The wavelet expansion of a signal f(t)f(t) as first formulated in (Reference)
is repeated here by
f
(
t
)
=
∑
k
=
-
∞
∞
∑
j
=
-
∞
∞
〈
f
,
ψ
j
,
k
〉
ψ
j
,
k
(
t
)
f
(
t
)
=
∑
k
=
-
∞
∞
∑
j
=
-
∞
∞
〈
f
,
ψ
j
,
k
〉
ψ
j
,
k
(
t
)
(1)where the {ψj,k(t)}{ψj,k(t)} form a basis or tight frame for the signal
space of interest (e.g., L2L2). At first glance, this infinite series
expansion seems to have the same practical problems in calculation that an
infinite Fourier series or the Shannon sampling formula has. In a
practical situation, this wavelet expansion, where the coefficients are
called the discrete wavelet transform (DWT), is often more easily
calculated. Both the time summation over the index kk and the scale
summation over the index jj can be made finite with little or no error.
The Shannon sampling expansion [8], [6] of a signal with
infinite support in terms of sinc(t)=sin(t)t(t)=sin(t)t expansion
functions
f
(
t
)
=
∑
n
=
-
∞
∞
f
(
T
n
)
sinc
(
π
T
t
-
π
n
)
f
(
t
)
=
∑
n
=
-
∞
∞
f
(
T
n
)
sinc
(
π
T
t
-
π
n
)
(2)requires an infinite sum to evaluate f(t)f(t) at one point because the sinc
basis functions have infinite support. This is not necessarily true for a
wavelet expansion where it is possible for the wavelet basis functions to
have finite support and, therefore, only require a finite summation over
kk in Equation 1 to evaluate f(t)f(t) at any point.
The lower limit on scale jj in Equation 1 can be made finite by adding the
scaling function to the basis set as was done in (Reference). By using the
scaling function, the expansion in Equation 1 becomes
f
(
t
)
=
∑
k
=
-
∞
∞
〈
f
,
φ
J
0
,
k
〉
φ
J
0
,
k
(
t
)
+
∑
k
=
-
∞
∞
∑
j
=
J
0
∞
〈
f
,
ψ
j
,
k
〉
ψ
j
,
k
(
t
)
.
f
(
t
)
=
∑
k
=
-
∞
∞
〈
f
,
φ
J
0
,
k
〉
φ
J
0
,
k
(
t
)
+
∑
k
=
-
∞
∞
∑
j
=
J
0
∞
〈
f
,
ψ
j
,
k
〉
ψ
j
,
k
(
t
)
.
(3)where j=J0j=J0 is the coarsest scale that is separately represented. The
level of resolution or coarseness to start the expansion with is arbitrary,
as was shown in Chapter: A multiresolution formulation of Wavelet Systems in (Reference), (Reference), and
(Reference). The space spanned by the scaling function contains all the
spaces spanned by the lower resolution wavelets from j=-∞j=-∞ up to the
arbitrary starting point j=J0j=J0. This means VJ0=W-∞⊕⋯⊕WJ0-1VJ0=W-∞⊕⋯⊕WJ0-1. In a practical case,
this would be the scale where separating detail becomes important. For a
signal with finite support (or one with very concentrated energy), the
scaling function might be chosen so that the support of the scaling
function and the size of the features of interest in the signal being
analyzed were approximately the same.
This choice is similar to the choice of period for the basis sinusoids in
a Fourier series expansion. If the period of the basis functions is
chosen much larger than the signal, much of the transform is used to
describe the zero extensions of the signal or the edge effects.
The choice of a finite upper limit for the scale jj in Equation 1 is more
complicated and usually involves some approximation. Indeed, for
samples of f(t)f(t) to be an accurate description of the signal, the signal
should be essentially bandlimited and the samples taken at least at the
Nyquist rate (two times the highest frequency in the signal's Fourier
transform).
The question of how one can calculate the Fourier series coefficients of a
continuous signal from the discrete Fourier transform of samples of the
signal is similar to asking how one calculates the discrete wavelet
transform from samples of the signal. And the answer is similar. The
samples must be “dense" enough. For the Fourier series, if a frequency
can be found above which there is very little energy in the signal (above
which the Fourier coefficients are very small), that determines the
Nyquist frequency and the necessary sampling rate. For the wavelet
expansion, a scale must be found above which there is negligible detail or
energy. If this scale is j=J1j=J1, the signal can be written
f
(
t
)
≈
∑
k
=
-
∞
∞
〈
f
,
φ
J
1
,
k
〉
φ
J
1
,
k
(
t
)
f
(
t
)
≈
∑
k
=
-
∞
∞
〈
f
,
φ
J
1
,
k
〉
φ
J
1
,
k
(
t
)
(4)or, in terms of wavelets, Equation 3 becomes
f
(
t
)
≈
∑
k
=
-
∞
∞
〈
f
,
φ
J
0
,
k
〉
φ
J
0
,
k
(
t
)
+
∑
k
=
-
∞
∞
∑
j
=
J
0
J
1
-
1
〈
f
,
ψ
j
,
k
〉
ψ
j
,
k
(
t
)
.
f
(
t
)
≈
∑
k
=
-
∞
∞
〈
f
,
φ
J
0
,
k
〉
φ
J
0
,
k
(
t
)
+
∑
k
=
-
∞
∞
∑
j
=
J
0
J
1
-
1
〈
f
,
ψ
j
,
k
〉
ψ
j
,
k
(
t
)
.
(5)This assumes that approximately f∈VJ1f∈VJ1 or equivalently,
∥f-PJ1f∥≈0∥f-PJ1f∥≈0, where PJ1PJ1 denotes the orthogonal
projection of ff onto VJ1VJ1.
Given f(t)∈VJ1f(t)∈VJ1 so that the expansion in Equation 5
is exact, one computes the DWT coefficients in two steps.
- Projection onto finest scale: Compute 〈f,φJ1,k〉〈f,φJ1,k〉
- Analysis: Compute 〈f,ψj,k〉〈f,ψj,k〉,
j∈{J0,...,J1-1}j∈{J0,...,J1-1} and 〈f,φJ0,k〉〈f,φJ0,k〉.
For J1J1 large enough, φJ1,k(t)φJ1,k(t) can be approximated by a
Dirac impulse at its center of mass since ∫φ(t)dt=1∫φ(t)dt=1. For large jj this gives
2
j
∫
f
(
t
)
φ
(
2
j
t
)
d
t
≈
∫
f
(
t
)
δ
(
t
-
2
-
j
m
0
)
d
t
=
f
(
t
-
2
-
j
m
0
)
2
j
∫
f
(
t
)
φ
(
2
j
t
)
d
t
≈
∫
f
(
t
)
δ
(
t
-
2
-
j
m
0
)
d
t
=
f
(
t
-
2
-
j
m
0
)
(6)where m0=∫tφ(t)dtm0=∫tφ(t)dt is the first moment of
φ(t)φ(t). Therefore the scaling function coefficients at the j=J1j=J1
scale are
c
J
1
(
k
)
=
〈
f
,
φ
J
1
,
k
〉
=
2
J
1
/
2
∫
f
(
t
)
φ
(
2
J
1
t
-
k
)
d
t
=
2
J
1
/
2
∫
f
(
t
+
2
-
J
1
k
)
φ
(
2
J
1
t
)
d
t
c
J
1
(
k
)
=
〈
f
,
φ
J
1
,
k
〉
=
2
J
1
/
2
∫
f
(
t
)
φ
(
2
J
1
t
-
k
)
d
t
=
2
J
1
/
2
∫
f
(
t
+
2
-
J
1
k
)
φ
(
2
J
1
t
)
d
t
(7)which are approximately
c
J
1
(
k
)
=
〈
f
,
φ
J
1
,
k
〉
≈
2
-
J
1
/
2
f
(
2
-
J
1
(
m
0
+
k
)
)
.
c
J
1
(
k
)
=
〈
f
,
φ
J
1
,
k
〉
≈
2
-
J
1
/
2
f
(
2
-
J
1
(
m
0
+
k
)
)
.
(8)For all 2-regular wavelets (i.e., wavelets with two vanishing moments,
regular wavelets other than the Haar wavelets—even in the MM-band case
where one replaces 2 by MM in the above equations, m0=0m0=0), one can
show that the samples of the functions themselves form a third-order
approximation to the scaling function coefficients of the signal
[2]. That is, if f(t)f(t) is a quadratic polynomial, then
c
J
1
(
k
)
=
〈
f
,
φ
J
1
,
k
〉
=
2
-
J
1
/
2
f
(
2
-
J
1
(
m
0
+
k
)
)
≈
2
-
J
1
/
2
f
(
2
-
J
1
k
)
.
c
J
1
(
k
)
=
〈
f
,
φ
J
1
,
k
〉
=
2
-
J
1
/
2
f
(
2
-
J
1
(
m
0
+
k
)
)
≈
2
-
J
1
/
2
f
(
2
-
J
1
k
)
.
(9)Thus, in practice, the finest scale J1J1 is determined by the sampling
rate. By rescaling the function and amplifying it appropriately, one
can assume the samples of f(t)f(t) are equal to the scaling function
coefficients. These approximations are made better by setting some of
the scaling function moments to zero as in the coiflets. These are
discussed in Section: Approximation of Scaling Coefficients by Samples of the Signal .
Finally there is one other aspect to consider. If the signal has finite
support and LL samples are given, then we have LL nonzero
coefficients 〈f,φJ1,k〉〈f,φJ1,k〉. However, the DWT will
typically have more than LL coefficients since the wavelet and scaling
functions are obtained by convolution and downsampling. In other words,
the DWT of a LL-point signal will have more than LL points. Considered
as a finite discrete transform of one vector into another, this situation
is undesirable. The reason this “expansion" in dimension occurs is that
one is using a basis for L2L2 to represent a signal that is of finite
duration, say in L2[0,P]L2[0,P].
When calculating the DWT of a long signal, J0J0 is usually chosen to give
the wavelet description of the slowly changing or longer duration features
of the signal. When the signal has finite support or is periodic, J0J0
is generally chosen so there is a single scaling coefficient for the
entire signal or for one period of the signal. To reconcile the
difference in length of the samples of a finite support signal and the
number of DWT coefficients, zeros can be appended to the samples of f(t)f(t)
or the signal can be made periodic as is done for the DFT.
