periodogram Py(w,). To make this possible we need N more data, and that can be real- 
ized by adding N 0’s to the original data,as x» ..... SHEN O,O,0, oo 00 We 
N N 
2N 
In this way the FFT. method can also be used for computing the correlation function 
from the spectrum ordinates at N frequency points as in Fig. 2.21. 
I 
| 
N 
FREQ. POINTS | 
FREQ. POINTS 
Fig. 2.21. Frequencies to calculate the spectrum. 
25.8 Filtering 
As was mentioned in Section 2.4.3, in sampling a continuous time series for compu- 
tation of the spectrum, we have to pay attention to aliasing, and the sampling interval Ar 
should be small enough to avoid the aliasing. We must also be careful about the leakage 
of power or the blurring effect through the spectral window, especially when the spectrum 
has sharp peaks or steep valleys, because the spectral window acts as a smoothing filter. 
Besides, sometimes we are especially interested in the spectrum over a certain range of 
frequencies. Then the filtering technique is helpful in dividing the power of the spectrum 
by the frequencies or in modifying the shape of the spectrum to a shape more easily han- 
died. 
Generally, for a continuous process, the filtering effect can be expressed as 
Y(t) = | g(t)X(t —T)dr, (2.119) 
where X(t), Y(t) are the original and filtered processes, respectively, and g(t) shows the 
filtering effect in the time domain. For a physically realizable filter g(t) = 0, for t < 0, the 
range of integral can be from 0 to ©. Then from the discussion in the preceding 
48 
