3.8 Properties of convolution integrals

We list several important properties and their proofs.

1. Commutative Property:
   $$x\left(t\right)*h\left(t\right)=h\left(t\right)*x\left(t\right)$$
   Lets start with
   $$\begin{array}{ccc}\hfill x\left(t\right)*h\left(t\right)& =& {\int }_{-\infty }^{\infty }x\left(\tau \right)h(t-\tau )d\tau \hfill \end{array}$$
   and make the substitution $\gamma =t-\tau$ . It follows that
   $$\begin{array}{ccc}\hfill x\left(t\right)*h\left(t\right)& =& {\int }_{-\infty }^{\infty }x(t-\gamma )h\left(\gamma \right)d\gamma d\tau \hfill \\ & =& h\left(t\right)*x\left(t\right)\hfill \end{array}$$
2. Associative Property:
   $$\left[x,\left(t\right),*,h_1,\left(t\right)\right]*h_2\left(t\right)=x\left(t\right)*\left[h_1,\left(t\right),*,h_2,\left(t\right)\right]$$
   To prove this property we begin with an expression for the left-hand side of [link]
   $${\int }_{-\infty }^{\infty }x\left(\tau \right)h_1(t-\tau )d\tau *h_2\left(t\right)$$
   where we have expressed $x\left(t\right)*h_1\left(t\right)$ as a convolution integral. Expanding the second convolution gives
   $${\int }_{-\infty }^{\infty }\left[{\int }_{-\infty }^{\infty },x,\left(\tau \right),h_1,(\gamma -\tau ),d,\tau \right]h_2(t-\gamma )d\gamma$$
   Reversing the order of integration gives
   $${\int }_{-\infty }^{\infty }x\left(\tau \right)\left[{\int }_{-\infty }^{\infty },h_1,(\gamma -\tau ),h_2,(t-\gamma ),d,\gamma \right]d\tau$$
   Using the variable substitution $\phi =\gamma -\tau$ and integrating over $\phi$ in the inner integral gives the final result:
   $${\int }_{-\infty }^{\infty }x\left(\tau \right)\left[{\int }_{-\infty }^{\infty },h_1,\left(\phi \right),h_2,(t-\tau -\phi ),d,\gamma \right]d\tau$$
   where the inner integral is recognized as $h_1\left(t\right)*h_2\left(t\right)$ evaluated at $t=t-\tau$ , which is required for the convolution with $x\left(t\right)$ .
3. Distributive Property:
   $$x\left(t\right)*\left[h_1,\left(t\right),+,h_2,\left(t\right)\right]=x\left(t\right)*h_1\left(t\right)+x\left(t\right)*h_2\left(t\right)$$
   This property is easily proven from the definition of the convolution integral.
4. Time-Shift Property: If $y\left(t\right)=x\left(t\right)*h\left(t\right)$ then $x(t-t_0)*h\left(t\right)=y(t-t_0)$ Again, the proof is trivial.