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