Adaptive Fourier Decomposition Based Signal Instantaneous Frequency Computation Approach
There have been different approaches to compute the
analytic instantaneous frequency with a variety of background reasoning
and applicability in practice, as well as restrictions. This paper presents an adaptive Fourier decomposition and (α-counting) based
instantaneous frequency computation approach. The adaptive Fourier
decomposition is a recently proposed new signal decomposition
approach. The instantaneous frequency can be computed through the so called mono-components decomposed by it. Due to the fast energy
convergency, the highest frequency of the signal will be discarded by the adaptive Fourier decomposition, which represents the noise of
the signal in most of the situation. A new instantaneous frequency
definition for a large class of so-called simple waves is also proposed
in this paper. Simple wave contains a wide range of signals for which
the concept instantaneous frequency has a perfect physical sense.
The α-counting instantaneous frequency can be used to compute the highest frequency for a signal. Combination of these two approaches one can obtain the IFs of the whole signal. An experiment is demonstrated the computation procedure with promising results.
 A. Bultheel, P.Gonzalez-Vera, E.Hendriksen, and O.Njastad, Orthogonal
Rational Functions, volume 5 of Cambridge Monographs on Applied and Computational Mathematics, Cambeidge University Press, 1999.
 P. Butzer and R. Nessel, Fourier Analysis and Approximation, Volumn 1:
One-dimensional Theory. Birkhauser,Basel,and Academic Press, New
 L. Cohen, Time-frequency analysis: theory and application, Prentice Hall,
Englewood Cliffs, NJ, 1995.
 J. B. Conway, Functions of One Complex Variable,2nd ed. Springer. 1978.
 J. B. Garnett , Bounded Analytic Functions, Academic Press, 1987.
 S. Mallat and Z. Zhang, Matching pursuits with time-frequency dictionaries,
IEEE Transactions on Signal Processing, 1993, 41:3397-3415.
 A. V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing,
3rd ed. Prentice Hall. 2010.
 B. Picinbono, "On instantaneous amplitude and phase of signals", IEEE
Transactions on Signal Processing, vol, 45, no. 3, pp. 552-560, 1997.
 T. Qian, Intrinsic Mono-components Decomposition of Functions: An
Advance of Fourier Theory. Math.Meth.Appl.Sci. 2010, 33:880-891.
 T. Qian, Mono-components for Decomposition of Signals.
Math.Meth.Appl.Sci. 2006, 29:1187-1198.
 T. Qian and Y. B. Wang, Adaptive Fourier Series - a Variation of
Greedy Algorithm. Advances in Computational Mathematics. 2010. DOI
 T. Qian, L.M. Zhang and Z. Li, Algorithm of Adaptive Fourier Transform.
IEEE Transactions on Signal Processing. vol. 59(12),pp. 5899-5906,
 W. Rudin, Real and complex analysis, New York, McGraw Hill (1996).
 J. S. Walker, Fourier Analysis and Wavelet Analysis. Notices of the
AMS. vol. 44(6), pp. 658-670, June/July, 19