240 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



this calculation, which involves treating cu as a complex variable and 

 integrating around a semicircular contour in the complex plane. 



The square of the absolute value of the spectrum is important in the 

 development of techniques for analyzing noise processes. This is the energy 

 density spectrum giving the distribution of signal energy with frequency. 

 This terminology is adopted because the function /(/) will normally be a 

 voltage or its equivalent, and its square will be proportional to power. The 

 integral of the square of/(/), then, will be proportional to the total energy. 

 In the development to be given below, it will be shown that (in this sense) 

 the square of the absolute value of the spectrum of/(/) gives a resolution of 

 the energy into frequency components. This development is obtained by 

 manipulating a general definition of the energy density spectrum as it is 

 derived from Equation 5-1 : 



!F(co)|2 = F(co)F*(co) 



= / f{ti)e-^'^'^dtA f{t2)ei'''^dt^ (5-5) 



= j_ j_ At,)/(t,)e-^"^'r'.^ dt^dt2. 



Making the substitution t = ti — ti and dr = dti and interchanging the 

 order of integration 



I^MP = l_^ e-^'^Ur j _J{t, + T)f{t,)dt, 



= j e-''''<p(T)dT. (5-6) 



The right-hand side of Equation 5-6 is of exactly the same form as 

 Equation 5-1; that is |F(aj)|^ is expressed as the spectrum of the function 

 <p(t) or its Fourier transform. If ^(t) satisfies the conditions prescribed for 

 the existence of a Fourier integral, then the inverse operation given by 

 Equation 5-2 is applicable, and <p{t) can be expressed by 



<p{r) = l_J(t + r)/(/) dt = ^l_^ \F(o:)\'- .^^ do:. (5-7) 



When T is set equal to zero, the following important special case is obtained. 



^(0) = /_y'w^^ = ^/_^ \F(-^)\' d^- (5-8) 



This relation is often referred to as ParsevaFs equality. It expresses the idea 

 that the total energy of /(/) is equal to the sum of the energies of each 

 component of the frequency representation oif{t). 



