Laplace Transform

Question 1

What is the Laplace Transform formula?

Answer 1

Question 2

What is s in the LT?

Answer 2

s = σ + jω

where σ is the growth factor and ω is the frequency of the oscillation

Question 3

What is the relation of Laplace and Fourier Transform?

Answer 3

Fourier transform can be considered a special case of the laplace where σ = 0

Question 4

How do we find the inverse Laplace of a function?

Answer 4

By using partial fraction expansion to express a rational function as a sum of lower-order rational functions whose inverse Laplace can be found using look up tables.

Question 5

What is the Laplace equivalent of differentiation?

Answer 5

Multiplication by s in the Laplace domain

Question 6

What is the Laplace Transform of dx/dt?

Answer 6

L(s) = sX(s) - x(0)

where x(0) is the initial condition (value of x at t = 0)

Question 7

What is integration in the Laplace domain?

Answer 7

Multiplication by 1/s

Question 8

How do we represent a time shift in the Laplace Domain?

Answer 8

Delaying x(t) by t⁰ is equivalent to a multiplication of the Laplace Transform X(s) by e^-st₀.

Question 9

How do we represent a frequency shift in the Laplace Domain?

Answer 9

A frequency shift of the Laplace Transform X(s) by s₀ is X(s - s₀),

This is equivalent to multiplying its time function by e^s₀t

Question 10

What is the equivalent of convolution in the Laplace Domain?

Answer 10

Convolution in the time domain is equivalent to multiplication of two transforms in the Laplace Domain.

x₁(t)*x₂(t) ==> X₁(s)X₂(s)