Fourier and wavelet transforms
Andres Garcia Saravia Ortiz de Montellano
23.11.2015
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Representation of functions
-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
-The Fourier representation looks different depending on the domain of the function \( f:A\rightarrow\mathbb{C} \), with \( A \) being:
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\( \mathbb{R} \): Real numbers
- -\( \mathbb{Z} \): Integer numbers
- -\( \mathbb{T}_p \): Circle of length \( p \)
- -\( \mathbb{P}_N \): Polygon of \( N \) sides
- -Continuous Fourier Transform (CFT)
-For functions defined on the real line \( \mathbb{R} \).
- -\( f:\mathbb{R}\rightarrow\mathbb{C} \)
- -\[ - \begin{aligned} - f(x) &= \int_{-\infty}^\infty F(s) e^{2\pi i s x} ds \\ - F(s) &= \int_{-\infty}^\infty f(x) e^{-2\pi i s x} ds \\ - \end{aligned} - \]
- -Discrete Time Fourier Transform (DTFT)
-For functions defined on the integers \( \mathbb{Z} \).
- -\( \phi:\mathbb{Z}\rightarrow\mathbb{C} \)
- -\[ - \begin{aligned} - \phi(n) &= \int_{0}^p \Phi(s) e^{2\pi isn/p} ds \\ - \Phi(s) &= \frac{1}{p}\sum_{n=-\infty}^\infty \phi(n) e^{-2\pi isn/p} \\ - \end{aligned} - \]
- -Fourier Series
-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 Fourier Transform (DFT)
-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
-\[ - \begin{aligned} - \int_{-\infty}^\infty f(x)\overline{g(x)}dx - &= \int_{-\infty}^\infty F(s)\overline{G(x)}ds,\ \ \mathbb{R} \\ - \sum_{n=-\infty}^\infty f(n)\overline{g(n)} - &= p \int_{0}^p F(s)\overline{G(x)}ds,\ \ \mathbb{Z} \\ - \int_{0}^p f(x)\overline{g(x)}dx - &= p \sum_{k=-\infty}^\infty F(k)\overline{G(k)},\ \ \mathbb{T}_p \\ - \sum_{n=0}^{N-1} f(n)\overline{g(n)} - &= N \sum_{k=0}^{N-1} F(k)\overline{G(k)},\ \ \mathbb{P}_N - \end{aligned} - \]
- -Discretization and periodization
-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
-\( p \)-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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Nyquist-Shannon sampling theorem
-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
-\( f(t) = a_1\sin(2\pi \omega_1 t) \)
- -
Example
-Sample at \( N \) points,
- -\[ -\begin{aligned} - \phi(n) &= f(n\Delta t + t_0),\ \ n = 0,1,\ldots,N-1 \\ - \Delta t &\equiv (t_f - t_0)/N -\end{aligned} - \]
- -\[ F(k) = \frac{1}{N}\sum_{n=0}^{N-1} f(n\Delta t) e^{-2\pi ikn/N} \]
- -\[ \Delta \omega = \frac{1}{N\Delta t} \]
- -Example
-Orthogonality relations
-\[ \int_{-\infty}^\infty e^{2\pi i (x-x')}dx' = \delta(x) \]
- -\[ \int_0^p e^{2\pi i x (k-l)/p}dx = - \begin{cases} - p &\mbox{if}\ \ k=l \\ - 0 &\mbox{otherwise} - \end{cases} - \]
- -\[ \sum_{n=0}^{N-1} e^{2\pi i n (k-l)/N} = - \begin{cases} - N &\mbox{if}\ \ k = mN\ ,m\in\mathbb{Z} \\ - 0 &\mbox{otherwise} - \end{cases} - \]
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