Fourier Transform Formula
The Fourier Transform is a mathematical operation that transforms a function of time, \(f(x)\), into its frequency domain representation, \(F(k) = \mathcal{F}[f(x)]\). The forward and inverse Fourier Transforms are defined as follows:
Fourier Transform Properties
The Fourier Transform has several important properties:
- \[\mathcal{F}[a f(x) + b g(x)] = a \mathcal{F}[f(x)] + b \mathcal{F}[g(x)]\]Linearity:
- \[\mathcal{F}[f(x - x_0)] = e^{-2\pi i k x_0} \cdot \mathcal{F}[f(x)]\]Time Shifting:
- \[\mathcal{F}[e^{2\pi i f_0 x} f(x)] = F(k - f_0)\]Frequency Shifting:
- \[\mathcal{F}[f(a x)] = \frac{1}{|a|} F\left(\frac{k}{a}\right)\]Scaling:
These properties are crucial in signal processing, image analysis, and various other fields.
Fourier Transform Table
The following table presents the Fourier transform for different functions:
| Function | ||
|---|---|---|
| 1 | \(\delta(t)\) | \(1\) |
| 2 | \(1\) | \(2\pi\delta(\omega)\) |
| 3 | \(\delta(t - t_0)\) | \(e^{-j\omega t_0}\) |
| 4 | \(e^{-j\omega_0 t}\) | \(2\pi\delta(\omega - \omega_0)\) |
| 5 | \(u(t)\) | \(\pi\delta(\omega) + \frac{j}{\omega}\) |
| 6 | \(\sum_{n=-\infty}^{\infty}\delta(t - nT)\) | \(\omega_0\sum_{n=-\infty}^{\infty}\delta(\omega - n\omega_0)\) |
| 7 | \(\text{sgn}(t)\) | \(\frac{2j}{\omega}\) |
| 8 | \(e^{j\omega_0 t}\) | \(2\pi\delta(\omega - \omega_0)\) |
| 9 | \(\cos(\omega_0 t)\) | \(\frac{\pi}{2}[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]\) |
| 10 | \(\sin(\omega_0 t)\) | \(\frac{-j\pi}{2}[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)]\) |
| 11 | \(e^{-at}u(t); a > 0\) | \(\frac{1}{a + j\omega}\) |
| 12 | \(te^{-at}u(t); a > 0\) | \(\frac{1}{(a + j\omega)^2}\) |
| 13 | \(e^{-|at|}; a > 0\) | \(\frac{2a}{a^2 + \omega^2}\) |
| 14 | \(e^{-|t|}\) | \(\frac{2}{\omega^2 + 1}\) |
| 15 | \(\frac{1}{\pi t}\) | \(-j\text{sgn}(\omega)\) |
| 16 | \(\text{rect}(t/\tau)\) | \(\tau\text{sinc}(\omega\tau/2)\) |
| 17 | \(\delta(t/\tau)\) | \(\tau\text{sinc}(\omega\tau/2)\) |
| 18 | \(\text{tri}(t/\tau)\) | \(\tau^2\text{sinC}^2(\omega\tau/4)\) |
| 19 | \(\sin(\omega_0 t)\pi(t)\) | \(P_a(\omega)\) |
| 20 | \(\cos(\omega_0 t)\pi(t)\) | \(\frac{\pi}{2}[P_a(\omega - \omega_0) + P_a(\omega + \omega_0)]\) |
Fourier Sine Transform
The Fourier Sine Transform is a variant of the Fourier Transform that specifically deals with odd functions and is often used in solving partial differential equations. It is defined as:
Fourier Cosine Transform
The Fourier Cosine Transform is another variant that deals with even functions and is also essential in solving differential equations and signal processing. It is defined as: