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.