Now, what if we __convolve__ *x*(*t*) with an impulse?

__Example 2__ Find *x*(*t*) *
*δ*(*t* - *t _{0}*) =

Now, *δ*(*t* - *τ* - *t _{0}*)
is an impulse at

So, if you convolve *x*(*t*) with a shifted impulse, you just get the function
*x*(*t*) shifted to where the impulse occurs.

In other words,

*x*(*t*) * *δ*(*t* -
*t _{0}*) =

We'll see this a lot. One important place we'll see this is when we discuss SAMPLING or discretizing a continuous time signal.

__NOW, IN EXAMPLE 3, YOU
WILL BE ASKED TO WORK THROUGH THE PROBLEM
STEP-BY-STEP.
FOR ADDITIONAL
PRACTICE IN PERFORMING CONVOLUTION, WORK EACH EXAMPLE TWICE.
THE FIRST TIME, FLIP-AND-SHIFT x(t), AND THE SECOND TIME
FLIP-AND-SHIFT h(t).__

__Example 3__ Since convolution is
commutative, do this example twice. One time pick the unit step function
to time reverse and the next time, pick the exponential.

From before:

Sifting property of impulse:

"This sifts out the value of
the function *x*(*t*) where impulse occurs"

__Example 4__

__Example 5__

__Example 6__