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Fourier Transform

The Fourier transform is a special case of the bilateral Laplace transform under certain conditions. The Laplace transform of a function is a complex function of a complex variable, while the Fourier transform is a complex function of a real variable. The Laplace transform is usually restricted to non-negative time domains, making it a holomorphic function, unlike the Fourier transform. The Fourier transform is equivalent to evaluating the bilateral Laplace transform with an imaginary argument, relating the two transforms and allowing the determination of frequency spectra.

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49 views2 pages

Fourier Transform

The Fourier transform is a special case of the bilateral Laplace transform under certain conditions. The Laplace transform of a function is a complex function of a complex variable, while the Fourier transform is a complex function of a real variable. The Laplace transform is usually restricted to non-negative time domains, making it a holomorphic function, unlike the Fourier transform. The Fourier transform is equivalent to evaluating the bilateral Laplace transform with an imaginary argument, relating the two transforms and allowing the determination of frequency spectra.

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vijay kumar honnali
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Fourier transform[edit]

Further information: Fourier transform § Laplace transform


The Fourier transform is a special case (under certain conditions) of the bilateral Laplace transform.
While the Fourier transform of a function is a complex function of a real variable (frequency), the
Laplace transform of a function is a complex function of a complex variable. The Laplace transform
is usually restricted to transformation of functions of t with t ≥ 0. A consequence of this restriction is
that the Laplace transform of a function is a holomorphic function of the variable s. Unlike the Fourier
transform, the Laplace transform of a distribution is generally a well-behaved function. Techniques of
complex variables can also be used to directly study Laplace transforms. As a holomorphic function,
the Laplace transform has a power series representation. This power series expresses a function as
a linear superposition of moments of the function. This perspective has applications in probability
theory.
The Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary
argument s = iω or s = 2πiξ[27] when the condition explained below is fulfilled,

This convention of the Fourier transform ( in Fourier transform § Other conventions)


requires a factor of 1/2π on the inverse Fourier transform. This relationship between the Laplace and
Fourier transforms is often used to determine the frequency spectrum of a signal or dynamical
system.
The above relation is valid as stated if and only if the region of convergence (ROC) of F(s) contains
the imaginary axis, σ = 0.
For example, the function f(t) = cos(ω0t) has a Laplace transform F(s) = s/(s2 + ω02) whose ROC
is Re(s) > 0. As s = iω0 is a pole of F(s), substituting s = iω in F(s) does not yield the Fourier
transform of f(t)u(t), which is proportional to the Dirac delta function δ(ω − ω0).
However, a relation of the form

holds under much weaker conditions. For instance, this holds for the above example provided that
the limit is understood as a weak limit of measures (see vague topology). General conditions relating
the limit of the Laplace transform of a function on the boundary to the Fourier transform take the
form of Paley–Wiener theorems.
Mellin transform[edit]
Main article: Mellin transform
The Mellin transform and its inverse are related to the two-sided Laplace transform by a simple
change of variables.
If in the Mellin transform

we set θ = e−t we get a two-sided Laplace transform.


Z-transform[edit]
Further information: Z-transform § Relationship to Laplace transform
The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal
with the substitution of

where T = 1/fs is the sampling interval (in units of time e.g., seconds) and fs is the sampling
rate (in samples per second or hertz).
Let

be a sampling impulse train (also called a Dirac comb) and

be the sampled representation of the continuous-time x(t)

The Laplace transform of the sampled signal xq(t) is

This is the precise definition of the unilateral Z-transform of the discrete function x[n]

with the substitution of z → esT.


Comparing the last two equations, we find the relationship between the unilateral Z-transform and
the Laplace transform of the sampled signal,

The similarity between the Z- and Laplace transforms is expanded upon in the theory of time scale
calculus.

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