We consider the frequency line 
split as 
, with
and 
. On 
we introduce a
window function 
such that 
and 
is supported in 
. We can clearly view 
as a window function over the interval 
and observe that
form an orthonormal basis of 
. Similarly
gives another basis, whose elements are not orthogonal to those of the first
one. If we define 
as an odd extension to 
of 
and
as an even extension we find 
permitting us
to write 
where
is the Fourier transform of the
wavelet 
(cf. [4]).
Thus, wavelet analysis corresponds to windowing frequency space in ``octave''
windows 
.
A natural extension is provided by allowing all dyadic windows in frequency space and adapted window choice. This sort of analysis is equivalent to wavelet packet analysis.
The actual fast wavelet packet analysis algorithms (wavelets being a special cases) permit us to perform an adapted Fourier windowing directly in time domain by successive filtering of a function into different regions in frequency. The dual version of the window selection provides an adapted subband coding algorithm.
The wavelet packet library is constructed by iterating the wavelet algorithm. This library contains the wavelet basis, Walsh functions, and smooth versions of Walsh functions called wavelet packets (cf. [5])
These waveforms are mutually orthogonal. Moreover, each of them is orthogonal to all of its integer translates and dyadic rescaled versions. The full collection of these wavelet packets (including translates and rescaled versions) provides us with a library of ``templates'' or ``notes'' which are matched ``efficiently'' to signals for analysis and synthesis (cf. [2]), Wavelet packet expansions correspond algorithmically to subband coding schemes and are numerically as fast as the FFT.
