# Laplace transform

 Page 1 / 1
Describes Laplace transforms.

## Introduction

The Laplace transform is a generalization of the Continuous-Time Fourier Transform . It is used because the CTFT does not converge/exist for many important signals, and yet it does for the Laplace-transform (e.g., signals with infinite ${l}_{2}$ norm). It is also used because it is notationally cleaner than the CTFT. However, instead of using complex exponentials of the form $e^{i\omega t}$ , with purely imaginary parameters, the Laplace transform uses the more general, $e^{st}$ , where $s=\sigma +i\omega$ is complex, to analyze signals in terms of exponentially weighted sinusoids.

## Bilateral laplace transform pair

Although Laplace transforms are rarely solved in practice using integration ( tables and computers ( e.g. Matlab) are much more common), we will provide the bilateral Laplace transform pair here for purposes of discussion and derivation. These define the forward and inverse Laplace transformations. Notice the similarities between the forwardand inverse transforms. This will give rise to many of the same symmetries found in Fourier analysis .

## Laplace transform

$F(s)=\int_{()} \,d t$ f t s t

## Inverse laplace transform

$f(t)=\frac{1}{2\pi i}\int_{c-i} \,d s$ c F s s t

We have defined the bilateral Laplace transform. There is also a unilateral Laplace transform ,
$F(s)=\int_{0()} \,d t$ f t s t
which is useful for solving the difference equations with nonzero initial conditions. This is similar to the unilateral Z Transform in Discrete time.

## Relation between laplace and ctft

Taking a look at the equations describing the Z-Transform and the Discrete-Time Fourier Transform:

## Continuous-time fourier transform

$ℱ(\Omega )=\int_{()} \,d t$ f t Ω t

## Laplace transform

$F(s)=\int_{()} \,d t$ f t s t
We can see many similarities; first, that :
$ℱ(\Omega )=F(s)$
for all $\Omega =s$

the CTFT is a complex-valued function of a real-valued variable $\omega$ (and 2 $\pi$ periodic). The Z-transform is a complex-valued function of a complex valued variable z.

## Visualizing the laplace transform

With the Fourier transform, we had a complex-valued function of a purely imaginary variable , $F(i\omega )$ . This was something we could envision with two 2-dimensional plots (real and imaginary parts or magnitude andphase). However, with Laplace, we have a complex-valued function of a complex variable . In order to examine the magnitude and phase or real andimaginary parts of this function, we must examine 3-dimensional surface plots of each component.

While these are legitimate ways of looking at a signal in the Laplace domain, it is quite difficult to draw and/or analyze.For this reason, a simpler method has been developed. Although it will not be discussed in detail here, the methodof Poles and Zeros is much easier to understand and is the way both the Laplace transform and its discrete-time counterpart the Z-transform are represented graphically.

## Using a computer to find the laplace transform

Using a computer to find Laplace transforms is relatively painless. Matlab has two functions, laplace and ilaplace , that are both part of the symbolic toolbox, and will find the Laplace and inverseLaplace transforms respectively. This method is generally preferred for more complicated functions. Simpler and morecontrived functions are usually found easily enough by using tables .

## Conclusion

The laplace transform proves a useful, more general form of the Continuous Time Fourier Transform. It applies equally well to describing systems as well as signals using the eigenfunction method, and to describing a larger class of signals better described using the pole-zero method.

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By Brooke Delaney By OpenStax By OpenStax By OpenStax By Rohini Ajay By OpenStax By Yasser Ibrahim By Anh Dao By Dindin Secreto By OpenStax