Proof: y(t)=x(t)*(h_1(t)*h_2(t))=(x(t)*h_1(t))*h_2(t)

Given:

1. $y(t)=x(t)*h(t)=\int_{k=-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$
2. $y(t)=x(t)*h(t)=h(t)*x(t)$ commutative property of convolution for continuous time

Steps:

1. $x(t)*(h_1(t)*h_2(t))=x(t)*(h_2(t)*h_1(t))$ commutative property of convolution for continuous time
2. $x(t)*(h_1(t)*h_2(t))=x(t)*\int_{-\infty}^{\infty}h_2(\tau)h_1(t-\tau)d\tau$
3. $x(t)*(h_1(t)*h_2(t))=\int_{-\infty}^{\infty}x(\mu)\int_{-\infty}^{\infty}h_2(\tau)h_1(t-\tau-\mu)d\tau d\mu$
4. $x(t)*(h_1(t)*h_2(t))=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x(\mu)h_2(\tau)h_1(t-\tau-\mu)d\tau d\mu$
5. $x(t)*(h_1(t)*h_2(t))=\int_{-\infty}^{\infty}h_2(\tau)\int_{-\infty}^{\infty}x(\mu)h_1(t-\tau-\mu)d\mu d\tau$
6. $x(t)*(h_1(t)*h_2(t))=h_2(t)*\int_{-\infty}^{\infty}x(\mu)h_1(t-\mu)d\mu$
7. $x(t)*(h_1(t)*h_2(t))=h_2(t)*(x(t)*h_1(t))$
8. $x(t)*(h_1(t)*h_2(t))=(x(t)*h_1(t))*h_2(t)$ commutative property of convolution for continuous time

## Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood