly defined as the 

 complex ratio of the network output to an input of the form exp {iwt). 

 These two functions are closely related. In fact, the transfer function is 

 the Fourier transform of the impulse response. This relation is made more 

 understandable by noting that an impulse function has a uniform spectrum 

 (see Paragraph S-S) and so represents an input of the required form where 

 all the frequency components occur simultaneously with differential ampli- 

 tudes. As an example, consider the single-section, low-pass, RC filter shown 

 in Fig. 5-1. Suppose that the driving point impedance is zero and the load 

 impedance is very large. Then the transfer function of this network is 

 readily recognized as 



Transfer function = ^/^^^c'^{l ^ = y^— (5-11) 



R 

 AAAAA/ 1— ^ Transfer Function: 



T 



X 



1+jRCc 

 Impulse Response= (l/RC)e 

 f>0 



Fig. 5-1 RC Filter. 

 *See Paragraph 5-3 for a definition of an impulse function and a discussion of its properties. 



