1. Rational Transform F(s)

F(s) =
e-2ss+1
Isolate the delay:
F(s) = e-2s ·
1s+1

2. s-plane & Poles

Pole at s = -1. The e-2s term is a pure delay.

3. Partial Fraction Expansion

e-2s · [
1s+1
]

4. Time-Domain Signal f(t)

f(t) = e-(t-2) u(t-2)

Pole Type → Time-Domain Signature

Pole Location Term in F(s) Time Domain f(t)
Real: s = -p
1s+p
 e-pt u(t)
Repeated: s = -p (x2)
1(s+p)2
 t e-pt u(t)
Complex: -σ ± jω
ω(s+σ)2+ω2
 e-σt sin(ωt) u(t)
(Numerator delay) e-as Shift right: f(t-a)

What to notice

  • The exponential term e-2s in the numerator represents a time shift, not a pole.
  • First find the inverse transform of the rational part (which is e-tu(t)).
  • Then shift every 't' to 't-2' to get the final delayed response.