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\title{Time Series Recap and Wavelets Intro}
\author{C. Kankelborg}
\date{\today}

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\noindent We have been doing time series analysis. Today we introduce the \emph{continuous wavelet transform} (follow the links below).
\begin{enumerate}
   \item DFT/FFT
   \begin{enumerate}
      \item Properties of the DFT
      \item FFT efficiency
   \end{enumerate}
   \item Power spectral estimation
   \begin{enumerate}
      \item Windowing
      \item Smoothing (by segmenting the data into many windows)
   \end{enumerate}
   \item Digital filtering\quad $\kappa(t) \ast f(t) = 
      \mathcal{F}^{-1} \left[ \widetilde{\kappa}(\omega)\, \widetilde{f}(\omega) \right]$
   \begin{enumerate}
      \item Example: Integration of data
      \item There are enough types of digital filters to do a whole course...
      \begin{enumerate}
         \item High pass, low pass, band pass and band stop filters
         \item Savitzky-Golay (\emph{Numerical Recipes} 2nd ed.\ \S\,14.8)
         \item Wiener filtering (\emph{Numerical Recipes} 2nd ed.\ \S\,13.3)
         \item Template matching
         \item Nonlinear filters, \emph{etc.}
      \end{enumerate}
   \end{enumerate}
   \item Next up: \textbf{Continuous Wavelet Transform}
   \begin{enumerate}
      \item Inspiration: fig.\,1 of \href{http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.116.061102}
         {GW discovery paper}
      \item Highlights of \href{http://paos.colorado.edu/research/wavelets/}
         {Torrence \& Compo's \emph{Practical Guide}}
      \begin{enumerate}
         \item CWT definition
         \item Mother wavelets and scaling
         \item White noise, red noise, and significance testing
         \item Reconstruction
      \end{enumerate}
      \item \href{http://solar.physics.montana.edu/kankel/ph567/examples/Octave/wavelets-tc/}
         {Wavelet Filtering Example}
   \end{enumerate}
\end{enumerate}



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