2.3 Mathematical Tools
47
L{ ˙
x(t)} = s X (s).
(2.82)
• If the convolution integral x 1 (t) ) x 2 (t) of two real-valued causal signals x 1 (t) and
x 2 (t) is defined as
x 1 (t) ) x 2 (t) :=
t
0
x 1 (t − t
) x 2 (t
) dt
=
t
0
x 1 (t
) x 2 (t − t
) dt
(2.83)
the corresponding Laplace transformation of the convolution integral computes as
the product of the corresponding Laplace transformed signals X 1 (s) and X 2 (s),
i.e.
L{x 1 (t) ) x 2 (t)} = X 1 (s) X 2 (s).
(2.84)
With these properties at hand the inversion x(t), a real-valued causal signal with
initial condition x(0) = 0, of the convolution integral z(t) = y(t) ) ˙
x(t) of a realvalued causal signal y(t) and the rate ˙
x(t), a typical problem in what follows, is
obtained in complex frequency domain by the Laplace transformed signals Z (s) and
Y (s) as X (s) = Z (s)/[s Y (s)] with a subsequent inverse Laplace transformation of
X (s) to the time domain, see the sketch in Fig. 2.13.
A brief list of real-valued causal signals x(t) in time domain and corresponding
Laplace transformed complex-valued signals X (s) in complex frequency domain
that are relevant in the sequel is given in Table 2.2.
2.3.3 Complex Representations
Complex Numbers and Harmonically Oscillating Signals
Complex numbers allow the condensed representation of twice the information that
is contained in an ordinary real number. Thereby a complex number x ∈ C allows
the following alternative representations
z(t) = y(t) ˙
x(t)
x(t) = L
−1
{X(s)}
Z(s) = L{z(t)} = s Y (s) X(s)
X(s) = Z(s)/[s Y (s)]
Laplace Transformation
Inverse Laplace Transformation
Inversion
Fig. 2.13 Inversion x(t) of a frequently occurring, typical convolution integral z(t) = y(t) ) ˙
x(t)