5-5] 



THE POWER DENSITY SPECTRUM 



251 



To find the average power spectrum, we must average over each variable 4 

 supposing it to be uniformly distributed between —T and -\-T: 



2j^l^«(co)p = 22^ 



H{o:)\'^ 



. I'T CT n n 



e-j'^itk-tDdt^ ... dtn. (5-39) 



The integrals of the terms in this sum will have two forms, depending upon 

 whether k = I or not. When k = I, the average value of each term is unity. 

 There are n such terms. When k t^ I, the average value of each term is 

 (sin coT/coTy. There are n(n — 1) of these terms. Thus, the average power 

 spectrum has the following form : 



«(w — 1) 



2T 



FM\' = m<^w 



— + 



2T^ 



{iry 



©(-7 



(5-40) 



As T -^ 00 , we note that the term involving the factor sin^ coT is of the same 

 form as the definition of an impulse function given by Equation 5-21. The 

 power density spectrum over all time, then, will have the following form: 



lim 



^\F{o^)\' 



|i7(co)|M7 + 2x7^5(0.)]. 



(5-41) 



The singular part of this spectrum corresponds to a concentration of 

 power at zero frequency or the d-c component. If h{t) has no such d-c 

 component, then //(O) will be zero and the impulse has no significance. The 

 continuous portion of the power spectrum is seen to be proportional to the 

 energy spectrum of A(/). The remarkable thing about this is that the form 

 of the spectrum is independent of the average number of functions per unit 

 time 7. 



As a concrete illustration, suppose that h{t) is given by the decaying 

 exponential defined in Equation 5-3. An element of such a noise process 

 might then look like the example shown in Fig. 5-G. The energy spectrum 

 of the exponential function has already been computed in Equation 5-10. 



Fig. 



5-6 Element of a Noise Process Composed of Identical Exponential Functions 

 with Random Time Origins. 



