Fourier Transform Calculator: Analyze Signals with DFT/FFT
Example: 1 0 -1 0 1 0 -1 0
Fourier Transform Results
Number of points:
Frequency resolution: Hz
Nyquist frequency: Hz
Time Domain Signal
Frequency Spectrum
| Index | Frequency (Hz) | Value |
|---|---|---|
Understanding Fourier Transform: A Comprehensive Guide
The Fourier Transform is a fundamental mathematical tool that decomposes a time-domain signal into its constituent frequencies. This powerful technique is widely used in signal processing, audio engineering, image analysis, and many scientific fields.
How to Use This Fourier Transform Calculator:
- Select input method - Enter values manually, define a mathematical function, or choose from sample signals
- Choose transform type - DFT for small datasets (≤1000 points) or FFT for larger datasets (power of 2)
- Set output format - View complex numbers, magnitude, phase, or individual components
- Click Calculate - See both time-domain and frequency-domain visualizations
Key Concepts in Fourier Analysis:
- Time Domain vs Frequency Domain: Understand signal representation in different domains
- Nyquist Theorem: Sampling rate must be at least twice the highest frequency
- Frequency Resolution: Δf = 1/T where T is the total sampling duration
- Aliasing: High frequencies appearing as low frequencies due to undersampling
- Leakage: Energy spreading across frequencies due to finite sampling
Common Fourier Transform Pairs:
- • Sine wave: Single frequency peak at the sine frequency
- • Square wave: Odd harmonics with amplitudes decreasing as 1/n
- • Impulse function: Flat spectrum (all frequencies equally present)
- • Constant signal: Only DC component (0 Hz) present
- • Gaussian pulse: Fourier Transform is also a Gaussian
Practical Applications of Fourier Transform:
- • Audio Processing: Equalizers, compression, noise reduction
- • Image Compression: JPEG uses Discrete Cosine Transform (DCT)
- • Medical Imaging: MRI reconstruction uses Fourier methods
- • Vibration Analysis: Identifying machinery faults
- • Wireless Communications: OFDM modulation
- • Seismology: Earthquake frequency analysis
Technical Notes:
This calculator implements both the Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) algorithms. The FFT is automatically selected when the input length is a power of 2 (32, 64, 128, etc.). For educational purposes, the DFT is used for other input lengths. In practice, FFT is preferred for its O(N log N) computational efficiency compared to DFT's O(N²) complexity.