Scribd
Upload
69 views3 pages

Example X (T) Step Function: Laplace Transforms

This document discusses properties of the Laplace transform including: linearity, taking the Laplace transform of derivatives, multiplying signals by exponentials, and examples of calculating Laplace transforms. It explains that differentiation in the time domain corresponds to multiplication by s in the Laplace domain, and multiplication by an exponential in time corresponds to shifting s in the Laplace domain. Examples are provided to demonstrate finding Laplace transforms by using properties like shifting s.

Uploaded by

lucky250
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOC, PDF, TXT or read online on Scribd
69 views3 pages

Example X (T) Step Function: Laplace Transforms

This document discusses properties of the Laplace transform including: linearity, taking the Laplace transform of derivatives, multiplying signals by exponentials, and examples of calculating Laplace transforms. It explains that differentiation in the time domain corresponds to multiplication by s in the Laplace domain, and multiplication by an exponential in time corresponds to shifting s in the Laplace domain. Examples are provided to demonstrate finding Laplace transforms by using properties like shifting s.

Uploaded by

lucky250
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOC, PDF, TXT or read online on Scribd
You are on page 1/ 3

Laplace Transforms

Example x(t)=Step Function

Example Exponential Function

We will now start studying the properties of the Laplace transform. Linearity of Laplace Transforms: If constants . This property can be derived in the following way: then for any

Laplace Transform of a Derivative: If

then

This property can be derived in the following way:

This property shows that the process of differentiation in the time-domain corresponds to multiplication by s in the Laplace domain plus the addition of the constant -x(0). Laplace Transforms of Higher Derivatives: If then

This property can be derived in the following way:

In general we have

Laplace Transform of a Signal Multiplied by an Exponential: If

then

. This property can be derived as follows.

This property shows that multiplication by an exponential in time corresponds to shifting in s. Example: Find the Laplace transform for the following functions

We can find the Laplace transform for t and sin2t using the Laplace transform table. In part (a) we need to shift s by 3 whereas we shift s by -1 in part (b). Therefore the Laplace transforms of the above two functions are given as

You might also like