1. Modeling A#

1.1. 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 \({S}_{1}\) and \({S}_{2}\) auto-spectra shown in Figure 1 below.

_images/10000000000004B100000384960B5731F68CBC1A.jpg

Figure 1: Auto-spectrum look \({S}_{1}\) and \({S}_{2}\)

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

1.2. Tested sizes and results#

1.2.1. Signal generation with interpolation and imposed duration#

Identification

Reference

% Tolerance

Type

Autospectrum standard deviation \({S}_{1}\)

\({\sigma }_{1}\)

1.0 10—5

Analytics

Autospectrum standard deviation \({S}_{2}\)

\({\sigma }_{2}\)

1.0 10—5

Analytics

Signal standard deviation 1

\({\sigma }_{1}\)

1.0 10—3

analytical

Signal standard deviation 2

\({\sigma }_{2}\)

1.0 10—3

analytical

Autospectrum standard deviation signal 1

\({\sigma }_{1}\)

1.0 10—3

analytical

Autospectrum standard deviation signal 2

\({\sigma }_{2}\)

1.0 10—3

analytical

Moment order 0 autospectrum signal 1

\({\sigma }_{1}^{2}\)

1.0 10—2

analytical

Moment order 0 autospectrum signal 1

\({\sigma }_{2}^{2}\)

1.0 10—2

analytical

Moment order 1 autospectrum signal 1

\(2.982305601621{10}^{5}\)

1.0 10—3

Non-regression

Moment order 1 autospectrum signal 1

\(2\pi {300}^{2}\)

1.0 10—2

analytical

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

1.2.2. Signal generation with interpolation, imposed number of points#

Identification

Reference

Tolerance

Type

Signal standard deviation 1

\({\sigma }_{1}\)

1.00%

Analytics

Signal standard deviation 2

\({\sigma }_{2}\)

1.00%

Analytics

1.2.3. Signal generation with interpolation, nothing imposed, truncation of 10-100Hz#

Identification

Reference

Tolerance

Type

Signal standard deviation 1

\(180\mathrm{\times }2\)

1.00%

Analytics

Signal standard deviation 2

\(90\mathrm{\times }2\)

1.00%

Analytics

1.2.4. Signal generation with interpolation, number of points and imposed duration#

Identification

Reference

Tolerance

Type

Signal standard deviation 1

\({\sigma }_{1}\)

0.10%

Analytics

Signal standard deviation 2

\({\sigma }_{2}\)

0.10%

Analytics

1.2.5. Signal generation with interpolation#

Identification

Reference

Tolerance

Type

Signal standard deviation 1

\({\sigma }_{1}\)

1.00%

Analytics

Signal standard deviation 2

\({\sigma }_{2}\)

1.00%

Analytics

1.2.6. Generating signals without interpolation#

Identification

Reference

Tolerance

Type

Signal standard deviation 1

\({\sigma }_{1}\)

1.00%

Analytics

Signal standard deviation 2

\({\sigma }_{2}\)

1.00%

Analytics

1.2.7. Signal generation without interpolation, imposed number of points#

Identification

Reference

Tolerance

Type

Signal standard deviation 1

\({\sigma }_{1}\)

0.10%

Analytics

Signal standard deviation 2

\({\sigma }_{2}\)

0.10%

Analytics