Chaptre 1

Propertys

• Commutativity : \(y(t) = 𝑥(𝑡) ∗ ℎ(𝑡) = ℎ(𝑡) ∗ 𝑥(𝑡)\)

• Associativity : \(y(t) =𝑥(𝑡) ∗ ℎ(𝑡) = 𝑥(𝑡) ∗ [ℎ1 (𝑡) ∗ ℎ2 (𝑡)] = [𝑥(𝑡) ∗ ℎ1 (𝑡)] ∗ ℎ2(𝑡)\)

• Distributivity : This property is the consequence of the linearity of the integrals. \(y(t)=𝑥(𝑡) ∗ ℎ1 (𝑡) + 𝑥(𝑡) ∗ ℎ2 (𝑡) = 𝑥(𝑡) ∗ [ℎ1 (𝑡) + ℎ2 (𝑡)]\)

• The neutral element : \(f\left(t\right)\ast\delta(t)=\int_{-\infty}^{+\infty}{f\left(\tau\right)\delta\left(t-\tau\right)d\tau=}f(t)\)