Differences

This shows you the differences between two versions of the page.

--- math105-s22:s:matthewk:start [2022/02/18 04:33]
matthewk
+++ math105-s22:s:matthewk:start [2026/02/21 14:41] (current)
@@ Line 26: / Line 26: @@
+====== Wavelet Transform and the Uncertainty Principle ======
+In Fourier theory, we have been focused on the orthonormal basis $\{e^{2\pi inx}\}$ to do most of the work for us. A natural question to ask, is what if we select a different basis in our analysis? The question turns out to open up the theory of the Wavelet Transform, and leads to insights on the famous uncertainty principle.
+Define $\psi : \mathbb{R} \rightarrow \mathbb{C}$. Define the family of functions $\{\psi_{jk} : j,k \in \mathbb{Z}\}$ as $\psi_{jk} = 2^{j/2} \psi(2^j x - k)$. We call $\psi$ an orthonormal wavelet if the family $\{\psi_{jk}\}$ is a complete orthnormal basis for $L^2(\mathbb{R})$.
+Thus, $f$ can be written: $$f = \sum_{j,k = -\infty}^{+\infty} c_{jk}\psi_{jk}$$
+We also define the integral wavelet transform:$$W_{\psi}[f(x)](a,b) = \int_{-\infty}^{+\infty} f(x) \bar{\psi}(\frac{x-b}{a});$$
+where $a = 2^{-j}, b = k2^{-j}$ and $c_{jk} = W_{\psi}[f](a,b)$.
+A simple example of an orthonormal wavelet is the Haar wavelet $\psi(x) = \psi_{0,0}(x) = \chi_{[0,1/2]} - \chi_{[1/2, 1]}$ where $\chi$ is the indicator function.
+We can see that wavelets allow us to localise to particular regions of our function $f$ by translating and dilating $\psi$ with $a$ and $b$. To understand how this localisation relates to the uncertainty principle, consider the orthonormal wavelet $\phi(x) = \chi_{[p,q]}(x)e^{2\pi inx}$ , where $p,q \in \mathbb{R}$ and $p \leq q$. We construct the family of wavelets by:
+$$\chi_{[p,q]}^{k} = \chi{[p+k(q-p), q+k(q-p)]}$$
+for $k \in \mathbb{Z}$. So we use the notation: $$\phi_{k,n} = \chi_{[p,q]}^k e^{2\pi inx}$$ Note that indeed: $\langle\phi_{k,n}, \phi_{l,m} \rangle = \delta_{kl}\delta_{nm}$.
+With this wavelet, we can analyse different sections of $f$ independently, and we can build up a picture of

Lecture Notes

User Tools

Site Tools

Differences

Page Tools