main.md

Fourier and wavelet transforms
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author:Andres Garcia Saravia Ortiz de Montellano
date: 23.11.2015

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![](sage_logo.png)


Representation of functions
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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,

$$f(x) = a_0 + a_1 e^{i w_1 x} + a_2 e^{i w_2 x} + \cdots$$


Fourier representations
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The Fourier representation looks different depending on the domain of the function $f:A\rightarrow\mathbb{C}$, with $A$ being:
<br/><br/>

$\mathbb{R}$: Real numbers

$\mathbb{T}_p$: Circle of length $p$

$\mathbb{Z}$: Integer numbers

$\mathbb{P}_N$: Polygon of $N$ sides

Continuous Fourier Transform (CFT)
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For functions defined on the real line $\mathbb{R}$. 

$f:\mathbb{R}\rightarrow\mathbb{C}$

$$
  \begin{aligned}
    f(x) &= \int_{-\infty}^\infty F(s) e^{2\pi i s x} ds \\
    F(s) &= \int_{-\infty}^\infty f(x) e^{-2\pi i s x} dx \\
  \end{aligned}
$$

Fourier Series
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For functions defined on the integers on an interval $\mathbb{T}_p=[0,p)$. 

$g:\mathbb{T}_p\rightarrow\mathbb{C}$ with $g(p) = g(0)$

$$
  \begin{aligned}
    g(x) &= \sum_{k=-\infty}^\infty G(k) e^{2\pi ikx/p} \\
    G(k) &= \frac{1}{p}\int_{0}^p g(x) e^{-2\pi ikx/p} dx \\
  \end{aligned}
$$


Discrete Time Fourier Transform (DTFT)
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For functions defined on the integers $\mathbb{Z}$. 

$\phi:\mathbb{Z}\rightarrow\mathbb{C}$

$$
  \begin{aligned}
    \phi(n) &= \int_{0}^p \Phi(s) e^{2\pi isn/p} ds \\
    \Phi(s) &= \frac{1}{p}\sum_{n=-\infty}^\infty \phi(n) e^{-2\pi isn/p} \\
  \end{aligned}
$$


Discrete Fourier Transform (DFT)
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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 $\gamma(N) = \gamma(0)$

$$
  \begin{aligned}
    \gamma(n) &= \sum_{k=0}^{N-1} \Gamma(k) e^{2\pi ikn/N} \\
    \Gamma(k) &= \frac{1}{N}\sum_{n=0}^{N-1} \gamma(n) e^{-2\pi ikn/N} \\
  \end{aligned}
$$

Parseval's identities
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$$
  \begin{aligned}
    \int_{-\infty}^\infty f(x)\overline{g(x)}dx 
      &= \int_{-\infty}^\infty F(s)\overline{G(x)}ds,\ \ \mathbb{R} \\
    \int_{0}^p f(x)\overline{g(x)}dx 
      &= p \sum_{k=-\infty}^\infty F(k)\overline{G(k)},\ \ \mathbb{T}_p \\
    \sum_{n=-\infty}^\infty f(n)\overline{g(n)}
      &= p \int_{0}^p F(s)\overline{G(x)}ds,\ \ \mathbb{Z} \\
    \sum_{n=0}^{N-1} f(n)\overline{g(n)} 
      &= N \sum_{k=0}^{N-1} F(k)\overline{G(k)},\ \ \mathbb{P}_N
  \end{aligned}
$$


Discretization and periodization
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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*

$$g(x) = \sum_{m=-\infty}^\infty f(x-mp) $$


Poisson relations
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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

$$\phi(n) = f\left(\frac{np}{N}\right)
\implies \Phi(s) = \sum_{m=-\infty}^\infty F\left(s - \frac{mN}{p}\right)$$


Poisson cube
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![](Poisson-cube.png)


Nyquist-Shannon sampling theorem
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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

$$2\sigma \Delta t <1$$

$$\sigma <  \frac{1}{2\Delta t} \equiv \omega_{Nyq}$$

Example: single sinusoid
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$$
  \begin{aligned}
    f(t) &= a_1\sin(2\pi \omega_1 t) \\
    F(\omega) 
     &= \int_{-\infty}^{\infty}a_1\sin(2\pi \omega_1 t)e^{-2\pi i \omega t} dt\\
     &= a_1\int_{-\infty}^{\infty} 
      \left(\frac{e^{2\pi i\omega_1 t} - e^{-2\pi i\omega_1 t}}{2i}\right)
      e^{-2\pi i \omega t}dt\\
     &= \frac{a_1}{2i}\left[\delta(\omega-\omega_1) 
        - \delta(\omega+\omega_1)\right]
  \end{aligned}
$$

$$
  P(\omega) \equiv \left|F(\omega)\right|^2= \left(\frac{a_1}{2}\right)^2\left[\delta(\omega-\omega_1) 
        - \delta(\omega+\omega_1)\right]^2$$


Example: single sinusoid
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![plot of chunk unnamed-chunk-1](main-figure/unnamed-chunk-1-1.png) 

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![plot of chunk unnamed-chunk-2](main-figure/unnamed-chunk-2-1.png) 


Example: gaussian noise
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![plot of chunk unnamed-chunk-3](main-figure/unnamed-chunk-3-1.png) 

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![plot of chunk unnamed-chunk-4](main-figure/unnamed-chunk-4-1.png) 


Example: Multiple sinusoids with noise
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![plot of chunk unnamed-chunk-5](main-figure/unnamed-chunk-5-1.png) 

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![plot of chunk unnamed-chunk-6](main-figure/unnamed-chunk-6-1.png) 

Example: Time-varying signal
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![plot of chunk unnamed-chunk-7](main-figure/unnamed-chunk-7-1.png) 

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![plot of chunk unnamed-chunk-8](main-figure/unnamed-chunk-8-1.png) 

Example: Time-varying signal 2
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![plot of chunk unnamed-chunk-9](main-figure/unnamed-chunk-9-1.png) 

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![plot of chunk unnamed-chunk-10](main-figure/unnamed-chunk-10-1.png) 


Short-time Fourier Transform (STFT)
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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


Wavelet transforms
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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

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

Wavelet transforms
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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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![plot of chunk unnamed-chunk-11](main-figure/unnamed-chunk-11-1.png) 


Definition of wavelet transforms
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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 

$$ 0 < C_{\psi}\equiv\int_{-\infty}^\infty
\frac{\left|\hat{\psi}(\omega)\right|^2}{\left|\omega\right|} d\omega
<\infty $$

Example: Time-varying signal CWT
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![plot of chunk unnamed-chunk-12](main-figure/unnamed-chunk-12-1.png) 

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Mexican hat CWT

![](tvs1.png)

Example: Time-varying signal CWT 2
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Haar CWT

![](tvs2.png)

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Morlet CWT

![](tvs3.png)


Example: Time-varying signal CWT 3
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Morlet

![](tvs4.png)

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Mexican-hat

![](tvs5.png)


Parseval's relation for wavelets
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$$
\int_{-\infty}^\infty\int_{-\infty}^\infty \mathcal{W}_\psi f (a,b)\overline{\mathcal{W}_\psi g (a,b)} \frac{da db}{a^2} = C_{\psi}\langle f,g\rangle
$$

where

$$
C_\psi \equiv \int_{-\infty}^\infty \frac{|\hat{\psi}(\omega)|^2}{|\omega|}d\omega 
$$


Inverse of a wavelet transform
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$$f(t) = \frac{1}{C_\psi}\int_{-\infty}^\infty\int_{-\infty}^\infty \mathcal{W}_\psi[f](a,b) \psi_{a,b}(t) \frac{da\ db}{a^2}$$

only when

$$ 0 < C_{\psi}\equiv\int_{-\infty}^\infty
\frac{\left|\hat{\psi}(\omega)\right|^2}{\left|\omega\right|} d\omega
<\infty $$


Discrete wavelet transform
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incremental: true

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 $\mathcal{W}_\psi[f](m,n)$ ?


Orthonormal basis
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Find $\psi_{m,n}$ such that they form a complete and orthonormal basis in $L^2(\mathbb{R})$:

$$f(t) = \sum_{m,n=-\infty}^\infty \langle f,\psi_{m,n}\rangle\ \psi_{m,n}(t)$$


Multiresolution analysis (MRA)
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![](MRA.png)

> *MRA is really an effective mathematical framework for hierarchical decomposition of an image (or signal) into components of different scales (or frequencies).*


Example application
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**Time-frequency analysis of solar $p$-modes**

*F. Baudin, A. Gabriel, D. Gibert (1994)*

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![](Baudin.png)


Example application 2
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**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)*

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![](kprl-example.png)


Thank you!
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![](coffee_is_essential.jpg)

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[https://github.molgen.mpg.de/saravia/wavelets-SAGE](https://github.molgen.mpg.de/saravia/wavelets-SAGE)

saravia@mps.mpg.de

ags3006@gmail.com

Orthogonality relations
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$$\int_{-\infty}^\infty e^{2\pi i (x-x')}dx' = \delta(x)$$

$$\int_0^p e^{2\pi i x (k-l)/p}dx =
  \begin{cases} 
    p &\mbox{if}\ \ k=l \\
    0 &\mbox{otherwise}
  \end{cases}
$$

$$\sum_{n=0}^{N-1} e^{2\pi i n (k-l)/N} =
  \begin{cases}
    N &\mbox{if}\ \ k = mN\ ,m\in\mathbb{Z} \\
    0 &\mbox{otherwise}
  \end{cases}
$$