# Is it possible to compute the wavelet transform of a unit step function?

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.

• More details needed? – Laurent Duval Jun 30 '20 at 21:54

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.