190325 commit

# ╔═╡ 477704cc-f022-4cc5-a4a4-679387b819bd

# ╠═╡ show_logs = false

md"""

cm"""

## A reminder of the CFT

***

- Given a continuous signal g(t) the Fourier transform and its inverse are:

- Given a continuous signal ``g(t)`` the Fourier transform and its inverse are:

```math

\hat{g}(f) \equiv \int_{-\infty}^\infty g(t) e^{-2\pi i f t} dt \quad  g(t) \equiv \int_{-\infty}^\infty \hat{g}(f) e^{+2\pi i f t} df

\mathcal{F}\{g\} = \hat{g} \quad    \mathcal{F}^{-1}\{\hat{g}\} = g

```

- g and ĝ are know as a Fourier pair: $g \Longleftrightarrow \hat{g}$.

- ``g`` and ``ĝ`` are know as a Fourier pair: ``g \Longleftrightarrow \hat{g}``.

- The Fourier Transform (FT) is a linear operator:

       \mathcal{F}\{A f(t)\} = A\mathcal{F}\{f(t)\}

```

- The FT of a sinusoid with frequency $f_0$ is a sum of delta functions at $\pm f_0$, where $\delta(f)\equiv\int_{-\infty}^\infty e^{-2\pi i x f}df$.

- The FT of a sinusoid with frequency ``f_0`` is a sum of delta functions at ``\pm f_0``, where ``\delta(f)\equiv\int_{-\infty}^\infty e^{-2\pi i x f}df``.

- We can write: $ \mathcal{F}\{e^{2\pi f_0 t}\} = \delta(f - f_0)$, and:

- We can write: `` \mathcal{F}\{e^{2\pi f_0 t}\} = \delta(f - f_0)``, and:

```math

\mathcal{F}\{\cos(2\pi f_0 t)\} = \frac{1}{2}\left[\delta(f - f_0) + \delta(f + f_0)\right] \quad

      \mathcal{F}\{\sin(2\pi f_0 t)\} = \frac{1}{2i}\left[\delta(f - f_0) - \delta(f + f_0)\right]

```

- Relations that can be derived from Euler’s formula: $e^{ix} = \cos x + i\sin x$

- Relations that can be derived from Euler’s formula: ``e^{ix} = \cos x + i\sin x``

- A time shift imparts a phase in the FT:  $  \mathcal{F}\{g(t - t_0)\} = \mathcal{F}\{g(t)\} e^{-2\pi i ft_0}$.

- A time shift imparts a phase in the FT:  ``  \mathcal{F}\{g(t - t_0)\} = \mathcal{F}\{g(t)\} e^{-2\pi i ft_0}``.

- And, as we know, the squared amplitude of the FT of a continuous signal is known as the power spectral density (PSD):  $ \mathcal{P}_g \equiv \left|\mathcal{F}\{g\}\right|^2 $.

- And, as we know, the squared amplitude of the FT of a continuous signal is known as the power spectral density (PSD):  `` \mathcal{P}_g \equiv \left|\mathcal{F}\{g\}\right|^2 ``.

    - Note that if $g$ is real-valued, it follows that $P_g$ is an even function, {i.e.} $\mathcal{P}_g(f) = \mathcal{P}_g(-f)$.

    - Note that if ``g`` is real-valued, it follows that ``P_g`` is an even function, {i.e.} ``\mathcal{P}_g(f) = \mathcal{P}_g(-f)``.

$(LocalResource("Pics/FTpairs.png"))

"""

# ╔═╡ 9d413d79-c9fb-4c30-ac02-92380510b607

# ╠═╡ show_logs = false

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	fg6 = Figure(size=(700,350))

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	    title="Peak scaling with number of data points (fixed S/N=10)",

	p99 = LombScargle.fapinv(lsboot7,0.99)

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