where ^(f ) is the Dirac delta function, S(f) and N(f ) are the signal 

 signal and noise power spectra into the square -law device,^ and the symbol 

 H denotes convolution. 



The wave form y(t) is averaged by a filter with transfer function 

 H(f )j thus the output power spectrum is 



Sz(f) = Sy(f) |H(f)|' Ci) 



The results of equations (5) and (6) contain the information 

 needed to determine the system output SWR, which must now be defined. The 

 output signal of interest is the shift in the average (d-c) voltage 

 level of the output due to the presence of signal. Assuming that the 

 input signal and noise spectra do not contain impulses (i.e., ignoring 

 the effect of lines), and assuming |h(o)|= 1, the output d-c voltage with 

 signal present is as + aN^(the S(f)term from equation 5a). Since the d-c 

 level with noise alone is On (equation 5b), the output signal voltage of 

 interest is Os . Returning to a power basis in order to write SNE's, 

 the output signal power is 0$ . 



The continuous portion of Sz (f ) is the "ripple", or noise, 

 present at the smoothing filter output which tends to obscure the shift in 

 d-c level due to the presence of signal. There are two possible definitions 

 of output noise: one would involve using the continuum portion of the 

 spectrum in equation 5a, i.e., the ripple due to the presence of both 

 signal and noise at the input; the other involves the continuum portion of 

 the spectrum in equation 5b, i.e., the ripple due to the presence of noise 

 alone at the input. For a complete description of processor performance, 

 both quantities are needed; however, in most passive sonar situations of 

 interest, the input Sim«l, and, in this case, the presence of a signal 

 has a negligible effect upon the output variation. Because of this effect, 

 as well as the resulting mathematical simplicity and the degree of arbi- 

 trariness associated with any definition of SNR, the output noise is 



*The range variable, R, has been suppressed since it exerts no influence 

 upon the following results. 



10 



