254 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



<p{r) = 3^2 = j Axr)f(x2)P2{x,,X2)^x,dx2. (5-50) 



The power spectrum of y is simply the Fourier transform of ^(t). 



A Square Law Device. As a specific example, suppose that the 

 nonlinear operation is provided by a square law device: 



y = x^ (5-51) 



This type of nonlinearity is often assumed to approximate the rectifying 

 action of second detectors in radar receivers. The mean and variance ofjy 

 are found by carrying out the operations indicated in Equations 5-48 and 

 5-49: 



y = X" = cr'^ 

 y = .^ = 3(,4 (5.52) 



The autocorrelation function is found by evaluating the following integral: 



\ —Xi^ + 2pXiX2 — Xi"^! /c cn\ 



exp 1^ 2.2(1 - p2) r^'^''' ^^-^^^ 



= (7^1 + 2p2). 



This integral is evaluated by completing the square of one of the variables 

 in the exponent and transforming to standard forms. The constant term in 

 <p{t) corresponds to the square of the average value of y and will contribute 

 an impulse function at zero frequency to the power spectrum of jy. 



In general, the squaring operation will provide a widening of the con- 

 tinuous noise spectrum as the various frequency components beat with 

 themselves to produce sum and diflPerence frequencies. To show this and to 

 illustrate this type of analysis generally, suppose the x process is similar to 

 the one defined in Paragraph 5-5 (Fig. 5-6) by a sum of exponential func- 

 tions. For simplicity, we assume that on the average only half of the 

 exponential functions are positive while the other half are negative, so that 

 the average value of the x process is zero. We assume further that the 

 variance is unity. The power spectrum and autocorrelation of the x process 

 will be given by Equations 5-42 and 5-43. There will be no d-c term, and 

 in order to have unit variance 7 = 2: 



<p(t) = a-'p(T) = .-IH (5-54) 



Nic) = y^, (5-55) 



1 + CO- 



