Non-Parametric Methods - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 130

From the Fourier transform, we obtain:

1

⎛ ⎞

( )

2

ES = S w

⎠

[5.57]

⎜ ⎟

xx xx

NU

⎝

1

2

If the function w is of unit integral over a period, it tends towards the

NU

Dirac comb Ξ 1 and the estimator is asymptotically unbiased. By applying Parseval's

1

N
k

−
=

1

∑

2

()

theorem, this integral is equal to wk . Thus, the normalization factor

0

NU

U is equal to:

N − 1

1

= ∑

2

()

[5.58]

U w k

N

k = 0

This procedure is also accompanied by a loss in resolution, the main lobe of the
Fourier transform of all the windows being larger than the rectangular window.

The Welsh periodogram combines the average periodogram and the modified
periodogram. We split the signal with N points available into K sections overlapping
L points, we calculate the modified periodogram on each segment, then we average
the modified periodograms obtained. The deterioration in the resolution is much less
and the variance is lower than for L = 0; however, the variance does not decrease

1
K

any further in , the different segments are correlated. On a Gaussian sequence,

with a Hanning window, we can show that the variance is minimum for L ≈ 0.65 K.
Usually L ≈ 0.5 K .

The correlogram, or windowed periodogram , consists of a prior estimation of an
autocorrelation function using formula [5.38], by applying a real even truncation
window w zero outside the interval { - ( L - 1), …, ( L - 1)} ( L < N ) and the Fourier
transform W :

L − 1

= ∑

− j 2 πνκ

() ()()

S ν r κ κ w e [5.59]

xx xx

( )

κ =− − L 1

It is necessary that w ( 0) = 1, so that the power spectral density obtained once
again gives the variance () r 0 . From the inverse Fourier transform, we obtain:

xx

SI

=

W [5.60]

xx xx