The discrete wavelet transform (DWT) is a
representation of a signal
xt∈
ℒ
2
x
t
ℒ
2
using an orthonormal basis consisting of a
countably-infinite set of wavelets. Denoting the
wavelet basis as
{
ψ
k
,
n
t|k∈ℤ∧n∈ℤ}
ψ
k
,
n
t
k
n
, the DWT transform pair is
xt=∑k=-∞∞∑n=-∞∞
d
k
,
n
ψ
k
,
n
t
x
t
k
n
d
k
,
n
ψ
k
,
n
t
(1)
d
k
,
n
=<
ψ
k
,
n
t,xt>=∫-∞∞
ψ
k
,
n
t¯xtdt
d
k
,
n
ψ
k
,
n
t
x
t
t
ψ
k
,
n
t
x
t
(2)
where
d
k
,
n
d
k
,
n
are the wavelet coefficients. Note the relationship
to Fourier series and to the sampling theorem: in both cases
we can perfectly describe a continuous-time signal
xt
x
t
using a countably-infinite (
i.e.,
discrete) set of coefficients. Specifically, Fourier series
enabled us to describe
periodic signals using
Fourier coefficients
{Xk|k∈ℤ}
X
k
k
, while the sampling theorem enabled us to describe
bandlimited signals using signal samples
{xn|n∈ℤ}
x
n
n
. In both cases, signals within a limited class are
represented using a coefficient set with a single countable
index. The DWT can describe
any signal
in
ℒ
2
ℒ
2
using a coefficient set parameterized by two countable
indices:
{
d
k
,
n
|k∈ℤ∧n∈ℤ}
d
k
,
n
k
n
.
Wavelets are orthonormal functions in
ℒ
2
ℒ
2
obtained by shifting and stretching a mother
wavelet,
ψt∈
ℒ
2
ψ
t
ℒ
2
. For example,
∀k,n,k∧n∈ℤ:
ψ
k
,
n
t=2-k2ψ2-kt-n
k
n
k
n
ψ
k
,
n
t
2
k
2
ψ
2
k
t
n
(3)
defines a family of wavelets
{
ψ
k
,
n
t|k∈ℤ∧n∈ℤ}
ψ
k
,
n
t
k
n
related by power-of-two stretches. As
kk increases, the wavelet
stretches by a factor of two; as
nn increases, the wavelet shifts
right.
When
∥ψt∥=1
ψ
t
1
, the normalization ensures that
∥
ψ
k
,
n
t∥=1
ψ
k
,
n
t
1
for all
k∈ℤ
k
,
n∈ℤ
n
.
Power-of-two stretching is a convenient, though somewhat
arbitrary, choice. In our treatment of the discrete wavelet
transform, however, we will focus on this choice. Even with
power-of two stretches, there are various possibilities for
ψt
ψ
t
, each giving a different flavor of DWT.
Wavelets are constructed so that
{
ψ
k
,
n
t|n∈ℤ}
ψ
k
,
n
t
n
(i.e., the set of all shifted
wavelets at fixed scale kk),
describes a particular level of 'detail' in the signal. As
kk becomes smaller
(i.e., closer to
-∞
), the wavelets become more "fine grained" and the
level of detail increases. In this way, the DWT can give a
multi-resolution description of a signal, very
useful in analyzing "real-world" signals. Essentially, the
DWT gives us a discrete multi-resolution description
of a continuous-time signal in
ℒ
2
ℒ
2
.
In the modules that follow, these DWT concepts will be
developed "from scratch" using Hilbert space principles. To
aid the development, we make use of the so-called
scaling function
φt∈
ℒ
2
φ
t
ℒ
2
, which will be used to approximate the signal
up to a particular level of detail. Like
with wavelets, a family of scaling functions can be
constructed via shifts and power-of-two stretches
∀k,n,k∧n∈ℤ:
φ
k
,
n
t=2-k2φ2-kt-n
k
n
k
n
φ
k
,
n
t
2
k
2
φ
2
k
t
n
(4)
given mother scaling function
φt
φ
t
. The relationships between wavelets and scaling
functions will be elaborated upon later via
theory and
example.
The inner-product expression for
d
k
,
n
d
k
,
n
,
Equation 2 is written for the general complex-valued
case. In our treatment of the discrete wavelet transform,
however, we will assume real-valued signals and wavelets.
For this reason, we omit the complex conjugations in the
remainder of our DWT discussions
