MATHEMATICAL ANALYSIS OF RANDOM NOISE 133 



where we have used (3.5-4) to evaluate the integral. The arc cosine is 

 taken to be between and tt. We therefore have for the correlation func- 

 tion of /(/), 



^(r) = 1 ([^0^ - ^T' + h cos-^ [^^]) (4.7-5) 



The power spectrum ]V(f) may be obtained from this by use of (4.6-1). 

 For this purpose it is convenient to write (4.7-5) in terms of a hypergeo- 

 metric function. By expanding and comparing terms it is seen that 



4 27r \ ;/,-/ 



= — + — + -— + terms mvolving xf/r , ^r , etc. 

 4 Zir Airxf/o 



(4.7-6) 



As will be discussed more fully in Section 4.8, a constant term A" in \1/(t) 

 indicates a direct current component of /(/) of ^4 amperes. Thus I{t) has 

 a dc component equal to 



r^ I = -^ X rms value of V(t) (4.7-7) 



LzttJ 'v'lir 



This agrees with (4.2-3) when the P of that equation is set equal to zero. 

 Integrals of the form 



Gn(f) = I ^T cos 27r/r dr 

 Jo 



which result w^hen (4.7-6) is put in (4.6-1) and integrated termwise are 

 discussed in Appendix 4C. From the results given there it is seen that if 

 we neglect i/'^ and higher powers we obtain an approximation for the con- 

 tinuous portion Wdf) of W{f): 



Wcif) = Gi(/) + ^-^ 



■KXf/Q 

 W{f) , 1 1 r , N ,. .. 



= -^ -f -— •- / w{x)w(J - X) dx 



where iv{—f) is defined as w{f). 



When VN(t) is uniform over a relatively narrow band extending from 

 fa to fb so that w(/) is equal to wo in this band and is zero outside it, we may 

 use the results for Filter c of Appendix 4C. The /o and jS given there are 

 related to fa and fb by 



/a = /o — 2 > /& = /o + 2 



