It shows the band-pass nature of ψ( t) and the time-frequency resolution of the wavelet transform.
Figure 5.3 displays a typical wavelet and its dilations.
The admissibility condition ensures that the continuous wavelet transform is complete if W f( a, b) is known for all a, b. Gaussian based (second derivative) wavelet function and its Fourier transform. In time and Fourier transform domains, the wavelet isįigure 6.1. (6.1) in terms of dilations and translations of a prototype or mother function ϕ( t). The continuous wavelet transform (CWT) is defined by Eq. The concept of frames is introduced to address these issues. A too-coarse grid could result in loss of information. A fine grid mesh would permit easy reconstruction, but with evident redundancy, i.e., oversampling. The question of reconstruction of the signal from its transform values naturally depends on the coarseness of the sampling grid.
The discrete wavelet transform (DWT) is then generated by sampling the wavelet parameters ( α, b) on a grid or lattice. In this section, we define the continuous wavelet transform and develop an admissibility condition on the wavelet needed to ensure the invertibility of the transform. The wavelet transform (WT) is another mapping from L 2( R) → L 2( R 2), but one with superior time-frequency localization as compared with the STFT. Haddad, in Multiresolution Signal Decomposition (Second Edition), 2001 6.1 The Wavelet Transform
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