Circuits, Systems, and Signal Processing

, Volume 37, Issue 8, pp 3515–3536 | Cite as

Breaking the Limits: Redefining the Instantaneous Frequency

  • Pushpendra SinghEmail author


The Carson and Fry 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 eliminates this problem by redefining the IF such that it is valid for all monocomponent, multicomponent signals of nonlinear and nonstationary nature. This is achieved by using the property of the multivalued inverse tangent function that provides base to ensure that the instantaneous phase is an increasing 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.


Analytic signal (AS) Hilbert transform (HT) An increasing (or a nondecreasing)  function Instantaneous frequency (IF) Linearly independent nonorthogonal yet energy preserving (LINOEP)  vectors Zero-phase filtering (ZPF) 



Author would like to show his gratitude to the Prof. S. D. Joshi (IITD), Prof. R. K. Pateny (IITD), and Dr. Kaushik Saha (CTO, Samsung R&D Institute India—Delhi) for sharing their wisdom and expertise during this research.


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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Engineering and Applied Sciences (SEAS)Bennett UniversityGreater NoidaIndia

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