Note

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# Create a spectral modelΒΆ

This use case illustrates how the User can define his own density
spectral function from parametric models. The library allows it thanks to
the object *UserDefinedSpectralModel* defined from:

a frequency grid with step , stored in the object

*RegularGrid*,a collection of hermitian matrices stored in the object

*HermitianMatrixCollection*, which are the images of each point of the frequency grid through the density spectral function.

The library builds a constant piecewise function on , where
the intervals where the density spectral function is constant are
centered on the points of the frequency grid, of length .
Then, it is possible to evaluate the spectral density function for a
given frequency thanks to the method [*computeSpectralDensity*]{}: if
the frequency is not inside the interval , an exception is thrown.
Otherwise, it returns the hermitian matrix of the
subinterval of that contains the given frequency.

In the following script, we illustrate how to create a modified low pass model of dimension with exponential decrease defined by: where

Frequency value should be positive,

for , the spectral density function is constant: ,

for , the spectral density function is equal to .

The frequency grid is with Hz. The figure draws the spectral density.

```
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
import math as m
ot.Log.Show(ot.Log.NONE)
```

Create the frequency grid:

```
fmin = 0.1
df = 0.5
N = int((10.0 - fmin) / df)
fgrid = ot.RegularGrid(fmin, df, N)
```

Define the spectral function:

```
def s(f):
if f <= 5.0:
return 1.0
else:
x = f - 5.0
return m.exp(-2.0 * x * x)
```

Create the collection of HermitianMatrix:

```
coll = ot.HermitianMatrixCollection()
for k in range(N):
frequency = fgrid.getValue(k)
matrix = ot.HermitianMatrix(1)
matrix[0, 0] = s(frequency)
coll.add(matrix)
```

Create the spectral model:

```
spectralModel = ot.UserDefinedSpectralModel(fgrid, coll)
# Get the spectral density function computed at first frequency values
firstFrequency = fgrid.getStart()
frequencyStep = fgrid.getStep()
firstHermitian = spectralModel(firstFrequency)
# Get the spectral function at frequency + 0.3 * frequencyStep
spectralModel(frequency + 0.3 * frequencyStep)
```

Draw the spectral density

```
# Create the curve of the spectral function
x = ot.Sample(N, 2)
for k in range(N):
frequency = fgrid.getValue(k)
x[k, 0] = frequency
value = spectralModel(frequency)
x[k, 1] = value[0, 0].real
# Create the graph
graph = ot.Graph(
"Spectral user-defined model", "Frequency", "Spectral density value", True
)
curve = ot.Curve(x, "UserSpectral")
graph.add(curve)
graph.setLegendPosition("topright")
view = viewer.View(graph)
plt.show()
```