6/15/2009 Negative frequency in IMF

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

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
The amplitude modulated part A(t) is the envelope and φ(t) the phaser of FM signal. What interests us is the frequency modulated signal, or in Hilbert’s definition, the derivative of phase.
             ω= dφ/dt

Bedrosian theorem states that Hilbert Transform is not the same for spectrum overlapping. That is, H(A(t)Q(t))=A(t)H(Q(t)) 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.