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Is it possible to find theoretically the continuous or discrete wavelet transform of a unit step function (consntly 0 for t<0 and constantly 1 for t>=0) or of a function starting from zero and approaching 1 in a finite time? I just wonder if this is possible because these signals don't have a finite energy. Thanks. E.

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  • $\begingroup$ More details needed? $\endgroup$ – Laurent Duval Jun 30 '20 at 21:54
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Finite energy functions is a class of choice, but wavelet transforms can exists in more generic cases.

For the continuous case: let us call $H(t)$ the Heaviside step function. Then $|H(t)\psi(t)|$ is dominated by $|\psi(t)|$. If $\psi(t)$ is $L_1$ or integrable (in addition to being $L_2$), then $\int_{-\infty}^{\infty}H(t)\psi(\frac{t-a}{b})\mathrm{d}t$ is defined. Moreover, since admissible wavelets have zero-average, the continuous wavelet coefficients can decay at infinity. With wavelet of finite support, they will be zero for all $\psi(\frac{t-a}{b})$ with support in $[0,\;+\infty[$.

For discrete wavelets, this would be about the same. As long as the filters are summable (or $\ell_1$), and this happens especially for FIR wavelet filters, you can compute coefficients. Special note: if a discrete filter is $\ell_1$, then it is $\ell_2$ as well, so the condition is milder than in the continuous case.

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