Laplace Transform

LAPLACE TRANSFORM

Continuous time systems are analyzed using Lapalce transform. Unstable system is also analyzed using Lapalce transform.

Types of Lapalce transform

There are 2 types of Laplace transform

(1) Unilateral (or) one sided Laplace transform

(2) Bilateral (or) two sided Laplace transform

Definition of Laplace Transform

Laplace transform of signal x(t) is defined as

σ ⇒ Real part of S & it is denoted as attenuation constant

ω ⇒ Imaginary part of "S" and it is denoted as angular frequency.

Laplace transform pair

x(t) is obtained from X(s) by taking inverse Laplace transform of X(s)

Inverse lapalce transform of X(s)

Relationship Between Fourier Transform and Laplace Transform

Laplace transform of x(t) is written as

(2) is Fourier transform of x(t) e ^{– σ t}

Hence Laplace transform is Fourier transform of x(t) e ^{– σ t}

If σ = 0, then S = j ω

Now Laplace transform of x(t) is given as

Region of convergence of Laplace transform (Roc)

The range of values of "σ" for which Laplace transform converges is known as Region of convergence.

Problems Based on Laplace Transform

1. Find Laplace transforms and Roc of x(t) = u(t)

Solution :

2. Find Laplace transform of δ(t)

Solution :

Shifting property of δ(t)

Compare (1) & (2)

3. Find Laplace transform of x(t) = e ^{a t} u(t) and plot its ROC.

Solution :

4. Find Laplace transform of x(t) = e ^{– a t} u(t)

Solution :

5. Find Laplace transform of x(t) = 3 - 6e ^{– 4 t}

Solution :

6. Determine Laplace transform of x(t) = e ^{– 2 t} u(t) + e ^{– 3 t} u(t) and sketch its ROC

Solution :