\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{Time domain description}
A general LTI system with input \(x\) and output \(y\) is described in the time domain with a linear differential equation of the form
\[
  \sum_{i=0}^n a_i y^{(i)} = \sum_{k=0}^m b_k x^{(k)}.
\]

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

\subsection{Transfer function}
By taking the Laplace transform of the differential equation of the system a and assuming all initial conditions to be zero, we obtain
\[
  Y \sum_{i=0}^n a_i s^i = X \sum_{k=0}^m b_k s^k,
\]
where \(Y\) and \(X\) are the Laplace transform of \(y\) and \(x\) respectively. We then define the \emph{transfer function} to be the ratio \(H = Y/X\), or
\[
  H(s) = \frac{\displaystyle\sum_{k=0}^m b_k s^k}{\displaystyle\sum_{i=0}^n a_i s^i}
    = \frac{\displaystyle\prod_{k=0}^m s - z_k}{\displaystyle\prod_{i=0}^n s - p_i},
\]
since polynomials can be expressed in terms of their roots. We say the roots of \(Y\) are \emph{zeroes} and those of \(X\) \emph{poles}, because of how they appear in the complex plane of \(H\).

\subsection{Frequency response}
%% TODO: Frequency response

\subsection{Stability}
%% TODO: Hurwitz

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}

For a periodic signal the Fourier transform is a bunch of weighted Dirac deltas (or a Fourier series), i.e.
\[
  \fourier\{f\} = \sum_i d_i \delta(\omega - \omega_i).
\]
The spectrum of a sinusoidal signal of frequency \(\omega_1\) is only one weighted delta \(d_1\delta(\omega - \omega_1)\). When a system introduces a \emph{nonlinear} distortion, with a clean sine input new higher harmonics are found in the output.

To measure the distortion of a signal in the English literature there is the \emph{total harmonic distortion} (THD) defined as
\[
  \text{THD} = \frac{1}{d_1}\sqrt{\sum_{i=1}^n d_i^2}.
\]
In the German literature there is the distortion factor (\emph{Klirrfaktor}, always between 0 and 1)
\[
  k = \sqrt{\frac{d_2 + d_3 + \cdots + d_n}{d_1 + d_2 + \cdots + d_n}}.
\]
Both are usually given in percent (\%) and are related with
\[
  (\text{THD})^2 = \frac{k^2}{1-k^2},
\]
thus THD \(\leq k\).

\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