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Unformatted text preview: 2 φ (t ) 1 0 1 2 2 1 0 1 2 t, second 2 w(t) 1 0 1 2 2 1 Im[W (f)] 0.5 0 0.5 1 10 8 6 4 2 0 2 4 6 8 f , Hz 10 1 0 1 2 t, second Figure 1. Haar scaling function and Haar wavelet with its Fourier transform MAE 591 RANDOM DATA C. W. Lee Fig.2 Demonstration of Haar wavelet analysis1 1 MATLAB > wavemenu MAE 591 RANDOM DATA C. W. Lee Table 1. Properties of wavelets Family Order Orthogonality Compact Support DWT/CWT Support Width Filters Length Regularity Symmetry/ Antisymmetry # of vanishing moments for w(t) Remarks Haar 2 = D2= db15 Yes Yes O/O 1 2 Discontinuous Yes 1 Daubechies N (dbN = D2N) Yes Yes O/O 2N1 2N ~0.2N Far from N  Crude wavelets (e.g. Mexican hat)  Scaling function does not exist.  Explicit expression for w(t) N even/ N odd Yes Gaussian3 N No No X/O Infinite Effective [5 5] Morlet4 No No X/O Infinite Effective [4 4]  Orthogonal and compactly supported wavelets.  Poor regularity 2 3 The oldest and simplest, compactly supported wavelet. It is defined as the derivatives of the Gaussian probability density function 4 x2 morl ( x ) = exp − cos 5 x 2 The notation ‘dbN’ is adopted in MATLAB, which is equivalent to D2N. 5 MAE 591 RANDOM DATA C. W. Lee Figure 3. Scaling and wavelet functions: (a) D4 and (b)D20. MAE 591 RANDOM DATA C. W. Lee Figure 4. Demonstration of D4 wavelet analysis MAE 591 RANDOM DATA C. W. Lee Figure 5 Scaling and wavelet functions of symlets (symN) Figure 6. Wavelet functions of (a) Gaussian and (b) Morlet wavelets. ...
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This note was uploaded on 11/21/2009 for the course ME . taught by Professor . during the Spring '09 term at Korea University.
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