Fourier Transform Calculator
Compute Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) with comprehensive frequency domain analysis. Our calculator supports signal generation, time series input, and advanced spectral analysis with window functions, providing detailed insights for digital signal processing, audio analysis, and engineering applications.
Last updated: December 15, 2024
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Signal Generation
Sampling Parameters
Fourier Transform Results
Transform Properties:
Signal Analysis:
Frequency Spectrum (First 10 bins):
Transform Formulas:
Calculation Steps:
- Step 1: Initialize Fourier Transform calculation
- Step 2: Transform type = FFT
- Step 3: Sampling rate = 100 Hz, Duration = 1 s
- Step 4: Generated sine signal with frequency 10 Hz
- Step 5: Signal amplitude = 1, phase = 0 radians
- Step 6: Computed FFT with 100 frequency bins
- Step 7: Peak frequency = 10.000 Hz
- Step 8: DC component = 0.000
- Step 9: Total energy = 50.000
Quick Example Result
For sine wave at 10 Hz, amplitude 1, sampling rate 100 Hz:
Peak Frequency = 10.0 Hz
How This Calculator Works
Our Fourier Transform calculator implements advanced digital signal processing algorithms to convert time-domain signals into their frequency-domain representations. The calculator supports both DFT and FFT algorithms, signal generation with various waveforms, time series data input, and comprehensive spectral analysis with window functions for professional signal processing applications in engineering, audio processing, and scientific research.
Fourier Transform Algorithms
Discrete Fourier Transform (DFT):
X[k] = Σ x[n] × e^(-j2πkn/N)Magnitude Spectrum:
|X[k]| = √(Re²[k] + Im²[k])Phase Spectrum:
φ[k] = arctan(Im[k]/Re[k])Shows time-domain signal transformation to frequency domain with magnitude and phase plots
Mathematical Foundation
The Fourier Transform, developed by Jean-Baptiste Joseph Fourier, is a fundamental mathematical tool that decomposes signals into their constituent frequencies. The Discrete Fourier Transform adapts this concept for digital signals, while the Fast Fourier Transform provides computational efficiency through the Cooley-Tukey algorithm. Window functions like Hann, Hamming, and Blackman reduce spectral leakage and improve frequency resolution in practical applications.
- Frequency decomposition: Breaking signals into sinusoidal components
- Complex representation: Real and imaginary parts encode magnitude and phase
- Sampling theorem: Nyquist criteria for accurate digital representation
- Window functions: Spectral leakage reduction and frequency resolution enhancement
Sources & References
- Digital Signal Processing Textbook - Oppenheim, Schafer & BuckComprehensive coverage of DFT/FFT theory and applications
- IEEE Signal Processing Society - Standards and Best PracticesProfessional standards for digital signal processing implementations
- MIT OpenCourseWare - Signals and SystemsAcademic foundation for Fourier analysis and signal processing
Need help with other signal processing tools? Check out our convolution calculator and Z-transform calculator.
Get Custom Signal Processing ToolsExample Calculation
Signal Parameters:
- Function: Sine wave
- Frequency: 10 Hz
- Amplitude: 1
- Sampling Rate: 100 Hz
- Duration: 1 second
Analysis Results:
- Signal length: 100 samples
- Transform type: FFT
- Peak frequency: 10.0 Hz
- DC component: 0.000
- Total energy: 50.0
Result: The FFT correctly identifies the peak frequency at 10.0 Hz with minimal DC component.
This demonstrates how Fourier Transform reveals the frequency content of time-domain signals for spectral analysis.
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