Enter signal parameters
This tool applies common continuous-time Fourier transform pairs using the convention X(ω) = ∫-∞∞ x(t)e-jωtdt.
Worked examples for common signal families
| Signal family | Inputs | Symbolic transform | Key observation |
|---|---|---|---|
| Gaussian | A = 1, a = 0.5, t₀ = 0 | X(ω) = √(2π)e-ω2/2 | Gaussian maps to Gaussian. |
| Rectangular pulse | A = 2, T = 3, t₀ = 0 | X(ω) = 6 sinc(1.5ω) | Wider pulse, narrower main lobe. |
| One-sided exponential | A = 4, a = 2, t₀ = 0 | X(ω) = 4/(2 + jω) | Complex spectrum with decaying envelope. |
| Cosine | A = 3, ω0 = 5, t₀ = 0 | X(ω) = 3π[δ(ω-5)+δ(ω+5)] | Two spectral lines appear. |
Core transform relationships
Continuous-time Fourier transform: X(ω) = ∫-∞∞ x(t)e-jωtdt
Inverse transform: x(t) = (1/2π) ∫-∞∞ X(ω)ejωtdω
Time-shift property: If x(t - t₀) has transform X(ω)e-jωt₀, the magnitude stays unchanged while the phase becomes linear.
Common symbolic pairs included here:
- Gaussian: e-at² ↔ √(π/a)e-ω2/(4a), a > 0
- Two-sided exponential: e-a|t| ↔ 2a/(a² + ω²), a > 0
- Rectangular pulse: A·rect(t/T) ↔ A·T·sinc(ωT/2)
- One-sided exponential: A·e-atu(t) ↔ A/(a + jω), a > 0
- Impulse: Aδ(t - t₀) ↔ A·e-jωt₀
- Cosine and sine create weighted impulse lines at ±ω0
The graph uses sampled values for visualization. For sine, cosine, and impulse terms, true transforms involve distributions, so spikes are shown as practical display approximations.
Simple workflow
- Choose the signal family that best matches your expression.
- Enter amplitude A and the main parameter such as a, T, or ω₀.
- Add a time shift t₀ when the signal is displaced in time.
- Set the frequency range and number of graph samples.
- Press Submit to display the symbolic transform above the form.
- Review the transform formula, DC value, peak estimate, and symmetry note.
- Inspect the graph and the computed sample table.
- Export the spectrum data as CSV or generate a PDF report.
Frequently asked questions
1) What does “symbolic” mean here?
It means the tool returns a closed-form transform pair for standard signal families instead of only producing numeric integration results. You get formulas, conditions, and interpretation notes.
2) Which Fourier transform convention is used?
The calculator uses the continuous-time engineering convention X(ω) = ∫ x(t)e-jωtdt with angular frequency ω measured in radians per second.
3) Why do cosine and sine show spikes?
Pure sinusoids transform into impulses at positive and negative carrier frequencies. A plotted graph cannot display ideal impulses exactly, so narrow spikes are used as a visual approximation.
4) Why does a time shift change phase only?
A time shift multiplies the spectrum by e-jωt₀. This factor has magnitude 1, so amplitude remains unchanged while phase rotates linearly with frequency.
5) What is sinc in the rectangular pulse result?
This page uses sinc(u) = sin(u)/u, with sinc(0) defined as 1. It naturally appears because the rectangular pulse integrates to a sinusoidal-over-argument expression.
6) Why must parameters like a or T be positive?
Positive values ensure the selected transform pair is valid and well behaved. Negative widths or unstable decay rates usually break the assumed standard formula.
7) Why is the one-sided exponential spectrum complex?
The one-sided exponential is not even in time, so its transform usually has both real and imaginary parts. That is why phase and complex values matter for this case.
8) Can I use this for arbitrary custom formulas?
This version focuses on widely used symbolic pairs and their shifted forms. For arbitrary expressions, you would typically need a full computer algebra system.