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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

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-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
Alfred Reply
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?
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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
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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
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salma
Commplementary angles
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greetings from Iran
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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.
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What is the expressiin for seven less than four times the number of nickels
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How do i figure this problem out.
how do you translate this in Algebraic Expressions
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why surface tension is zero at critical temperature
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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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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