702 Z. Yang et al. / Digital Signal Processing 20 (2010) 699–714
Fig. 3. Theasteriskmark‘∗’ represents a local maximum point of f
l
(t) and the circle mark ‘◦’ represents an oblique local maximum point of f (t).
Fig. 4. (a)Theobliqueextremumpointξ
1
(circle) is an extremum point since f (t
a
) = f (t
b
); (b) the oblique extremum point ξ
3
(circle) is different from
the extremum point
ξ
2
(asterisk) since f (t
a
) = f (t
b
);(c)ξ
4
(circle) is an oblique extremum point but there is no extremum point on (t
a
, t
b
).
So we should choose other more powerful characteristic points to define the envelopes. As we know that an inflection point
is where the signal changes the concavity (see [23] for the precise definition). It is a separation between the concave and
convex of a signal and the joint point between various oscillation modes. With inflection points, we can depict the charac-
teristics of a signal more locally, since even a gentle oscillation may generate a new inflection point. However, if we define
the local mean by inflection points directly, we cannot guarantee the local symmetry. Motivated by the extremum point and
based on inflection points, we propose the concept of ‘oblique extremum point’ whose precise definition is given as follows.
Definition 1 (Oblique extremum point). Let
(t
a
, f (t
a
)) and (t
b
, f (t
b
)) (t
a
< t
b
) be two consecutive inflection points of a curve
f
(t) and l(t) be the straight line connecting these two inflection points. Denote
f
l
(t) = f (t) −l(t), t ∈[t
a
, t
b
], (2)
then the local maximum (or minimum) point ξ of f
l
(t) is called an oblique local maximum (or minimum) point of f (t).
A typical oblique extremum point is shown in Fig. 3, where
(ξ, f
l
(ξ)) (asterisk) is a local maximum point of f
l
(t) and
(ξ, f (ξ)) (circle) is an oblique local maximum point of f (t). From Definition 1, it is easy to check that the oblique extremum
point is an extremum point if and only if f
(t
a
) = f (t
b
). An illustration is given in Fig. 4. In Fig. 4(a) the oblique extremum
point
ξ
1
is just an extremum point due to the fact that f (t
a
) = f (t
b
).Butif f (t
a
) = f (t
b
) theobliqueextremumpointneed
not be an extremum point even if it exists nearby, as shown in Fig. 4(b) or not, as shown in Fig. 4(c). The latter case attracts
our particular attention, because in the monotonic segment from t
a
to t
b
of the wave a gentle oscillation mode is included
which cannot be extracted by EMD. To extract finer scale of a signal, oblique extremum points will be employed to improve
the EMD algorithm instead. It will be shown that there is a unique oblique extremum point between any two consecutive
inflection points. Before this, let us clarify two concepts on extremum points and inflection points, which are practically
useful in EMD and our algorithm.
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