Descargar - UTP

48
an alternative way for communications system
assessment. This work is structured as follows:
In second part the usage of a I/O scheme is
explained. A simple example is provided in order
to test the validity of the problem. In third
section the model is adapted to the problem of
computing a PSCC spectrum by having welldefined
input and transfer functions. In fourth
section results are presented and finally conclusions
regarding the results of this report are
presented.
II. WHY I/O?
In linear control and system identification
theory there is well-defined correspondence
between I/O system elements through a
transfer function or kernel (at the MIMO case,
for instance). Mathematically speaking the
realization of such correspondence is given by
means integral equations and convolutions.
In this framework, both I/O observables are
described by determinism variables because
the phenomenon involves determinism operators
as well. This concept is exemplified by the
master equation where the operator
which performs the transition in
the determinism manner. However, a similar
picture would occurs in the case of using stochastic
variables and parameters. If a variable
transition is allowed, then under the I/O view
the existence of a transfer function stochastic
should encompass such action. For instance,
if a PSCC is perceived as output, then would
have to have its input associated function in
the following way,
(2)
The function JK is expressed in its general
form as a matrix. Under this scenario it is assumed
the existence of several I/O functions
as indicates us Eq. (2), and for the simple
MIMO case, a one-to-one operation is required
to model the PSCC uniquely. To validate this
approach, a simple example is presented.
A. A Simple Example
The most simple case which would be used
to extract an approximate PSCC curve would
have to be the replacement of the transfer
function by a delta function,
returning a shape exactly like the PD because
the effect in using a Delta function does
not change the initial form on it. In reality for
this simple case, one can test a Delta function
with different argument which would make the
departure from an ordinary one to one much
more phenomenological,
(3)
resulting in a distribution that does not coincides
with that of the P D necessarily. Contrarily
to the trivial example as given in (3), to note
now appears a quantity in (5). The resulting
shape obtained in (5) is plotted in Fig. 1
together to its input as well. In blue the PSCC
curve is plotted against , whereas in red the
P D one as a simple exponential. Clearly one observes
that (5) satisfies a relation a SISO type.
In order to be much more precise, one option
would be the case where the transfer function
argument depends on the difference
It is because the basic definition at the Laplace space
which is actually a linear relation between I/O.
When is postulated a nonlinear one then one
would expect an expression as Y (s) = G(s)X(s) +
G 2 (s)X 2 (s) thereby providing a phenomenology at
the nonlinear regime. Due to the stochastic nature
of the phenomena a possible justification
for using the I/O mechanism would consist in
the incorporation of quadratic terms like Eq. (1)
by keeping a certain coherence with the randomness
of variables in the sense that stochastic
would have its equivalence with nonlinearity.
III. MODEL ADAPTATION FOR THE CASE
OF THE PSCC
A. Input Selection
For the present analysis, we invoke to a simulation
in order to built the corresponding input
function namely the PDF function. We have
appealed to an algorithm Monte-Carlo-like in
order to generate the PDF curves. Instead of
exponential profiles as commonly is derived
from statistical foundations
Fig. 1. PSCC versus curves with =0.5, from the usage
of Eq. (5). In red is plotted the P D input function denoted
by a simple exponential function whereas the blue one
displays the Eq. 5. It is observed the correlation between
I/O functions through the Eq. (4) by which a Delta function
plays the role of a transfer function.
CIENCIA, CULTURA Y TECNOLOGÍA - UNIVERSIDAD TECNOLÓGICA DEL PERÚ
(6)
(4)
(5)