020326 commit

*.pptx

*.webpm

*.jl

Plutos

    "- [Vio et al. (2010) - \"Unevenly-sampled signals: a general formalism for the Lomb-Scargle periodogram\"](https://ui.adsabs.harvard.edu/abs/2010A%26A...519A..85V/abstract)\n",

    "- [Glynn et al. (2006) - \"Detecting periodic patterns in unevenly spaced gene expression time series using Lomb–Scargle periodograms\"](https://academic.oup.com/bioinformatics/article-pdf/22/3/310/48839224/bioinformatics_22_3_310.pdf)\n",

    "- [Emmanoulopoulos et al. (2013) - \"Generating artificial light curves: revisited and updated\"](https://ui.adsabs.harvard.edu/abs/2013MNRAS.433..907E/abstract)\n",

    "- [Perig et al. (2019) - \"Periodicity in Volcanic Gas Plumes: A Review and Analysis\"](https://ui.adsabs.harvard.edu/abs/2019Geosc...9..394P/abstract)"

    "- [Perig et al. (2019) - \"Periodicity in Volcanic Gas Plumes: A Review and Analysis\"](https://ui.adsabs.harvard.edu/abs/2019Geosc...9..394P/abstract)\n",

    "- [Scargle & Wagner (2026) - \"Studies in Astronomical Time Series Analysis: The Double Lomb-Scargle Periodogram and Super Resolution\"](https://ui.adsabs.harvard.edu/abs/2026arXiv260104552S/abstract)"

   ]

  },

  {

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**What is this?**

*This jupyter notebook is part of a collection of notebooks on various topics discussed during the Time Domain Astrophysics course delivered by Stefano Covino at the [Università dell'Insubria](https://www.uninsubria.eu/) in Como (Italy). Please direct questions and suggestions to [stefano.covino@inaf.it](mailto:stefano.covino@inaf.it).*

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**This is a `textual` notebook**

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![Time Domain Astrophysics](Pics/TimeDomainBanner.jpg)

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## PDS and ACF are Fourier dual

***

- We can write power spectral density (or power density or power density spectrum), $S(ω)$, as:

$$ S(ω) = \lim_{τ\to\infty}\frac{|X(ω)|^2}{τ}$$

- And the autocorrelation function of power (or periodic) signal $x(t)$ with any time period $T$ is given by:

$$ R(τ) = \lim_{T\to\infty}\frac{1}{T}\int^{T/2}_{−T/2}x(t)\ x^∗(t−τ) dt$$

- Where, $τ$ is called *the delayed parameter*.

- We want to prove that the power spectral density function $S(ω)$ and the autocorrelation function $R(τ)$ of a power signal form a Fourier transform pair, i.e.:

$$ R(τ) \Leftrightarrow S(ω)$$

- The autocorrelation function of a power signal $x(t)$ in terms of exponential Fourier series coefficients is given by,

$$ R(τ) = \sum_{n=−\infty}^{+\infty}C_nC_{−n} e^{jnω_0τ}$$

- Where, $C_n$ and C$_{−n}$ are the exponential Fourier series coefficients.

- Since $C_nC_{−n}=|C_n|^2$ we have:

$$ R(τ) = \sum_{n=−\infty}^{+\infty}|C_n|^2 e^{jnω_0τ}$$

- By taking the Fourier transform on both sides we get:

$$F[R(τ)] = F[\sum_{n=−\infty}^{+\infty}|C_n|^2 e^{jnω_0τ}] = \int_{-\infty}^{+\infty} [\sum_{n=−\infty}^{+\infty}|C_n|^2 e^{jnω_0τ}] e^{-jωτ} dτ$$

- By interchanging the order of integration and summation on RHS of the above expression, we also have:

$$F[R(τ)] = \sum_{n=−\infty}^{+\infty}|C_n|^2 \int_{-\infty}^{+\infty} e^{jnω_0τ} e^{-jωτ} dτ = \sum_{n=−\infty}^{+\infty}|C_n|^2 \int_{-\infty}^{+\infty} e^{-jτ(ω-nω_0)} dτ$$

- Since $\int_{-\infty}^{+\infty} e^{-jτ(ω-nω_0)} dτ = 2πδ(ω−nω_0)$, we have:

$$F[R(τ)] = 2π \sum_{n=−\infty}^{+\infty}|C_n|^2 δ(ω−nω_0)$$

- The RHS is the power spectral density (PSD) of the power function $x(t)$. Therefore:

$$F[R(τ)]=S(ω)$$

- Hence, it proves that the autocorrelation function $R(τ)$ and PSD function $S(ω)$ of a power signal form the Fourier transform pair.

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### Credits

***

This notebook contains material obtained from https://www.tutorialspoint.com/power-spectral-density-psd-and-autocorrelation-function#.

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## Course Flow

<table>

  <tr>

    <td>Previous lecture</td>

    <td>Next lecture</td>

  </tr>

  <tr>

    <td><a href="Lecture-SpectralAnalysis.ipynb">Spectral analysis</a></td>

    <td><a href="Lecture-SpectralAnalysis.ipynb">Spectral analysis</a></td>

  </tr>

 </table>

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**Copyright**

This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the examples, unless obtained from other properly quoted sources, under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Stefano Covino, Time Domain Astrophysics - Lecture notes featuring computational examples, 2025*.

%% Cell type:code id:226679b8-f75b-456d-ab17-6043f9066819 tags:

``` julia

```

This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the examples, unless obtained from other properly quoted sources, under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Stefano Covino, Time Domain Astrophysics - Lecture notes featuring computational examples, 2026*.