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2 Fundamentals of Information, Signal and System Theory

transformability can be achieved by an additional convergence term. This leads us to

the Laplace transform.

2.4.2 Continuous Laplace Transform

The Laplace¹⁵ transformation also belongs to the class of integral transformations and

is an extension of the Fourier transformation. The integral core is similar to that of the

Fourier transform (cf. Equation 2.49):

K(ξ, t) = K(p, t) = e^−pt

(2.55)

where p is a complex variable,

p = σ + jω .

(2.56)

Here σ is a real number and j is the imaginary unit. Equation 2.55 inserted into Equa-

tion 2.48 yields the mathematical operation of the Laplace transformation:

S(p) = ∫s(t)e^−ptdt .

(2.57)

If we separate the variable p in Equation 2.57 into its components according to Equa-

tion 2.56 the similarity of the Laplace transform with the Fourier transform becomes

even clearer:

S(σ, jω) = ∫s(t)e^−jωte^−σtdt .

(2.58)

The term

e^−σt

(2.59)

corresponds to an additional damping term, whereby also signals become transform-

able which do not satisfy the Dirichlet condition according to Equation 2.54. Thus

Equation 2.54 for the Laplace transform can be extended to

∫|s(t)e^−σt|dt < ∞.

(2.60)

The damping effect of Equation 2.59 occurs only when the argument of the exponen-

tial function is negative overall. Since we are dealing with causal signals in signal

processing (s(t) = 0, ∀t < 0), so the negative time domain is not considered, σ must

be > 0. This region in the complex p-plane is called the convergence region of the

Laplace transform. There, polynominals with arbitrary power of t are transformable.

For exponentially increasing signals of type s(t) = u(t)e^at with a > 0, one further re-

strictions for the convergence region of σ have to be considered and the dampening

15 Pierre-Simon Laplace (1749–1827), french mathematician, physicist and astronomer, Minister of the

Interior under Napoleon.