Fourier Transform Family - Interactive Visualization

The Fourier Transform Family

Core Concept

"Projecting signals onto sinusoidal bases of different frequencies to find the weight (coefficient) of each basis"

Transform Hierarchy (Theory → Practice)

Name	Input Signal	Output	Complexity	Implementation
Continuous FT	Continuous, Aperiodic	Continuous Spectrum	—	Nearly Impossible
Fourier Series	Continuous (Periodic)	Discrete Spectrum	—	Difficult
DTFT	Discrete, Aperiodic	Continuous (Periodic)	—	Nearly Impossible
DFT	Discrete, Finite	Discrete Spectrum	O(N²)	Slow but Possible
FFT	Same as DFT	Same as DFT	O(N log N)	★ Fast & Easy ★

Twiddle Factor (旋转因子)

W_N = e^-j2π/N

The key to understanding FFT - equally spaced points on the unit circle

W_N^N = 1 Periodicity: W^N = 1

W_N^N/2 = -1 Symmetry: W^(N/2) = -1

W_N/2 = W_N² Reductions: W_{N/2} = W²

Time Domain ↔ Frequency Domain

Explore how different time-domain signals look in the frequency domain

Signal Type

Frequency: 1 Hz

Amplitude: 1

Harmonics (for approximation): 5

Sample Rate: 100 Hz

Window Size (N): 256

DFT Formula

X[k] = Σ_n=0^N-1 x[n] · e^-j2πkn/N

Time Domain Signal

Frequency Spectrum (Magnitude)

Phase Spectrum

Fourier Series Approximation

Watch how periodic signals are constructed from harmonics

Wave Type

Number of Harmonics: 1

Show Individual Harmonics

Animate Phases

Fourier Series Formula

x(t) = a₀ + Σ_n=1^∞ (a_ncos(nω₀t) + b_nsin(nω₀t))

Signal Approximation (Green = Current, Red = Target)

Individual Harmonics

Fourier Coefficients

FFT Butterfly Diagram

Visualize the Cooley-Tukey radix-2 FFT algorithm

FFT Size (N): 8

Animation Speed

Show Complex Values

Highlight Stage

Butterfly Operation

A' = A + W_N^k · B
B' = A - W_N^k · B

Bit-Reversal Permutation

Signal Flow Graph

DFT vs FFT Performance

Compare computational complexity: O(N²) vs O(N log N)

Max N to Test: 4096

Theoretical Operation Count

N	DFT (N²)	FFT (N log₂N)	Speedup

Execution Time (ms)

Operation Count

Speedup Ratio

Real-time Audio Spectrum Analysis

Visualize audio frequency content in real-time using FFT

Audio Source

Oscillator Frequency: 440 Hz

Oscillator Wave Type

FFT Size: 2048

Time Constant Smoothing: 0.8

Display Mode

Frequency Spectrum

Time Waveform

Peak Frequency -- Hz

Peak Amplitude -- dB

Real-World Applications (2025)

🎵

Audio Processing

MP3/AAC compression, echo cancellation, pitch shifting, pitch detection

★★★★★ Essential

📡

Wireless Communication

OFDM (4G/5G/6G/WiFi), frequency domain equalization

★★★★★ Core

🖼️

Image Processing

JPEG compression, denoising, sharpening, super-resolution

★★★★☆ Important

🏥

Medical Imaging

MRI reconstruction, CT algorithms

★★★★★ Critical

📡

Radar/Sonar

Pulse compression, target detection

★★★★☆ Core

🌍

Seismic Processing

Tomography, denoising

★★★★ Very Important

🧮

Large Integer/Poly Multiplication

Cryptography, computer algebra (Schönhage–Strassen)

★★★★☆ Core

🤖

Machine Learning

FNet, time-series acceleration, attention alternatives

★★★→★★★★ Growing