MATHEMATICAL AX A LYSIS OF RANDOM NOISE 97 



where S{f) is given by (2.1-2) we find that we need the result 



Limit -J [ dt, f dt.J'"^'''-'''^ I-{h)l\t-^ 

 7'-»M T Jo Jo 



+00 (3.9-31) 



= I w(f,)w(f-fOdf, 



where / > and /(/) is a noise current with w(/) as its power spectrum. 

 This may be proved by using (3.9-7) and 



»CO -.+00 



8 / \P'{t) cos lirfr dr = I w(x)w{f — x) dx 

 Jo J-« 



which is given by equation (4C-6) in Appendix 4C. 



An expression for the mean square fluctuation of A (/) in terms of w(f) may 

 be obtained by setting r equal to zero in (3.9-29) 



(A{t) -Ay = ^(0) -A 



•+»\, w{h)w{^) (3.9-32) 



Jo J-oo OC 



^' + 4t'(/i - /2)' 



The same result may be obtained by integrating Wdf), (3.9-30), from 

 to cc : 



r df r+°° 



/ 2 ■ ' 2.2 dfMfiMf-fi) (3.9-33) 



Jo cc -f- 47r / J_oo 



Although this differs in appearance from (3.9-32) it may be transformed 

 into that expression by making use of w(— /) = w(f). 



Suppose that /(/) is the current through an ideal band pass filter so that 

 -d)(f) is zero except in the band /a <f<fb where it is wo . Then, if 3fa > fb , 



A = - (fb - fa) (3.9-34) 



a 



^ 2wl{f, - fa - f) <f<f,-f, 



/+00 

 W{x)w{f - X)dx =1 Wlif - 2fa) 2fa<f<fb+ fa 



[wl(2ft -f) fb+fa </< 2/fc 



and is zero outside these ranges. The power spectrum l'l'c(/) may be ob- 

 tained immediately from (3.9-30) by dividing these values by a + 47r"/". 

 From (3.9-33) 



{A{t) - If = 2wl I 

 Jo 



a" + 47r-/- 



, p". (/ - 2/.) ^, /•- (2/,-/) 



