CHAPTER 4 
CONCLUSION FOR PART I 
In Part I, the nonparametric procedure for analyzing the irregular time series was 
discussed. At the beginning of Chapter 2 the basic procedure for obtaining the spectrum 
through periodograms was summarized and then a few considerations for improving this 
method were introduced in Chapter 3. The conclusions derived from the discussion are as 
follows: 
1. The spectral analysis technique with the use of periodograms is well established 
and affords us a powerful approach for analyzing irregular phenomena. 
2. In sample computations of the correlation function as well as of the spectra, 
however, care is necessary in sampling the data. The sampling time interval, the length 
of record, maximum length of lag, and proper windows must be chosen to get consistent 
estimates and avoid aliasing, blurring by the windows, or loss of reliability. 
3. Spectral—lag window pairs have been proposed by many scholars, and were 
shown that we must be careful of the effect of windows on the reliability, variability, and 
resolution of the results. 
4. Computation of the spectrum through periodograms by use of the Finite Fourier 
Transform is the same as that computed through the correlation function. 
5. The Fast Fourier Transform (F.F.T.) method is a convenient algorithm for reduc- 
ing the number of operations in the computation, but the considerations of the windows 
are also necessary and important in applying this method. 
6. The correlation functions can also be conveniently calculated by the F.F.T. 
method if proper precautions are taken. 
7. In connection with the choice of windows, the use of filters before applying the 
window is worth considering. 
8. Not only the spectrum functions but also the correlation functions (correlo- 
grams) should be investigated carefully in estimating the character of the process. 
9. In applying the spectral window in cross spectral analysis or in response analy- 
sis, the shift of the origin of the cross correlation (shift of the output record) should be 
considered to minimize the leakage of power through the use of spectral windows in 
computing cross spectrum. 
10. Cross spectral analysis is essential in the analysis of the response process of a 
dynamic system to get full information on the frequency response functions of the sys- 
tem. Cross spectral analysis is effective in reducing the effect of noise that contaminates 
the output. 
11. The coherency function is a good index to the extent to which the response can 
be expressed by linear relations. To make the coherency function useful the computations 
of all spectra must be done properly according to the preceding items 4~10 in getting the 
coherencies. 
12. The impulse response function may be obtained from the cross and auto corre- 
lations of output and input, without the trouble of choosing a window. This procedure 
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