TM No. 377 



where AT is the sampling interval of the time series. This interval may 

 be the preset real time sampling interval of the instrument sensor. It 

 can,, moreover,, be the selected time intervsl over which a continuous 

 analog record was sampled,, or the interval at which data were interpolated 

 from an unequally spaced digital record,, This was the case for the wave 

 meter records processed, as discussed in the previous sections „ The use 

 of digital data (rather than continuous records) can introduce into the 

 spectrum the phenomenon known as "aliasing ". For example, the harmonic 

 oscillations of the frequencies 



-i 



f ^ (IT)' - ^ (^T)-V 4. 2(Atrl %.,. (111=3) 



can appear , under certain conditions., as a single frequency contribution to 

 the ^spectra. If the frequency f is slightly lower than the sampling frequency 

 G&T)* 1 , then it cannot be distinguished from the low frequaney(AT>"-fo Thus, 

 the high frequency f appears under the alias of the low frequency £.r)"- >?■ . 

 To minimize the aliasing effect and to obtain the best possible spectrum, 

 the sampling interval AT should be small enough tc include all frequencies 

 containing quantities of energy that contribute significantly to the 

 variance of the motions studied. Note that the response time of the instru= 

 ment sensor defines the lower limit of AT 3 or, in effect, the highest 

 I^yquist freq:iency one can choose. Assuming that the sensor response time 

 (as defined in appendix A) is -veil below the chosen AT , one can incorporate 

 an appropriate low-pass filter, either in the output of the sensor or into 

 the data record prior to analysis', and use a sampling rate at which the 

 Nyquist frequency coincides with the filter cutoff frequency. Filtering 

 direct output data is not good practice, in general, because one is irre« 

 versibly excluding information (at higher frequencies) which later may be 

 desired. 



Another method of assimilating the spectral information at a desired 

 frequency band is to assess the sensor response, and choose what might be 

 the natural cutoff frequency based on physical grounds. This choice can 

 be governed by a trial spectrum, using the highest frequency resolution at= 

 tainable (governed by the sensor) and examining the energy at the upper 

 limit of the frequency range. This should guide one in the optimum choice 

 of the Nyquist frequency. 



The next consideration is' that of the resolution desired in the spectral 

 estimates. The frequency resolution may be defined as; 



where tA is the total number of units on the frequency scale, fv^ is actually 

 the maximum lag at which the auto-covariance function is computed if the 



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