Time analysis and frequency analysis are both well-established ways in
engineering to gain more knowledge about a physical phenomena. Time and
frequncy analysis can be combinen in a joint time and frequency distribution. A
simple method to gain a joint distribution is to window segments of the data at
different time locations and calculate its Fourier transform. By doing this a
set of 'local' spectra are gained and joined to present a time-frequency
distribution. This method is well known as the Short-Time Fourier Transform.
The Short-Time Fourier Transform has the disatvantage that i does not localize
time and frequency phenomena very well. Instead the time-frequency information
is scattered which depends on the length of the window. This can be attended to
by altering the length of the window bu a certain balance between good time and
good frequency localization is unavoidable.
To cope with this disadvantage, the Wavelet Transform uses dilated and
translated functions, which are local in time, and frequency, which results in
good frequency resolutin for low-frequency phenomena and good time resolution
for high-frequency phenomena. The advantage of the Wavelet Transform is its
efficient fast transform in discrete time. But still, there is no complete
solution to the localization problem.
Adaptive Time-Frequency Analysis can be advantageous for solving the
localization problem. The functionality of methods is hereby adapted to the
time-frequency content of the signa.
The Adaptive Wavelet Packets Transform is based upon the Wavelet Transform but
is a more general way to gain a time-frequency distribution. It is even
possible to gain a time-frequency distribution similar to the Short-Time
Fourier Transform. The energy levels in the frequency bands determine the
frequency resolution. Much energy located in a small frequency band will result
in good frequency resolution for the specific band. Other frequency areas will
be analyzed with as good time resolution as possible. Sine wave with constant
frequency precedes time phenomena. The method is implemented using a fast
Quadrature Mirror Filter bank tree which form determines resolution of the
analysis.
In the Adaptive Window Short-Time Fourier Transform, the time phenomena precede
sine waves in the analysis. Good time resolu...
