250 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



Thus, the noise power or mean square vakie is equal to the sum of the power 

 components at all frequencies. Equation 5-34 can be regarded as a general- 

 ization of Parseval's equality given in Equation 5-8. 



At the end of Paragraph 5-2 it was pointed out that the absolute square 

 of the transfer function of a network acts as a transfer function relating the 

 input and output energy spectra. We have just defined power density 

 spectra as the average of the energy spectra of the elements of the process 

 divided by the observation time to give power. Thus the same relation 

 must hold between the input and output power density spectra, A^i(co) and 

 A^o(co), of a noise process being transmitted through a network with a 

 transfer function y(co): 



Noio:) = \Y(w)['N iic^). (5-35) 



We might note at this time that it is normal practice not to use a bar to 

 indicate specifically that the power density spectrum of a noise process is an 

 average value unless the averaging takes place explicitly in the derivation of 

 the power spectrum. Thus, the power density spectrum of a noise process 

 would normally be denoted by A^(co) rather than N{co)- 



In order to illustrate some of these ideas, we shall make up a noise process 

 and compute its power spectrum and autocorrelation function. We suppose 

 the process to be composed of the sum of identical functions A(/) which occur 

 at random times. Initially, we consider only functions which originate in 

 the finite range — T to + T. We denote the average density of these 

 functions by y and suppose that there are ITy = n functions in the finite 

 range of interest. Denoting the origin of the ^-th function by 4, our 

 approximation to a random process is given by the following expression. 



fn{t) =i:,h{t- /,) (5-36) 



Denoting the Fourier transform o( h{t) by H{co), the Fourier transform of 

 /„(/) is given by 



F„(co) = [ h{t - t,)e-i'^^dt = //(co) i; ^-'•"'^. (5-37) 



The power spectrum is simply the absolute square of F(oo) divided by the 

 observation time: 



^\FM\-^ = ^[//(c.)E.--^-][//*(co)2:%^-']. 



= ^j^{^)\'zi:^^''''"'-'''- (5-38) 



ZI 1 1 



