Time Domain Signal
f(t)

Complex s-plane
s = σ + jω

Weighted Signal Area
f(t)e^{-σt}

What to notice

  • Move the σ slider to change the real part of the complex frequency variable s.
  • Watch how the e^{-σt} term reweights the signal in the third panel.
  • If the weighted signal completely decays, its integral is finite. This means the Laplace transform exists for that specific σ.
  • The Fourier transform is simply the Laplace transform evaluated exactly on the jω axis (σ = 0).
  • If the jω axis lies inside the shaded Region of Convergence (ROC), the Fourier transform exists.

System Analysis Status

Laplace Transform exists at current σ?
Yes
Fourier Transform exists (σ = 0 in ROC)?
Yes
BIBO Stable (if this is impulse response h(t))?
Yes

Note: For causal LTIC systems, BIBO stability requires all poles to be in the strict left-half plane, which guarantees the ROC includes the jω-axis.