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 , and .
A wavelet is a (complex) function with a zero integral
over the real line, i.e.
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.