Problems with STFT Uncertainity Principle: Improved space resolution Degraded frequency resolution Improved frequency resolution Degraded space resolution Problem: the same and t throught the entire plane! STFT is redundant representation Not good for compression
Solution: Frequency Scaling Smaller frequency make the window more narrow Bigger frequency make the window wider More narrow time window for higher frequencies here s is scaling factor
New partition of the space-frequency plane Coordinate, t Frequency,
New partition of the plane Discrete wavelet transform Short-time Fourier transform Wavelet functions are localized in space and frequency Hierarchical set of of functions
Five steps to calculate WT 1.Take a wavelet and compare it to a section at the start of the original signal. 2.Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal. 3.Shift the wavelet to the right and repeat steps 1 and 2 until you’ve covered the whole signal. 4.Scale (stretch) the wavelet and repeat steps 1 through 3. 5.Repeat steps 1 through 4 for all scales.
Scaling function and Wavelets Wavelet function: Scaling function : The functions (t) and (t) are orthonormal The most important property of the wavelets: To obtain WT coefficients for level j we can process WT coefficients for level j+1. The most important property of the wavelets: To obtain WT coefficients for level j we can process WT coefficients for level j+1. where