Now, what if we convolve x(t) with an impulse?
Example 2 Find x(t) * δ(t - t0) = y(t),
Now, δ(t - τ - t0) is an impulse at τ = t - t0. So, by the SIFTING property, we sift out the value of x(t) where the impulse occurs. This is just x(t - t0).
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 - t0) = x(t - t0)
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
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.
Sifting property of impulse:
"This sifts out the value of the function x(t) where impulse occurs"