2.3 Definition and Classification of Signals

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25

time t/s

time t/s

time t/s

time t/s

Fig. 2.14: Dirac series of the normal distribution δϵ(t) as a boundary transition to the delta-

distribution δ(t): Top left the density distribution of the normal distribution δϵ(t) and top right the

distribution function of the normal distribution Φϵ(t) for μ = 0, ϵ →0. The lower left shows the

delta-distribution used in discrete signal distribution δ(t) applied in discrete signal processing and

on the right the Heaviside-function H(t) as the density distribution of the Delta-distribution.

the function value f(0). The shift of the delta-distribution by a leads to

∞

∫

−∞

f(t) δ(t −a) dt =

∞

∫

−∞

f(t) δ(a −t) dt = f(a)

(2.24)

and thus hides all values of the function f(t) at the places t

̸= a. For the case of the

constant function f(t) = 1, the weight of the delta-distribution is again obtained:

∞

∫

−∞

δ(t −a) dt = 1 .

(2.25)

Another important signal form results from the limit value consideration of the cumu-

lative sum of the density function of the normal distribution (cf. Figure 2.14), i.e. the

cumulative integral from −∞to t:

Φϵ(t) =

1

√2πϵ

t

∫

−∞

e^−t2

2ϵ dt ,

(2.26)

the so-called distribution function of the normal distribution. In the limit value con-

sideration ϵ →0 one obtains the so-called Heaviside-function (cf. Figure 2.14)⁹, it is

also called step or unit step function. It has a function value of zero for arguments less

9 After the British mathematician and physicist Oliver Heaviside