Transforms of Heaviside's Unit Step Function and Unit Impulse function (Dirac delta function)

TRANSFORMS OF UNIT STEP FUNCTION AND IMPULSE FUNCTION

PROBLEMS BASED ON UNIT STEP FUNCTION (OR) HEAVISIDE'S UNIT STEP FUNCTION

Example 1: Define the unit step function.

Solution:

The unit step function, also called Heaviside's unit function is defined as

This is the unit step functions at t = a

It can also be denoted by H(t − a).

Example 2: Give the Laplace transform of the unit step function.

Solution:

The L.T. of the unit step function is given by

Example 3: Find the Laplace transform of the unit step function

Solution:

Example 4: Define the Dirac delta function (or) Unit Impulse function.

Solution:

The Dirac delta function is denoted by δ(t‒a). It exists only at t = a at which it is infinitely great. It is defined by

 δ(t − a) = 0,  t ≠ a

  = ∞,  t = a

such that, _‒∞ʃ^∞ δ(t − a) dt = 1

It is also called the unit impulse function.

Example 5: Give the Laplace transform of the Dirac Delta function.

Solution:

 L[δ (t − a)] = e^‒as

Example 6: Define the Impulse function and state its application.

Solution:

The impulse function finds applications in problems where a large force is applied for a very short time or a large force acts over a very small area as in the loading of a beam or a shock applied to an electrical circuit.

Note: The Dirac delta function is the derivative of the unit step function.

Example 7: Find the Laplace transform of t U(t − 9)

Solution:

 t U(t = 9) = [(t − 9) + 9]U(t − 9)

= (t −9)U(t −9) + 9U(t − 9)

L[t U(t − 9)] = L[(t −9)U(t −9)] + 9L[U(t −9)]

= e^−9s L(t) + 9 [e^−9s / s]

= e^−9s [1/s²] + 9 [e^−9s / s] 

= e^−9s/s² (1+9s)

Example 8: Find L^‒1[1]

Solution:

We know that, L [δ(t − a)] = e^‒as

Put a = 0,

  L[δ (t)] = 1

⇒ L^‒1(1) = δ(t).