The discrete wavelet transform In the case of the wavelet transform, the natural way to sample the time-frequency plane is to take samples on the non-uniform grid (lattice) defined by $${(t,a) = (nt_0\ a_0^{-m},a_0^{-m})\ ;\ t_0>0,\ a_0>0\ ;\ m,n \in {\cal Z}}.$$ Then, the discrete wavelet transform (DWT) is defined as $$T_x[n,m;\Psi]=a_0^{m/2} \int_{-\infty}^{+\infty} x(u)\ \Psi^*_{n,m}(u)\ du\ ;\ m,n \in {\cal Z}$$ where $\Psi_{n,m}(u)=\Psi(a_0^m u-nt_0)$. The common choice $(a_0=2,t_0=1)$ corresponds to a dyadic sampling of the time-frequency plane (one set of coefficients per octave) (see fig. 3.14). Figure 3.14: Sampling of the time-frequency plane. Different forms of sampling: Shannon, Fourier, Gabor, Wavelet Thanks to such a sampling, it is now possible to obtain for the family $\{\Psi_{n,m}(u)\ ;\ m,n \in {\cal Z}\}$ an orthonormal basis with a wavelet $\Psi$ well localized in time and in frequency (the Balian-Low obstruction is no longer valid). This is strongly related to the multiresolution analysis theory (we will not develop it here ; see for more details the tutorial of the Wavelet Toolbox). The main drawback of such a sampling is the loss of time-covariance. Indeed, a signal analyzed by the DWT will not have the same pattern on the dyadic grid whatever its initial position is. As for the Gabor representation, a solution halfway between the over-complete family of wavelets ${\Psi_{n,m}(u)}$ used by the CWT and an orthonormal basis of wavelets obtained on the dyadic grid and for a particular choice of $\Psi$ is given by the theory of frames (see [Dau92] for an overview of this theory with application to the wavelet transform).
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