\section{Signals}
\subsection{Classification}
\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}
  \[
    \begin{array}{l l}
      \toprule
      \text{\bfseries Characteristic} & \text{\bfseries Symbol and formula} \\[6pt]
	    \text{\itshape Class 1 Signals} \\
      \midrule
      \text{Normalized energy} & E_n = \lim_{T\to\infty} \int_T |x|^2 \,dt \\[6pt]
	    \text{\itshape Class 2a Signals} \\
      \midrule
      \text{Normalized power}  & P_n = \lim_{T\to\infty} \frac{1}{T} \int_T |x|^2 \,dt \\[12pt]
      \text{Linear mean}       & X_0 = \frac{1}{T} \int_T x\, dt \\[12pt]
      \text{Mean square}       & X^2 = \frac{1}{T} \int_T x^2 \, dt \\[12pt]
      n\text{-th order mean}   & X^n = \frac{1}{T} \int_T x^n \, dt \\[12pt]
      \text{Rectified value}   & |\bar{X}| = \frac{1}{T} \int_T |x| \,dt \\[12pt]
      \text{Variance}          & \sigma^2 = \frac{1}{T} \int_T \left(x - X_0\right)^2 dt \\[12pt]
                               & = X^2 - X_0 \\[6pt]
      \text{Root mean square}  & X_\text{rms} = \sqrt{X^2} \\
      \bottomrule
    \end{array}
  \]
\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 \(T\).
\[
  p(a) = \frac{1}{T}\frac{dt}{dx} \in [0,1]
\]