Signal plus noise = yjia + x)^ + y'^ cos ccj + 



5-7] NARROW BAND NOISE 261 



'-^ -(5-73) 

 a -^ xj 



Here an envelope function modulates a carrier frequency with random 

 phase modulation depending upon x and y. We note in passing that a 

 frequency discriminator would be sensitive to this phase modulation and 

 that studies similar to those which we shall make of the video envelope can 

 also be made of a discriminator output. 



We first determine the probability density function of the envelope which 

 is denoted by r: 



Envelope = r = -yjia + x)'- -j- y^- (5-74) 



The random variables x and y are assumed to represent independent 

 Gaussian noise processes with zero means and equal variances. The 

 differential probability that they will be found in the differential area (jxdy 

 is given by their joint probability times this differential area: 



1 \ —x^ 



dp = Pi(x)Pi{y)dxdy = ^ — -, exp 



2^2 ^^^ 2(7^ 



dxdy. (5-75) 



In order to determine the probability density function of the video en- 

 velope, this expression will be transformed to polar coordinates and the 

 average value for all angles found. This transformation is represented 

 as follows: 



a -{- X = r cos 6 



y ^ rsmd (5-76) 



dx dy ^ r dr dd. 



Substituting these relations into the expression in Equation 5-75 and 

 integrating over the variable d gives 



dp = Pi{r)dr = — exp 



2a' 



1 P'^ 

 dr ^ / exp [ar cos e/a^dd. (5-77) 



2xJDo 



The integral in this expression can be recognized as a representation of a 

 zero-order Bessel function of the first kind with imaginary argument^ 

 denoted by loi^r/a^). The probability density function of r is thus of the 

 following form: 



Pi(r) = —exp 



2(7^ 



- \h{ar/a'). (5-78) 



A curve showing Pi{r) for some representative values of S /N is given in 

 Fig. 5-11. In the two extremes of very small and very large values of the 

 signal-to-noise ratio, Pi(r) approaches the following forms: 



^J. L. Lawson and G. E. Uhlenbeck, op. cit., p. 173. 



