|
|
|
|
|
|
|
Does IMF
(Intrinsic Mode Function) have negative frequency? If not, why do I have
negative frequency all the time?
|
On
instantaneous frequency calculation of Intrinsic Mode function
Why
do we have negative frequency all the time?
Negative
frequency arises often when doing the instantaneous frequency calculation on
IMF. The Intrinsic Mode Function characterized by its smoothly oscillatory
nature enjoys good properties of Hilbert Transform. The major one is the ruling
out of negative frequency which arises often for non-IMF signals. The merit of
IMF lies on its symmetric envelope and equal numbers of extrema and
zero-crossing, making it a perfect input to Hilbert Transform without worrying
on negative frequency occurrence. All signal can be decomposed into a collection of IMFs using EMD
(Empirical Mode Decomposition). The finding of the clever connection between
EMD and Hilbert Transform had won Dr. Huang the Member of Science Foundation.
But we
still have the negative frequency from time to time. There are several reasons:
one is the IMF you have does not exactly meet the criterion of equal number of
extrema and zero-crossing. That is, you can see signal goes up to the zero line
and do at least one oscillation before it goes down to the zero crossing. The
violation occurs very often when you are dealing with wave like signal, or
signal having large variation on its amplitude. The small amplitude part is not
that sensitive to variation on the scale defined by large amplitude especially
when EMD is done on the whole signal, not to address the detail of small
variation in term of amplitude. The other is reason comes from Bedrosian
Theorem, stating that Hilbert Transform is not quite the same on IMF if its
envelope spectrum has overlapping on its frequency modulated part of the
signal. The condition holds if envelope of IMF varies slowly without any part
of it close to zero. For IMF with large envelope variation, such condition
results in frequency out of our expectation. One of them is frequency
negativity.
Annihilation
of negative frequency requires us to clear up the question: what is our
definition of frequency? Frequency is intuitively defined as number of occurrences
during a period of time. As the “period of time” reduces to zero, the
definition of occurrences become vague: number of extrema, zero crossing, or number
of time passing through a predefined value? Hilbert defined the occurrence as
number of revolutionary circle generating the signal as its x-axis projection.
Such definition is normally refers to “instantaneous frequency” as the period
of time in counting the occurrences goes to zero.
Bearing
in mind Hilbert’s definition on instantaneous frequency we come to the point
that would make Hilbert “unhappy”, as Dr. Huang had put it that way for
amusing. What if the signal of interest could not be regarded as generated from
the conceptually revolutionary circle? Or the signal can be generated from the
circle if we need to reverse sometimes the revolution? That would make the
frequency negative. Unfortunately mostly signal does. It was until Dr. Huang
proposed its preprocessing EMD approach that the awkward condition can be
relieved: Do EMD before Hilbert Transform. After EMD, each IMF represents a
clear projection from the revolutionary circle that have exactly all good prerequisite
you need for doing Hilbert Transform. Such preprocessor is perfect and
theoretically it produces no negative frequency. Hilbert would then be “happy”
again. It is therefore Dr. Huang, as it would amusingly and clearly explain the
theory, would call the method “Happy Hilbert Transform,” in comparison to
Hilbert-Huang Transform as we officially call the method.
Knowing
the history and the definition of instantaneous frequency, we now come to the
point of resolving the frequency negativity. A IMF signal can be written as
AM-FM signal which form is
|
|
(1.1)
|
Where
|
|
(1.2)
|
The
amplitude modulated part A(t) is the envelope and
the phaser of FM signal. What interests
us is the frequency modulated signal, or in Hilbert’s definition, the derivative
of phase.
|
|
(1.3)
|
Bedrosian
theorem states that Hilbert Transform is not the same for spectrum overlapping.
That is,
provided spectrum of A(t) not overlapping
with Q(t). Question is, the frequency of interest is the phase derivative of
Q(t). The intention leads us to the work around: normalizing IMF enable us to
get rid of the enveloped effect A(t).
What Dr.
Huang proposed for more accurately calculate the instantaneous frequency is to do
normalization of IMF. The process is as followed:
1.
Take absolute value of IMF.
2.
Find extrema.
3.
Based on these extrema, construct
envelope.
4.
Normalize IMF using the envelope.
The FM part of signal becomes almost equal amplitude.
5.
Repeat process 2-4 after the amplitude
of normalized IMF retains a straight line with identical value.
6.
Find the instantaneous frequency
on the normalized IMF.