The Haar basis is perhaps the simplest example of a DWT basis,
and we will frequently refer to it in our DWT development.
Keep in mind, however, that the Haar basis is only
an example; there are many other ways of
constructing a DWT decomposition.
For the Haar case, the mother
scaling function is
defined by
Equation 1 and
Figure 1.
φt=1if0≤t<10otherwise
φ
t
1
0
t
1
0
(1)
From the mother scaling function, we define a family of
shifted and stretched scaling functions
φ
k
,
n
t
φ
k
,
n
t
according to
Equation 2 and
Figure 2
φ
k
,
n
t=∀k,n,k∈ℤn∈ℤ:2-k2φ2-kt-n=2-k2φ12kt-n2k
φ
k
,
n
t
k
n
k
n
2
k
2
φ
2
k
t
n
2
k
2
φ
1
2
k
t
n
2
k
(2)
which are illustrated in
Figure 3 for
various
kk and
nn.
Equation 2 makes clear the principle that incrementing
nn by one shifts the pulse one
place to the right. Observe from
Figure 3 that
{
φ
k
,
n
t|n∈ℤ}
φ
k
,
n
t
n
is orthonormal for each
kk (
i.e.,
along each row of figures).