2.3 Definition and Classification of Signals

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matically all power signals, because they have an infinite signal energy at a finite av-

erage power:

0 < P = lim

T→∞

1

2T

T

∫

−T

s(t) ⋅s^∗(t)dt < ∞.

(2.36)

The instantaneous power of real signals at time t is:

P(t) = lim

T→0

1

2T

t+T

∫

t−T

|s(t)|²dt = |s(t)|² .

(2.37)

The class of power signals includes periodically continued energy signals, such as

a sine-/cosine signal, or stochastic signals, such as noise with infinite energy. For ex-

ample, the energy for a DC signal s(t) = A, ∀t, 0 < A < ∞is infinite, and consequently

there is no energy signal but a power signal:

E = A²

∞

∫

−∞

dt = A² lim

θ→∞

θ

∫

−θ

dt →∞.

(2.38)

2.3.6 Deterministic and Stochastic Signals

Analytical signals are of great importance in the theory of signal processing. This is

due, among other things, to the fact that these signals are mathematically exactly

predictable, i.e. completely deterministic. The predictability of signals is therefore

an important property. Exactly predictable, so-called deterministic signals can be ex-

pressed in an analytical mathematical context and predicted for all times and places.

Stochastic random signals, on the other hand, cannot be fully expressed as an analytic

function and consequently cannot be predicted exactly (cf. Figure 2.17). As we will see

in subsection 5.3.2, for deterministic signals x(t) there always exists a spectral func-

tion X(f) which can be calculated via the Fourier series or the Fourier-transformation.

In the case of stochastic signals, however, this cannot be specified, since Fourier series

and Fourier-transformation require the exact knowledge of the time function for all

times t.

Definition: A deterministic signal can be described exactly in analytical form and

predicted at all future times, while a stochastic signal does not fulfil this condition or

only imperfectly.

Statistical Moments, Expected Value and Variance

The behaviour of stochastic signals can be characterised using statistical moments

such as expected value E(∙), variance Var(∙) and standard deviation σ(∙) as follows:

The first statistical moment or expected value of a random variable X is the value that