ANALOG EQUIPMENT FOR PROCESSING RANDOMLY FLUCTUATING DATA 



349 



plitude per unit bandwidth (e(iuivalent to average 

 power per unit bandwidth in the electrical system) in- 

 stead of simple sine-wa\'e amplitudes. 



The power spectrum then represents the distril)ution 

 of energy o\'er the frequency spectrum. I'or example, 

 the portion of the total energy included in the frequency 

 band/i to/2 in the figure is represented by the area under 

 the cur\-c between /i and /;. 



One extremely useful feature of the power spectra 

 concept is the simple input-output spectra relationship 

 illustrated in Fig. 6. For linear systems, the power 

 spectral density of the system's output Go{J) is e(|ual to 

 the power spectral density of the input Gi{j) times the 

 squared absolute value of the transfer function }'(/). 

 Thus, we ha\e a relationship among the three variables 

 such that if twf) of the three are known, the third may 

 be determinetl. 



This sort of input-oiitput-transfer function relation- 

 ship is one which is familiar to the electrical engineer 

 who is often fortunate enough to be able to work with 

 strictly sinusoidal signals. It should be noted that the 

 power spectra concept now enables the aeronautical 

 engineer, who often has no choice but to work with ran- 

 domly fluctuating type of "signals," to perform the same 

 sort of "circuit analysis" as the electrical engineer is ac- 

 customed to do. 



For example, again consider the problem of an air- 

 plane flying through rough air; the airplane can be con- 

 sidered as a mechanism having a tran.sfer function Y{j), 

 the turbulence can be considered to be the input Gi{f), 

 and the airplane's response can be considered as the out- 

 put G,Aj). 



Thus, by measuring an airplane's transfer function 

 and its response to atmospheric turbulence, it is pos- 

 sible to use the plane as an instrument for measuring 

 turbulence, as Clementson [4] and Summers [(')] did. Or, 

 if a specific turbulence spectrum is assumed and the air- 

 plane's transfer function is known, it is possible to pre- 

 dict the plane's response to the turbulence. 



Also, by using the technicjues developed by Rice [3], it 

 is possible in many instances to use power spectra to 

 calculate such things as the number of zero crossings and 

 the number of times the amplitude of a randomly vary- 

 ing quantity exceeds certain levels. 



Power spectral density may be numerically calculated 

 by a procedure outlined by Tukey [2 ]. A'ery briefly sum- 

 marized, this procedure is as follows: First, read the 

 time-history record point by point; second, calculate 

 the autocorrelation function of the data sample by eval- 

 uating the integral 



Inpui 



Constant Bandwidth 

 Tunable Filter 



Jo 



I{t):IV + t)(H 



for se\-eral discrete values of time lag r; and thii'd, de- 

 termine the power spectral density of the data by taking 

 the Fourier cosuie transform of the autocorrelation 

 function. 



This process is equivalent to passing a tunable, con- 

 stant-bandwidth filter of known characteristics over the 



Squaring fie 



Averaging 



Circuits 



-^ffi 



Recorder 

 Fig. 7 Power spectral density analyzer 



data and measuring the time average of the Sfjuare of 

 the filter's output. Except for scjuaring of the filter's 

 output and .some difference in the filter's characteristics, 

 the results are almost identical to those obtained from 

 the familiar spectrum analyzers that have been used for 

 years by communications and sound engineers. 



This suggests then that there are two types of elec- 

 tronic analog equipment which might be used to de- 

 termine power spectra: one type parallels the numer- 

 ical process by first determining the autocorrelation 

 function and then taking its transform to get pfiwer 

 spectral density; the other omits the autocorrelation 

 function entirely and measures the spectrum directly 

 by means of a scanning electrical filter. 



Since scanning filter types of analj^zers are compara- 

 tively simple electronic devices and are commercially 

 a\'ailable, the NACA uses the direct spectrum measure- 

 ment approach. 



This type of analyzer is illustrated in Fig. 7 and oper- 

 ates in the following manner. The data sample stored 

 on a continuous loop of magnetic tape is applied to the 

 iiandpass filter. Any fre(|uency components in the 

 data which fall within the filter's pass band will be 

 passed by the filter, squared, averaged, and then applied 

 to a direct-writing recorder. The filter is initially set at 

 the low end of the frequency range and slowly scans 

 ujiward through the spectrum until the entire freciuency 

 range of interest has been covered; at the same time the 

 recorder paper is moving under the stylus so that a con- 

 tinuous plot of power spectral density against frequency 

 is obtained. 



The scanning speed of the analyzer is normally con- 

 servatively adjusted so that about three jjasses of the 

 data sample on the loop are made diu'ing the time re- 

 <iuired for the filter to scan one filter bandwidth. Under 

 these conditions, the time required for complete analy- 

 sis of a typical record is 10 to 15 min. Faster scanning 

 .speeds may be used, but .some ".smearing" of the spec- 

 trum might result. 



