10.2. Laplace Transform of the Dirac Delta Function

Try these first

1. 1.

   What is the value of the Dirac Delta function δ(t - a) when t = a?

   Hint

   Think about the nature of the delta function.

2. 2.

   What does the Laplace transform of δ(t) equal?

   Hint

   Consider the special case when a = 0 in the transform.

3. 3.

   What is the Laplace transform of δ(t - 2)?

   • 2e^{-2s}
   • e^{-2s}
   • 1

   Hint

   Use the standard formula of Laplace transform for δ(t - a).

4. 4.

   True or False: The area under the curve of the Dirac Delta function δ(t - a) is always equal to 1.

   • True
   • False

   Hint

   Reflect on the properties of integrals involving the delta function.

5. 5.

   Given the function δ(t - 5), derive the system response when this impulse is applied to the system modeled by the equation dy/dt + 2y = δ(t - 5).

   Hint

   Remember to handle the transform with impulse appropriately.

6. 6.

   Illustrate how the impulse response can be utilized to design a control system, referencing δ(t) as an example.

   Hint

   Think about how a system suite deals with feedback and transient states.