In this paper, we develop an effective and robust
adaptive time-frequency analysis method for signals
with intra-wave frequency modulation. To handle
this kind of signals effectively, we generalize our
data-driven time-frequency analysis by using a
shape function to describe the intra-wave frequency
modulation. The idea of using a shape function in
time-frequency analysis was first proposed by Wu
(Wu 2013 Appl. Comput. Harmon. Anal. 35, 181–199.
(doi:10.1016/j.acha.2012.08.008)). A shape function
could be any smooth 2π-periodic function. Based
on this model, we propose to solve an optimization
problem to extract the shape function. By exploring
the fact that the shape function is a periodic function
with respect to its phase function, we can identify
certain low-rank structure of the signal. This low-rank
structure enables us to extract the shape function from
the signal. Once the shape function is obtained, the
instantaneous frequency with intra-wave modulation
can be recovered from the shape function. We
demonstrate the robustness and efficiency of our
method by applying it to several synthetic and
real signals. One important observation is that this
approach is very stable to noise perturbation. By using
the shape function approach, we can capture the intrawave
frequency modulation very well even for noisepolluted
signals. In comparison, existing methods
such as empirical mode decomposition/ensemble
empiricalmode decomposition seem to have difficulty
in capturing the intra-wave modulation when the
signal is polluted by noise.