The Wavelet Transform

We mentioned in the introduction to Part D the shortcomings of the windowed Fourier transform. This chapter gives another approach to the time-frequency issue of Fourier analysis. The role played in the windowed Fourier transform by the family of functions $${\omega _{v,b}}(t) = \omega (t - b){e^{ + 2i\pi vt}},\quad b,v \in \mathbb{R}$$ is played in the wavelet transform (WT) by a family (16)$${\psi _{a,b}}(t) = {\left| a \right|^{ - 1/2}}\psi (\frac{{t - b}}{a}),\quad a,b \in \mathbb{R},\;a \ne 0$$ where Ψ(t) is called the mother wavelet. The function Ψ _{a, b} is obtained from the mother wavelet Ψ by successively applying a change of time scale (accompanied by a change of amplitude scale in order to keep the energy constant) and a time shift (see Fig. D2.1).