Instantaneous Frequency, Its Standard Deviation And Multicomponent Signals

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Abstract
We consider instantaneous frequency and its variance using the bilinear joint time-frequency distribu-tions. It is well known that these distributions give the instantaneous frequency as the time derivative of the phase. We show that they also lead to a reasonable definition for the standard deviation of instantaneous frequency, namely σw2(t) )= ((A'(t))/(A(t)))2 where A(t) is the amplitude of the signal. We demonstrate the relationship with the bandwidth of the spectrum. We also derive the corresponding quantities for the short-time Fourier spectrum and show the relation to and consistency with the above definition. The concept of local spread of frequencies is used to define and clarify the meaning of multicomponent signals. It is argued that the breaking up of a signal into components is a local phenomenon and that the criteria for a meaningful decomposition is that the standard deviations of instantaneous frequency of each part about their own individual instantaneous frequencies be well separated and small in comparison to the standard deviations of the signal. In addition, we consider the new distributions of Choi and Williams which dramatically enhance the interpretive value and use of bilinear distributions. These distributions suppress the interference terms while preserving the desirable characteristics of the distributions. This is particularly the case for multicomponent signals. We show that if a class of distributions yields a certain expectation value, then the cross terms of the different distributions within that class contribute the identical value towards expectation values. Hence, even though the cross terms may be reduced, they none the less contribute an identical amount towards an expectation value.

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