18-06-2013, 04:01 PM
Short Time Fourier Transform
Short Time Fourier.pptx (Size: 152.74 KB / Downloads: 15)
Need a local analysis scheme for a time-frequency representation (TFR).
Windowed F.T. or Short Time F.T. (STFT)
Segmenting the signal into narrow time intervals (i.e., narrow enough to be considered stationary).
Take the Fourier transform of each segment.
For each time location where the window is centered, we obtain a different FT
Each FT provides the spectral information of a separate time-slice of the signal, providing simultaneous time and frequency information
Steps for STFT
Choose a window function of finite length
Place the window on top of the signal at t=0
Truncate the signal using this window
Compute the FT of the truncated signal, save results.
Incrementally slide the window to the right
Go to step 3, until window reaches the end of the signal
PRINCIPLES OF WAELET TRANSFORM
Split Up the Signal into a Bunch of Signals
Representing the Same Signal, but all Corresponding to Different Frequency Bands
Only Providing What Frequency Bands Exists at What Time Intervals
COMPUTATION OF CWT
The wavelet is placed at the beginning of the signal, and set s=1 (the most compressed wavelet);
The wavelet function at scale “1” is multiplied by the signal, and integrated over all times; then multiplied by ;
Shift the wavelet to t=, and get the transform value at t= and s=1;
Repeat the procedure until the wavelet reaches the end of the signal;
Scale s is increased by a sufficiently small value, the above procedure is repeated for all s;
Each computation for a given s fills the single row of the time-scale plane;
CWT is obtained if all s are calculated.
Motivation for Wavelet Analysis
Signals of practical interest are usually non-stationary, meaning that their time-domain and frequency-domain characteristics vary over short time intervals (i.e., music, seismic data, etc)
Classical Fourier analysis (Fourier transforms) assumes a signal that is either infinite in extent or stationary within the analysis window.
Non-stationary analysis requires a different approach: Wavelet Analysis
Wavelet analysis also produces better solutions to important problems such as the transform compression of images (jpeg versus jpeg2000)