Exponential Fourier Series Calculator

Turn periodic samples into clean complex series terms. Choose harmonics view magnitude and phase instantly. Download coefficient reports and share results with classmates easily.

Calculator

If blank, you can infer T from x-range.
Computes cₖ for k = −N…N.
Higher values create smoother plots.
Enter one point per line. Use comma or spaces. Keep x increasing across one period.

Example data table

Example: a repeating waveform with period 6.

xf(x)Meaning
01Start of the period
11High segment continues
2-1Signal flips sign
3-1Low segment continues
41Return to high
51Approach the period end

Formula used

Exponential form
f(x) ≈ Σk=-NN ck ei k ω₀ x,   ω₀ = 2π/T
ck = (1/T) ∫x₀x₀+T f(x) e-i k ω₀ x dx
Integrals are approximated using the trapezoidal rule on your samples.
Real form
f(x) ≈ a₀/2 + Σn=1N [an cos(nω₀x) + bn sin(nω₀x)]
an = (2/T) ∫ f(x) cos(nω₀x) dx,  bn = (2/T) ∫ f(x) sin(nω₀x) dx
For real signals: c₀ = a₀/2 and cₙ = (aₙ − i bₙ)/2.

How to use this calculator

  1. Paste one period of samples as x, f(x) pairs.
  2. Enter period T, or enable inference from x-range.
  3. Choose N harmonics to control approximation detail.
  4. Press Compute Series to view coefficients and reconstruction.
  5. Use Download CSV or Download PDF for reporting.

Tip: More samples across the period usually improve accuracy.

Related Calculators

continuous fourier transform calculatorlinear convolution calculator

Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.