Jan Mark de Haan MEE-98-01, pp. 106. Dept. of Signal Processing, 1998.
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 resolution is gained where the time-frequency concentration is highest for short windows. Other time intervals will be analyzed with a longer window, to gain better frequency resolution. The method is implemented using a set of Fast Fourier Transform calculations.