PROPERTIES OF SINE WA VE PLUS NOISE 129 



The integration with respect to 6' may be performed by setting RB' equal to 

 X and using 



/ 



+ 00 



X e ax = {(3/ a) yK/a) e 



The integral in R' reduces to a similar integral except that the factor x in 

 the integrand is absent. Performing these two integrations and using the 

 definition of B leads to 



0' = w--?!] dd / dR{R -Q cos d) 



exp^-^^(R' -2QRcosd + Q')^ 



We may integrate at once with respect to R. When this is done cos d dis- 

 appears and the integration with respect to 6 becomes easy. Thus 



d' = (bi/bo) exp [-Q'/i2bo)] = {bi/boje-" (5.7) 



When the noise power spectrum is equal to Wo in the band extending from 

 /o — 13/2 to/o + 13/2 and is zero outside the band, bi = 27r(/o — fq)bQ. 

 Hence, from (3.11), 

 ave. instantaneous frequency = /g + 0V(27r) 



= /o + a. - /o)(i - o ^^-^^ 



In the remainder of this section we assume the power spectrum of the 

 noise current to be symmetrical about the sine wave frequency /g. In this 

 case bi and c are zero, B is equal to ^0^2 and (5.4) becomes 



Pie') = iibo/b^y^K^ + zT"'e-'+"' 



[(y + l)/o(y/2) + y/i(y/2)] (5.9) 



(bo/b2Y''il + z')-'''e-\F,(^; 1; y^ 



where iFi denotes a confluent hypergeometric function^^ and 



s2 = boeyb2, y = (t)6,=0 = P/(1 + 2^) (5.10) 



When the noise power spectrum is constant in the band extending from 



fq - 13/2 to/g + 13/2 (see Table 2, Section 4) 



(^,2/^,o)l/2 = 3-'!'(3ir, z = 3''-d'/{M (5.11) 



12 The relation used above follows from equation (66) (with misprint corrected) of VV. R. 

 Bennett's paper cited in connection v.ith equation (1.2). 



