A wavelet tour of signal processing the Sparse way

Stephane Mallat

history

• 1910 - Haar - a piecewise constant function
• 1980 - Stromberg - a piecewise linear function - an orthonormal basis, better approximation of smooth function
• Morlet, Grossmann - continuous wavelet transform
• Meyer - a family of orthonormal wavelet bases with function infinitely continuously differentiable
• Daubechies - wavelets of compact support

Time-Frequency localisation trade-off

Keeping $\Vert f_s{\Vert }^2=\Vert f{\Vert }^2$, by denoting

$$f_s(t)=\frac{1}{\sqrt{s}}f\left(\frac{t}{s}\right)$$

then

$${\hat{f}}_s(\omega )=\sqrt{s}\hat{f}(s\omega )$$

it shows that we lose in frequency while we gain in time.

$\mathcal{F}\left(\frac{1}{\sqrt{s}}{e}^{-\frac{t^2}{s}}\right)=\frac{1}{\sqrt{2}}{e}^{-\frac{s{\omega }^2}{4}}$