where &ψ is a constant that depends on the choice of
wavelet and is given by
Ψ
2
∞
& =
G <∞
∫
(4)
-∞
The condition (4), known as the admissibility condition,
restricts the class of functions that can be wavelets.
The continuous wavelet transform is a redundant
representation. Frequently, we use its discrete version
called the discrete wavelet transform (DWT) which is
very useful in signal processing. In DWT, the scaling
parameter D is taken to be of the form 2 and E to be of
the form N 2 , where N, M ∈ Ζ. With these values of D and
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E, the integral in (3) becomes
∞
:I (N 2 2 ) = 2
∫ I (W)
2- W - N )GW
-
2
(5)
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-∞
Discretize now the function I(W), and assume, for
The wavelet transform is a mathematical transformation
simplicity, the sampling rate to be 1, the integral in (5)
that represents a signal in terms of shifted and dilated
can be written as
version of a single function called mother wavelet. A
wavelet transform has a window whose bandwidth varies
:I (N 2 2 ) = 2 -
∑ I (Q)
2 - Q - N )
2
in proportion to the centre frequency of the wavelet. It
provides the local scale of the signal over time. The
continuous wavelet transform of a signal I(W) with respect
One of the important properties of (10) is its time-variant
to some analysing wavelet ψ is defined as (Chui 1992)
nature. The DWT of a function shifted in time is quite
different from the DWT of the original function.
∞
:I (E D) =
∫ I (W)
(W) GW
There is a strong relationship between wavelet transform
(3)
and multiresolution analysis (MRA). In the context of the
-∞
where
MRA, one attempts to decompose a signal I(W) as
W -E
1
W =
D>0
,
D
D
0
I (W) =
X
(W) +
∑ ∑Z
(W)
∑
(6)
0
=-∞
The parameters E and D are called translation and
dilatation parameters, respectively. The normalization
where M0 is an integer, ψ(W) and ϕ (W) are the wavelet and
factor D - 1/2 is included to ensure ||ψ || = ||ψ||, where || . ||
scaling functions, respectively, Z represent the wavelet
denotes the norm. A necessary and sufficient condition
or detail coefficients of I(W) and X are called the scaling
for (3) to be invertible is that ψ(W) satisfies the wavelet
function or approximation coefficients. They are given by
admissibility condition
∞
Z
∫ I (W)
(W)GW
∞
=
G <∞
-1
2
∫
Ψ(ω)
-∞
-∞
∞
X =
∫ I (W)
(W)GW
where Ψ(ω) is the Fourier transform of ψ(W) If ψ(W) is
0
sufficiently smooth and decay at infinity, which is usually
-∞
In the representation given by (6), M indexes the scale or
the case, the admissibility condition can be written as
resolution of analysis and N indexes the spatial location of
∞
analysis. For a special choice of the wavelet ψ(W) centred
W GW = 0
∫
at time zero and frequency I0, the wavelet coefficient Z
-∞
measures the signal content around time 2 N and
The original signal can be reconstructed from its integral
frequency 2 - I0. The scaling coefficient X measures the
wavelet transform as
local mean around time 2 N and (6) may be interpreted as
a multistage filtering decomposition of the signal I(W)
∞
∞
1
1
[:I (E D)]
I (W) =
GE
(W) GD
where the first term represents the low frequency
∫
∫
&
D
2
-∞
-∞