Fourier Transform Properties

A complete reference of the key properties of the Fourier transform. If $f(t) \leftrightarrow F(\omega)$ and $g(t) \leftrightarrow G(\omega)$ , the following relationships hold.

Convention: $\mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t)\,e^{-j\omega t}\,dt$

Properties Reference Table

#	Property	Time Domain	Frequency Domain	Notes
1	Linearity	$a\,f(t) + b\,g(t)$	$a\,F(\omega) + b\,G(\omega)$	Superposition; $a,b \in \mathbb{C}$
2	Time Shifting	$f(t - t_0)$	$e^{-j\omega t_0}\,F(\omega)$	Delay → phase shift; magnitude unchanged
3	Frequency Shifting	$e^{j\omega_0 t}\,f(t)$	$F(\omega - \omega_0)$	Modulation; shifts spectrum
4	Time Scaling	$f(at)$	$\frac{1}{\|a\|}\,F\!\left(\frac{\omega}{a}\right)$	Compress time → expand freq; $a \neq 0$
5	Time Reversal	$f(-t)$	$F(-\omega)$	Special case of scaling with $a = -1$
6	Conjugation	$f^*(t)$	$F^*(-\omega)$	For real $f$ : $F(-\omega) = F^*(\omega)$
7	Duality	$F(t)$	$2\pi\,f(-\omega)$	Swap time ↔ frequency roles
8	Convolution	$f(t) * g(t)$	$F(\omega)\,G(\omega)$	Convolution ↔ multiplication
9	Multiplication	$f(t)\,g(t)$	$\frac{1}{2\pi}\,F(\omega) * G(\omega)$	Windowing; dual of convolution
10	Time Differentiation	$\frac{d^n f}{dt^n}$	$(j\omega)^n\,F(\omega)$	Derivatives become multiplication
11	Freq. Differentiation	$(-jt)^n\,f(t)$	$\frac{d^n F}{d\omega^n}$	Multiply by $t$ → differentiate in $\omega$
12	Integration	$\int_{-\infty}^{t} f(\tau)\,d\tau$	$\frac{F(\omega)}{j\omega} + \pi F(0)\,\delta(\omega)$	Requires $F(0)$ term for DC
13	Parseval's Theorem	$\int_{-\infty}^{\infty}\|f(t)\|^2\,dt$	$\frac{1}{2\pi}\int_{-\infty}^{\infty}\|F(\omega)\|^2\,d\omega$	Energy conservation
14	Cross-Correlation	$R_{fg}(\tau) = \int f^*(t)\,g(t+\tau)\,dt$	$F^*(\omega)\,G(\omega)$	Correlation theorem
15	Cosine Modulation	$f(t)\cos(\omega_0 t)$	$\frac{1}{2}[F(\omega - \omega_0) + F(\omega + \omega_0)]$	AM modulation; spectrum copies

Property Details

The Convolution Theorem

Perhaps the single most important property. It states that convolution in one domain corresponds to pointwise multiplication in the other:

\mathcal{F}\{f * g\} = F \cdot G

\mathcal{F}\{f \cdot g\} = \frac{1}{2\pi}(F * G)

This is the foundation of fast filtering: convolving two length- $N$ sequences directly costs $O(N^2)$ , but via FFT it costs $O(N \log N)$ .

Parseval's Theorem (Energy Conservation)

Total energy computed in the time domain equals total energy computed in the frequency domain:

E = \int_{-\infty}^{\infty}|f(t)|^2\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}|F(\omega)|^2\,d\omega

For the DFT the discrete analog is:

\sum_{n=0}^{N-1}|x[n]|^2 = \frac{1}{N}\sum_{k=0}^{N-1}|X[k]|^2

Differentiation Property

Differentiation in time becomes multiplication by $j\omega$ in frequency. This converts differential equations into algebraic equations — the core idea behind using transforms to solve ODEs and PDEs.

\frac{df}{dt} \leftrightarrow j\omega\,F(\omega), \qquad \frac{d^2f}{dt^2} \leftrightarrow (j\omega)^2\,F(\omega)

Each derivative multiplies the high-frequency content by $\omega$ , amplifying noise — this is why numerical differentiation is ill-conditioned.

Symmetry Properties of Real Signals

When $f(t)$ is real-valued, its Fourier transform has conjugate symmetry:

F(-\omega) = F^*(\omega)

This means:

Magnitude spectrum is even: $|F(-\omega)| = |F(\omega)|$
Phase spectrum is odd: $\angle F(-\omega) = -\angle F(\omega)$
Real part is even, imaginary part is odd

Additionally, if $f(t)$ is both real and even, then $F(\omega)$ is purely real and even.

Continue Exploring

See the complete table of transform pairs, or grab the printable cheat sheet.

Fourier Pairs Table Cheat Sheet Try Convolution Tool