2.4 Signal Processing Transformations

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This requirement is fulfilled by the Fourier-transformationandtheLaplace-transformation.

Both belong to the class of integral transformations¹¹. The structure of the transform-

ation operation is the same for all integral transformations. The quantity s(t) to be

transformed is multiplied by a function K(ξ, t) (integral kernel) and integrated over

the entire range of the variable t. The result is a new function S(ξ):

T {s(t)} = S(ξ) = ∫s(t)K(ξ, t)dt .

(2.48)

In this section the basics of four important integral transformations are presented,

namely the Fourier- and Laplace-transformation, the wavelet-transformation and the

convolution. A detailed description can be found, for example, in [30, 45, 82].

2.4.1 Continuous Fourier-Transformation

The best known integral transformation is the Fourier-transformation¹²., where a time-

dependent quantity s(t) is transformed into the frequency domain S(ω). The integral

core of the Fourier-transformation K(ξ, t) in Equation 2.48 is the eigenfunction of lin-

ear differential equations:

K(ξ, t) = K(ω, t) = e^−jωt .

(2.49)

Equation 2.49 inserted into Equation 2.48 yields the mathematical operation of the

Fourier-transform:

T {s(t)} = S(jω) = ∫s(t)e^−jωtdt .

(2.50)

According to the Euler formula

e^−jωt = cos(ωt) −j sin(ωt)

(2.51)

Equation 2.49 represents a linear combination of a cosine and sine function with the

angular frequency ω as a variable in the image domain. In Equation 2.50 the product

of the quantity to be transformed s(t) and cosine and sine functions with the angular

frequency ω is formed and then integrated.

This is done for all possible values of ω. The result is a complex function in the

image domain, S(jω). Accordingly, the Fourier-transformation yields large function

values for such ω values where the product of s(t) and the corresponding cosine or

sine function comes with a large area. This is the case when s(t) has a large similarity

to Equation 2.51. Accordingly, the Fourier-transformation S(jω) can be interpreted as

a measure of the similarity of the transformed quantity s(t) with cosine and sine func-

tions of the respective angular frequency ω. As already mentioned in subsection 2.3.3,

11 Other examples of integral transformations are the wavelet and Radon-transformation.

12 Jean Baptiste Fourier (1768–1830), important French mathematician and politician