IEEE SIGNAL PROCESSING MAGAZINE
[
124
]
NOVEMBER 2005
avoid confusion with the often-used acronym CWT for the (dif-
ferent) continuous wavelet transform.
BACKGROUND
This article aims to reach two different audiences. The first is the
wavelet community, many members of which are unfamiliar
with the utility, convenience, and unique properties of complex
wavelets. The second is the broader class of signal processing folk
who work with applications where the DWT has proven some-
what disappointing, such as those involving complex or modulat-
ed signals (radar, speech, and music, for example) or higher-
dimensional, geometric data (geophysics and imaging, for exam-
ple). In these problems, the complex wavelets can potentially
offer significant performance improvements over the DWT.
THE WAVELET TRANSFORM
AND MULTISCALE ANALYSIS
Since its emergence 20 years ago, the wavelet transform has
been exploited with great success across the gamut of signal
processing applications, in the process, often redefining the
state-of-the-art performance [102], [112]. In a nutshell, the
DWT replaces the infinitely oscillating sinusoidal basis functions
of the Fourier transform with a set of locally oscillating basis
functions called
wavelets
. In the classical setting, the wavelets
are stretched and shifted versions of a fundamental, real-valued
bandpass wavelet
ψ(
t
)
. When carefully chosen and combined
with shifts of a real-valued low-pass scaling function
φ(
t
)
, they
form an orthonormal basis expansion for one-dimensional (1-D)
real-valued continuous-time signals [27]. That is, any finite-
energy analog signal
x
(
t
)
can be decomposed in terms of
wavelets and scaling functions via
x
(
t
)
=
∞
X
n
=−∞
c
(
n
)φ(
t
−
n
)
+
∞
X
j
=
0
∞
X
n
=−∞
d
(
j
,
n
)
2
j
/
2
2
j
t
−
n
).
(
1
)
The scaling coefficients
c
(
n
)
and wavelet coefficients
d
(
j
,
n
)
are
computed via the inner products
c
(
n
)
=
Z
∞
−∞
x
(
t
t
−
n
)
dt
,(
2
)
d
(
j
,
n
)
=
2
j
/
2
Z
∞
−∞
x
(
t
)ψ(
2
j
t
−
n
)
.(
3
)
They provide a time-frequency analysis of the signal by measur-
ing its frequency content (controlled by the scale factor
j
) at dif-