Proof: y(t)=x(t)*(h_1(t)*h_2(t))=(x(t)*h_1(t))*h_2(t)
Given:
- $ y(t)=x(t)*h(t)=\int_{k=-\infty}^{\infty}x(\tau)h(t-\tau)d\tau $
- $ y(t)=x(t)*h(t)=h(t)*x(t) $ commutative property of convolution for continuous time
Steps:
- $ x(t)*(h_1(t)*h_2(t))=x(t)*(h_2(t)*h_1(t)) $ commutative property of convolution for continuous time
- $ x(t)*(h_1(t)*h_2(t))=x(t)*\int_{-\infty}^{\infty}h_2(\tau)h_1(t-\tau)d\tau $
- $ x(t)*(h_1(t)*h_2(t))=\int_{-\infty}^{\infty}x(\mu)\int_{-\infty}^{\infty}h_2(\tau)h_1(t-\tau-\mu)d\tau d\mu $
- $ x(t)*(h_1(t)*h_2(t))=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x(\mu)h_2(\tau)h_1(t-\tau-\mu)d\tau d\mu $
- $ x(t)*(h_1(t)*h_2(t))=\int_{-\infty}^{\infty}h_2(\tau)\int_{-\infty}^{\infty}x(\mu)h_1(t-\tau-\mu)d\mu d\tau $
- $ x(t)*(h_1(t)*h_2(t))=h_2(t)*\int_{-\infty}^{\infty}x(\mu)h_1(t-\mu)d\mu $
- $ x(t)*(h_1(t)*h_2(t))=h_2(t)*(x(t)*h_1(t)) $
- $ x(t)*(h_1(t)*h_2(t))=(x(t)*h_1(t))*h_2(t) $ commutative property of convolution for continuous time
