Fourier Transform

Definition, properties, transform pairs, and applications

Basic (university years 1–2)

Overview

The Fourier transform is a mathematical operation that turns a function of time (or space) into a function of frequency. It is one of the most important analytical tools in signal processing, image processing, physics, and engineering. It originates from Joseph Fourier's (1768–1830) study of heat conduction and underpins much of modern science and technology.

Through the Fourier transform, a complex waveform can be expressed as a superposition of simple sines and cosines, and the frequency content of a signal can be extracted explicitly. This is applied in acoustics, image compression, communication systems, quantum mechanics, and many other fields.

A finite-duration signal and its continuous amplitude spectrum, where each frequency component appears as a broadened hump rather than a discrete line — Figure 1: A finite-duration signal (left) and its continuous spectrum (right). For a signal of finite duration, each frequency component appears as a broadened hump (a continuous band) (an infinitely long pure sinusoid would instead give a line spectrum, a delta function). This also contrasts with the discrete line spectrum of a periodic signal's Fourier series.

Definition

Definition: Fourier transform

The Fourier transform $\hat{f}(\omega)$ of a function $f(t)$ is

$$\hat{f}(\omega) = \mathcal{F}[f(t)](\omega) = \displaystyle\int_{-\infty}^{\infty} f(t) e^{-i\omega t} \, dt$$

where:

$\hat{f}(\omega)$: the Fourier transform of $f(t)$ (also written $F(\omega)$; this article uses $\hat{f}(\omega)$ throughout)
$t$: the time variable (or a space variable)
$\omega$: the angular frequency [rad/s] ($\omega = 2\pi\nu$, where $\nu$ is the ordinary frequency [Hz])
$i = \sqrt{-1}$: the imaginary unit
$\mathcal{F}$: the symbol for the Fourier transform (the operation that maps a function $f$ to its transform $\hat{f}$)

Why does integrating reveal the frequency?

The Fourier transform multiplies the signal by $e^{-i\omega t}$ — a "ruler" for the frequency $\omega$ — and integrates. If the signal contains a component $e^{i\omega_0 t}$ of frequency $\omega_0$, then $$\displaystyle\int_{-\infty}^{\infty} e^{i\omega_0 t}\,e^{-i\omega t}\,dt = \displaystyle\int_{-\infty}^{\infty} e^{i(\omega_0 - \omega) t}\,dt$$ builds up to a large value when $\omega = \omega_0$ (the integrand is $1$) and cancels to nearly $0$ when $\omega \neq \omega_0$ (the positive and negative parts cancel; precisely, it equals $2\pi\,\delta(\omega - \omega_0)$). In this way the Fourier transform sifts out the strength of each frequency component in the signal.

Inverse Fourier transform

Definition of the inverse transform

The original function $f(t)$ is recovered from the frequency-domain function $\hat{f}(\omega)$ by:

$$f(t) = \mathcal{F}^{-1}[\hat{f}(\omega)](t) = \dfrac{1}{2\pi} \displaystyle\int_{-\infty}^{\infty} \hat{f}(\omega) e^{i\omega t} \, d\omega$$

This inverse transform recovers the original function when $f$ satisfies suitable conditions (roughly, that it is absolutely integrable and reasonably smooth). At points where $f$ is continuous it returns $f(t)$, and at a jump discontinuity it converges to the average of the left and right limits, $\dfrac{f(t^+)+f(t^-)}{2}$. This is the transform version of the convergence theorems for Fourier series, and the mechanism comes from the Dirichlet kernel. The same kernel is responsible for the Gibbs phenomenon — the roughly 9% overshoot near a discontinuity that never vanishes. The precise convergence conditions and inversion in the $L^2$ (norm) sense are treated in the intermediate definition of the Fourier transform and the Plancherel theorem.

A note on conventions

The definition of the Fourier transform differs in where the normalization factor $1/(2\pi)$ is placed — on the forward transform or the inverse. This article uses the angular frequency $\omega$ throughout and shows the three placements below.

$1/(2\pi)$ on the inverse (this article's definition)

$$\begin{aligned} \hat{f}(\omega) &= \displaystyle\int_{-\infty}^{\infty} f(t)\,e^{-i\omega t}\,dt \\[2pt] f(t) &= \dfrac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty} \hat{f}(\omega)\,e^{i\omega t}\,d\omega \end{aligned}$$

$1/(2\pi)$ on the forward

$$\begin{aligned} \hat{f}(\omega) &= \dfrac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty} f(t)\,e^{-i\omega t}\,dt \\[2pt] f(t) &= \displaystyle\int_{-\infty}^{\infty} \hat{f}(\omega)\,e^{i\omega t}\,d\omega \end{aligned}$$

$1/\sqrt{2\pi}$ on both sides (symmetric form)

$$\begin{aligned} \hat{f}(\omega) &= \dfrac{1}{\sqrt{2\pi}}\displaystyle\int_{-\infty}^{\infty} f(t)\,e^{-i\omega t}\,dt \\[2pt] f(t) &= \dfrac{1}{\sqrt{2\pi}}\displaystyle\int_{-\infty}^{\infty} \hat{f}(\omega)\,e^{i\omega t}\,d\omega \end{aligned}$$

In every case the product of the forward and inverse prefactors equals $1/(2\pi)$. (Using the ordinary frequency $\nu = \omega/(2\pi)$ [Hz] moves the $2\pi$ into the exponent and removes the prefactor, but this article keeps everything in the angular frequency $\omega$ [rad/s].)

Basic properties

Linearity

$$\mathcal{F}[af(t) + bg(t)] = a\hat{f}(\omega) + b\hat{g}(\omega)$$

Time shift

$$\mathcal{F}[f(t - t_0)] = e^{-i\omega t_0} \hat{f}(\omega)$$ (a time shift appears as a phase shift)

Frequency shift (modulation theorem)

$$\mathcal{F}[e^{i\omega_0 t} f(t)] = \hat{f}(\omega - \omega_0)$$ (the mathematical description of carrier modulation)

Scaling

$$\mathcal{F}[f(at)] = \dfrac{1}{|a|} \hat{f}\left(\dfrac{\omega}{a}\right)$$ (compression in time corresponds to stretching in frequency)

Differentiation

$$\mathcal{F}\left[\dfrac{d^n f}{dt^n}\right] = (i\omega)^n \hat{f}(\omega)$$ (differentiation becomes polynomial multiplication in the frequency domain)

Integration

$$\mathcal{F}\left[\displaystyle\int_{-\infty}^{t} f(\tau) d\tau\right] = \dfrac{\hat{f}(\omega)}{i\omega} + \pi \hat{f}(0) \delta(\omega)$$

Parseval's theorem (energy conservation)

$$\displaystyle\int_{-\infty}^{\infty} |f(t)|^2 \, dt = \dfrac{1}{2\pi} \displaystyle\int_{-\infty}^{\infty} |\hat{f}(\omega)|^2 \, d\omega$$ (the energy in the time domain equals the energy in the frequency domain)

Convolution theorem

Convolution in the time domain becomes a product in the frequency domain: $$\mathcal{F}[f * g] = \hat{f}(\omega) \cdot \hat{g}(\omega)$$ where the convolution is $(f * g)(t) = \displaystyle\int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau$

By this theorem, convolution (computing the response of a linear system) reduces to a simple multiplication in the frequency domain. For example, filtering a signal is a convolution in the time domain, but in the frequency domain it is just a multiplication that amplifies or attenuates each frequency component.

Duality

$$\mathcal{F}[\hat{f}(t)] = 2\pi f(-\omega)$$

Here $\hat{f}(t)$ denotes the transform $\hat{f}(\omega)$ with its variable renamed to $t$. In other words, transforming twice returns the original function, reflected in sign.

Important transform pairs

Rectangular function

$$f(t) = \begin{cases} 1 & (|t| < a) \\ 0 & (|t| > a) \end{cases} \quad \Leftrightarrow \quad \hat{f}(\omega) = \dfrac{2\sin(a\omega)}{\omega} = 2a \, \text{sinc}(a\omega)$$

A rectangular pulse of width 2a and its Fourier transform, the sinc-shaped spectrum 2 sin(a omega)/omega, with a main lobe and many side lobes — Figure 2: A rectangular pulse of width $2a$ (left) and its Fourier transform $2\sin(a\omega)/\omega$ (right). A signal with sharp discontinuities produces a broadband spectrum made of a main lobe and slowly decaying side lobes.

Gaussian

$$f(t) = e^{-at^2} \quad \Leftrightarrow \quad \hat{f}(\omega) = \sqrt{\dfrac{\pi}{a}} e^{-\omega^2/(4a)}$$ (the Fourier transform of a Gaussian is a Gaussian)

Exponential decay

$$f(t) = e^{-a|t|} \quad (a > 0) \quad \Leftrightarrow \quad \hat{f}(\omega) = \dfrac{2a}{a^2 + \omega^2}$$ (Lorentzian)

Delta function

$$f(t) = \delta(t) \quad \Leftrightarrow \quad \hat{f}(\omega) = 1$$ (contains all frequency components)

Constant function

$$f(t) = 1 \quad \Leftrightarrow \quad \hat{f}(\omega) = 2\pi \delta(\omega)$$ (only the DC component)

Sine and cosine

$$\cos(\omega_0 t) \quad \Leftrightarrow \quad \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$$ $$\sin(\omega_0 t) \quad \Leftrightarrow \quad i\pi[\delta(\omega + \omega_0) - \delta(\omega - \omega_0)]$$

Examples

Example 1: Frequency spectrum of a rectangular pulse

Problem: Find the Fourier transform of a rectangular pulse of width $2a$.

Solution: $$f(t) = \begin{cases} 1 & (|t| < a) \\ 0 & (|t| > a) \end{cases}$$ $$\hat{f}(\omega) = \displaystyle\int_{-a}^{a} e^{-i\omega t} dt = \left[ \dfrac{e^{-i\omega t}}{-i\omega} \right]_{-a}^{a} = \dfrac{e^{-i\omega a} - e^{i\omega a}}{-i\omega} = \dfrac{2\sin(a\omega)}{\omega}$$ This has the shape of a $\text{sinc}$ function, with a main lobe and many side lobes.

Example 2: Application to a differential equation

Problem: Solve the differential equation $y'' + \omega_0^2 y = f(t)$ using the Fourier transform.

Solution: Taking the Fourier transform of both sides, $$(i\omega)^2 \hat{y}(\omega) + \omega_0^2 \hat{y}(\omega) = \hat{f}(\omega)$$ $$\hat{y}(\omega) = \dfrac{\hat{f}(\omega)}{\omega_0^2 - \omega^2}$$ The inverse transform then yields $g(t)$ (the transfer-function method).

Example 3: Transform of a Gaussian

Problem: Find the Fourier transform of $f(t) = e^{-t^2/2}$.

Solution: $$\hat{f}(\omega) = \displaystyle\int_{-\infty}^{\infty} e^{-t^2/2} e^{-i\omega t} dt$$ Completing the square in the exponent: $$-\dfrac{t^2}{2} - i\omega t = -\dfrac{1}{2}(t + i\omega)^2 - \dfrac{\omega^2}{2}$$ $$\hat{f}(\omega) = e^{-\omega^2/2} \displaystyle\int_{-\infty}^{\infty} e^{-(t + i\omega)^2/2} dt = \sqrt{2\pi} e^{-\omega^2/2}$$ (using the Gaussian integral; a rigorous justification shifts the integration contour in the complex plane, but the result is as shown)

Applications

Signal processing

Used for frequency analysis of speech and audio, spectrograms, filter design (low-pass, high-pass, band-pass), and noise removal.

Image processing

The two-dimensional Fourier transform analyzes the frequency content of images. It is used in JPEG compression (via the discrete cosine transform), edge detection, and image filtering.

Communication engineering

It is the basis of modulation and demodulation (AM, FM, OFDM), channel analysis, and equalizer design.

Quantum mechanics

The momentum representation of a wave function is the Fourier transform of its position representation: $$\tilde{\psi}(p) = \dfrac{1}{\sqrt{2\pi\hbar}} \displaystyle\int \psi(x) e^{-ipx/\hbar} dx$$ The uncertainty principle $\Delta x \Delta p \geq \hbar/2$ follows from the bandwidth theorem of the Fourier transform.

Partial differential equations

In solving the heat equation, the wave equation, and the Schrödinger equation, it turns spatial derivatives into algebraic operations.

Optics

Diffraction and interference (Fraunhofer diffraction) are described by the Fourier transform. A lens performs a Fourier transform.

Probability and statistics

The characteristic function (the Fourier transform of a probability distribution) turns convolution (the distribution of a sum of independent variables) into a product.

Discrete Fourier transform (DFT)

For actual numerical computation, one uses the discrete Fourier transform (DFT) of discrete samples $\{f_n\}_{n=0}^{N-1}$: $$F_k = \displaystyle\sum_{n=0}^{N-1} f_n e^{-2\pi i kn/N}, \quad k = 0, 1, \ldots, N-1$$

The $1/N$ normalization

Like the $1/(2\pi)$ of the continuous case, the DFT has a choice of where to put the factor $1/N$. With the definition above (no factor on the forward), a constant (DC) signal $f_n = 1$ transforms to $F_0 = N$. To read off the mean or amplitude directly (a unit-height DC component as $F_0 = 1$), put $1/N$ on the forward:

$$F_k = \dfrac{1}{N}\displaystyle\sum_{n=0}^{N-1} f_n e^{-2\pi i kn/N}$$

The $1/N$ may sit on the forward, on the inverse, or split as $1/\sqrt{N}$ on each (the product of the two factors must equal $1/N$); most libraries such as NumPy put it on the inverse by default. By contrast, the unit impulse $f_n = \delta_{n,0}$ transforms to a flat $1$ at every frequency, with no factor.

Fast Fourier transform (FFT)

An algorithm that computes the DFT in $O(N \log N)$. Made practical by the Cooley–Tukey algorithm (1965), it is the foundation of modern digital signal processing.

Implementation examples

Python (NumPy/SciPy)

import numpy as np
from scipy.fft import fft, ifft, fftfreq
import matplotlib.pyplot as plt

# Sample signal (sum of two sinusoids)
N = 1000
T = 1.0 / 800.0  # sampling interval
t = np.linspace(0, N*T, N)
f = np.sin(50 * 2*np.pi*t) + 0.5*np.sin(120 * 2*np.pi*t)

# FFT
F = fft(f)
freq = fftfreq(N, T)

# Power spectrum
power = np.abs(F)**2

# Plot
plt.figure(figsize=(12, 4))
plt.subplot(121)
plt.plot(t[:100], f[:100])
plt.title('Signal')
plt.subplot(122)
plt.plot(freq[:N//2], power[:N//2])
plt.title('Power Spectrum')
plt.show()

# Inverse transform
f_reconstructed = ifft(F)
print(f"Reconstruction error: {np.max(np.abs(f - f_reconstructed)):.2e}")

MATLAB

% Sample signal
Fs = 1000;  % sampling frequency
t = 0:1/Fs:1;
f = sin(2*pi*50*t) + 0.5*sin(2*pi*120*t);

% FFT
F = fft(f);
freq = (0:length(F)-1)*Fs/length(F);

% Power spectrum
power = abs(F).^2;

% Plot
subplot(1,2,1); plot(t(1:100), f(1:100)); title('Signal');
subplot(1,2,2); plot(freq(1:end/2), power(1:end/2)); title('Power Spectrum');

Extensions

Multidimensional Fourier transform

The Fourier transform of an $n$-dimensional function: $$\hat{f}(\boldsymbol{\omega}) = \displaystyle\int_{\mathbb{R}^n} f(\mathbf{x}) e^{-i \boldsymbol{\omega} \cdot \mathbf{x}} d\mathbf{x}$$ (applied in image processing and CT scanning)

Laplace transform

An extension of the Fourier transform ($s = \sigma + i\omega$): $$\mathcal{L}[f(t)](s) = \displaystyle\int_0^{\infty} f(t) e^{-st} dt$$ (standard in control theory and circuit analysis)

Z-transform

The discrete-time counterpart of the Laplace transform: $$F(z) = \displaystyle\sum_{n=-\infty}^{\infty} f[n] z^{-n}$$ (the basis of digital filter design)

Wavelet transform

A transform that localizes both time and frequency. Effective for analyzing non-stationary signals.

Generalized Fourier series

The Fourier transform can be understood as a generalization of orthonormal basis expansion in an inner-product space (the functional-analytic viewpoint).

References

Bracewell, R. N. (2000). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill.
Stein, E. M., & Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press.
Oppenheim, A. V., & Willsky, A. S. (1997). Signals and Systems (2nd ed.). Prentice Hall.
Brigham, E. O. (1988). The Fast Fourier Transform and Its Applications. Prentice Hall.
Weisstein, E. W. "Fourier Transform." From MathWorld--A Wolfram Web Resource.

Frequently asked questions

Q. What is the difference between the Fourier transform and the Fourier series?

The Fourier series decomposes a periodic function into a sum of discrete frequency components. The Fourier transform decomposes an aperiodic function (or the limit as the period $\to\infty$) into an integral of continuous frequency components. Its definition is $\hat{f}(\omega)=\int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt$.

Q. What are the main properties (theorems) of the Fourier transform?

The representative ones are linearity, the time-shift theorem (a time shift becomes a phase change), the modulation theorem, the differentiation theorem (differentiation becomes multiplication by $i\omega$), the convolution theorem (convolution in the time domain becomes a product in the frequency domain), and Parseval's theorem (energy is preserved).

Q. What kinds of problems is the Fourier transform used for?

It is used widely: solving partial differential equations (heat equation, wave equation), signal processing (filtering, spectral analysis), image processing (JPEG compression), quantum mechanics (the complementarity of position and momentum), and communication engineering (modulation and demodulation).