Properties The wavelet transform is covariant by translation in time and scaling, which means that $$y(t) = \sqrt{\vert a_0\vert}\ x(a_0(t-t_0)) \ \Rightarrow\ T_y(t,a;\Psi) = T_x(a_0^*(t-t_0),a/a_0;\Psi).$$ The corresponding group of transforms is called the affine group (to be compared to the Weyl-Heisenberg group). The signal $x$ can be recovered from its continuous wavelet transform according to the formula $$x(t) = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} T_x(s,a;\Phi)\ \Psi_{s,a}(t)\ ds\ \frac{da}{a^2}$$ where $\Phi$ is the synthesis wavelet, if the following admissibility condition is verified by $\Phi$ and $\Psi$: $$\int_{-\infty}^{+\infty} \Psi(\nu)\ \Phi^*(\nu)\ \frac{d\nu}{\vert\nu\vert}\ =\ 1.$$ Time and frequency resolutions, like in the STFT case, are related via the Heisenberg-Gabor inequality. However, in the present case, these two resolutions depend on the frequency: the frequency resolution (resp. time resolution) becomes poorer (resp. better) as the analysis frequency grows.
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