\section{LTI systems}

\subsection{Properties}
Let \(\mathcal{S}\) denote a system.
\begin{table}[H]
  \begin{tabularx}{\linewidth}{p{.3\linewidth} X}
    \toprule
    \bfseries Property & \bfseries Meaning \\
    \midrule
    static \(\leftrightarrow\)\newline dynamic & Static means that it is memoryless (in the statistical sense), whereas dynamic has memory. Static systems depend only on the input \(u\), dynamic systems on \(du/dt\) or \(\int u\,dt\). \\
    causal \(\leftrightarrow\)\newline acausal & Causal systems use only informations from the past, i.e. \(h(t < 0) = 0\). Real systems are always causal. \\
    linear \(\leftrightarrow\)\newline nonlinear & The output of a linear system does not have new frequency that were not in the input. For linear system the superposition principle is valid: \(\mathcal{S}(\alpha_1 x_1 + \alpha_2 x_2) = \alpha_1 \mathcal{S} x_1 + \alpha_2 \mathcal{S} x_2\). \\
    time invariant \newline\(\leftrightarrow\) time variant & Time invariant systems do not depend on time, but for ex. only on time differences. \\
    \midrule
    SISO, MIMO & Single input single output, multiple input multiple output. \\
    BIBO & Bounded input bounded output, i.e. there are some \(A\), \(B\) such that \(|x| < A\) and \(|y| < B\) for all \(t\), equivalently \(\int_\mathbb{R} |h|\,dt < \infty\).\\
    \bottomrule
  \end{tabularx}
\end{table}

\subsection{Impulse response}
%% TODO: impulse response

\subsection{Stability}
Let \(\mathcal{S}\) be a system with impulse response \(h(t)\) and transfer function \(H(s)\).
\begin{table}[H]
  \centering
  \begin{tabularx}{\linewidth}{lX}
    \toprule
    Stable & All poles are on the LHP\footnote{Left half plane, where \(\mathrm{Re}(s) < 0\).}. \\
    Marginally stable & There are no poles in the RHP but a simple pole on the \(j\)-axis. \\
    Instable & There are poles in the RHP or poles of hider order on the \(j\)-axis. \\
    \bottomrule
  \end{tabularx}
\end{table}
\subsection{Distortion}
\subsection{Stochastic inputs}

\iffalse
\begin{figure}
\begin{tikzpicture}[
    system/.style = {draw, thick, inner sep = 4mm, outer sep = 1mm}
  ]
  \matrix[row sep=3mm, column sep=1cm] (M) {
    \node (x) {\(x(t)\)}; &
    \node (g) {\(g(t) = y_\delta (t)\)}; &
    \node (y) {\(y(t) = g(t) * x(t)\)}; \\

    &
    \node (h) {\(h(t)\)}; &
    \node (yw) {\(y_\omega(t) = h(t) * x(t)\)}; \\

    \node (in) {Input}; &
    \node[system, fill=white] (sys) {LTI-System \(\mathcal{S}\)}; &
    \node (out) {Response}; \\

    \node (X) {\(X(s)\)}; &
    \node (G) {\(G(s) = 1/p(s)\)}; &
    \node (Y) {\(Y(s) = G(s) \cdot X(s)\)}; \\

    \node (Xw) {\(X(\omega)\)}; &
    \node (H)  {\(H(\omega) = G(j\omega)\)}; &
    \node (Yw) {\(Y_\omega (\omega) = H(\omega) \cdot X(\omega)\)}; \\
  };

  \draw[thick, ->] (in) to (sys);
  \draw[thick, ->] (sys) to (out);

  \begin{pgfonlayer}{background}
    \coordinate (T1) at ($(x.north west) - (.8,-.1)$);
    \coordinate (T2) at ($(yw.south east) + (.8,-.1)$);

    \coordinate (B1) at ($(X.north west) - (0,-.1)$);
    \coordinate (B2) at ($(Y.south east) + (0,-.1)$);

    \coordinate (F1) at ($(Xw.north west) - (0,-.1)$);
    \coordinate (F2) at ($(Yw.south east) + (0,-.1)$);

    \fill[color=blue!20] (T1) rectangle (T2);
    \fill[color=magenta!20] (B1 -| T1) rectangle (B2 -| T2);
    \fill[color=red!20] (F1 -| T1) rectangle (F2 -| T2);
    % \fill[top color=blue!20, bottom color=magenta!20]
      % (T1) rectangle (B2);
  \end{pgfonlayer}
\end{tikzpicture}
\end{figure}
\fi


