Calculus

Laplace Transform

The Laplace transform sends a signal f(t) to a function F(s) of a complex variable. Plot |F(s)| as a landscape over the complex plane and the algebra turns geometric: poles are spikes, the region of convergence is a half-plane, and the Fourier transform is just the slice above the imaginary axis.

Signal f(t), t ≥ 0

Decay a 0.60

Frequency ω 2.50

Shade the region of convergence Slice above the imaginary axis

F(s)—

Poles—

ROC—

Fourier transform—

Drag the background to orbit. Height is |F(s)| over the complex plane s = σ + jω, clipped at the top so the poles do not run off to infinity. Red pins mark the poles, the amber curve is the slice at σ = 0.

What to observe

Lower the decay a towards zero. The pole at s = −a slides right until it sits on the imaginary axis, and the whole landscape lifts near it: a signal that never dies out makes the transform blow up on the axis.
Push a negative. The signal now grows, the ROC half-plane retreats to the right of the axis, and the amber slice no longer lives inside it. That is exactly when the Fourier transform stops existing while the Laplace transform is still perfectly fine.
Switch to e^−atsin ωt and raise ω. The single spike splits into a conjugate pair at s = −a ± jω, and the amber slice grows two resonant humps at those frequencies: poles near the axis are exactly what a sharp resonance looks like from the frequency side.
Compare e^−at with t·e^−at. Same pole location, but the double pole makes a taller, broader spike: repeated poles are polynomial factors of t in time.

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