Digital Signal Processing Reference
In-Depth Information
The multispectrum
S
n
of order
n
is the multidimensional Fourier transform of the
moment of order
n
: for all values of
[
T
]
∈ℜ
n
−
1
ννν
=
:
1
n
−
1
T
()
=
∑
n
()
2πνκ
j
S
ν
M
κ
e
[5.62]
n
n
κ
∈
Z
From a recording of finite length (
x
(
k
))
0≤
k
≤
N-
1
of characterization
x
of a random
signal
x
, the spectrum
S
n
can be estimated from the natural estimator:
T
()
=
∑
n
()
2πνκ
j
S
ν
M
κ
e
[5.63]
n
n
κ
∈
Z
Frequency (Hz)
Figure 5.3.
Estimated power spectral density
(
Hamming's window of 64 samples,
covering 32 samples
)
: a
)
from Bartlett's periodogram; b
)
from Welsh's periodogram
(
Hamming's window
)
; c
)
from Welsh's periodogram
(
Blackman's window
)
where
M
is a biased estimator of the moment of order
n
:
n
n
−
1
1
()
∑ ∏
x
() (
)
M
κ
=
x k
k
+
κ
[5.64]
n
i
N
k
i
=
1
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