s-plane (Drag the 'X')

Time-Domain Mode
h(t) vs t

System Stability & Characteristic Mode

Behavior:
Decaying Oscillation
BIBO Stability:
BIBO Stable
Time-Domain Formula:
h(t) = e-0.16t cos(4.85t) u(t)
Recall: The pole location p = σ + jω directly determines the characteristic mode. The real part (σ) governs exponential growth/decay, and the imaginary part (ω) governs the oscillation frequency.

What to notice

  • Drag the poles in the s-plane to see the time-domain mode change instantly.
  • For a causal impulse response, poles in the Left Half-Plane (LHP) produce decaying modes and are BIBO stable.
  • Poles in the Right Half-Plane (RHP) produce growing signals that eventually blow up (Unstable).
  • Moving a pole vertically changes the frequency of oscillation without affecting the decay/growth rate.
  • Poles exactly on the jω-axis cause sustained modes. They are marginal as natural modes, but not BIBO stable as impulse responses.
  • A pole exactly at the origin (s=0) produces u(t), which is not absolutely integrable and is not BIBO stable.