Fourier Transform Formula
The Fourier transform converts a time-domain signal into its frequency components.
Learn the formula, inverse transform, and real-world applications.
The Formula
Inverse: f(t) = (1/2π) × ∫₋∞^∞ F(ω) × e^(iωt) dω
The Fourier transform, developed from the work of French mathematician Joseph Fourier (1768–1830), is one of the most powerful tools in mathematics, physics, and engineering. It converts a function of time f(t) into a function of frequency F(ω), revealing which frequency components make up any signal — no matter how complex.
Think of it like a musical spectrum analyzer. When you play a chord on a piano, a single complicated sound wave enters your ear. The Fourier transform is the mathematical equivalent of separating that wave back into its individual pure notes, showing exactly how much of each frequency is present.
The inverse Fourier transform goes in the other direction — it reconstructs the original time-domain signal from its frequency components. These two operations are perfect inverses of each other, meaning no information is lost when transforming between the two domains. The factor e^(−iωt) = cos(ωt) − i×sin(ωt) encodes both the cosine and sine components at each frequency ω.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| f(t) | The original signal as a function of time | varies |
| F(ω) | The Fourier transform — the signal in the frequency domain | varies |
| t | Time variable | s (seconds) |
| ω | Angular frequency (ω = 2πf where f is frequency in Hz) | rad/s |
| i | Imaginary unit (i² = −1) | dimensionless |
| e | Euler's number ≈ 2.71828 | dimensionless |
Example 1
Find the Fourier transform of a rectangular pulse: f(t) = 1 for |t| ≤ T/2, and 0 otherwise.
F(ω) = ∫₋T/2^T/2 1 × e^(−iωt) dt
Integrating: F(ω) = [e^(−iωt) / (−iω)] from −T/2 to T/2
F(ω) = (e^(iωT/2) − e^(−iωT/2)) / iω = 2×sin(ωT/2) / ω
F(ω) = T × sinc(ωT/2π) — a rectangular pulse in time becomes a sinc function in frequency. This is why audio processing uses windowed functions.
Example 2
What is the Fourier transform of a pure sine wave at frequency f₀?
f(t) = sin(2πf₀t) = (e^(i2πf₀t) − e^(−i2πf₀t)) / 2i
The Fourier transform of e^(iω₀t) is 2π×δ(ω − ω₀), where δ is the Dirac delta function
F(ω) contains only two spikes at ω = ±2πf₀ — confirming that a pure sine wave has exactly one frequency component, as expected
When to Use It
Use the Fourier transform when:
- Analyzing the frequency content of audio, radio, seismic, or electrical signals
- Filtering noise from a signal (by suppressing unwanted frequency components)
- Solving linear differential equations in physics and engineering
- Compressing images and audio (JPEG and MP3 both rely on Fourier-related transforms)
- Processing medical imaging data in MRI and CT scanners