Analyze real and imaginary spectra for common signals. Adjust limits, samples, frequencies, and parameters instantly. See transforms, compare behavior, and understand integral structure better.
The page uses a single-column structure, while the input fields switch to 3, 2, or 1 columns by screen size.
These sample settings show how different signal models influence the transform and help you test the calculator quickly.
| Signal | A | Primary Shape Parameter | ω₀ | Interval | Suggested ω Range | Use Case |
|---|---|---|---|---|---|---|
| Gaussian Pulse | 1.0 | σ = 0.75 | 0 | -10 to 10 | -15 to 15 | Smooth spectrum study |
| Rectangular Pulse | 1.0 | W = 2.0 | 0 | -10 to 10 | -25 to 25 | Sinc-shaped spectrum |
| One-Sided Exponential | 2.0 | α = 1.5 | 0 | -2 to 12 | -20 to 20 | Decay response analysis |
| Damped Cosine | 1.0 | α = 0.8 | 5.0 | -10 to 10 | -15 to 15 | Resonance-like peaks |
F(ω) = ∫ f(t)e-iωt dt
F(ω) = ∫ f(t)cos(ωt) dt - i∫ f(t)sin(ωt) dt
∫ g(t) dt ≈ Δt [0.5g(t₀) + g(t₁) + ... + 0.5g(tₙ)]
|F(ω)| = √(Re² + Im²)Phase = atan2(Im, Re)
It numerically evaluates the continuous Fourier transform integral for several common signal models. It returns the complex value at a chosen ω, estimates the full spectrum over a range, displays magnitude and phase, and plots both the signal and transform behavior.
A computer must approximate the infinite integral over a practical time window. Wider limits usually improve accuracy for slowly decaying signals, while shorter limits can still work well for compact pulses or strongly damped signals.
Use more samples when the signal oscillates quickly, has narrow features, or when you need smoother plots. Start around 1501 samples, then increase to compare stability. If the answer barely changes, your grid is probably adequate.
They show how strongly the signal aligns with cosine and sine components at a chosen angular frequency. Together they form the full complex transform, from which magnitude and phase are computed.
Choose Gaussian for smooth localized pulses, rectangular for abrupt windows, exponential for decays that start at a time, cosine for pure oscillation, damped cosine for oscillation with loss, and sinc for a broad time signal with a compact spectral character.
Fourier transforms can turn hard integrals into easier algebraic forms. You rewrite the integrand as a transform pair, use known transforms, apply convolution or differentiation properties, then evaluate the transformed expression and convert back if needed.
No. A Fourier transform is a linear integral transform over an ordinary variable. A path integral sums contributions over an entire space of trajectories or functions. They can both use oscillatory exponentials, but they are different mathematical constructions.
The numeric result uses a finite interval and discrete sampling, while the closed form assumes the exact continuous integral over its ideal domain. Differences shrink when you widen the interval, increase samples, and choose parameters suited to the model.
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.