from the “tail” of autocovariance R(n), which shows more erratic fluctuations than the 
Teal value of R(r) and is Statistically less reliable, as discussed at the end of Section 2.3. 
As the inverse Fourier transfer of Eq. 2.91 
oh 
w(r) = | e’? W@) dp (2.92) 
I 
PaW, esl, s2 2 coe 28 (NTs [w(r) = 0 at Irl>N-1). 
Since w(r) are real and even, 
1 N-1 
W@)=— Ss w(r) cos ro. (2.93) 
21 r=-(N-1) 
W(@), which is the Fourier pair with w(r), is called the spectral window function, and 
w(r) the lag window function. 
wo(r), given by Eq. 2.88, is one of the lag windows and is called a “truncated 
window” or “do nothing window.” Its Fourier transform Wo(@) is from Eq. 2.73 
M 
Wo(@) = = cose = (2.94) 
—M 
This function, shown in Fig. 2.12, is also referred to as Dirichlet’s kernel function 
Dy(@). It has a high peak at @ = 0 and rather deep valleys on both sides of the main 
lobe. This sometimes causes harm to the computation of the spectrum and results in 
some spurious negative values for the spectrum ordinates. Many studies have been made 
to obtain good windows, and many different windows with different characteristics have 
been proposed and claimed to be good from different points of view. 
Several windows described by Priestley? are summarized in Fig. 2.13 and Fig. 2.14 
in the form of pairs of lag window w(r) and spectral window W(@). 
25.4 Effect of Windows 
Some of the proposed window pairs are compared in Fig. 2.15. They were designed 
mostly under the following conditions: 
1. Wy(P) = 0 is desirable to avoid the effect of large negative lobes. 
31 
