A JOINT TIME–FREQUENCY EMPIRICAL MODE DECOMPOSITION FOR
NONSTATIONARY SIGNAL SEPARATION
N. Stevenson, M. Mesbah, B. Boashash
Perinatal Research Centre
University of Queensland
Herston, QLD, 4029, Australia
email: n.stevenson@uq.edu.au
H.J. Whitehouse
Linear Measurements, Inc.
4174 Sorrento Valley Blvd,
San Diego, CA 92121
ABSTRACT
This paper outlines the application of the empirical mode de-
composition (EMD) to a frequency domain representation of
a signal. The application to frequency domain representa-
tions, as opposed to time domain representations, may pro-
vide more useful decomposition in the case where signal com-
ponents overlap in frequency. The combined use of the EMD
for both time and frequency domain representations of a sig-
nal is capable of generating more desirable signal decompo-
sitions for nonstationary signals. The results of the time–
frequency EMD on two example signals shows improvement
in nonstationary signal separation with little loss of desirable
decomposition performance measures such as IMF orthogo-
nality.
1. INTRODUCTION
The empirical mode decomposition (EMD) has been proposed
by Huang et al., in [1], as a tool for analysing nonlinear and
nonstationary time series. The EMD decomposes a signal into
a series of intrinsic mode functions (IMFs). These IMFs are
optimised for instantaneous frequency (IF) estimation using
the analytic associate of a signal. To this end, the EMD uses
an iterative estimate of the local or short–time mean, based
on the signal envelope, to generate IMFs that have an equal
number of extrema and zero–crossings.
The use of the local mean has the implication that the
EMD is an adaptive decomposition technique, as it decom-
poses a signal based on a set of signal dependent basis func-
tions. This is contrary to well known signal decomposition
methods, such as Fourier, wavelet or Wigner–Ville. It is the
behaviour of the EMD when decomposing signals of a non-
stationary nature that is of interest to the time–frequency (TF)
signal processor. Of particular interest is the representation of
the signal using the EMD as
s(t)=
Q
i=1
IMF
i
(t)+ r(t), (1)
where, Q is the number of IMFs and r(t) is the decomposition
residue. This representation is identical to that given in [2] to
define signal components in a TF signal processing context,
s(t)=
Q
i=1
a
i
(t) cos(θ
i
(t)) + r(t), (2)
where a
i
(t) cos(θ
i
(t)) = IMF
i
(t), a
i
(t) is the envelope and
θ
i
(t) is the phase of the i
th
component or IMF. According to
this decomposition, an IMF is equivalent to a signal compo-
nent. In the context of the EMD, this representation suggests
that the EMD is capable of separating signals with different
frequencies at a single time instant, but it cannot separate sig-
nals with identical frequency content at different times.
The representation in (2) defines each component as ex-
isting for the entire signal duration. In practice, however, a
component can additionally be defined with limits in the time
domain. This is a more complete definition of nonstationary
signal components and can be outlined as,
s(t)=
L
k=1
a
k
(t)rect
t − τ
k
b
k
cos(θ
k
(t)) + r(t), (3)
where rect((t − τ
k
)/b
k
) is the rectangular function of width
b
k
, centre τ
k
and L ≥ Q. The advantage of such a definition is
that a signal component can be localised in both the time (us-
ing cos(θ
k
(t))) and frequency (using rect((t − τ
k
)/b
k
)) axes
of the TF plane. This representation also results in monotonic
components with respect to time and frequency. This has
applications in kernel design for reduced interference time–
frequency distributions (TFDs), [3, pp. 168].
The aim of this paper is to adjust the EMD so that it pro-
vides IMFs that fit the definition of (3) rather than (2). That
is, force the EMD to separate signals with identical frequency
content at different times, in addition to separating signals
with different frequencies at a single time instant. The pro-
posed solution involves using the discrete cosine transform
(DCT), [4], as an additional processing step in the EMD.
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