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consider an alternative, utilizing overlapping segments, introduced by Welch
(1967). It is sometimes called the Welch overlapping segment analysis (WOSA).
While the estimates on individual segments are now correlated, the extra e
ect-
ive record length more than makes up for the inflation of the variance due to the
common parts of the data entering the estimate.
If we again break the data into segments of length M , the spectral density estim-
ator for the n th segment is given by (2.349) as
ff
2
I ,
= | H ( f )
|
S gg, n ( f )
(2.356)
with H ( f ) the Fourier transform of
h ( t )
= w( t )g( t ),
(2.357)
the windowed form of the original time sequence g( t ). The expected value of this
sample spectral density estimate is
E S gg, n ( f )
I E H ( f ) H ( f )
1
=
w( t )w ( u ) E g( t )g ( u ) dt du
1
I
e i f ( t u )
=
−∞
−∞
τ) E g( t )g ( t
τ) dt d τ, (2.358)
1
I
e i f τ w( t )w ( t
=
−∞
−∞
on making the variable change τ =
u . Once again, recognising that the window
w( t ) is real and even, the expected value of the sample spectral density estimate
takes the form
E S gg, n ( f )
t
−∞ w( t )w(τ
1
I
e i f τ φ gg (τ)
=
·
t ) dt
·
d τ.
(2.359)
−∞
The latter integral is the Fourier transform of the product of two functions of lag τ,
the autocorrelation φ gg (τ) and the convolution of the window with itself. Using the
convolution results (2.269) and (2.274), the expected value of the sample spectral
density estimate is
E S gg, n ( f )
1
I
f 1 ) W 2 ( f 1 ) df 1 ,
=
S gg ( f
(2.360)
−∞
where S gg ( f ), the Fourier transform of the autocorrelationφ gg (τ), is the true power
spectral density of g( t ). As before, we see that the estimator, on average, produces
a spectral density estimate that is smoothed or convolved with the square of the
frequency window, W 2 ( f ). For the Parzen window, the half-power points of W 2 ( f )
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