MATHEMATICAL ANALYSIS OF RANDOM NOISE 99 



Now suppose that we have a noise current In plus a sine wave. By com- 

 bining our representation (2.8-6) for /jv with the idea of ipp being random 

 mentioned above we are led to the representation 



/(/) = I = 1, + h^ 



.If 



= P cos (Upt — (^p) + ^ Cn COS {C0,J — iPn), (3.10-4) 



1 



c; = 2w{fn)^f 



where (p^ and v'l , • • • (fM are independent random angles. 



If we note / at the random times ti , to • • • how are the observed values 

 distributed? Since Ip and /jv may be regarded as independent random 

 variables and since the characteristic function for the sum of two such vari- 

 ables is the product of their characteristic functions we have from (3*. 1-6) 

 and (3.10-2) 



ave. e'" = ave. e'^^^^^+^v) 



/-./'oA (3.10-5) 



= MPz) e.xp [^~-) 



which gives the characteristic function of /. The probabiHty density of I 



• 37 



is 



1 f ^°° ,--/-(^o-'^/2) j^^p^^ ^2 ^ ^^^^ r ^-(/-p cos e^v-2,0 ^Q (3_io_6) 



27r J-oo 'K\' iTrxpo Jo 



In the same way the two-dimensional probability density of (/i , 72), 

 where /i = /(/) is a sine wave plus noise (3.10-4) and I2 = I{t + r) is its 

 value at a constant interval r later, may be shown to be 



{^l - ^IV 



r-^ r Bid) 1 



where 



B{d) = Uih - P cos ef + (A - P cos {6 + co,,r))'] 



- l^Prill - P cos d){l2 - P cos {d + WpT)) 



The characteristic function for 1\ and I1 is 

 ave. g'"^i+"'^2 _ j^[p-^^ii _|_ ^2 _|_ 2uv cos Wpx) 



X exp — y {ll' + V) — lArWZ^ 



'^ A different derivation of this expression is given bv W. R. Bennett, Jour. Aeons. Soc. 

 Amer., Vol. 15, p. 165 (Jan. 1944); B.S.T.J., Vol. 23, p. 97 (Jan. 1944). 



