View source for Associative Property for Continous Time (ECE301Summer2008asan) - Rhea

[[Category: ECE]]
[[Category: ECE 301]]
[[Category: Summer]]
[[Category: 2008]]
[[Category: asan]]
[[Category: Bonus]]
Proof: y(t)=x(t)*(h_1(t)*h_2(t))=(x(t)*h_1(t))*h_2(t)

Given:
#<math>y(t)=x(t)*h(t)=\int_{k=-\infty}^{\infty}x(\tau)h(t-\tau)d\tau</math>
#<math>y(t)=x(t)*h(t)=h(t)*x(t)</math> commutative property of convolution for continuous time

Steps:
#<math>x(t)*(h_1(t)*h_2(t))=x(t)*(h_2(t)*h_1(t))</math> commutative property of convolution for continuous time
#<math>x(t)*(h_1(t)*h_2(t))=x(t)*\int_{-\infty}^{\infty}h_2(\tau)h_1(t-\tau)d\tau</math>
#<math>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</math>
#<math>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</math>
#<math>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</math>
#<math>x(t)*(h_1(t)*h_2(t))=h_2(t)*\int_{-\infty}^{\infty}x(\mu)h_1(t-\mu)d\mu</math>
#<math>x(t)*(h_1(t)*h_2(t))=h_2(t)*(x(t)*h_1(t))</math>
#<math>x(t)*(h_1(t)*h_2(t))=(x(t)*h_1(t))*h_2(t)</math> commutative property of convolution for continuous time