(the TW portion of Eqn. ll) more than coTrrpensates for the decreasing 



r- 1 



input SNR. Intuitively, one would suspect the value of c,o to be 2 

 (corresponding to a relatiAai signal spectral slope of -1.5db/oct); for 

 once, intuition proves correct. For 5 = -|, equation 18 shows that no 

 relative maximum exists, an'J tl e infinite bandwidth solution again re- 

 sults. Appendix C demonstrates that § = ^ is the critical value, and 

 that for values of § incrementally larger than ^, finite bandwidth 

 solutions result. 



One could use equation 18 to determine optimum bandwidths for 

 various values of §(§ / 1, 5 > i); however, it is not only the optimum 

 value of bandwidth that is of concern, but also the penalty that is paid 

 for not operating at the optimum. This is simply the evaluation of the 

 non-optimum system discussed in Section 1, Such an evaluation can be 

 accomplished by examining a normalized yJ[^) as a function of p. for 

 various values of 5 in the region where optimum bandwidths exist. This 

 normalized "Ypd-^), given by yJ\^)/y^i\i'o), is plotted in Figure h for I 

 equal to 1 through t in integer values . 



Referring to equation 16, it can be seen that yJ[i:)/yJ\^o) 

 is simply the ratio of the output SNR at a normalized bandwidth M^ to 

 the output SNR for the optimum bandwidth, M^o , corresponding to the 

 particular value of 5. The quantity yJ[i.)/yJ\i.o) can thus be conven- 

 iently referred to as the relative output SER since it measures the SNR 

 out of a non-optimum system relative to the output SNR that would be ©"b- 

 tained had an optimum handwidth been employed. 



The graphs in Figure h are intuitively satisfying since the 

 peaks are rather broad, indicating little loss associated with picking 

 a non-optimum band. Often, an octave band (m- = l) is used in acoustic 

 analysis systems. Figure h shows that for signal spectra into the 

 square-law device with slopes beitween -3db/oct and -l8db/oct, the loss 

 in output SNR associated with an octave band is less than 3d-b, Thus, 

 Figure h- would indicate that an octave is a good compromise band. It 

 must be stressed, however, that this section has dealt with extremely 

 simple, and consequently, somewhat artificial, spectral characteristics. 

 The results are enlightening and meaningful, but simply in pointing the 

 way toward a solution; they do not provide a comprehensive answer to the 



15 



