Vakman's analysis inL2(ℝ)
Jianfeng Huang, Lihua Yang · International Journal of Computer Mathematics · 2010
The traditional way to define the instantaneous frequency of a real-valued signal is based on the analytic signal obtained by means of the Hilbert transform. Vakman showed the justification of this method by proving that under some fundamental conditions the imaginary part signal operator ℐ must be the Hilbert transform. This paper shows that his arguments are not valid in the finite-energy signal space L 2(ℝ) – an important space of signals. In this paper, we propose substitutive conditions and establish the L 2(ℝ) version of Vakman's analysis, which is a meaningful supplement to Vakman's theory.