Fourier Series Visualizer
Interactive tool for signal analysis and function decomposition
Visualize how periodic functions can be represented as infinite sums of sines and cosines. Perfect for students, engineers, and researchers studying signal processing, harmonic analysis, and mathematical physics.
Controls
Function vs Fourier Series
Fourier Coefficients
How to Use This Tool
1. Select a Function
Choose from predefined functions (square wave, triangle, sawtooth) or enter a custom mathematical expression using variable x.
2. Adjust Parameters
Control the number of terms, period, and series type to see how they affect the approximation quality.
3. Animate Convergence
Use the Animate button to see how adding more terms progressively improves the approximation.
4. Analyze Results
View Fourier coefficients, mathematical formulas, and RMS error to understand the decomposition.
Mathematical Background
Fourier Series Formula
Any periodic function f(x) can be represented as:
f(x) = a₀/2 + Σ[aₙcos(nωx) + bₙsin(nωx)]
Coefficients Calculation
Uses Simpson's rule numerical integration to compute Fourier coefficients aₙ and bₙ with high accuracy.
Applications
Signal processing, image compression, solving differential equations, acoustics, and electromagnetic field analysis.
Custom Function Examples
Linear Function
x Simple linear ramp function
Parabola
x^2 Quadratic function
Sine Harmonics
sin(x) + sin(3*x)/3 Fundamental plus third harmonic
Exponential Decay
exp(-abs(x)) Double-sided exponential
Gaussian Pulse
exp(-x^2) Bell curve function
Absolute Value
abs(sin(x)) Rectified sine wave
Features & Applications
Key Features
- ✓ Real-time coefficient calculation using Simpson's rule
- ✓ Interactive animation showing series convergence
- ✓ Support for custom mathematical expressions
- ✓ Coefficient visualization with bar charts
- ✓ Mathematical formula display with MathJax
- ✓ RMS error analysis for approximation quality
- ✓ CSV data export for further analysis
Educational Applications
- → Engineering signal analysis and filtering
- → Physics wave mechanics and acoustics
- → Mathematics harmonic analysis courses
- → Computer science algorithm visualization
- → Research in frequency domain analysis
- → Understanding JPEG compression principles
About Fourier Series Analysis
Fourier series analysis is a fundamental mathematical technique that decomposes periodic functions into combinations of simple sine and cosine waves. This powerful method, developed by Joseph Fourier in the early 19th century, has revolutionized fields ranging from signal processing to quantum mechanics.
Our interactive Fourier Series Visualizer provides an intuitive way to understand this complex mathematical concept. By visualizing how different harmonic components combine to recreate the original function, users can gain deep insights into frequency domain analysis and the nature of periodic signals.
The tool supports both predefined functions (square waves, triangle waves, sawtooth waves) and custom mathematical expressions, making it versatile for educational purposes, research, and practical applications in engineering and physics. The real-time coefficient calculation and interactive animation features make complex mathematical concepts accessible and engaging.
Whether you're a student learning about harmonic analysis, an engineer working with signal processing, or a researcher exploring frequency domain techniques, this visualizer provides the interactive experience needed to truly understand Fourier series decomposition and its practical applications in science and technology.
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Fourier Series Visualizer
Interactive tool for visualizing Fourier series decomposition with real-time coefficient calculation, animation, and function approximation for signal analysis.
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