\if^)- -^ lOW^->f ,1*1 ('''") 



y, ca)^ /rr^''^'V')]^ ; 5 = 1 07^) 



and |i = w/fi is the normalized bandwidth variable, which gives the sys- 

 tem bandwidth as a multiple of fi . Thus, M^ = 1 corresponds to an octave 

 band. 



In any given sonar situation, with an assiomed constant rela- 

 tive signal spectral slope §, T, A, B, f i , and § are all constant. Thus, 

 the yAV') describe the relative SNR performance as a function of band- 

 width. From the expressions in equation 17, it can be seen that the 

 possibility exists for Yc(M') to have a relative maximum at some value 

 |J. = IJ-o , depending upon the value of §, Considering the I / 1 case, set- 



dyf 



ting T-* = results in the following equation for determinant Mo: 



^ |4-/<(2t-l) = (li-/A)^ ^ 5:^1 (,8) 



Thus if § = 2, iJio = 1; consequently, for a relative signal spectral 

 slope of -6db/oct, the bandwidth which maximizes the output SNR is an 

 octave wide . 



there does not always exist an optimum finite bandwidth, if 

 5=0, i.e., the signal spectrum into the square-law device is also 

 white, then equation l8 results in Mo = 0, which is obviously not the 

 relative maximum sought. Referring to equation (ll), if both signal and 

 noise are white, then the SNR into the square-law device (^in Eqn. ll) 

 is a constant as W is varied, and equation (ll) indicates that the out- 

 put SNR will increase linearly with W. Thus, the bandwidth which would 

 maximize the output SNR in this case, § = 0, would be infinitely wide, 

 Natxarally, such a spectrum will not occur physically as various limiting 

 mechanisms will eventually cause the high frequency response to fall 

 off; however, mathematically, no finite M-o exists. 



Looking at equations (ll) and (l8), it can be seen that an 

 infinite bandwidth solution will result until some critical value of 

 ?, So-'O, is used for the relative signal spectral slope. Thus, in 

 range ^ 5 ^ §o , even though the signal spectrum is falling off 

 compared to the noise spectrum, and, as a consequence, the SNR into the 

 square-law device is decreasing as W is increased, signal processing 



1^ 



