Digital Signal Processing Reference
In-Depth Information
mt
of the upper envelope and lower envelope, that
()
2)
Calculate the average value
is
1
[
]
mt
()
=
x t x t
()
+
()
(2)
max
min
2
mt
from the initial signal
()
xt
, that is
()
And minus the
ht xt mt
1
()
=−
()
()
(3)
ht
could satisfy the two conditions of IMF. If not, take
1
()
ht
as
1
()
Test whether
ht
satisfy the two
1
()
the new initial signal and repeat the steps above until
conditions, then the first IMF can be written as
ct ht
()
=
()
(4)
1
1
ct
from signal
1
()
xt
, the result is written as
()
rt
1
()
3)
Minus
rt xt ct
()
=−
()
()
(5)
1
1
And then take
1
()
rt
as the new signal and repeat steps 1), 2), 3) until
n
rt
becomes
()
monotone function or constant.
By the decomposition above, the arbitrary signal
xt
could be decomposed as the
()
linear combination of IMFs below
n
xt
()
=
c t r t
()
+
()
(6)
i
n
i
=
1
3.2
The IF and Hilbert Spectrum
xt
, its analytic signal is given by
()
As for a real signal
ˆ
Xt xt jxt
()
=+
()
()
(7)
where
ˆ()
xt
means the Hilbert transform of the real signal
xt
, that is
()
1
x
( )
τ
∞
ˆ
()
xt
=
d
τ
(8)
π
t
−
τ
−∞
The formula (7) could be rewritten in the polar coordinate form
Xt ate
φ
()
=
()
jt
()
(9)
where
2
ˆ
2
at
()
=
x t x t
()
+
()
ˆ
()
(10)
xt
φ
( )
t
=
arctan
xt
()
