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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 ω t , with purely imaginary parameters, the Laplace transform uses the more general, s t , where s σ ω is complex, to analyze signals in terms of exponentially weighted sinusoids.

The laplace transform

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 t f t s t

Inverse laplace transform

f t 1 2 s c c F s s t

We have defined the bilateral Laplace transform. There is also a unilateral Laplace transform ,
F s t 0 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

Ω t f t Ω t

Laplace transform

F s t f t s t
We can see many similarities; first, that :
Ω F s
for all Ω s

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

Plots

Visualizing the laplace transform

With the Fourier transform, we had a complex-valued function of a purely imaginary variable , F ω . 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.

Real and imaginary sample plots

The Real part of H s
The Imaginary part of H s
Real and imaginary parts of H s are now each 3-dimensional surfaces.

Magnitude and phase sample plots

The Magnitude of H s
The Phase of H s
Magnitude and phase of H s are also each 3-dimensional surfaces. This representation is more common than real and imaginary parts.

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 .

Laplace transform definition demonstration

LaplaceTransformDemo
Interact (when online) with a Mathematica CDF demonstrating the Laplace Transform. To Download, right-click and save target as .cdf.

Interactive demonstrations

Khan lecture on laplace

See the attached video on the basics of the Unilateral Laplace Transform from Khan Academy

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.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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