Fourier Series Approximation

Approximating periodic functions by summing sinusoidal waves

f(t) = (4/π)[sin(t) + (1/3)sin(3t) + (1/5)sin(5t) + ...]
Target Waveform
Approximation
Harmonics
Amplitude Spectrum
Current Terms: 1, 3, 5, 7, 9

Fourier Coefficients

Harmonic Frequency Coefficient Contribution

Statistics

Total Harmonics
5
Approximation Error (MSE)
0.123
Convergence Rate
1/n

What is Fourier Series Approximation?

Fourier series approximation is a method of representing periodic functions as a sum of sine and cosine functions. For common waveforms like square and sawtooth waves, we can approximate them by summing harmonics of different frequencies and amplitudes.

Key Concepts:

Harmonic Addition

By superimposing the fundamental wave and harmonics (frequencies that are integer multiples of the fundamental), we gradually approach the target waveform. The amplitude of each harmonic determines its contribution to the final waveform.

Convergence

As the number of harmonics increases, the approximate waveform gets closer to the target waveform. Square wave converges at 1/n, sawtooth wave at 1/n².

Gibbs Phenomenon

Near discontinuities, overshoot and oscillation persist even with many harmonics. This is an inherent property of Fourier series, with overshoot of about 9% of the jump.

Fourier Series for Common Waveforms:

Square Wave

f(t) = (4/π)[sin(t) + (1/3)sin(3t) + (1/5)sin(5t) + ...]

Odd harmonics only, coefficient 1/n

Sawtooth Wave

f(t) = (2/π)[sin(t) - (1/2)sin(2t) + (1/3)sin(3t) - ...]

All harmonics, alternating signs, coefficient 1/n

Triangle Wave

f(t) = (8/π²)[sin(t) - (1/9)sin(3t) + (1/25)sin(5t) - ...]

Odd harmonics only, coefficient 1/n², faster convergence

Applications:

How to Use:

  1. Select target waveform type (square, sawtooth, triangle, etc.)
  2. Adjust 'Number of Harmonics' slider to observe approximation changes
  3. View amplitude spectrum to understand frequency component contributions
  4. Study coefficient table to understand Fourier coefficient calculation
  5. Check 'Animate Approximation Process' to observe harmonic superposition