256 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



An application of this result is made in Paragraph 5-7 in course of a discus- 

 sion of the effect of the second detector in a pulse radar. 



A Synchronous Detector. Another example which is of interest is 

 that of a product demodulator or synchronous detector. Such a device or 

 an approximation to such a device is a common component in many types 

 of radar systems. It will provide an example of a time-dependent operator. 

 In operation, a -product demodulator simply multiplies the signal or noise by 

 a sinusoid. Thus if the input is x{t) , the output would be x(t) cos Ww/. When 

 x(t) possesses a component at the angular frequency Wm, the dc in the output 

 gives a measure of the phase between the input component and the refer- 

 ence. We again assume that ;c is a Gaussian noise process with zero mean, 

 autocorrelation ^(r), and power density spectrum 7V(co). The autocorre- 

 lation of the output is given by 



yiy2 = ■'''1-V2 cos a)mt cos COm (/ + t) 



(5-61) 



= (i) <P{t) [cos OOmT + cos C0m{2t + t)]. 



The autocorrelation of the output evidently varies with time periodically 

 at the angular frequency 2ajm- The spectrum of the output will likewise vary 

 periodically. In most cases, however, the angular frequency 2aJm is outside 

 the range of practical interest, and we can use the time average of the 

 autocorrelation or spectrum for our purposes. On taking the time average, 

 the periodic component disappears: 



1 [T 



Xr) = yiy2 = hm ;p^ / yiy2dt = (Dv'W cos w^r. (5-62) 



Zl 2r 



The wavy bar is used to indicate a time average. Bearing in mind that the 

 autocorrelation function and power density spectrum (p{t) and A^(co) of the 

 input noise are Fourier transforms, the Fourier transform of the expression 

 above is easily computed to give the output power density spectrum in 

 terms of that of the input: 



/:.' 



Nyiw) = h ^(r) cos co„t^-J"Vt 



-I v'(t)[^-'<"-"'"' + ^-'("+"'«)]^r (5-63) 



A product demodulation, then, operates to shift the input power density 

 spectrum N{o}) into sidebands about the modulating frequency aj„ and the 

 image of the modulating frequency —ojm- 



