The difference between Fourier Series, Fourier Transform and Laplace Transform

In summary, the Laplace Transform, Fourier Transform, and Fourier Series are all mathematical tools used to analyze functions. The Laplace Transform is defined for positive values of t, the Fourier Transform for all real values of t, and the Fourier Series for a finite interval. The coefficients in the Fourier Series are given by an integral and can be simplified using the exponential function. Real Fourier series use sine and cosine functions instead of the exponential function.
  • #1
mathman
Science Advisor
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Mathematically, these are three distinct, although related beasts.
Laplace transform (function f(x) defined from 0 to inf) integral of f(x)e-xt, defined for t>=0.
Fourier transform (function f(x) defined from -inf to inf) integral of f(x)e-itx defined for all real t.
Complex Fourier series (function f(x) defined on finite interval - simplify by making it (0,1)) Coeficients (cn) are given by integral of f(x)e-2(pi)inx, where n ranges over all integers. The series terms are cne2(pi)nx
Real Fourier series use sin and cos instead of exp function.
 
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  • #2
can someone help me to explain the difference between Fourier Series, Fourier Transform and Laplace Transform

-thanx
 

What is the difference between Fourier Series, Fourier Transform and Laplace Transform?

The main difference between Fourier Series, Fourier Transform and Laplace Transform lies in their applications and domains. Fourier Series is used to represent periodic signals in the time domain as a sum of sinusoidal functions. Fourier Transform is used to analyze non-periodic signals in the frequency domain by decomposing them into their constituent frequency components. Laplace Transform is used to solve differential equations in the time domain by transforming them into algebraic equations in the s-domain.

How are Fourier Series, Fourier Transform and Laplace Transform related?

Fourier Series, Fourier Transform and Laplace Transform are all mathematical tools used to analyze signals and systems. They are related by the fundamental concept of representing a signal or function in terms of its constituent components, whether they be in the time or frequency domain. Laplace Transform is considered to be an extension of Fourier Transform, and Fourier Series can be seen as a special case of Fourier Transform when dealing with periodic signals.

What are the mathematical equations for Fourier Series, Fourier Transform and Laplace Transform?

The mathematical equations for Fourier Series, Fourier Transform and Laplace Transform are as follows:

Fourier Series:
f(x) = a0/2 + Σ(an*cos(nx) + bn*sin(nx))
Where a0, an and bn are the coefficients of the series and n is an integer.

Fourier Transform:
F(ω) = ∫f(t)*e^(-jωt) dt
Where F(ω) is the Fourier Transform of f(t) and ω is the frequency variable.

Laplace Transform:
F(s) = ∫f(t)*e^(-st) dt
Where F(s) is the Laplace Transform of f(t) and s is the complex frequency variable.

What are the applications of Fourier Series, Fourier Transform and Laplace Transform?

Fourier Series is primarily used in signal processing, communications, and control theory to analyze and represent periodic signals. Fourier Transform is used in a wide range of applications, including image processing, audio signal analysis, and spectral analysis of non-periodic signals. Laplace Transform is mainly used in control theory, circuit analysis, and solving differential equations in physics and engineering.

How can I determine which transform to use for a particular problem?

The choice of which transform to use depends on the nature of the problem at hand. If the signal or function is periodic, then Fourier Series is the appropriate choice. For non-periodic signals, Fourier Transform can be used to analyze the frequency components. Laplace Transform is useful for solving differential equations in the time domain. However, in some cases, a combination of these transforms may be needed to fully understand and analyze the problem. It is important to have a good understanding of the properties and applications of each transform to determine the most suitable one for a given problem.

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