NAFI TR-lUUO 



defined to be the output variation due to noise alone at the input.* 

 Thus, the output SNR is, from equations 5 and 6, 



This expression can be simplified a great deal by one furth«r 

 assumption often satisfied in practice. Assume that the output filter 

 bandwidth is very narrow compared to the input bandwidth of the system; 

 under these conditions, the auto-convolution of N(f) will have a very broad 

 peak in the region of f=0 when compared to |H(f)l^. The value of this peak 

 will be the auto-convolution evaluated at f=0 i.e., 



r/v'{Vcif 



.«0 



since N(-§) = N(5), i.e., the power spectra of real functions are real, and 

 even. Utilizing this assumption that the output filter response is suffi- 

 ciently narrow so that lH(f)|2 ^ before the value of N(f) HN(f) has varied 

 appreciably from its value at f=0, and further recognizing cts^ ^^ "^^^ input 

 sigAal power, equation 7 can be approximated by 



Equation 8 is a rather general expression for the output SNR of a square-law 

 detector-averager. The only restrictive assumptions used to derive this 

 •quation are 



*Thi8 definition of output SNR corresponds to the following statistical 

 definition: 



where Z is the output mean, a^^ is the output variance, and N and S + N 

 denote the presence of noise alone or signal plus noise, respectively. 



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