Table of Contents
- Transformation of a function by integrating with a kernel.
- \[ \mathcal{T}[f](u) = \int_\Omega f(t)K(u, t)\,dt. \]
1. Fourier Transform
- Change of basis in of functions. #worthavideo
- A Transition from the Fourier series to Fourier transform is equivalent to that from PMF to PDF.
- https://youtu.be/R_4GuyJTzMo?si=gsr-ynWOoHeVwJVp
1.1. Phasor
- \[ z=Ae^{i\phi} \]
- A complex number representing the amplitude \(A\) and the phase \(\phi\) of a sinusoid.
- \[ ze^{i\omega t}=Ae^{i\phi} e^{i\omega t}= Ae^{i(\omega t + \phi)} \]
1.2. Definition
- \[ \mathcal{F}[f](\omega)=\int_P f(t) e^{-i\omega t}\, dt \] where \(e^{-i\omega t}\) part is called the kernel.
- The transformed function is also called \(\hat{f}(\omega)\)
- \(\hat{f}\) maps a frequency to its phasor.
1.3. Examples
- \[
\hat{f}(\omega)=\mathcal{F}[\sin(\omega_0t)](\omega)=\frac{1}{2i}(\delta(\omega-\omega_0)-\delta(\omega + \omega_0))
\]
where \(\delta\) is the .
Note \[
\begin{align*} \mathcal{F}^{-1}[\hat{f}](t)&=\frac{1}{2i}\left[\int_{-\infty}^{\infty}\delta(\omega-\omega_0)e^{i\omega t}\,d\omega-\int_{-\infty}^{\infty}\delta(\omega+\omega_0)e^{i\omega t}\,d\omega\right]\\ &=\frac{e^{i\omega_0 t}-e^{-i\omega_0 t}}{2i} \end{align*}\]
- Decaying exponential transforms into a Lorentzian
distribution?
- \[ \mathcal{F}[e^{-\alpha t}](\omega) = \frac{1}{\alpha + i\omega} \]
- Square pulse transforms into a sinc function.
- \[
f(t)=\begin{cases}A&-\frac{\tau}{2}
- \(\tau\cdot\Omega = 4\pi\), where \(\Omega\) is the width of the center lobe at the line \(y=0\).
- \[
f(t)=\begin{cases}A&-\frac{\tau}{2}
- Gaussian function transforms into a Gaussian function.
- \[ e^{\frac{-t^2}{2\sigma^2}}\ \overset{\mathcal{F}}{\to}\ \sigma\sqrt{2\pi}e^{-2\pi^2\sigma^2k^2} \]
1.4. Properties
- Fourier transform of a real function is Hermitian: \(\hat{f}(-\omega)=\hat{f}^*(\omega)\).
- Even functions have real Fourier transform and odd functions have imaginary Fourier transform.
- Periodicity
- \(\mathcal{F}[f]=\hat{f}\)
- \(\mathcal{F}^2[f(t)]=f(-t)\)
- \(\mathcal{F}^3[f]=\mathcal{F}^{-1}[f]\)
- \(\mathcal{F}^4[f]=f\)
- \(\mathcal{F}[f*g]=\mathcal{F}[f]\cdot\mathcal{F}[g]\) where \(*\) is the .
- Shifting
- Modulation
- Frequency shifting
- \[ f(t)\cdot e^{i\omega_0t}\ \overset{\mathcal{F}}{\to}\ \hat{f}(\omega-\omega_0) \]
- \(f(t)\) can be thought of as an envelope function, or the function in the rotating frame.
- Modulation
1.5. k-Space
- \(k\) is for the spacial frequency.
- Fourier transform of a spacial information results in a \(k\)-space.
- For 2D case, the kernel is \(e^{-i(k_xx+k_yy)}\).
- \(e^{i(k_xx+k_yy)}\) is a sinusoidal surface with the fringe progression in the direction of the vector \([k_x\ k_y]^{\mathrm{T}}\).
- In a discrete case, \(k\)-space is quantized with \(\Delta k_{x_i} = 1/\mathrm{FOV}_{x_i}\).
- The pixel shape is the shape of \(k\)-space and the reciprocal of the image shape is the pixel shape of the \(k\)-space.
- DC centered view of \(k\)-space is the normal result of Fourier transform, and from we get
1.6. Dirichlet Kernel
1.6.1. Definition
- \[ D_n(x) = \frac{\sin((n+1/2)x)}{\sin(x/2)}. \]
1.6.2. Properties
- The convolution of Dirichlet kernel \(D_n\) with any function with
period \(2\pi\) is the \(n\)th degree Frourier series approximation
of \(f\).
- \[ (D_n\ast f)(x) = \int_{-\pi}^\pi f(y)D_n(x-y)\,dy = 2\pi \sum_{k=-n}^n \hat{f}(k)e^{ikx} \] where \(f(x+2\pi) = f(x)\).
2. Laplace Transform
2.1. Definition
- \[ \mathcal{L}\{f\}(s) := \int_0^\infty f(t)e^{-st}dt. \]
- It transforms the complex-valued real function to a complex-valued
complex function.
- time domain into s-domain.
- The Laplace transform of a function is often written in a capital letter.
2.2. Properties
- Linearity
- Frequency-Domain Derivative
- \[ \mathcal{L}\{t^nf\} = (-1)^n(\mathcal{L}f)^{(n)} \]
- (Time-Domain) Derivative
- \[ \mathcal{L}\{f^{(n)}\} = s^n\mathcal{L}f - \sum_{k=1}^ns^{n-k}f^{(k-1)}(0^-) \]
- Frequency Shifting
- \[ \mathcal{L}\{e^{at}f\} = \mathcal{Lf}(s-a) \]
- Convolution
- \[ \mathcal{L}\{f\ast g\} = (\mathcal{L}f)\cdot(\mathcal{L}g) \]
2.3. Z-Transform
- Discrete-time equivalent of Laplace transform
- Complex valued frequency domain, the z-domain.
2.3.1. Definition
- \[ \mathcal{Z}\{x[n]\}(z) = \sum_{n=-\infty}^{\infty}x[n]z^{-n}. \]
- See
3. Mellin Transform
- Multiplicative version of two-sided 2
3.1. Definition
- \[ \mathcal{M}\{f\}(s) = \varphi(s) = \int_0^\infty x^{s-1}f(x)\,dx. \]
- The inverse transform is given by \[ \mathcal{M}^{-1}\{\varphi\}(x) = f(x) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\varphi(s)\,ds. \]