5-6] NONLINEAR AND TIME-DEPENDENT OPERATIONS 253 



power spectrum. The factor 2 is introduced to account for contributions 

 from negative frequencies. Forming this product and substituting //e for 7 

 in Equation 5-41, yields the following expression for the mean square 

 noise current: 



(KTp = \H{W'i2^F = leHC^FI^ = 2e/AF. (5-46) 



Comparison with Equation 5-44 indicates that the noise process model used 

 does indeed give the correct expression for shot noise. The forms of the 

 functions h{t) are not significant in this derivation as long as their spectra 

 are wide compared with AF. Similar discussions can be made in connection 

 with many physical phenomena which generate noise by means of some 

 random mechanism. 



5-6 NONLINEAR AND TIME-DEPENDENT OPERATIONS 



In tracing signals and noise through radar systems, we find that the 

 operations of many components are either nonlinear or time-dependent. 

 Examples of such operations are rectification by second detectors, auto- 

 matic gain control, time and frequency discrimination, phase demodulation, 

 and sampling or gating. In this paragraph, procedures which can be used 

 in the analysis of such operations will be discussed briefly and illustrated 

 with a few examples. 



A basic case is provided by a nonlinear device which has no energy storing 

 capacity; that is, it is assumed to operate instantaneously. We suppose 

 that the input to this device is a Gaussian noise process denoted by x\ the 

 output noise process is denoted by jy. The functional relation between these 

 processes is denoted by 



v=/W (5-47) 



The process y will be random but not in general Gaussian. The average 

 values of_y and jy^ can be found as the weighted averages of/(x) and/^(^): 



y =W) = vi^/-y^^^'"'^'"^^^^ ^^'^^^ 



7 = a/+y^=n^) = j^ f nx)e-^'''~^' dx. (5-49) 



The power spectrum of the y process can be found by first finding its 

 autocorrelation function and then computing the Fourier transform of this 

 function. The autocorrelation of y is the average value of the product 

 yiy^ = f{xv)f{xi)- The average will have to be computed relative to the 

 joint Gaussian probability density function expressed by Equation 5-23. 

 If this probability density function is denoted by P2{xi,Xi), the autocorre- 

 lation of jy is given by 



