ANALOG EQUIPMENT FOR PROCESSING RANDOMLY FLUCTUATING DATA 



351 





Dual Recorder 

 Fig. 10 Cross spectral density analyzer 



' Input Spectrum 

 ' Output Spectrum 



• Co- Spectrum 

 Q.uad. Spectru 



_ 



whose power siiectra are represented on Fi";;. 9 as PSDi 

 and PSDj. The cross spectrum between the two func- 

 tions will indicate only those frequencies in PSDi which 

 are also contained in PSD2 and which bear a specific, 

 nonrandoni phase relationship to the fretiueneies in 

 PSDi. For example, PSD, and PSD. may include 

 power in the same freciuency band, but, if the corre- 

 sponding freciuency components are entirely independ- 

 ent of one another, the phase between them would be 

 random and the cross spectral density would be zero 

 throughout the band. Howe\'er, if some of the fre- 

 quency' components in P8l)i bear a iletinite phase or 

 time relationship to the corresponding components in 

 PSD2, this relationship will be indicated in the cross 

 spectrum. 



One useful application of the cross spectriun is in the 

 determination of the phase response of a linear system 

 subjected to a random-type input. Recall that the 

 power spectra of the input and output made possible 

 calculation of the absolute amplitude of a system's 

 transfer function but did not furnish any knowledge of 

 the phase resjionse. By using cro.ss spectra, both the 

 amplitude and the phase response may be calculated. 

 As shown by Lee [9] the transfer function Y(f) ec|uals the 

 input-output cro.ss spectrum G',,//) di\ided by the power 

 spectrum of tlie input Gu{f). 



An interesting featiu'e of this relationship is the fact 

 that the equation liolds true even though other inde- 

 pendent random noises are present in the output. This 

 is true since the input-output cross spectrum will ignore 

 the presence of uncorrelated random fluctuatidus in 

 the output. 



The cross spectra has other useful, practical applica- 

 tions to sy.stems with multijile inputs and outputs, Init 

 these are too involved to di.scuss here. A good illus- 

 tration of this usage, however, is included in Simimers' 

 paper on atmospheric turbulence measurements [6]. 



The numerical process for determining cross specti-a is 



< o 



— Phase f^, 



— — Transfer Function / « 



20 



30 40 



Frequency, cps 



50 



Fig. 1 1 Cross spectra of wing bending moment 



even more lengthy and expensive than that rc(|iiiii'il 

 for determining ])ower spectra. However, the numer- 

 ical i-esults can be tluplicated l)y the analog process 

 illustrated in Fig. 10. The two fluctuating data sam- 

 ples recorded on a continuous loop of dual-channel tape 

 are simultaneously apjilied to two synchronized filters 

 of the same type used with the power spectral density 

 analyzer. However, instead of S(|uaring and averaging 

 the outputs of the filters as would he done to determine 

 power spectra, the two outputs are fed into a multi- 

 plier whf)se output is a\-eraged and automatically 

 plotted against frequency to fin-nish the co-power spectral 

 density. At the same time, one of the filter outputs is 

 run through a 90° phase shifter and again multiplied 

 by the output of the other filter. The product is then 

 averaged and plotted to give the (|uadrature power 

 spectral density. 



To illustrate the use of the cross spectrum analyzer, 

 Fig. 11 shows some data secured from a buffeting air- 

 foil in a supersonic wind tunnel. The double-peaked 

 spectnnu is the power spectral density of the fluctuat- 

 ing aerodynamic forces existing at a particular span- 

 wise station and the single-peaked spectrimi i.s the power 

 spectral density of the fluctuating bending moment 

 existing at the root of the wing. The cross spectral 

 densities between these two quantities as determined 

 Ijy the analyzer are shown in the center of the figure. 

 The calculated amplitude and phase response shown at 

 the bottom of the figure indicate that the wing is sim- 



