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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 (-f_c, \dots, f_c) with step \delta f, stored in the object RegularGrid,

  • a collection of hermitian matrices \in \mathbb{M}_d(\mathbb{C}) 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 [-f_c,f_c], where the intervals where the density spectral function is constant are centered on the points of the frequency grid, of length \delta f. 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 [-f_c,f_c], an exception is thrown. Otherwise, it returns the hermitian matrix of the subinterval of [-f_c,f_c] that contains the given frequency.

In the following script, we illustrate how to create a modified low pass model of dimension d=1 with exponential decrease defined by: S: \mathbb{R} \rightarrow \mathbb{R} where

  • Frequency value f should be positive,

  • for f < 5 Hz, the spectral density function is constant: S(f)=1.0,

  • for f > 5 Hz, the spectral density function is equal to S(f) = \exp \left[- 2.0 (f - 5.0)^2 \right].

The frequency grid is ]0, f_c] = ]0,10] with \delta f = 0.2 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)

[[ (2.50622e-15,0) ]]



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()
Spectral user-defined model