Non-Parametric Methods - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 131

The windowed periodogram can be interpreted as a smoothing of the
periodogram, which reduces the oscillations caused by the secondary lobes of
Bartlett's window and present in the periodogram. If the exact calculation of the
variance of the windowed periodogram is difficult, an approximate calculation
shows that this variance is approximately multiplied, in relation to the non-
windowed periodogram, by the coefficient between 0 and 1 defined by:

L − 1

1

∑

2

()

w κ

N κ

( )

=− − L 1

To ensure a positive estimation, it is sufficient to consider a positive Fourier
transform truncation window. As a result, Bartlett's window is often used.

N

Blackman and Tuckey, the sources of the method, recommend L ≈ .

10

As an example, in Figures 5.2 and 5.3 are drawn the power spectral densities
estimated using a recording of 128 points of a real value signal sampled at 1 kHz,
sum of a sine wave of amplitude 100, frequency 100 Hz, of a sine wave of
amplitude 100, frequency 109 Hz, of a sine wave of amplitude 7, frequency 200 Hz,
and a white noise of variance 25, and represented in Figure 5.1.

Figure 5.1. Signal

In the standard periodogram (Figure 5.2(a)), we mainly observe the lobes of
Dirichlet's kernel centered on the frequencies of two predominant sine waves,
which mask the spectral contribution of the sine wave of weak amplitude. The use
of Hamming's window for the modified periodogram, Figure 5.2(b) reduces the
amplitude of the secondary lobes (at the cost of the enlargement of the main lobe,
the two predominant sine waves are less apart), and makes it possible to view the
contribution of the 3 rd sine wave. With a Blackman's window (Figure 5.2(c)), with a
larger bandwidth, but with weaker secondary lobes, these two effects are even more
prominent. The averaging effect of Bartlett's or Welsh's periodogram on 3 segments
of 64 points is overlapped by 32 points, Figure 5.3 reducing the variance, thus the
oscillations of the spectrum in the zones describing the bandwidth contributions of