TM No. 377 



lags and AT = 0.2 second; then, according to equation (III-6), the frequency 

 band upon which each of the 50 spectral density estimates is centered (i.e., the 

 frequency resolution) is 50 mcps. Thus, the units of $uj are i n cm sec"l. 

 The abscissa is shown as units of frequency, in millicycles per second (mcps), 

 and period (sec). Using equation (III-IO), the number of degrees of freedom 

 DF = k2. From the confidence limit plot of figure III- 3? the range of the 80 

 percent confidence limits lies below I.38 and above 0.73 times the value obtained. 

 The vertical bar (shown on each graph) delineates the range R of the 80 percent 

 confidence limits when the middle crossline is placed on the curve. This is best 

 done using a pair of dividers. 



Since &T = 0.2 second and M = 50, the limiting frequency or Nyquist 

 frequency f n is given by equation (III-2): 



£ N - 2^7 = 2 5°° mc P s ° 

 The frequency of resolution &■£ is given by equation (III-U): 



Af =• fn^T = 5 ° mcps- 



The spectrum displays two peaks, at frequencies of 100 and 250 mcps (10 and 

 h sec periods). The energy is concentrated in the band between 50 and 500 mcps. 



The use of a logarithmic ordinate provides equal resolution at all ranges of 

 the values of ^j . Because of this effect, there is a distortion in the relative 

 magnitudes of the energy distribution with frequency. To illustrate this: ($<jj 

 for serial 069 was plotted using a linear ordinate (see figure V-lU, bottom graph). 

 Observe the strong damping of the higher frequency energy on the linear scale. The 

 two peaks stand out more than in the logarithmic plot of serial 069 (in appendix B). 



To resolve more highly the low frequency range from 250 to 500 mcps, one need 

 only apply the relationship in equation (III-U), increasing the number of lags M. 

 By letting M = 100, the frequency range is resolved into twice as many divisions 

 (i.e., A-p = 25 mcps) over the same range from - 2500 mcps. The new spectrum 

 is produced in figure V-lU (upper curve). For convenience, the f scale has been 

 stretched out and cut off at 1250 mcps, since negligible energy lies beyond. The 

 two peaks are much more clearly defined. Now the peaks occur at about 125 and at 

 275 mcps, or at about 9-0 and 3.6 second periods. 



This increase in resolution cannot be obtained indefinitely by increasing M 

 values. One must keep equation (III-IO) in mind. With the degrees of freedom 

 DF = 21, the upper and lower ratios for 80 percent confidence are now 1.U0 and 

 0.6^, respectively (figure III-3). These values can be considered near enough to 



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