Start something

I won't ever make it in time but whatever

1 files changed, 72 insertions, 2 deletions

diff --git a/tex/signals.tex b/tex/signals.tex
index 17a5a72..1cb86ce 100644
--- a/tex/signals.tex
+++ b/tex/signals.tex
@@ -1,16 +1,42 @@
 \section{Signals}
 \subsection{Classification}
-%% TODO
+\begin{figure}[h]
+  \centering
+  \begin{tikzpicture}[
+      nodes = {
+        thick,
+        draw = black,
+        fill = lightgray!20,
+        align = center,
+        inner sep = 2mm,
+        outer sep = 1mm,
+      },
+      sibling distance = 3cm,
+    ]
+    \node {All signals}
+      child {node {Class 1 \\ \(0 < E_n < \infty\)}}
+      child {
+        node {Class 2 \\ \(0 < P_n < \infty\)}
+        child {node {Class 2a \\ periodic}}
+        child {node {Class 2b \\ stochastic}}
+      }
+    ;
+  \end{tikzpicture}
+\end{figure}
 
 \subsection{Properties}
+For class 2b signals the formula for class 2a signals can used by taking \(\lim_{T\to\infty} f_\text{2a}(T)\) (if the limits exists).
+The notation \(\int_T\) is short for an integral from \(-T/2\) to \(T/2\).
 \begin{table}[h]
   \everymath={\displaystyle}
   \[
     \begin{array}{l l}
       \toprule
-      \text{\bfseries Characteristic} & \text{\bfseries Symbol and formula} \\
+      \text{\bfseries Characteristic} & \text{\bfseries Symbol and formula} \\[6pt]
+	    \text{\itshape Class 1 Signals} \\
       \midrule
       \text{Normalized energy} & E_n = \lim_{T\to\infty} \int_T |x|^2 \,dt \\[6pt]
+	    \text{\itshape Class 2a Signals} \\
       \midrule
       \text{Normalized power}  & P_n = \lim_{T\to\infty} \frac{1}{T} \int_T |x|^2 \,dt \\[12pt]
       \text{Linear mean}       & X_0 = \frac{1}{T} \int_T x\, dt \\[12pt]
@@ -24,3 +50,47 @@
     \end{array}
   \]
 \end{table}
+
+\subsection{Correlation}
+\paragraph{Autocorrelation}
+The \emph{autocorrelation} is a measure for how much a signal is coherent, i.e. how similar it is to itself.
+For class 1 signals the autocorrelation is
+\[
+  \varphi_{xx}(\tau) = \lim_{T\to\infty} \int_T x(t) x(t - \tau) \,dt,
+\]
+whereas for class 2a and 2b signals
+\begin{gather*}
+  \varphi_{xx}(\tau) = \frac{1}{T} \int_T x(t) x(t - \tau) \,dt \quad\text{(2a)}, \\
+  \varphi_{xx}(\tau) = \lim_{T\to\infty} \frac{1}{T} \int_T x(t) x(t - \tau) \,dt \quad\text{(2b)}.
+\end{gather*}
+Properties of \(\varphi_{xx}\):
+\begin{itemize}
+  \item \(\varphi_{xx}(0) = X^2 = (X_0)^2 + \sigma^2\)
+  \item \(\varphi_{xx}(0) \geq |\varphi_{xx}(\tau)|\)
+  \item \(\varphi_{xx}(\tau) \geq (X_0)^2 - \sigma^2\)
+  \item \(\varphi_{xx}(\tau) = \varphi_{xx}(\tau + nT)\) (periodic)
+  \item \(\varphi_{xx}(\tau) = \varphi_{xx}(-\tau)\) (even, symmetric)
+\end{itemize}
+The Fourier transform of the autocorrelation \(\Phi_{xx}(j\omega) = \fourier \varphi_{xx}(t)\) is called \emph{energy spectral density} (ESD) for class 1 signals or \emph{power spectral density} (PSD) for class 2 signals.
+
+\paragraph{Cross correlation}
+The \emph{cross correlation} measures the similarity of two different signals \(x\) and \(y\). For class 1 signals
+\[
+  \varphi_{xy}(\tau) = \lim_{T\to\infty} \int_T x(t) y(t-\tau) \,dt.
+\]
+Similarly for class 2a and 2b signals
+\begin{gather*}
+  \varphi_{xy}(\tau) = \frac{1}{T} \int_T x(t) y(t - \tau) \,dt \quad\text{(2a)}, \\
+  \varphi_{xy}(\tau) = \lim_{T\to\infty} \frac{1}{T} \int_T x(t) y(t - \tau) \,dt \quad\text{(2b)}.
+\end{gather*}
+Properties of \(\varphi_{xy}\):
+\begin{itemize}
+  \item For signals with different frequencies \(\varphi_{xy}\) is always 0.
+  \item For stochastic signals \(\varphi_{xy} = 0\)
+\end{itemize}
+
+\subsection{Amplitude density}
+The amplitude density is the probability that a signal has a certain amplitude during a time interval \(T\).
+\[
+  p(a) = \frac{1}{T}\frac{dt}{dx} \in [0,1]
+\]