05032025 commit

using InteractiveUtils

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import Pkg; Pkg.activate(".")

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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*.

"""

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using InteractiveUtils

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import Pkg; Pkg.activate(".")

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"""

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md"""

cm"""

## Discrete Fourier transform

***

- In the real world, we only have discrete measurements, the time series, commonly called, in astronomy, “light curves”. They consist of $N$ measurements $x_k$ taken (now) at equally-spaced times $t_k$ from $0$ to $T$.

- In the real world, we only have discrete measurements, the time series, commonly called, in astronomy, “light curves”. They consist of ``N`` measurements ``x_k`` taken (now) at equally-spaced times ``t_k`` from ``0`` to ``T``.

- In this case we can define the discrete Fourier transform (and its inverse) as:

```math

a_j =            \sum\limits_{k=0}^{N-1} x_k e^{-2\pi ijk/N} \quad\quad (j=-N/2,...,N/2-1)

a_j = \sum\limits_{k=0}^{N-1} x_k e^{-2\pi ijk/N}   \quad   (j=-N/2,...,N/2-1)

```

```math

x_k = {1\over N} \sum\limits_{k=-N/2}^{N/2-1} a_j e^{2\pi ijk/N} \quad\quad (k=0,...,N-1)

```

- Since the data are equally spaced, the times are $kT/N$ and the frequencies are $j/T$.

- Since the data are equally spaced, the times are ``kT/N`` and the frequencies are ``j/T``.

- The time step is $δt = T/N$ and the frequency step is $δν = 1/T$.

- The time step is ``δt = T/N`` and the frequency step is ``δν = 1/T``.

- As the discrete time series has a time step $δt$ and a duration $T$, there are limitations to the frequencies that can be examined:

    - The lowest frequency is $1/T$, corresponding to a sinusoid with a period equal to the signal duration.

    - The highest frequency that can be sampled, is called *Nyquist frequency*: $\nu_\rm{Nyq} = \frac{1}{2\delta T} = \frac{1}{2}\frac{N}{T} $.

- As the discrete time series has a time step ``δt`` and a duration ``T``, there are limitations to the frequencies that can be examined:

    - The lowest frequency is ``1/T``, corresponding to a sinusoid with a period equal to the signal duration.

    - The highest frequency that can be sampled, is called *Nyquist frequency*: ``\nu_\rm{Nyq} = \frac{1}{2\delta T} = \frac{1}{2}\frac{N}{T} ``.

- At the zero frequency, the FT value is just the sum of the signal values:

a_0 = \sum\limits_{k=0}^{N-1} x_k e^{-2\pi i0k/N} = \sum\limits_{k=0}^{N-1} x_k

```

- *Parseval’s theorem* applies also to the discrete case and one can see that the variance of a signal is $1/N$ times the sum of the $a_j$ over all indices besides zero (also known as *Plancherel Theorem*):

- *Parseval’s theorem* applies also to the discrete case and one can see that the variance of a signal is ``1/N`` times the sum of the ``a_j`` over all indices besides zero (also known as *Plancherel Theorem*):

```math

Var(x_k) = \sum_k (x_k - \bar{x})^2 = \sum_k x_k^2 + \sum_k \bar{x}^2 - \sum_k 2\bar{x}x_k = \sum_k x_k^2 + N \bar{x}^2 -  2N\bar{x}^2 = \sum_k x_k^2 - N\bar{x}^2 = \sum_k x_k^2 - \frac{1}{N}(\sum_k x_k)^2 = \frac{1}{N} \sum_j |a_j|^2 - \frac{1}{N}a_0^2 \Longrightarrow Var(x_k) = \frac{1}{N}\sum_{j=-\frac{N}{2}}^{j=\frac{N}{2}-1} |a_j|^2, \quad j\ne0

Var(x_k) = \sum_k (x_k - \bar{x})^2 = \sum_k x_k^2 + \sum_k \bar{x}^2 - \sum_k 2\bar{x}x_k = \sum_k x_k^2 + N \bar{x}^2 -  2N\bar{x}^2 = 

```

```math

= \sum_k x_k^2 - N\bar{x}^2 = \sum_k x_k^2 - \frac{1}{N}(\sum_k x_k)^2 = \frac{1}{N} \sum_j |a_j|^2 - \frac{1}{N}a_0^2 =

```

```math

\Longrightarrow Var(x_k) = \frac{1}{N}\sum_{j=-\frac{N}{2}}^{j=\frac{N}{2}-1} |a_j|^2,  j\ne0

```

"""

- In the following exercize, we are going to analyse weather data spanning about 20 years in France obtained from the US National Climatic Data Center.

- Data are imported http://www.ncdc.noaa.gov/cdo-web/datasets#GHCND. The number "-9999" is used for N/A values. And we need to parse dates contained in the DATE column

- Data are imported [http://www.ncdc.noaa.gov/cdo-web/datasets#GHCND](http://www.ncdc.noaa.gov/cdo-web/datasets#GHCND). The number "-9999" is used for N/A values. And we need to parse dates contained in the DATE column

"""

# ╔═╡ 3d75fecb-157b-408a-aa32-2d2096a766ba

	temp_psd = abs.(temp_fft).^2

	# 1 is the sampling frequency

	temp_freq = fftfreq(length(temp_psd), 1) * 365;

end

	temp_freq = fftfreq(length(temp_psd), 1) * 365

end;

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md"""

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*.

"""

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