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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 `julia` notebook**

%% Cell type:code id:41d298ca-3a5c-457d-ab57-308a7de2d3c0 tags:

``` julia

import Pkg; Pkg.activate(".")

```

%% Output

      Activating project at `/mnt/chromeos/GoogleDrive/MyDrive/Teaching/Insubria/Docs_2025_26/Lectures/Lecture - Spectral Analysis`

%% Cell type:code id:6fff2b26-f553-42dd-a7fd-292dbcdbfbed tags:

``` julia

Pkg.instantiate()

```

%% Output

    Precompiling packages...

       3751.8 ms  ✓ StructUtils

       1833.7 ms  ✓ Interpolations → InterpolationsUnitfulExt

       1036.7 ms  ✓ StructUtils → StructUtilsTablesExt

       2383.2 ms  ✓ FFMPEG_jll

       4454.8 ms  ✓ Cairo

       4604.1 ms  ✓ DSP

       1529.4 ms  ✓ DSP → OffsetArraysExt

       6901.0 ms  ✓ JSON

       7864.2 ms  ✓ Distributions

       1474.2 ms  ✓ ColorBrewer

       1719.4 ms  ✓ Distributions → DistributionsTestExt

       2840.3 ms  ✓ Distributions → DistributionsChainRulesCoreExt

       1687.3 ms  ✓ KernelDensity

      50750.9 ms  ✓ DataFrames

       2007.5 ms  ✓ Latexify → DataFramesExt

     166724.3 ms  ✓ Makie

       6654.0 ms  ✓ Polynomials → PolynomialsMakieExt

      51968.9 ms  ✓ CairoMakie

      18 dependencies successfully precompiled in 237 seconds. 309 already precompiled.

%% Cell type:code id:8fa2633c-c139-4196-8a1e-3273bfe990a4 tags:

``` julia

using CairoMakie

```

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

%% Cell type:markdown id:a7b36f9a tags:

# A few important Fourier Transforms

***

## The rectangular function

***

- We want to compute the FT of the rectangular function: e.g. a function $g(t)$ with a given amplitude and extending from $t=-T/2$ to $t=T/2$. For $|t|>T/2$, $g(t)=0$.

- We will write the square pulse or box function as $rect_T(t)$, indicating that the rectangle function is equal to 1 for a period of $T$ (from $-T/2$ to $+T/2$) and 0 elsewhere,

%% Cell type:markdown id:8781bea3 tags:

 $$\mathfrak{F} \{  g(t) \} = G(f) = \int_{-\infty}^\infty g(t) e^{-2 \pi i f t} dt = \int_{-T/2}^{T/2} A e^{-2 \pi i f t} dt = \frac{A}{-2 \pi i f} e^{-2 \pi i f t}\Big|_{-T/2}^{T/2} = \frac{A}{-2 \pi i f} [ e^{-\pi i f T} - e^{\pi i f T} ] = $$

 $$ = \frac{AT}{\pi f T} \frac{ e^{-\pi i f T} - e^{\pi i f T} }{2i} = \frac{AT}{\pi f T} \sin (\pi f T) \equiv AT {\rm sinc} (fT)$$

 - where we have defined the "sinc" function defined as ${\rm sinc} (t) = \sin(\pi t) / \pi t$.

%% Cell type:code id:d6bcbb4b-132a-4ebf-9af6-acc1d871f456 tags:

``` julia

T=2.

#f(x) = (x >= -T/2 && x <= T/2) ? 1 : 0

function f(x; T=2)

    if x >= -T/2 && x <= T/2

        return 1

    else

        return 0

    end

end

xrng = range(start=-5.,stop=5.,step=0.01)

fg1 = Figure(size=(1000,500))

ax1 = Axis(fg1[1, 1],

    xlabel = "t",

    )

T=1

lines!(xrng,map(x -> f(x;T=T),xrng),label="T = "*string(T))

T=3

lines!(xrng,map(x -> f(x;T=3),xrng),label="T = "*string(T))

axislegend()

frng = range(start=-10.,stop=10.,step=0.01)

ax2 = Axis(fg1[1, 2],

    xlabel = "f",

    )

T=1

lines!(frng,T*sinc.(T*frng),label="T = "*string(T))

T=3

lines!(frng,T*sinc.(T*frng),label="T = "*string(T))

xlims!(-5,5)

axislegend()

fg1

```

%% Output

%% Cell type:markdown id:15b3a2a7-1cbc-4615-8c20-c8c35e5c92b7 tags:

## The Gaussian function

***

- We want now to compute the FT of the Gaussian function: $g(t) = e^{-\pi t^2}$.

- Let G(f) be the Fourier Transform of g(t), so that:

$$G(f) = \mathfrak{F} \{  g(t) \} = \int_{-\infty}^\infty g(t) e^{-2 \pi i f t} dt = \int_{-\infty}^\infty e^{-\pi t^2} e^{-2 \pi i f t} dt$$

- Take the derivative of both sides of equation with respect to f, so that:

$$\frac{dG(f)}{dt} = \int_{-\infty}^\infty e^{- \pi t^2} (-2 i \pi t) e^{-2 \pi i f t} dt$$

- Writing $u = e^{-2 \pi i f t}  \rightarrow du = (-2 \pi i f)e^{-2 \pi i f t} dt$ and $v = i e^{-\pi t^2} \rightarrow dv = -2 \pi i t e^{-\pi t^2} dt $, and given the formula for integration by parts: $\int udv = uv - \int vdu$ we have (the uv term becomes zero, because the limits are evaluated from $-\infty$ to $\infty$, where the product is zero):

$$\frac{dG(f)}{dt} = \int_{-\infty}^\infty ie^{- \pi t^2} (-2 i \pi f) e^{-2 \pi i f t} dt = -2 \pi f \int_{-\infty}^\infty e^{- \pi t^2} e^{-2 \pi i f t} dt = -2 \pi f G(t)$$

- This is a first order simple differential equation for $G(f)$. The solution for this differential equation is given by:

$$G(f) = G(0) e^{-\pi f^2} $$

- All we need to do now to find $G(f)$ is figure out what $G(0)$ is.

- $G(0)$ is simply the average value of $g(t)$, because if you substitute $f=0$ into the equation for $G(f)$ the complex exponential term goes away.

    - The integral has actually an [elegant solution](https://en.wikipedia.org/wiki/Gaussian_integral). The result is:

$$G(0) = \int_{-\infty}^\infty e^{-\pi t^2} dt = 1$$

- Therefore:

$$G(f) = \mathfrak{F} \{  g(t) \} = e^{-\pi f^2} $$

%% Cell type:markdown id:0cd961f2 tags:

### Credits

***

This notebook contains material obtained by https://towardsdatascience.com/a-proof-of-the-central-limit-theorem-8be40324da83.

%% Cell type:markdown id:05e93b1d tags:

## Course Flow

***

<table>

  <tr>

    <td>Previous lecture</td>

    <td>Next lecture</td>

  </tr>

  <tr>

      <td><a href="Lecture-StatisticsReminder.ipynb">Reminder of frequentist statistics</a></td>

    <td><a href="Lecture-StatisticsReminder.ipynb">Reminder of frequentist statistics</a></td>

  </tr>

 </table>

%% Cell type:markdown id:591bd355 tags:

**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, 2026*.