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2 Fundamentals of Information, Signal and System Theory

than zero and a function value of one otherwise. The Heaviside-function is continuous

everywhere except at t = 0:

H(t) =

{

{

{

1

t ≥0

0

t < 0

.

(2.27)

In the technical literature, there is also a different nomenclature. In systems theory,

the symbol 1(t) or u(t) is used after the English term unit step function. The Heaviside-

unction is the integral of the Dirac-distribution

H(t) :=

t

∫

−∞

δ(s) ds ,

(2.28)

and consequently the derivative of the Heaviside-function is the diracian delta-

distribution.

2.3.3 Even and Odd Signals

The consideration of symmetries of a signal can be helpful in the decomposition and

description of a signal. Periodic signals can basically be decomposed into an even and

an odd signal component. A signal is called even if it is axisymmetrical to the ordinate.

Odd signals have point symmetry to the origin (cf. Figure 2.15).

time t/s

time t/s

Fig. 2.15: An odd signal su(t) of two square pulses (left) and an even signal sg(t) of two square

pulses (right).

Mathematically, this symmetry can be expressed as follows:

s(t) =

{

{

{

s(−t)

even

−s(−t)

odd

,

∀t .

(2.29)

As we will see later in the introduction of the Fourier-transformation in subsection

2.4.1, even signal components can only be expressed by a linear combination of even

harmonic functions such as the cosine and odd signal components can only be ex-

pressed by a linear combination of odd harmonic functions such as the sine.