Fourier Transform
The Fourier Transform decomposes a time-domain signal into its constituent frequency components, expressing it as a superposition of sinusoids of different frequencies, amplitudes, and phases. It provides the frequency-domain representation of a signal and is fundamental to signal processing, communications, image analysis, and solving differential equations. The transform is invertible, meaning the original signal can be perfectly reconstructed from its frequency spectrum.
Key Formula
F(ω) = integral from -∞ to +∞ of f(t) × e^(-jωt) dt
LaTeX: F(\omega) = \int_{-\infty}^{\infty} f(t)\, e^{-j\omega t}\, dt
| Symbol | Meaning | Unit |
|---|---|---|
| F(ω) | Frequency-domain representation (spectrum) | Complex amplitude |
| f(t) | Time-domain signal | Varies (V, A, etc.) |
| ω | Angular frequency | rad/s |
| j | Imaginary unit (√−1) | Dimensionless |
| t | Time variable | s |
Worked Example
Problem
Find the Fourier Transform of a rectangular pulse: f(t) = 1 for |t| ≤ T/2 = 0.5 s, and 0 otherwise. What is F(ω)?
Solution
Step 1: Apply the Fourier Transform definition. F(ω) = ∫_{-T/2}^{T/2} 1 · e^{-jωt} dt Step 2: Evaluate the integral. F(ω) = [-e^{-jωt} / (jω)]_{-T/2}^{T/2} F(ω) = (e^{-jωT/2} - e^{jωT/2}) / (-jω) Step 3: Use Euler's formula: sin(x) = (e^{jx} - e^{-jx}) / (2j). F(ω) = (2/ω) · sin(ωT/2) Step 4: With T = 1 s, express as a sinc function. F(ω) = T · sinc(ωT/2π) = 1 · sinc(ω/2π) At ω = 0: F(0) = T = 1 s (area of pulse).
Answer
F(ω) = sin(ω/2) / (ω/2) = sinc(ω/2π), with F(0) = 1 s (for a unit-amplitude pulse of width 1 s)
Key Fourier Transform Pairs
| Signal f(t) | Transform F(ω) | Notes |
|---|---|---|
| δ(t) (impulse) | 1 | All frequencies equally |
| 1 (constant) | 2πδ(ω) | DC component only |
| e^{-at}u(t), a>0 | 1/(a + jω) | Decaying exponential |
| rect(t/T) | T·sinc(ωT/2π) | Rectangular pulse |
| cos(ω₀t) | π[δ(ω−ω₀)+δ(ω+ω₀)] | Pure tone — two impulses |
| Gaussian e^{-t²/2} | √(2π) e^{-ω²/2} | Self-dual under FT |
Interactive Tools
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Related Terms
Signal Processing
Signal processing is the analysis, manipulation, and synthesis of signals — including audio, video, sensor data, and communications waveforms — to extract information or transform them for a desired purpose. It encompasses filtering, compression, modulation, spectral analysis, and noise reduction using both analog and digital techniques. Signal processing underpins technologies such as telecommunications, medical imaging, radar, speech recognition, and multimedia systems.
Laplace Transform
The Laplace Transform converts a time-domain function into the complex frequency domain (s-domain), enabling differential equations to be solved as algebraic equations. It generalises the Fourier Transform by including a real exponential damping term, making it applicable to a broader class of signals including those that grow with time. It is the primary mathematical tool in control systems engineering, circuit analysis, and linear system theory for analysing stability, transient response, and frequency behaviour.
Signal-to-Noise Ratio
Signal-to-Noise Ratio (SNR) is the ratio of the power of a desired signal to the power of background noise, expressed in decibels (dB), that quantifies the quality of a signal in a communication or measurement system. A higher SNR indicates a cleaner, more detectable signal relative to noise, while a low SNR indicates the signal is buried in noise. SNR is a critical parameter in audio engineering, telecommunications, radar, medical imaging (MRI), and data acquisition systems, directly determining the fidelity, range, and reliability of signal transmission and detection.
Named after French mathematician and physicist Jean-Baptiste Joseph Fourier (1768–1830), who introduced the concept in his 1822 treatise "Théorie analytique de la chaleur" (The Analytical Theory of Heat) to solve the heat equation. He proposed that any periodic function could be expressed as a sum of sinusoids — a revolutionary idea initially resisted by contemporaries such as Lagrange.