Modeling A
==============


Characteristics of modeling
-----------------------------------

This test case proposes a number of analytical tests of the random signal generation operator GENE_FONC_ALEA. This operator generates realizations of a stationary Gaussian random process characterized by its power spectral density (DSP). We consider a two-dimensional DSP with :math:`{S}_{1}` and :math:`{S}_{2}` auto-spectra shown in Figure 1 below.


.. image:: images/10000000000004B100000384960B5731F68CBC1A.jpg
    :width: 4.0264in
    :height: 3.502in


.. _RefImage_10000000000004B100000384960B5731F68CBC1A.jpg:

**Figure 1: Auto-spectrum look** :math:`{S}_{1}` **and** :math:`{S}_{2}`

The interspectrum is written as :math:`{S}_{12}(f)\mathrm{=}\rho {({S}_{1}(f){S}_{2}(f))}^{\mathrm{0,5}}{e}^{i2\pi fT}` where :math:`T\mathrm{=}\mathrm{0,025}` and :math:`{\rho }^{2}\mathrm{=}\mathrm{0,8}` is the correlation coefficient. The standard deviations are :math:`{\sigma }_{1}\mathrm{=}\sqrt{(603.)}` and :math:`{\sigma }_{2}=\sqrt{(600.)}` respectively.

We are also testing the construction of DSP via POST_DYNA_ALEA, the operator for statistical post-processing of DSP.


* Random signals are drawn by operator GENE_FONC_ALEA


* DSP are estimated with the CALC_INTE_SPEC operator


* We perform statistical post-processing of DSP with the POST_DYNA_ALEA operator


* INFO_FONCTION is used to estimate the standard deviation of a given signal


Tested sizes and results
------------------------------


Signal generation with interpolation and imposed duration
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**% Tolerance**", "**Type**"
    "Autospectrum standard deviation :math:`{S}_{1}` "," :math:`{\sigma }_{1}` ", "1.0 10—5", "Analytics"
    "Autospectrum standard deviation :math:`{S}_{2}` "," :math:`{\sigma }_{2}` ", "1.0 10—5", "Analytics"
    "Signal standard deviation 1"," :math:`{\sigma }_{1}` ", "1.0 10—3", "analytical"
    "Signal standard deviation 2"," :math:`{\sigma }_{2}` ", "1.0 10—3", "analytical"
    "Autospectrum standard deviation signal 1"," :math:`{\sigma }_{1}` "," ", "1.0 10—3", "analytical"


.. csv-table::

    "Autospectrum standard deviation signal 2"," :math:`{\sigma }_{2}` "," ", "1.0 10—3", "analytical"
    "Moment order 0 autospectrum signal 1"," :math:`{\sigma }_{1}^{2}` "," ", "1.0 10—2", "analytical"
    "Moment order 0 autospectrum signal 1"," :math:`{\sigma }_{2}^{2}` "," ", "1.0 10—2", "analytical"
    "Moment order 1 autospectrum signal 1"," :math:`2.982305601621{10}^{5}` "," ", "1.0 10—3", "Non-regression"
    "Moment order 1 autospectrum signal 1"," :math:`2\pi {300}^{2}` "," ", "1.0 10—2", "analytical"


We also test (with respect to analytical values and in non-regression) the RMS values of the estimated autospectra.


Signal generation with interpolation, imposed number of points
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**", "**Type**"
    "Signal standard deviation 1"," :math:`{\sigma }_{1}` "," 1.00% ", "Analytics"
    "Signal standard deviation 2"," :math:`{\sigma }_{2}` "," 1.00% ", "Analytics"



Signal generation with interpolation, nothing imposed, truncation of 10-100Hz
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**", "**Type**"
    "Signal standard deviation 1"," :math:`180\mathrm{\times }2` "," 1.00% ", "Analytics"
    "Signal standard deviation 2"," :math:`90\mathrm{\times }2` "," 1.00% ", "Analytics"



Signal generation with interpolation, number of points and imposed duration
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**", "**Type**"
    "Signal standard deviation 1"," :math:`{\sigma }_{1}` "," 0.10% ", "Analytics"
    "Signal standard deviation 2"," :math:`{\sigma }_{2}` "," 0.10% ", "Analytics"



Signal generation with interpolation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**", "**Type**"
    "Signal standard deviation 1"," :math:`{\sigma }_{1}` "," 1.00% ", "Analytics"
    "Signal standard deviation 2"," :math:`{\sigma }_{2}` "," 1.00% ", "Analytics"



Generating signals without interpolation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**", "**Type**"
    "Signal standard deviation 1"," :math:`{\sigma }_{1}` "," 1.00% ", "Analytics"
    "Signal standard deviation 2"," :math:`{\sigma }_{2}` "," 1.00% ", "Analytics"



Signal generation without interpolation, imposed number of points
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**", "**Type**"
    "Signal standard deviation 1"," :math:`{\sigma }_{1}` "," 0.10% ", "Analytics"
    "Signal standard deviation 2"," :math:`{\sigma }_{2}` "," 0.10% ", "Analytics"