Common Fourier Transform Pairs
A comprehensive table of Fourier transform pairs showing the time-domain function f(t) and its frequency-domain counterpart F(\omega). All pairs use the angular frequency convention.
Convention: F(\omega) = \int_{-\infty}^{\infty} f(t)\,e^{-j\omega t}\,dt
Basic Pairs
Fundamental pairs that form the building blocks of Fourier analysis.
| # | Name | f(t) | F(\omega) | Notes |
|---|---|---|---|---|
| 1 | Impulse | \delta(t) | 1 | All frequencies equally |
| 2 | Constant | 1 | 2\pi\,\delta(\omega) | DC component only |
| 3 | Shifted Impulse | \delta(t - t_0) | e^{-j\omega t_0} | Phase shift = time delay |
| 4 | Complex Exponential | e^{j\omega_0 t} | 2\pi\,\delta(\omega - \omega_0) | Single frequency |
| 5 | Signum | \text{sgn}(t) | \dfrac{2}{j\omega} | ±1 function |
| 6 | Unit Step | u(t) | \pi\,\delta(\omega) + \dfrac{1}{j\omega} | Heaviside step |
Common Functions
Widely-used signals and their transforms — the pairs you'll reference most often.
| # | Name | f(t) | F(\omega) | Notes |
|---|---|---|---|---|
| 7 | Rectangular Pulse | \text{rect}\!\left(\frac{t}{\tau}\right) | \tau\,\text{sinc}\!\left(\frac{\omega\tau}{2\pi}\right) | Width τ centered at 0 |
| 8 | Sinc Function | \text{sinc}(Wt) | \frac{1}{W}\,\text{rect}\!\left(\frac{\omega}{2\pi W}\right) | Dual of rect |
| 9 | Gaussian | e^{-\alpha t^2} | \sqrt{\frac{\pi}{\alpha}}\,e^{-\omega^2/(4\alpha)} | Gaussian ↔ Gaussian |
| 10 | One-Sided Exponential | e^{-\alpha t}\,u(t),\ \alpha > 0 | \dfrac{1}{\alpha + j\omega} | Causal decay |
| 11 | Two-Sided Exponential | e^{-\alpha|t|},\ \alpha > 0 | \dfrac{2\alpha}{\alpha^2 + \omega^2} | Lorentzian spectrum |
| 12 | Triangular Pulse | \Lambda\!\left(\frac{t}{\tau}\right) | \tau\,\text{sinc}^2\!\left(\frac{\omega\tau}{2\pi}\right) | rect ∗ rect |
| 13 | Cosine | \cos(\omega_0 t) | \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)] | Two impulses |
| 14 | Sine | \sin(\omega_0 t) | \frac{\pi}{j}[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)] | Odd symmetry |
| 15 | Comb (Impulse Train) | \displaystyle\sum_{n=-\infty}^{\infty}\delta(t - nT) | \frac{2\pi}{T}\displaystyle\sum_{k=-\infty}^{\infty}\delta\!\left(\omega - \frac{2\pi k}{T}\right) | Comb ↔ Comb |
| 16 | Damped Cosine | e^{-\alpha t}\cos(\omega_0 t)\,u(t) | \dfrac{\alpha + j\omega}{(\alpha + j\omega)^2 + \omega_0^2} | RLC circuit response |
| 17 | Damped Sine | e^{-\alpha t}\sin(\omega_0 t)\,u(t) | \dfrac{\omega_0}{(\alpha + j\omega)^2 + \omega_0^2} | Oscillatory decay |
Discrete Fourier Transform Pairs
For a sequence of length N, the DFT is:
| # | Name | x[n] | X[k] | Notes |
|---|---|---|---|---|
| 18 | Discrete Impulse | \delta[n] | 1 \text{ for all } k | Flat spectrum |
| 19 | Constant | 1 \text{ for all } n | N\,\delta[k] | DC only |
| 20 | DFT Sinusoid | e^{j2\pi m n/N} | N\,\delta[k - m] | Single bin |
| 21 | Rectangular Window | \begin{cases}1 & 0 \le n \le L{-}1\\0 & \text{else}\end{cases} | e^{-j\omega(L-1)/2}\dfrac{\sin(\omega L/2)}{\sin(\omega/2)} | Dirichlet kernel |
| 22 | Discrete Cosine | \cos\!\left(\frac{2\pi m n}{N}\right) | \frac{N}{2}[\delta[k{-}m] + \delta[k{+}m]] | Two bins (symmetric) |
Key Observations
Duality
Many pairs are symmetric: the Gaussian transforms to a Gaussian, the comb function transforms to a comb. If f(t) \leftrightarrow F(\omega), then F(t) \leftrightarrow 2\pi f(-\omega).
Uncertainty Principle
A signal cannot be simultaneously localized in both time and frequency. Narrower in time → wider in frequency and vice versa. The Gaussian achieves the theoretical minimum time-bandwidth product: \Delta t \cdot \Delta \omega \ge \frac{1}{2}.
Convergence
Some pairs (like the constant and signum) only exist in the distributional sense using Dirac deltas. Practically, these are handled via limiting sequences or the theory of tempered distributions.
Explore More Reference
See how these pairs behave under various operations, or grab the one-page cheat sheet.