Wavelet Transforms - an overview

The wavelet transform (WT) is another mapping from L2(R) → L2(R2), but one with superior time-frequency localization as compared with the STFT. 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 discrete wavelet transform (DWT) is then generated by sampling the wavelet parameters (α, b) on a grid or lattice. The question of reconstruction of the signal from its transform values naturally depends on the coarseness of the sampling grid. A fine grid mesh would permit easy reconstruction, but with evident redundancy, i.e., oversampling. A too-coarse grid could result in loss of information. The concept of frames is introduced to address these issues.

The wavelet transform (WT) is another mapping from L2(R) → L2(R2), but one with superior time-frequency localization as compared with the STFT. 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 discrete wavelet transform (DWT) is then generated by sampling the wavelet parameters (α, b) on a grid or lattice. The question of reconstruction of the signal from its transform values naturally depends on the coarseness of the sampling grid. A fine grid mesh would permit easy recons