The uniqueness of the instantaneous frequency Based on Intrinsic Mode Function

It has been claimed that any expression of a(t) cosθ(t) with a(t) as the instantaneous amplitude and cosθ(t) as the carrier varying along with the phase θ(t) could not be uniquely defined. However, based on the fact that a(t) cosθ(t) with all its variational forms have the same numerical value at any given time, we propose the existence of a unique true intrinsic amplitude function ai(t) and phase function θi(t) that ai(t) cosθi(t) satisfying the envelope–carrier relationship is the only expression making physical sense. A constructive method is also presented to find such amplitude-phase pair uniquely from any Intrinsic Mode Function (IMF). As a result, we can treat any IMF in the form of ai(t) cosθi(t) as the unique defined amplitude-phase pair, from which the instantaneous frequency (IF) can also be determined.

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