Laplace & Fourier: When to Use?

In summary, Laplace transforms are used to resolve a function into moments in the complex plane, while Fourier transforms resolve a function into oscillations in the frequency domain. There is a common mathematical underpinning to all of these transforms, though it depends on the signal in question.
  • #1
amaresh92
163
0
greetings,

Can anyone tell me when we should use Laplace transform and Fourier transform? It seems both of them are equal except σ .

thanks in advanced.
 
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  • #2
The Laplace transform resolves a function into moments in the complex plane. It is best used in studies that involve the poles and zeros of a response, such as the exponential responses to an impulse.

The Fourier transform resolves a function into oscillations in the frequency domain. It is best used for the study of harmonic content and phase in repeating signals.

Laplace maps to the complex plane while Fourier maps to the frequency domain. You use the one that takes you from the time domain to your preferred domain for analysis or manipulation.
 
  • #3
Balun, you can just as well (or better) say that "Laplace maps to the entire complex plane while Fourier maps to the [itex]j \omega[/itex] axis of the complex plane."

it is true that the double-sided Laplace transform is identical to the continuous Fourier transform with the substitution that [itex]s = j \omega[/itex].

there are some signals that, without handwaving, are hard to F.T. while they are easy to L.T. e.g. the unit step function, [itex]u(t)[/itex]. there are issues of convergence and the conditions to get either the Laplace integral or the Fourier integral to converge to a finite result. but where these issues are identical for both the F.T. and L.T. (such as for [itex]x(t) = e^{-\alpha t} u(t)[/itex]), then the transformed result must be the same between the F.T. and L.T. with the substitution of [itex]s = j \omega[/itex].

a similar relationship exists between the Z Transform and the Discrete-Time Fourier Transform. one exists in the entire complex z plane while the other exists on the unit circle of the same z plane, where [itex]z = e^{j \omega}[/itex].

there is a common mathematical underpinning to all of these transforms. it depends on whether your signal in the time or frequency domain is discrete or not. whether you can get the integral to converge or not. whether you define your signals and linear systems to exist before [itex]t=0[/itex] or not (that affects the double or single-sidedness of the integral or summation).
 
  • #4
rbj; I can agree with everything you write.
I am not a mathematician so it is mostly beyond me.
After three days, amaresh92's question deserved a reply.
So I threw in an answer in the hope that it would stimulate some response.
Now 4 days later, I am enlightened by your interesting reply. Thank you.
 
  • #5


Greetings,

I can provide some insight on when to use Laplace transform and Fourier transform. Both of these transforms are commonly used in mathematics and engineering to analyze and solve problems involving differential equations. However, there are some key differences between them.

The main difference between Laplace and Fourier transforms is the type of function they are used for. Laplace transform is typically used for functions that are defined for all positive time values, while Fourier transform is used for functions that are defined for all time values. This means that Laplace transform is more suitable for studying transient behavior, while Fourier transform is more suitable for studying steady-state behavior.

Additionally, Laplace transform is useful for solving initial value problems, while Fourier transform is useful for solving boundary value problems. Laplace transform also has the advantage of being able to handle functions with singularities, while Fourier transform cannot.

In summary, Laplace transform is best used for studying transient behavior and solving initial value problems, while Fourier transform is best used for studying steady-state behavior and solving boundary value problems. It is important to understand the specific problem you are trying to solve in order to determine which transform is most appropriate.

I hope this helps answer your question. Best of luck in your studies.

Sincerely,

 

1. What is the difference between Laplace and Fourier transforms?

The Laplace transform is used to convert a time-domain function into a frequency-domain function. It is often used to solve differential equations in engineering and physics. The Fourier transform is used to decompose a function into its individual frequency components. It is commonly used in signal processing and image analysis.

2. When should I use a Laplace transform?

The Laplace transform is useful for solving differential equations with initial conditions. It is also used in control theory and circuit analysis to understand the behavior of systems over time. Additionally, it can be used to find the steady-state response of a system.

3. When is a Fourier transform more appropriate?

The Fourier transform is often used when analyzing signals and systems that are periodic or have a repeating pattern. It is also helpful in understanding the frequency content of a signal and identifying dominant frequencies. Additionally, it is commonly used in image processing and filtering applications.

4. Can Laplace and Fourier transforms be used interchangeably?

No, Laplace and Fourier transforms are not interchangeable. They have different applications and provide different insights into a function or system. However, they are closely related and can be used together in certain situations, such as solving differential equations with initial conditions.

5. What are some real-world applications of Laplace and Fourier transforms?

Laplace and Fourier transforms have numerous applications in fields such as engineering, physics, and mathematics. Some examples include analyzing the behavior of electronic circuits, solving differential equations in physics, understanding the frequency content of signals in communication systems, and processing images in medical imaging and astronomy.

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