# 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.

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years? Kala Reply lim x to infinity e^1-e^-1/log(1+x) given eccentricity and a point find the equiation Moses Reply 12, 17, 22.... 25th term Alexandra Reply 12, 17, 22.... 25th term Akash College algebra is really hard? Shirleen Reply Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table. Carole I'm 13 and I understand it great AJ I am 1 year old but I can do it! 1+1=2 proof very hard for me though. Atone hi Adu Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily. Vedant hi vedant can u help me with some assignments Solomon find the 15th term of the geometric sequince whose first is 18 and last term of 387 Jerwin Reply I know this work salma The given of f(x=x-2. then what is the value of this f(3) 5f(x+1) virgelyn Reply hmm well what is the answer Abhi If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10 Augustine how do they get the third part x = (32)5/4 kinnecy Reply make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be AJ how Sheref can someone help me with some logarithmic and exponential equations. Jeffrey Reply sure. what is your question? ninjadapaul 20/(×-6^2) Salomon okay, so you have 6 raised to the power of 2. what is that part of your answer ninjadapaul I don't understand what the A with approx sign and the boxed x mean ninjadapaul it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared Salomon I'm not sure why it wrote it the other way Salomon I got X =-6 Salomon ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6 ninjadapaul oops. ignore that. ninjadapaul so you not have an equal sign anywhere in the original equation? ninjadapaul hmm Abhi is it a question of log Abhi 🤔. Abhi I rally confuse this number And equations too I need exactly help salma But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends salma Commplementary angles Idrissa Reply hello Sherica im all ears I need to learn Sherica right! what he said ⤴⤴⤴ Tamia hii Uday hi salma hi Ayuba Hello opoku hi Ali greetings from Iran Ali salut. from Algeria Bach hi Nharnhar A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!