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Last Updated: Mar 27, 2024
Difficulty: Easy

Laplace transform of a unit step function

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Prerita Agarwal
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23 Jul, 2024 @ 01:30 PM

Introduction

Unit Step Function

The unit step function at time t, u(t), is defined as

 

 

That is, u or H is a function of time t,

  • When time is negative, u have a value zero, 
  • When time is positive, u have a value one.

 

 

At t = 0, 

  1. In some textbooks u(0) = 1, and somewhere it is 0.5.
  2. We are considering the second case, but it won't make much difference.

 

The thing is, u(t) is discontinuous at 0, therefore not differentiable at 0. The unit step function is also known as a Heaviside unit step function.

Shifted Unit Step Function

A shifted unit step function has value 0 up to the time a and has value 1 afterward.

 

Laplace Transformation

A function is considered piecewise continuous if it has a finite number of breaks and does not blow up anywhere. If the function f(t) is a piecewise continuous function, then the Laplace transform is used to define it. A function's Laplace transform is denoted by Lf(t) or F. (s). The Laplace transform helps solve differential equations by converting them into algebraic problems.

Laplace transform of a function f(t) is given by the equation:
 

Laplace transform of a unit step function

Step 1: Formula of Laplace transform for f(t).
 

 

Step 2: Unit Step function u(t):

 

 

Step 3: Now, as the limits in Laplace transform goes from 0 -> infinity, u(t) function = 1 in the interval 0 -> infinity. Hence Laplace transform equation for u(t):

Solving the above integral equation gives,

Hence, Laplace transform of a unit step function is

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Laplace transform of a shifted unit step function

Similarly, the Laplace transform of a shifted unit step function is the solution of the below-mentioned equation

Hence, the solution of the above equation is,

FAQs

1. What do you mean by unit step function?
The unit step function, also known as the Heaviside step function, is a step function named after Oliver Heaviside (1850–1925), whose value is zero for negative arguments and one for positive arguments. It is generally denoted by H or sometimes u. Except for a discontinuity at t = 0, the unit step function is level throughout. As a result, except at t = 0, the derivative of the unit step function is 0 at all positions t. The derivative of the unit step function is infinite when t = 0.

2. Why do we use Laplace transform?
The Laplace transform is used for solving linear ordinary differential equations that use integral transforms. Differential equations are solved using the Laplace transform. It is widely accepted in a variety of fields. The Laplace transform reduces a linear differential equation (LDE) to an algebraic equation, which may then be solved using basic algebraic identities. It has many applications in physics, control engineering, electrical engineering, optics, mathematics, signal processing, etc.

3. What is the Laplace transform of a unit step and unit impulse function?
The Laplace transforms of certain signals are as follows: The Laplace transform of a unit step input that starts at time t=0 and rises to the constant value 1 is 1/s. The Laplace transform of a unit impulse input that starts at time t=0 and rises to the value 1 is 1.    

Key Takeaways

In this article we learned about unit step functions, shifted unit step functions. We later found the Laplace Transform of a unit step function and Laplace transform of a shifted unit step function.

Results:

  1. A unit step function u is,
    • 0, when t has negative values. 
    • 1, when t has positive values.
  2. The Laplace transform of a unit step function is L(s) = 1/s.
  3. A shifted unit step function u(t-a) is,
    • 0, when t has values less than a. 
    • 1, when t has values greater than a.
  4. The Laplace transform of a shifted unit step function is L(s) = e-as/s.


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Topics covered
1.
Introduction
1.1.
Unit Step Function
1.2.
Shifted Unit Step Function
1.3.
Laplace Transformation
2.
Laplace transform of a unit step function
3.
Laplace transform of a shifted unit step function
4.
FAQs
5.
Key Takeaways