Fundamentals - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 7

The example of Figure 1.3 shows some outcomes of a same physical signal
(Doppler echo signal from a 38 GHz traffic radar carried by a motor car). Every
realization contains the same relevant information: same vehicle speed, same
pavement surface, same trajectory, etc. However, there is no reproducibility of
outcomes, and there are fairly large morphological differences between them. At the
very most we cannot avoid noticing that they have a “family likeness”. The
probabilistic description is nothing but one of the approaches that helps account for
this “family likeness”, i.e. by the membership of the signal to a well-defined
probability space.

,

,

,

Figure 1.3. Example of 3 outcomes of an actual random signal

The laws are probability sets

() () () ( )

Pxt ⎡ ∈ D xt , ∈ D , … , xt ∈ D ⎤ = LDD , , … , D tt , , , … [1.6] , t

⎣ ⎦

1 1 2 2 n n 1 2 n 1 2 n

whose most well-known examples are the probability densities in which
[

[

Dxx x = , +

i i i i

[ ] (

= …………

)

P p x , , x , t , , t dx , , dx [1.7]

1 n 1 n 1 n

] ]

and the distribution functions, with D =−∞ , x

i i