From 08522b830cd1dd1d0d1d5a060574f50f388205a7 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Thu, 19 Aug 2021 18:46:28 +0200
Subject: Continue working

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 tex/lti.tex | 48 +++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 47 insertions(+), 1 deletion(-)

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@@ -18,10 +18,33 @@ Let \(\mathcal{S}\) denote a system.
   \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
+%% 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
@@ -33,9 +56,32 @@ Let \(\mathcal{S}\) be a system with impulse response \(h(t)\) and transfer func
     \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}[
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