09-08-2012, 12:10 PM
Introduction to Wavelet
Introduction to Wavelet (1).ppt (Size: 318 KB / Downloads: 189)
OVERVIEW
Wavelet
A small wave
Wavelet Transforms
Convert a signal into a series of wavelets
Provide a way for analyzing waveforms, bounded in both frequency and duration
Allow signals to be stored more efficiently than by Fourier transform
Be able to better approximate real-world signals
Well-suited for approximating data with sharp discontinuities
“The Forest & the Trees”
Notice gross features with a large "window“
Notice small features with a small
Historical Development
Pre-1930
Joseph Fourier (1807) with his theories of frequency analysis
The 1930s
Using scale-varying basis functions; computing the energy of a function
1960-1980
Guido Weiss and Ronald R. Coifman; Grossman and Morlet
Post-1980
Stephane Mallat; Y. Meyer; Ingrid Daubechies; wavelet applications today
Mathematical Transformation
Why
To obtain a further information from the signal that is not readily available in the raw signal.
Raw Signal
Normally the time-domain signal
Processed Signal
A signal that has been "transformed" by any of the available mathematical transformations
Fourier Transformation
The most popular transformation
FREQUENCY ANALYSIS
Frequency Spectrum
Be basically the frequency components (spectral components) of that signal
Show what frequencies exists in the signal
Fourier Transform (FT)
One way to find the frequency content
Tells how much of each frequency exists in a signal
STATIONARITY OF SIGNAL
Stationary Signal
Signals with frequency content unchanged in time
All frequency components exist at all times
Non-stationary Signal
Frequency changes in time
One example: the “Chirp Signal”
NOTHING MORE, NOTHING LESS
FT Only Gives what Frequency Components Exist in the Signal
The Time and Frequency Information can not be Seen at the Same Time
Time-frequency Representation of the Signal is Needed
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