Key Fourier Transforms Overview
This table provides a concise overview of the essential Fourier transforms for common functions in signal analysis. Each row presents a function in the time domain along with its corresponding Fourier transform in the frequency domain.
| Function in the Time Domain \( f(t) \) | Fourier Transform \( F(f) \) |
|---|---|
| \( \delta(t) \) | 1 |
| 1 | \( \delta(f) \) |
| \( e^{j 2 \pi f_0 t} \) | \( \delta(f - f_0) \) |
| \( e^{j -2 \pi f_0 t} \) | \( \delta(f + f_0) \) |
| \( \cos(2 \pi f_0 t) \) | \( \frac{1}{2} \left[ \delta(f - f_0) + \delta(f + f_0) \right] \) Proof |
| \( \sin(2 \pi f_0 t) \) | \( \frac{1}{2j} \left[ \delta(f - f_0) - \delta(f + f_0) \right] \) Proof |
| \( u(t) \) | \( \frac{1}{j 2 \pi f} + \frac{1}{2} \delta(f) \) |
| \( e^{-a t} u(t) \), \( a > 0 \) | \( \frac{1}{a + j 2 \pi f} \) |
| \( e^{a t} u(-t) \), \( a > 0 \) | \( \frac{1}{a - j 2 \pi f} \) |
| \( \text{sinc}(t) = \frac{\sin(\pi t)}{\pi t} \) | \( \text{rect}(f) \) |
| \( \text{rect}(t) \) | \( \text{sinc}(f) \) |
| \( e^{-(t/T)^2} \) | \( T \sqrt{\pi} e^{-\pi^2 f^2 T^2} \) |
| \( \frac{1}{\pi (t^2 + 1)} \) | \( e^{-2 \pi |f|} \) |
| \( \text{tri}(t) \) | \( \text{sinc}^2(f) \) |
| \( t \cdot e^{-a t} u(t) \) | \( \frac{1}{(a + j 2 \pi f)^2} \) |
This collection serves as a valuable resource for understanding how various signals behave in terms of frequency and has practical applications in signal processing.
And so forth.
