  The Carson and Fry (1937) introduced the concept of variable frequency as a
generalization of the constant frequency. The instantaneous frequency (IF) is
the time derivative of the instantaneous phase and it is well-defined only when
this derivative is positive. If this derivative is negative, the IF creates
problem because it does not provide any physical significance. This study
proposes a mathematical solution and eliminate this problem by redefining the
IF such that it is valid for all monocomponent and multicomponent signals which
can be nonlinear and nonstationary in nature. This is achieved by using the
property of the multivalued inverse tangent function. The efforts and
understanding of all the methods based on the IF would improve significantly by
using this proposed definition of the IF. We also demonstrate that the
decomposition of a signal, using zero-phase filtering based on the well
established Fourier and filter theory, into a set of desired frequency bands
with proposed IF produces accurate time-frequency-energy (TFE) distribution
that reveals true nature of signal. Simulation results demonstrate the efficacy
of the proposed IF that makes zero-phase filter based decomposition most
powerful, for the TFE analysis of a signal, as compared to other existing
methods in the literature.
