5-6] NONLINEAR AND TIME-DEPENDENT OPERATIONS 255 



From Equation 5-50, the autocorrelation function of the y process will be 



<pAt) = jT^ = 1 + 2^-21^1. {S-SG) 



The Fourier transform of this expression gives the power spectrum of the 

 y process : 



NyiiS) = lirdico) + 



4 + 



(5-57) 



Thus, in this case, the form of the continuous spectrum remained the same, 

 but the bandwidth was doubled. 



Another case which is very common in radar applications corresponds to 

 the assumption of a uniform spectrum of finite bandwidth for the x process. 

 Such an assumption normally represents a simplifying approximation to 

 the more complicated forms which actual spectra might take. Such a 



2W 



-2ttW 2irW 



Fig. 5-7a Uniform Spectrum {x Process). 



spectrum is shown in Fig. 5-7a. The autocorrelation function corresponding 

 to this spectrum will be 



■2wW 



r2wW 



■J -2^W 



in IttIVt 



ItvWt 

 The autocorrelation function of the y process will now be 



Ysin iTrWrV- 

 \ 1-kWt J ■ 



(5-58) 



(5-59) 



At the end of Paragraph 5-3 it was indicated that the Fourier transform of 

 a triangular function is of the same form as the trigonometric term above. 

 Thus the continuous part of the power spectrum of y will be triangular. 

 This spectrum is pictured in Fig. 5-7b and it is represented symbolically by 



Ny(a>) = 2Tra^8(o^) + i<r'/fV)il - |a;|/47r/F), Ico] < 4w^. (5-60) 



cr" (Impulse Strength) 



Fig. 5-7b Triangular Spectrum (y = x"^ Process). 



