(Blank Abstract)
Main concepts
The
discrete wavelet transform (DWT) is a
representation of a signal
$x(t)\in {\mathcal{L}}_{2}$ using an orthonormal basis consisting of a
countably-infinite set of
wavelets . Denoting the
wavelet basis as
$\{{\psi}_{k,n}(t)\colon k\in \mathbb{Z}\land n\in \mathbb{Z}\}$ , the DWT transform pair is
$x(t)=\sum_{k=()} $∞
∞
n
∞
∞
d
k
,
n
ψ
k
,
n
t
${d}_{k,n}={\psi}_{k,n}(t)\cdot x(t)=\int_{()} \,d t$∞
∞
ψ
k
,
n
t
x
t where
$\{{d}_{k,n}\}$ are the wavelet coefficients. Note the relationship
to Fourier series and to the sampling theorem: in both caseswe can perfectly describe a continuous-time signal
$x(t)$ using a countably-infinite (
i.e. ,
discrete) set of coefficients. Specifically, Fourier seriesenabled us to describe
periodic signals using
Fourier coefficients
$\{X(k)\colon k\in \mathbb{Z}\}$ , while the sampling theorem enabled us to describe
bandlimited signals using signal samples
$\{x(n)\colon n\in \mathbb{Z}\}$ . In both cases, signals within a limited class are
represented using a coefficient set with a single countableindex. The DWT can describe
any signal
in
${\mathcal{L}}_{2}$ using a coefficient set parameterized by two countable
indices:
$\{{d}_{k,n}\colon k\in \mathbb{Z}\land n\in \mathbb{Z}\}$ .
Wavelets are orthonormal functions in
${\mathcal{L}}_{2}$ obtained by shifting and stretching a
mother
wavelet ,
$\psi (t)\in {\mathcal{L}}_{2}$ . For example,
$\forall k, n, (k\land n)\in \mathbb{Z}\colon {\psi}_{k,n}(t)=2^{-\left(\frac{k}{2}\right)}\psi (2^{-k}t-n)$
defines a family of wavelets
$\{{\psi}_{k,n}(t)\colon k\in \mathbb{Z}\land n\in \mathbb{Z}\}$ related by power-of-two stretches. As
$k$ increases, the wavelet
stretches by a factor of two; as
$n$ increases, the wavelet shifts
right.
When
$(\psi (t))=1$ , the normalization ensures that
$({\psi}_{k,n}(t))=1$ for all
$k\in \mathbb{Z}$ ,
$n\in \mathbb{Z}$ .
Power-of-two stretching is a convenient, though somewhat
arbitrary, choice. In our treatment of the discrete wavelettransform, however, we will focus on this choice. Even with
power-of two stretches, there are various possibilities for
$\psi (t)$ , each giving a different flavor of DWT.
Wavelets are constructed so that
$\{{\psi}_{k,n}(t)\colon n\in \mathbb{Z}\}$ (
i.e. , the set of all shifted
wavelets at fixed scale
$k$ ),
describes a particular level of 'detail' in the signal. As
$k$ 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, theDWT gives us a
discrete multi-resolution description
of a continuous-time signal in
${\mathcal{L}}_{2}$ .
In the modules that follow, these DWT concepts will be
developed "from scratch" using Hilbert space principles. Toaid the development, we make use of the so-called
scaling function
$\phi (t)\in {\mathcal{L}}_{2}$ , which will be used to approximate the signal
up to a particular level of detail . Like
with wavelets, a family of scaling functions can beconstructed via shifts and power-of-two stretches
$\forall k, n, (k\land n)\in \mathbb{Z}\colon {\phi}_{k,n}(t)=2^{-\left(\frac{k}{2}\right)}\phi (2^{-k}t-n)$
given mother scaling function
$\phi (t)$ . The relationships between wavelets and scaling
functions will be elaborated upon later via
theory and
example .
The inner-product expression for
${d}_{k,n}$ ,
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 theremainder of our DWT discussions