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2 Fundamentals of Information, Signal and System Theory

consequently their transmission requires energy ¹⁰. However, this refers to the trans-

mission process and not to the signal itself. If one assumes an energised form of the

signals, the following relationships can be defined for energy and power signals of

signal processing, following electrical engineering: The energy of a complex-valued,

time-continuous and dimensionless signal s(t) is determined by

E(st) =

∞

∫

−∞

s(t) ⋅s^∗(t)dt =

∞

∫

−∞

|s(t)|²dt .

(2.31)

The integral is to be calculated section by section over all time intervals in which s(t) is

defined. For dimensionally affected quantities, such as the electrical voltage U in volts

(V), the signal energy E(st) then has the dimension V²s as in electrical engineering.

Based on the signal energy, one distinguishes two classes of signals, the so-called

energy signals with finite but non-zero energy and the power signals with infinite en-

ergy but finite mean power. Accordingly, a signal s(t) is an energy signal if it satisfies

the following condition:

E(st) =

∞

∫

−∞

s(t) ⋅s^∗(t)dt =

∞

∫

−∞

|s(t)|²dt < ∞.

(2.32)

The energy signal is therefore a square-integrable function and has a non-vanishing

energy E. Typical energy signals are all signals with finite signal values that are

switched on and off at some point. This class of signals can be assigned to oscil-

lation and decay processes or to time-limited pulse-shaped signals. The following

examples are intended to clarify the concept of an energy signal: A square-wave pulse

s(t) = A rect(T) with the amplitude A and the width T is, according to the above

definition, an energy signal with the energy

E = A²

T

∫

0

rect²(T)dt = A²T < ∞.

(2.33)

The energy calculation for a delta-distribution,

δϵ(t) = ^θ

2ϵ ^{rect (2ϵ}

θ ^{) ,}

θ →∞,

(2.34)

on the other hand, does not lead to an energy signal, since:

E(δϵ) = ^θ2

4ϵ²

2ϵ

θ ^{= θ}

2ϵ ^→∞,

für

θ →∞, 0 < ϵ < ∞.

(2.35)

The limiting process from the rectangular function to the delta distribution thus leads

to a change in the energy properties of the signal. Typical non-energy signals are auto-

10 for example, the transmitting power of the sender and the energy input of the sender/receiver