MFI TR-1^^0 

 III. DERIVATION OF CONVENTIONAL SYSTEM E'fflATION 



In order to proceed with the performance evaluation, it is nec- 

 essary to determine the relationship between the system output SNR and 

 the signal and noise spectra into the square-law device. Compared to the 

 remainder of this paper, this section is rather mathematical. The reader 

 should have a basic understanding of Fourier transforms and their manipu- 

 lation as well as a familiarity with the fundamental concepts of signal 

 processing; the second chapter of Woodward's monograph provides an adequate 

 coverage of this material (lO) . Those not so endowed can skip to the next 

 section with little loss in continuity. 



This section is concerned with only two blocks of the processing 

 functions shown in Figure 2; these are the square-law device and the out- 

 put smoothing filter, shown in Figure 3. The input, x(t) , consists of 

 either signal plus noise or noise alone: 



Assume that s(t) and n(t) are members of a zero mean stationary Gaussian 

 random process. Then the correlation function at the output of the square- 

 law device is (ih , p. 255) . 



R^(r) -'- ^x'^ -f- a^Kx (rj 



Further assume that s(t) is statistically independent of n(t) , and that 

 they have variances as"* and a^ "^ ; then 



Realizing that the power spectral density of a random process is the 

 Fourier transform of its correlation function, and that multiplication 

 in the t domain corresponds to convolution in the frequency domain, the 

 power spectral density of y(t) is 



bv^Cfjs <! or 



8 



