Examples:

1) Consider a signal

y(t) = x₁(t - 2) * x₂ (-t + 3)

where

x₁(t) = e^-2t u(t) & x₂ = e^-3t u(t)

given that

e^-at u(t) ↔ 1 / (s + a) , Re{s} > a

Use properties of the Laplace transform to determine the Laplace transform Y(s) of y(t).

Solution :

x₁(t) ↔ 1 / (s + 2) , Re{s} > 2

x₂(t) ↔ 1 / (s + 3) , Re{s} > 3

x₁(t - 2) ↔ e^-2s / (s + 2) , Re{s} > 2 (using time shifting property)

x₂(t + 3) ↔ e^3s / (s + 3) , Re{s} > 3 (using time shifting property)

x₂(-t + 3) ↔ e^-3s / (-s + 3) , Re{s} > -3 (using time reversal property)

y(t) ↔ (e^-2s / (s + 2))(e^-3s / (-s + 3)) (using convolution property)

2) Consider a signal x[n] = −αⁿ u[-n - n_o] + δ[n - 2]

Use properties of z-transform to determne the z-transform of the above signal.

Solution :

δ[n - 2] ↔ z^-2 , R₁ = all z except origin 

u[n] ↔ 1/(1 - z^-1) , |z| > 1

u[n - k] ↔ z^-k / (1 - z^-1) , |z| > 1 using time shifting property

u[-n - k] ↔ z^k / (1 - z) , |z| < 1 using time reversal property

αⁿ u[-n - k]↔ (α^-1z)^k / (1 - (α^-1z)) , R₂ = |α^-1z| < 1 using modulation property

X(z) = z^-2 + ((α^-1z)^k) / (1 - (α^-1z)) , R₁ ∩ R₂ ⊂ R

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