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2 Fundamentals of Information, Signal and System Theory

time t/s

time t/s

Fig. 2.17: Examples of signals s1(t) to s3(t) originate from a deterministic (exactly predetermined)

process and can be completely described analytically as a polynomial or harmonic function (left).

The random signal s4(t), on the other hand, comes from a random process and cannot be fully ex-

pressed by analytical functions (right).

the random variable is most likely to take. If xi is a real discrete random variable with

the values (xi)i∈ℕand with the respective probabilities (pi)i∈ℕ(with ℕas the set of

natural numbers), the expected value (1. moment) of the time series is given by:

E(X) = μX = ∑

i∈ℕ

xipi = ∑

i∈ℕ

xiP(X = xi) .

(2.39)

The expectation E(X) is thus the weighted mean μX of X weighted by the probabilit-

ies P and thus the most probable value for a realisation of X (1. moment). With equal

probability of N realisations pi = p = 1/N, the expected value is equal to the mean

μX = μ of X. For integrable expected values, i.e. E(X) = μX < ∞, the second moment

or variance is the expected value of the squared deviation of the random variable X

from the mean μX.

Var(X) = E((X −μ)²) = ∑

i∈ℕ

(xi −μX)²P(X = xi) .

(2.40)

The variance is a quadratic quantity that gives the mean squared deviation of a ran-

dom variable from the expected value of X. It is thus the expected value of the squared

deviation (2. moment). The associated non-squared quantity is the standard deviation

σX, which is defined as the square root of the variance:

σ(X) = ^√var(X) .

(2.41)

Both the variance and the standard deviation are positive quantities. Thus, Var(∙) ≥0

and σ(∙) ≥0 hold.