\chapter{Spectral analyses}

This is a stub that should be filled :)

% \section{The Fourier space}

\section{The Fourier Transform}
Complex numbers... magnitude and phase

\subsection{Fast Fourier transform}

\section{Power spectrum}
\[ S_{x,x} = |X(f)|^2 \]

Parceval theorem:
\[ \int_{-\infty}^{+\infty} x(t)^2 dt = \int_{-\infty}^{+\infty} |X(f)|^2 df \]

Autocorrelation:

Wiener-Kinchin theorem:
\[ {\cal F}\{Corr(x,x)\} = |X(f)|^2 \]

\section{Spectrogram}

\section{Cross spectrum}
\[ S_{x,y} = X(f)Y^*(f) \]
is complex valued (magnitude and phase)!

Correlation theorem:
\[ {\cal F}\{Corr(x,y)\} = X(f)Y^*(f) = S_{x,y} \]

\section{Transfer function}

\section{Coherence function}

\subsection{Forward and reverse filter}