Biomedical Engineering Reference
In-Depth Information
1
Introductory concepts
1.1 Stochastic and deterministic signals, concepts of stationarity
and ergodicity
A signal is a physical quantity that can carry information. Physical and biological
signals may be classified as deterministic or stochastic. The stochastic signal con-
trary to the deterministic one cannot be described by a mathematical function. An
example of a deterministic signal may be the time course of a discharging capac-
ity or the position of a pendulum. Typical random process may be the number of
particles emitted by the radioactive source or the output of a noise generator. Phys-
iological signals can be qualified as stochastic signals, but they usually consist of a
deterministic and a random component. In some signals, the random component is
more pronounced while in others the deterministic influences prevail. An example
of the stochastic signal, where random component is important may be EEG. An-
other class of signals can be represented by an ECG which has a quite pronounced
deterministic component related to propagation of the electrical activity in the heart
structures, although some random component coming from biological noise is also
present.
A process may be observed in time. A set of observations of quantity
x
in function
of time
t
forms the time series
x
. In many cases the biophysical time series can be
considered as a realization of a process, in particular the stochastic process.
If
K
will be the assembly of
k
events (
k
(
t
)
∈
K
) and to each of these events we assign
function
x
k
(
t
)
called realization of the process ξ
(
t
)
, the stochastic process can be
defined as a set of functions:
ξ
(
t
)=
{
x
1
(
t
)
,
x
2
(
t
)
,...,
x
N
(
t
)
}
(1.1)
where
x
k
are the random functions of variable
t
.
In the framework of the theory of stochastic processes the physical or biophysical
process can be described by means of the expected values of the estimators found by
the ensemble averaging over realizations. The expected value of stochastic process is
an average over all realizations of the process weighted by the probabilities of their
occurrence. The mean value
μ
x
(
t
)
in the time
t
1
can be
found by summation of the actual values of each realization in time
t
1
weighted by
(
t
1
)
of the stochastic process ξ
(
t
)
1
