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\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 \(T\).
\[
p(a) = \frac{1}{T}\frac{dt}{dx} \in [0,1]
\]
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