10.3. Graphical Interpretation

Try these first

1. 1.

   What does the Dirac Delta Function represent?

   Hint

   Think of how it is used in engineering.

2. 2.

   What is the integral of δ(t − a) over its range?

   Hint

   This is one of its defining properties.

3. 3.

   What value does the integral of δ(t − a) over all time evaluate to?

   • 0
   • 1
   • a

   Hint

   Focus on the defining property of the function.

4. 4.

   True or False: The Laplace Transform of δ(t) equals 0.

   • True
   • False

   Hint

   Recall the Laplace Transform results we've discussed.

5. 5.

   Prove that the integral from -∞ to ∞ of δ(t − a) equals 1, using a continuous function.

   Hint

   Select an appropriate f(t) that you can evaluate easily.

6. 6.

   Using the properties of the Laplace Transform, derive the transform of kδ(t − a) where k is a constant.

   Hint

   Break the problem down to the basic form of the singular impulse first.