998
[6]
G. Strang, Linear Algebra and
Its
Applications.
1976.
New York: Academic,
Localization of the Complex Spectrum: The
S
Transform
R. G. Stockwell,
L.
Mansinha, and R.
P.
Lowe
Abstract-The
S
transform, which is introduced in this correspondence,
is an extension of the ideas of the continuous wavelet transform (CWT)
and is based on a moving and scalable localizing Gaussian window. It
is shown here to have some desirable characteristics that are absent in
the continuous wavelet transform.
The
S
transform is unique in
that
it provides frequency-dependent resolution while maintaining a
direct
relationship with the Fourier spectrum. These advantages of the
S
transform are due to the fact that the modulating sinusoids are fixed
with respect to the time axis, whereas the localizing scalable Gaussian
window dilates and translates.
IEEE
TRANSACTIONS
ON
SIGNAL
PROCESSING, VOL.
44,
NO.
4,
APRIL
1996
I.
INTRODUCTION
In geophysical data analysis and in many other disciplines, the
concept of
a
stationary time series is a mathematical idealization
that
is
never realized and is not particularly useful in the detection
of signal arrivals. Although the Fourier transform of the entire time
series does contain information about the spectral components in
a
time series, for a large class of practical applications, this informa-
tion
is
inadequate. An example from seismology is an earthquake
seismogram. The first signal to amve from an earthquake is the
P
(primary) wave followed by other
P
waves traveling along different
paths. The
P
arrivals are followed by the
S
(secondary) wave
and by higher amplitude dispersive surface waves. The amplitude
of these oscillations can increase by more than two orders
of
magnitude within a few minutes of the arrival of the
P.
The
spectral components of such a time series clearly have
a
strong
dependence on time. It would be desirable to have a joint time-
frequency representation (TFR). This correspondence proposes a
new transform (called the
S
transform) that provides
a
TFR with
frequency-dependent resolution while, at the same time, maintaining
the direct relationship, through time-averaging, with the Fourier
spectrum. Several techniques of examining the time-varying nature
of the spectrum have been proposed in the past; among them are the
Gabor transform
[7],
the related short-time Fourier transforms, the
continuous wavelet transform (CWT)
181,
and the bilinear class of
time-frequency distributions known
as
Cohen’s class [4], of which
the Wigner distribution
191 is
a
member.
Manuscript received November 28,
1993;
revised September 22. 1995.
This work was
supported
by
the Canadian Network
for
Space Research,
one
of
fifteen Networks
of
Centres
of
Excellence supported by the Govemment
of
Canada, by the Natural Sciences and Engineering Research Council
of
Canada, and by Imperial Oil Resources Ltd. The associate editor coordinating
the review
of
this paper and approving it for publication was Dr. Boualem
Boashash.
R. G. Stockwell and R. P. Lowe are with the Canadian Network for Space
Research and Department
of
Physics, The University
of
Western Ontario,
London, Ont., Canada N6A 5B7 (e-mail: stockwell@uwo.ca).
L.
Mansinha is with the Department
of
Earth Sciences, The University
of
Westem Ontario, London, Ont., Canada N6A 5B7.
Publisher Item Identifier
S
1053-587X(96)02790-9.
11.
THE
S
TRANSFORM
There are several methods of arriving at the
S
transform. We
consider it illuminating to derive the
S
transform
as
the “phase
correction”
of
the CWT. The CWT
W(T,~)
of
a
function
h(t)
is
defined by
00
Tl-(r.
d)
=
h(t)w(t
-
T,
d)
dt
(1)
lo
where
~(t.
d)
is a scaled replica of the fundamental mother wavelet.
The dilation
d
determines the “width’ of the wavelet
w(t.
d)
and thus
controls the resolution. Along with
(1),
there exists
an
admissibility
condition on the mother wavelet
w
(t,
d) 151 that
w(t,
d) must have
zero mean. Refer to
Rioul and Vetterli
[lo]
and Young
1111
for
reviews of the literature.
The
S
transform of
a
function
h(t)
is defined
as
a
CWT with a
specific mother wavelet multiplied by the phase factor
S(r,
f)
=
er2+W(T,
d)
(2)
where the mother wavelet is defined
as
Note that the dilation factor
d
is the inverse of the frequency
f.
The wavelet in
(3)
does not satisfy the condition of zero mean for
an admissible wavelet; therefore,
(2)
is not strictly
a
CWT. Written
out explicitly, the
S
transform
is
If the
S
transform is indeed a representation
of
the local spectrum,
one would expect a simple operation of averaging the local spectra
over time to give the Fourier spectrum. It is easy to show that
cc
1,
S(T.
f)dT
=
H(f)
(5)
(where
H(f)
is the Fourier transform of
h(t)).
It follows that
h(t)
is exactly recoverable from
S(T.
f).
Thus
This is clearly distinct from the concepts of the CWT.
The
S
transform provides an extension of
instantaneous frequency
(IF)
121 to broadband signals. The 1-D function of the variable
T
and
fixed parameter
fl
defined by S(T,
fl)
is called a voice
(as
with the
CWT).
The voice for
a
particular frequency
fl
can be written as
S(T.
fl)
=
A(T,
fl)ezQ(T’fl).
(7)
Since a voice isolates a specific component, one may use the phase
in
(7)
to determine the IF
as
defined by Bracewell 121.
(8)
Thus, the absolutely referenced phase information leads to
a
gener-
alization of the
IF
of Bracewell to broadband signals. The validity
of
(8) can easily be seen for the simple case of
h(t)
=
cos(2~wt),
where the function
@(T,
f)
=
2n(w
-
f)~.
The linear property of the
S
transform ensures that for the case of
additive noise (where one can model the data
as
data(t)
=
signal(t)
+
noise(t)), the
S
transform gives
S{data}
=
S{signal)
+
S{noise}.
la
IF(T.fl)
=
%%{27rflT+
@(T,fl)l.
1053-587X/96$05.00
0
1996
IEEE
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