added a few things to spectral

spectral/lecture/spectral.tex

@@ -2,18 +2,35 @@
 
 This is a stub that should be filled :)
 
-\section{The Fourier space}
+% \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{Forward and reverse filter}
-
 \section{Coherence function}
+
+\subsection{Forward and reverse filter}