What is the difference between the Fourier Transform and the Fourier Series?

The Fourier Series decomposes a periodic function into a discrete set of sinusoidal harmonics (frequencies that are integer multiples of a fundamental). The Fourier Transform handles non-periodic functions defined on all of ℝ and produces a continuous spectrum of frequencies. You can think of the Fourier Transform as the limit of Fourier Series when the period goes to infinity.

Fourier Transform — Definition, Formula & Examples

The Fourier Transform is a mathematical operation that converts a function of time (or space) into a function of frequency, revealing which frequencies are present and how strong each one is. It essentially decomposes any signal into a sum of sinusoidal waves.

Given an integrable function $f(t)$ on $\mathbb{R}$ , its Fourier Transform is the function $\hat{f}(\omega) = \int_{-\infty}^{\infty} f(t)\, e^{-i\omega t}\, dt$ , where $\omega$ is the angular frequency. The transform defines a mapping $f \mapsto \hat{f}$ from the time domain to the frequency domain, and under suitable conditions (e.g., $f \in L^1(\mathbb{R})$ ), $\hat{f}$ is continuous and bounded.

Key Formula

\hat{f}(\omega) = \int_{-\infty}^{\infty} f(t)\, e^{-i\omega t}\, dt

Where:

$f(t)$ = The original function in the time domain
$\hat{f}(\omega)$ = The Fourier Transform of f, a function of angular frequency
$\omega$ = Angular frequency (radians per unit time)
$i$ = The imaginary unit, where i² = −1
$t$ = The time variable

How It Works

You feed a time-domain signal

f(t)

into the Fourier Transform integral, and the output

\hat{f}(\omega)

tells you the amplitude and phase of each frequency component

\omega

present in the original signal. The magnitude

|\hat{f}(\omega)|

gives the strength of the frequency

\omega

, while the argument

\arg(\hat{f}(\omega))

gives its phase shift. To recover the original function, you apply the inverse Fourier Transform:

f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(\omega)\, e^{i\omega t}\, d\omega

. In practice, many signals are sampled discretely, so the Discrete Fourier Transform (DFT) and its fast algorithm (FFT) are used instead.

Worked Example

Problem: Find the Fourier Transform of the decaying exponential f(t) = e^{-at} for t ≥ 0 and f(t) = 0 for t < 0, where a > 0.

Step 1: Set up the integral: Since f(t) = 0 for t < 0, the integral reduces to an integral from 0 to infinity.

\hat{f}(\omega) = \int_{0}^{\infty} e^{-at}\, e^{-i\omega t}\, dt = \int_{0}^{\infty} e^{-(a + i\omega)t}\, dt

Step 2: Evaluate the integral: This is a standard exponential integral. Because a > 0, the integrand decays to 0 as t → ∞.

\hat{f}(\omega) = \left[\frac{e^{-(a+i\omega)t}}{-(a+i\omega)}\right]_0^{\infty} = 0 - \frac{1}{-(a+i\omega)} = \frac{1}{a + i\omega}

Step 3: Find the magnitude spectrum: The magnitude tells you how strong each frequency component is.

|\hat{f}(\omega)| = \frac{1}{\sqrt{a^2 + \omega^2}}

Answer: The Fourier Transform is

\hat{f}(\omega) = \dfrac{1}{a + i\omega}

, with magnitude spectrum

\dfrac{1}{\sqrt{a^2 + \omega^2}}

Another Example

Problem: Find the Fourier Transform of the rectangular pulse f(t) = 1 for |t| ≤ 1 and f(t) = 0 otherwise.

Step 1: Set up the integral: The function is nonzero only on [−1, 1].

\hat{f}(\omega) = \int_{-1}^{1} e^{-i\omega t}\, dt

Step 2: Evaluate: Integrate the complex exponential directly.

\hat{f}(\omega) = \left[\frac{e^{-i\omega t}}{-i\omega}\right]_{-1}^{1} = \frac{e^{-i\omega} - e^{i\omega}}{-i\omega} = \frac{2\sin(\omega)}{\omega}

Step 3: Recognize the sinc form: This result is the well-known sinc function (up to convention).

\hat{f}(\omega) = 2\,\mathrm{sinc}(\omega) \quad \text{where } \mathrm{sinc}(x) = \frac{\sin x}{x}

Answer: The Fourier Transform of the rectangular pulse is

\hat{f}(\omega) = \dfrac{2\sin(\omega)}{\omega}

Visualization

Why It Matters

The Fourier Transform is central to courses in signals and systems, differential equations, and mathematical physics. Engineers use it daily to analyze audio, filter noise from signals, and compress images (JPEG relies on a related transform). In quantum mechanics, the position and momentum representations of a wave function are Fourier Transform pairs of each other.

Common Mistakes

Mistake: Forgetting the negative sign in the exponent and writing e^{iωt} instead of e^{-iωt} for the forward transform.

Correction: Convention matters. The most common convention uses e^{-iωt} for the forward transform and e^{+iωt} for the inverse. Mixing these up flips the sign of the phase spectrum and leads to incorrect results.

Mistake: Confusing the different normalization conventions (using ω vs. f, or placing the 1/(2π) factor on the forward vs. inverse transform).

Correction: Multiple valid conventions exist. Some textbooks use ordinary frequency f with the pair

\hat{f}(\nu) = \int f(t) e^{-2\pi i \nu t} dt

, which places no extra constant on either transform. Always check which convention your course uses and stay consistent.

Related Terms

Fourier Series — Decomposes periodic functions into discrete frequencies
Laplace Transform — Generalization using complex exponential decay
Inverse Fourier Transform — Recovers time-domain function from frequency domain
Convolution — Multiplication in frequency domain equals convolution in time
Complex Exponential — The basis function used in the transform kernel
Dirac Delta Function — Its transform is constant across all frequencies