Continuous wavelet transform

Idea: be sensitive to irregularities instead of oscillations

• Let  $\psi(t)$ be an admissible “mother” wavelet, and its the dilated and translated versions

$$ \psi_{a,b}(t) = \frac{1}{\sqrt{a}}\psi\left( \frac{t-b}{a} \right) $$

• The continuous wavelet transform is given by:

$$ C_x(a,b) = \langle x(t),\psi_{a,b}(t)\rangle = \frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty} x(t) \psi\left(\frac{t-b}{a}\right)\mathrm{d} t $$

• $|C_x(a,b)|$ is called the magnitude scalogram

Properties

• It is a time-scale transform
• It is invertible

$$ x(t)= \frac{1}{c_\psi}\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} X(a,b) \psi\left(\frac{t-b}{a}\right)\frac{\mathrm{d} a\ \mathrm{d} b}{a^2} $$

• We have energy preservation