Digital Signal Processing Reference
In-Depth Information
Parameterized
transfer function
Figure 4.1.
Modeling “system” approach
This approach was historically decisive for signal processing, then for images:
all identification tools, essentially developed by the control engineer, and to a lesser
degree in numerical analysis where it can be used to solve the main problem, which
is that of estimation of parameters of the model which help adjust
x
(
k
)
to the
physical signal. The difference (and additional difficulty) with the problem of
identification of the systems is that the virtual input
u
(
k
)
is not physically accessible.
Thus we will have to base ourselves only on the observation of
x
(
k
)
to build the
estimators. The hypotheses, which we will make on the transfer function of the
system, will help classify the models: we will discuss the linear or non-linear,
stationary or non-stationary models, etc., according to the corresponding properties
of the system. As for the inputs, they will be selected for their “neutrality”, that is to
say they would limit the class
x
(
k
)
generated as much as possible: the most current
would be the null mean white noise, but we will also find excitations derived from a
series of periodic pulses (such as voice models), from a finite set of pulses (multi-
pulse models), or even a single pulse, etc.
4.2. Linear models
When the system appearing in Figure 4.1 is linear, the very large set of results on
this class of systems can be used: this is a very strong motive to accept the limitation
to this particular case and we will discover that it will provide a class of very
efficient models. The system could be invariant by translation or otherwise. In the
first case, it could lead to models of stationary signals, if
u
(
k
)
itself is stationary; in
the second case, we would have access to non-stationary models otherwise known as
“evolutionary”.
4.2.1.
Stationary linear models
If we suppose that the system appearing in Figure 4.1 is invariant and linear,
with an impulse response {
h
(
k
)},
and thus of transmittance in
z
:
()
(
)
()
hz
ZThk
[4.4]
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