author:Andres Garcia Saravia Ortiz de Montellano
date: 23.11.2015
incremental: true
Let $f:\mathbb{R}\rightarrow\mathbb{C}$
Maclaurin representation:
$$f(x) = a_0 + a_1 x + a_2 x^2 +\cdots $$
Fourier representation:
$$f(x) = a_0 + a_1 \cos{(w_1 x)} + \cdots + b_1 \sin{(w_1 x)} + \cdots$$
equivalently,
$$f(x) = a_0 + a_1 e^{i w_1 x} + a_2 e^{i w_2 x} + \cdots$$
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The Fourier representation looks different depending on the domain of the function $f:A\rightarrow\mathbb{C}$ , with $A$ being:
$\mathbb{R}$ : Real numbers
$\mathbb{T}_p$ : Circle of length $p$
$\mathbb{Z}$ : Integer numbers
$\mathbb{P}_N$ : Polygon of $N$ sides
For functions defined on the real line $\mathbb{R}$ .
$f:\mathbb{R}\rightarrow\mathbb{C}$
$$
\begin{aligned}
f(x) &= \int_{-\infty}^\infty F(s) e^{2\pi i s x} ds \\
F(s) &= \int_{-\infty}^\infty f(x) e^{-2\pi i s x} dx \\
\end{aligned}
$$
For functions defined on the integers on an interval $\mathbb{T}_p=[0,p)$ .
$g:\mathbb{T}_p\rightarrow\mathbb{C}$ with $g(p) = g(0)$
$$
\begin{aligned}
g(x) &= \sum_{k=-\infty}^\infty G(k) e^{2\pi ikx/p} \\
G(k) &= \frac{1}{p}\int_{0}^p g(x) e^{-2\pi ikx/p} dx \\
\end{aligned}
$$
For functions defined on the integers $\mathbb{Z}$ .
$\phi:\mathbb{Z}\rightarrow\mathbb{C}$
$$
\begin{aligned}
\phi(n) &= \int_{0}^p \Phi(s) e^{2\pi isn/p} ds \\
\Phi(s) &= \frac{1}{p}\sum_{n=-\infty}^\infty \phi(n) e^{-2\pi isn/p} \\
\end{aligned}
$$
For functions defined on a polygon with $N$ vertices $\mathbb{P}_N = {0,1,2,\ldots, N-1}$ .
$\gamma:\mathbb{P}_N\rightarrow\mathbb{C}$ with $\gamma(N) = \gamma(0)$
$$
\begin{aligned}
\gamma(n) &= \sum_{k=0}^{N-1} \Gamma(k) e^{2\pi ikn/N} \\
\Gamma(k) &= \frac{1}{N}\sum_{n=0}^{N-1} \gamma(n) e^{-2\pi ikn/N} \\
\end{aligned}
$$
$$
\begin{aligned}
\int_{-\infty}^\infty f(x)\overline{g(x)}dx
&= \int_{-\infty}^\infty F(s)\overline{G(x)}ds,\ \ \mathbb{R} \
\int_{0}^p f(x)\overline{g(x)}dx
&= p \sum_{k=-\infty}^\infty F(k)\overline{G(k)},\ \ \mathbb{T}p \
\sum {n=-\infty}^\infty f(n)\overline{g(n)}
&= p \int_{0}^p F(s)\overline{G(x)}ds,\ \ \mathbb{Z} \
\sum_{n=0}^{N-1} f(n)\overline{g(n)}
&= N \sum_{k=0}^{N-1} F(k)\overline{G(k)},\ \ \mathbb{P}_N
\end{aligned}
$$
A continuous function $f$ can be made discrete $\phi$ by $h$-sampling
$$\phi(n) = f(nh), n\in\mathbb{Z},\ h>0 $$
From a function $f$ , under some conditions, we can construct a periodic function $g$ by $p$-summation
$$g(x) = \sum_{m=-\infty}^\infty f(x-mp) $$
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$p$ -summation
$$g(x) = \sum_{m=-\infty}^\infty f(x-mp)
\implies G(k) = \frac{1}{p}F\left(\frac{k}{p}\right)$$
$p/N$ -sampling
$$\phi(n) = f\left(\frac{np}{N}\right)
\implies \Phi(s) = \sum_{m=-\infty}^\infty F\left(s - \frac{mN}{p}\right)$$
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Assume that $f(t)$ is $\sigma$-bandlimited
$$|\omega|>\sigma \implies F(\omega)=0$$
Sample it with an interval $\Delta t$
$$g(n) \equiv f(n\Delta t)$$
Then, $f(t)$ can be uniquely reconstructed only when
$$2\sigma \Delta t <1$$
$$\sigma < \frac{1}{2\Delta t} \equiv \omega_{Nyq}$$
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$$
\begin{aligned}
f(t) &= a_1\sin(2\pi \omega_1 t) \\
F(\omega)
&= \int_{-\infty}^{\infty}a_1\sin(2\pi \omega_1 t)e^{-2\pi i \omega t} dt\\
&= a_1\int_{-\infty}^{\infty}
\left(\frac{e^{2\pi i\omega_1 t} - e^{-2\pi i\omega_1 t}}{2i}\right)
e^{-2\pi i \omega t}dt\\
&= \frac{a_1}{2i}\left[\delta(\omega-\omega_1)
- \delta(\omega+\omega_1)\right]
\end{aligned}
$$
$$
P(\omega) \equiv \left|F(\omega)\right|^2= \left(\frac{a_1}{2}\right)^2\left[\delta(\omega-\omega_1)
- \delta(\omega+\omega_1)\right]^2$$
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Make Fourier Transform of short segments in the timeseries using a window function centered at $\tau$ , $w(t-\tau)$
$$\hat{F}(\omega, \tau) =
\int_{-\infty}^\infty f(t) w(t-\tau) e^{-2\pi i \omega t} dt$$
Problem: Choose an appropriate window width
Frequency vs. time resolution
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Idea:
Narrow windows for large frequencies (good time resolution)
Wide windows for small frequencies (good frequency resolution)
Choose a localized wave (mother wavelet) $\psi(t)$ and define
$$\psi_{a,b}(t)=\frac{1}{\sqrt{a}}\psi\left(\frac{t-b}{a}\right)$$
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Example: Mexican-hat wavelet
$$\psi(t) = \left(1-t^2\right)e^{-t^2/2}$$
$$\psi_{a,b}(t)=\frac{1}{\sqrt{a}}\psi\left(\frac{t-b}{a}\right)$$
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Given $f\in L^2(\mathbb{R})$ , we define its continuous wavelet transform with respect to the wavelet $\psi$ as
$$\mathcal{W}{\psi}[f] (a,b) = \int {-\infty}^\infty f(t)\overline{\psi_{a,b}(t)}dt$$
For $\mathcal{W}_\psi[f]$ to be invertible we require
$$ 0 < C_{\psi}\equiv\int_{-\infty}^\infty
\frac{\left|\hat{\psi}(\omega)\right|^2}{\left|\omega\right|} d\omega
<\infty $$
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Mexican hat CWT
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Haar CWT
Morlet CWT
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Morlet
Mexican-hat
$$
\int_{-\infty}^\infty\int_{-\infty}^\infty \mathcal{W}\psi f (a,b)\overline{\mathcal{W} \psi g (a,b)} \frac{da db}{a^2} = C_{\psi}\langle f,g\rangle
$$
where
$$
C_\psi \equiv \int_{-\infty}^\infty \frac{|\hat{\psi}(\omega)|^2}{|\omega|}d\omega
$$
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$$f(t) = \frac{1}{C_\psi}\int_{-\infty}^\infty\int_{-\infty}^\infty \mathcal{W}\psif \psi {a,b}(t) \frac{da\ db}{a^2}$$
only when
$$ 0 < C_{\psi}\equiv\int_{-\infty}^\infty
\frac{\left|\hat{\psi}(\omega)\right|^2}{\left|\omega\right|} d\omega
<\infty $$
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Change the continuous version
$$\psi_{a,b}(t) = \frac{1}{\sqrt{a}} \psi\left(\frac{t-b}{a}\right),\ \ a,b\in\mathbb{R},\ a\neq 0 $$
to a discrete version
$$\psi_{m,n}(t)=2^{-m/2} \psi \left( 2^{-m} t - n \right),\ \ \ n,m \in \mathbb{Z}$$
When can we recover $f(t)$ from $\mathcal{W}_\psif $ ?
Find $\psi_{m,n}$ such that they form a complete and orthonormal basis in $L^2(\mathbb{R})$ :
$$f(t) = \sum_{m,n=-\infty}^\infty \langle f,\psi_{m,n}\rangle\ \psi_{m,n}(t)$$
incremental: true
MRA is really an effective mathematical framework for hierarchical decomposition of an image (or signal) into components of different scales (or frequencies).
Time-frequency analysis of solar $p$ -modes
F. Baudin, A. Gabriel, D. Gibert (1994)
Wavelets: a powerful tool for studying rotation, activity, and pulsation in Kepler and CoRoT stellar light curves
J. P. Bravo, S. Roque, R. Estrela, I. C. Leão, and J. R. De Medeiros (2014)
https://github.molgen.mpg.de/saravia/wavelets-SAGE
saravia@mps.mpg.de
ags3006@gmail.com
$$\int_{-\infty}^\infty e^{2\pi i (x-x')}dx' = \delta(x)$$
$$\int_0^p e^{2\pi i x (k-l)/p}dx =
\begin{cases}
p &\mbox{if}\ \ k=l \\
0 &\mbox{otherwise}
\end{cases}
$$
$$\sum_{n=0}^{N-1} e^{2\pi i n (k-l)/N} =
\begin{cases}
N &\mbox{if}\ \ k = mN\ ,m\in\mathbb{Z} \\
0 &\mbox{otherwise}
\end{cases}
$$
