Wavelet Transform
Wavelet Decomposition of Discrete Signals
Wavelet Packet Transform
This section gives an overview of the theory of wavelets. For a more
detailed description see for example [6], and [7].
A wavelet 
is a (complex) function with a zero integral
over the real line, i.e.
We define
The functions 
are called wavelets and
is sometimes called mother wavelet.
The continuous wavelet transform according to the wavelet
of some function 
is then defined through
In contrast to the windowed Fourier transform which maps a function
onto to a time-frequency space, the continuous wavelet transform maps
a function onto a time-scale space.
The discrete wavelet transform of f is defined as
with 
and 
. Since
the discrete wavelet transform is a sampled continous wavelet
transform. The most common choice is 
and 
because it leads to fast and simple
algorithms for the computation of the discrete wavelet transform. For
and some function f we define
We call the wavelet orthogonal if the functions
build an orthonormal basis of 
.
For 
we define a closed subspace 
of through
Let 
be the orthogonal projection onto this
subspace. One can then write
Any reasonable orthogonal wavelet is associated with a
multiresolution analysis which consists
of a ladder of closed subsets 
of
and a function 
such that
We call 
the scaling function of the multiresolution
analysis. For each is the
orthogonal sum of 
and 
, i.e.
Let 
be the orthogonal projection onto
. Then we have for each
One can find a highpass filter 
and a lowpass filter 
such that
For a given discrete signal 
with finite energy we
can find a function such that
The wavelet decomposition of the discrete signal x over J octaves
consists of the sequence of signals 
and of the
signal 
where 
is defined through
and 
is defined through
We can interpret as a coarser
approximation at the resolution J of x and as the
detail information that gets lost if we go from the
approximation 
to the coarser approximation 
.
Using the filters 
and 
defined above,
and 
can be computed
recursively:
This means that resp. are
obtained from by applying the lowpass filter G
resp. the highpass filter H to and then subsampling the result by a
factor of 2. Note that if the signal x has finite length then
and are half
as long as .
The signal x can be reconstructed from its wavelet decomposition by
the following recursion
In the computation of the wavelet decomposition of a discrete signal
x we only use the signals as inputs for the filter scheme defined
above. The signals are not processed any
further. However, one can show that there are signals that are not well
represented by the sequence and
. Therefore we extend the sequence of signals
and to a tree 
of sequences. The tree is computed by the following recursion:
We call the whole tree b the wavelet packet transform of the signal
x over J octaves.