The random signal output- is fed into the analyzer system consisting of an oscillator, 

 analyzer, and power integrator. These three units transform the random time signal into 

 a continuous plot of the average energy density versus frequency, which is traced out on 

 a Moseley X-Y Recorder. This system also provides for estimation of the integrated spec- 

 tra. The block diagram of Figure 7 illustrates an integrated spectrum curve superimposed 

 on the energy density curve. The small inset curves in the lower right hand corner of 

 the X-Y Recorder represent the spectral outputs of the filter used to resolve the oridinate 

 scales of the graphs of average energy density and the integrated spectrum. Details and 

 circuit diagrams are given in "Instruction Booklets for TP-625 Wave Analyzer System, " 

 Technical Products Company, Los Angeles, California. A discussion of the problems 

 associated with filter bandwidth, loop periods, time constants, and frequency scanning 

 rates is given in Reference 4. 



Figure 8 is an example of the power spectrum of a submarine roll angle recording 

 and the power spectrum of the filter used in the analysis. This illustrates a typical 

 power spectrum of the filter used to calibrate the ordinate scale of the energy density 

 graphs. The broken line curve is the digital spectrum of the same recording which is 

 superimposed for comparison. The digital spectrum superimposed is one of four spectra 

 shown on Figure 9 (See Section D on comparison of spectral estimates). 



This analog spectrum was estimated by the wave analyzer in the Hydrographic 

 Office, and all the digital spectra contained in this report were estimated by a 

 Burroughs 205 Electronic Computer, also located in the Hydrographic Office. 



Such spectral estimates are distributed with a chi square distribution with 2N degrees 

 of freedom, where N is the number of elemental bands covered by the filter. If the 

 spectral density is not a fast changing function of frequency interval equal to the band- 

 width of the filter, then N is approximately equal to the effective bandwidth of the 

 filter divided by the elemental bandwidth. The elemental bandwidth is l/T, where T is 

 the loop period, and the effective bandwidth used was 6 cycles per second (Reference 4). 

 Thus, for the analog spectra in this report: 



effective bandwidth 

 Number of degrees of freedom = 2N = 2 . , , , , ,,,, (11) 



elemental bandwidth 



Confidence intervals for the estimated spectra can be determined from a table of the 

 chi square distribution. 



The 90-percent confidence intervals for the digital spectra are given by computing 

 the number of degrees of freedom according to: 



2 Tn 



Number of degrees of freedom = T~~ (12) 



19 



