where Tn is the length of the record in seconds, m is the number of lags, and At is the 

 sampling interval . For example, the spectral estimates in Figures 4 and 5 are distrib- 

 uted with a chi square distribution with 70 degrees of freedom. The 90-percent confi- 

 dence intervals can be determined from a table of chi square distribution. For 70 

 degrees of freedom, the true spectral density P.. (f) is bounded by the estimated spectral 

 density P p (f) according to: 



P E (f) P E (0 



T30~ < P T (f) < 0.76 or °. 77P E ( f )< P T ( f )<l -32P E (f) (13) 



Thus, if a single observed estimate with 70 degrees of freedom is observed to be 

 5 ft 2/sec , then we have 90-percent confidence that the true long-run value lies 

 between 5/1 .30 = 3.85 ft 2 /sec and 5/0.76 = 6.58 ft 2 />ec . 



D. Some Considerations in Estimating Spectra 



1. Resolution and Statistical Stability 



In estimating power spectra, whether by digital or by analog methods, a 

 choice must be made between a high resolution of the spectral estimates over a chosen 

 frequency band and the statistical stability of the spectral estimate in that band. The 

 two considerations are mutually opposed in that fulfillment of one Implies some sacrifice 

 of the other. For example: 



a. Width of the "spectral window" determines the resolution of any peaks in 

 the spectrum. Now, if the spectral window (i.e., the filter in the wave analyzer) is 

 chosen too narrow, too much detail is obtained, and a truly significant peak may be 

 overlooked in the resultant "blurring." However, if the spectral window is made too 

 wide, the resulting spectral estimate may be too smooth, and here again some physically 

 significant hump in the spectrum may be smoothed over. 



b. The statistical stability of the estimate depends on the width of the spectral 

 window and the length of the sample. When high resolution is desired, fewer Fourier 

 components are averaged over the narrow spectral window, and the spectral estimate will 

 have wider fluctuations over the totality of ensemble estimates. Conversely, less reso- 

 lution with a wider filter gives a value which is expected to deviate less from the true 

 but unknown spectrum. 



2. Aliasing 



a. If a signal g(t) contains no frequencies above f cycles per second, it is 



completely determined by giving its ordinate by a series of points spaced _! seconds 



2f 



21 



