"- Another important factor is the choice of a set of scaling parameters $s$, such that we adequately sample all the frequencies present in our time series. \n",
"- We typically first choose the smallest resolvable scale, $s_0$, as some multiple of our time resolution of the inpu data, $\\deltat$. \n",
"- We typically first choose the smallest resolvable scale, $s_0$, as some multiple of our time resolution of the input data, $\\deltat$. \n",
"- The larger scales (longer periods) are (often) chosen as power-of-two multiples of this smallest scale.\n",
"- A common choice for the scake scould then be: $s_j = s_0 2^{j\\delta_j}$, for $j = 1, ..., J$.\n",
"- And therefore the maximum scale becomes: $J = \\delta_j^{-1} \\log_2(N\\delta t/s_0)$."
"- And therefore the maximum scale becomes: $J = \\delta_j^{-1} \\log_2(N\\delta t/s_0)$.\n",
"\n",
"> The difference between period and scale is essentially that the scale refers to the width of the wavelet. As the scale increases and the wavelet gets wider, it includes more of the time series, and the finer details get smeared out.\n",
"> The scale can be defined as the distance between oscillations in the wavelet (e.g. for the Morlet), or it can be some average width of the entire wavelet (e.g. for the Marr or Mexican hat).\n",
"\n",
"- The period (or inverse frequency) is the approximate Fourier period that corresponds to the oscillations within the wavelet. For all wavelets, there is a one-to-one relationship between the scale and period. The relationship can be derived by finding the wavelet transform of a pure cosine wave with a known Fourier period, and then computing the scale at which the wavelet power spectrum reaches its maximum.\n"
]
},
{
@@ -1087,7 +1092,7 @@
"### Credits\n",
"***\n",
"\n",
"This notebook contains material obtained from https://www.astroml.org/book_figures/chapter10/fig_chirp2_PSD.html, https://github.com/alsauve/scaleogram/blob/master/doc/scale-to-frequency.ipynb, https://paos.colorado.edu/research/wavelets/wavelet1.html"
"This notebook contains material obtained from https://www.astroml.org/book_figures/chapter10/fig_chirp2_PSD.html, https://github.com/alsauve/scaleogram/blob/master/doc/scale-to-frequency.ipynb, https://paos.colorado.edu/research/wavelets/wavelet1.html, https://paos.colorado.edu/research/wavelets/faq.html"
This is a repository with material (notebooks, papers, etc.) for the **Time Domain Astrophysics** course delivered at the *Università dell'Insubria* by Stefano Covino.
