242 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



Similarly, the impulse response of the network is recognized as a decaying 

 exponential. If the RC time constant is assumed to be unity, the impulse 

 response and transfer function are identical with the functions given as an 

 example in Equations 5-3 and 5-4 which make up a Fourier transform pair. 

 We denote the transfer function of a network and the input and output 

 spectra by Y(o3), Fi((i:), and Fdco), respectively. Since the input spectrum 

 gives the resolution of the input into components of the form exp (jW) and 

 the transfer function indicates how each such component is modified by 

 transmission through the network, it is clear that the output spectrum 

 should be given by the product of these two functions. This can be rigor- 

 ously demonstrated.^ 



Fo(a)) = y(a;) F, (a;). (5-12) 



The relation between the input and output energy density spectra is 

 easily found by multiplying each side of this equation by its conjugate: 



|Fo(a,)|2 = \Y{coy\F,ico)\\ (5-13) 



Thus the input and output energy density spectra are related by the 

 absolute square of the transfer function, which might appropriately be 

 called the energy transfer function or, if power spectra are being considered, 

 the power transfer function. 



It is often convenient to express the time history of the output of a 

 network purely in terms of the time history of the input and the impulse 

 response of the network. This relation is easily determined by substituting 

 for y(co) and Fi(aj) in Equation 5-12 their expressions as Fourier transforms 

 o{ y{t) and/i(/), the filter impulse response and the input to the filter: 



/:/- 



Fo(co) = j_ j_^J'(/0/-:(/2)^-^"^''+'^>^/i^/2. (5-14) 



Substituting r = /] + /2 and dr = dti, and interchanging the order of 

 integration, 



^o( 



0;)=/ e-^'^'drl fi{t->)y{r - t2)dt.. (5-15) 



The right-hand side of this expression is again in the form of Equation 5-1 ; 

 that is, Fo{oi) is given as a Fourier transform. Thus we can formally make 

 an inverse transformation of both sides to obtain the desired relation 

 between the input and output time histories: 



/o(r) = / //(/)v(r - t)dt. (5-16) 



5See M. 1'". Gardner and J. 1"". Barnes, Transirn/s in Lineur Systems, Vol. 1, pp. 233-236, 

 John Wiley & Sons, Inc., New York, 1942. 



