(Blank Abstract)
Main concepts
The
discrete wavelet transform (DWT) is a
representation of a signal
using an orthonormal basis consisting of a
countably-infinite set of
wavelets . Denoting the
wavelet basis as
, the DWT transform pair is
where
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
using a countably-infinite (
i.e. ,
discrete) set of coefficients. Specifically, Fourier seriesenabled us to describe
periodic signals using
Fourier coefficients
, while the sampling theorem enabled us to describe
bandlimited signals using signal samples
. 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
using a coefficient set parameterized by two countable
indices:
.
Wavelets are orthonormal functions in
obtained by shifting and stretching a
mother
wavelet ,
. For example,
defines a family of wavelets
related by power-of-two stretches. As
increases, the wavelet
stretches by a factor of two; as
increases, the wavelet shifts
right.
When
, the normalization ensures that
for all
,
.
Wavelets are constructed so that
(
i.e. , the set of all shifted
wavelets at fixed scale
),
describes a particular level of 'detail' in the signal. As
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
.
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
, 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
given mother scaling function
. The relationships between wavelets and scaling
functions will be elaborated upon later via
theory and
example .
The inner-product expression for
,
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
