2.3 Definition and Classification of Signals

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23

const

time t/s

time t/s

Fig. 2.12: Examples of periodic signals s1(t) and s2(t) from exactly-periodic processes (left): the

quasi-periodic ECG signal s3(t), on the other hand (right), comes from a process that is subject to

uncertainties. Its period T1, T2, T3, . . . changes with time.

Aperiodic signals are defined by the absence of a period, i.e. by the condition T0 →

∞. To this category belong, on the one hand, arbitrary transient signals and, on the

other hand, special functionals of system theory such as impulses and step functions,

which model the on,/off switching or redirection of processes. These signals include

both the purely monotonic increasing exponential functions s1(t) = e^−0,25t or their

products with harmonic functions s2(t) = sin(2πt) e^−0,25t as well as the density func-

tion of the "standard" normal distribution (cf. Figure 2.13)

N(μ, σ) =

1

σ^√2π

e⁻¹

2 ^{( t−μ}

σ ⁾

2

.

(2.19)

time t/s

time t/s

Fig. 2.13: The two transient signals s1(t) and s2(t) (left) and the pulse signal of the normal distribu-

tion N(μ = 0, σ = 1) (right).