Exercises Module 2

Module 2: Fundamentals on Gaussian Random Fields

Exercise 1 - White Noise Field and its Power Spectrum

1) Generate a white noise field with variance \(\sigma^2=1\).

2) Calculate its power spectrum: show numerically that the power spectrum is constant and equal to \(P(k)=\sigma^2\)

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Exercise 2 - Incresing the variance of the Gaussian Random Field in Fourier Space

1) Starting from the previous generated white noise field, modify the Fourier components of the field to produce a new Gaussian field with variance \(\sigma^2=4\).

2) Calculate the inverse Fourier transform of this field and verify that the distribution of values in real space follows the desired Gaussian distribution.

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Exercise 3 - P(k) Gaussian Field from White Noise

1) Generalize the previous exercise by impossing a power spectrum \(1/k^3\).

2) Plot the resulting field in real space.

3) Plot the histogram of the field values in real space. Does the histrogram corresponds to a Gaussian distribution? Why not?

4) Average several realization of the same field and check if you recover a Guassian distribution.

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Exercise 4 (optional) - A General Homogeneous and Isotropic Random Field Generator

1) Generalize the exercises and create a simple function that returns a Gaussian Random Field given an arbitrary power spectrum index

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