From ee2f2515612f7d3355a5f277e88ce6c5501afcc8 Mon Sep 17 00:00:00 2001
From: Naoki Pross <np@0hm.ch>
Date: Sat, 2 Oct 2021 19:46:36 +0200
Subject: Review notes from lecture 1

---
 DigSig1.tex       | 96 ++++++++++++++++++++++++++++++++++++++++++++++++++++---
 tex/docmacros.sty | 15 +++++++++
 2 files changed, 107 insertions(+), 4 deletions(-)
 create mode 100644 tex/docmacros.sty

diff --git a/DigSig1.tex b/DigSig1.tex
index 9c6cbad..2ca6207 100644
--- a/DigSig1.tex
+++ b/DigSig1.tex
@@ -1,15 +1,20 @@
 % !TeX program = xelatex
 % !TeX encoding = utf8
 % !TeX root = DigSig1.tex
+% vim: set ts=2 sw=2 et:
 
-%% TODO: publish to CTAN
-\documentclass[]{tex/hsrzf}
+\documentclass[margin=small]{tex/hsrzf}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % Packages
 
-%% TODO: publish to CTAN
 \usepackage{tex/hsrstud}
+\usepackage{tex/docmacros}
+
+%% Font configuration
+\usepackage{fontspec}
+% \usepackage{gfsbaskerville}
+\setmainfont[Ligatures = TeX]{TeX Gyre Pagella}
 
 %% Language configuration
 \usepackage{polyglossia}
@@ -44,13 +49,96 @@
 \maketitle
 \tableofcontents
 
-\section{License}
+\section*{License}
 \doclicenseThis
 
 \twocolumn
 \setcounter{page}{1}
 \pagenumbering{arabic}
 
+
+\section{Probability and stochastics}
+
+\subsection{Random variables}
+
+A \emph{random variable} (RV) is a function \(x : \Omega \to \mathbb{R}\).
+The \emph{distribution function} of a RV is a function \(F_x : \mathbb{R} \to [0,1]\) that is always monotonically increasing and given by
+\[
+  F_x(\alpha) = \Pr{x \leq \alpha}.
+\]
+The probability density function (PDF) is
+\[
+  f_x(\alpha) = \frac{dF_x}{d\alpha}.
+\]
+The \emph{expectation} of a RV is
+\[
+  \E{x} = \int_\mathbb{R} \alpha f_x(\alpha) \,d\alpha,
+\]
+and in the case of a discrete RV
+\[
+  \E{x} = \sum_k \alpha_k \Pr{x = \alpha_k}.
+\]
+In general it holds that
+\[
+  \E{g(x)} = \int_\mathbb{R} g(\alpha) f_x(\alpha) \,d\alpha,
+\]
+for example
+\begin{align*}
+  \E{x^2} &= \int_\mathbb{R} \alpha^2 f_x(\alpha) \,d\alpha \\
+  \E{|x|} &= \int_\mathbb{R} |\alpha| f_x(\alpha) \,d\alpha \\
+    &= \int_0^\infty \alpha \left[ f_x(\alpha) + f_x(-\alpha) \right] \,d\alpha
+\end{align*}
+The \emph{variance} of a RV is
+\[
+  \sigma^2 = \Var{x} = \E{(x - \E{x})^2} = \E{x^2} - \E{x}^2,
+\]
+where \(\sigma\) is called the \emph{standard deviation}.
+The variance is sometimes also called the \emph{second moment} of a RV, the \emph{\(n\)-th moment} of a RV is \(\E{x^n}\).
+
+\subsection{Jointly distributed RVs}
+
+\section{Analog signals}
+
+\paragraph{Notation} \(\Omega = 2\pi f\) is used for physical analog frequencies (in radians / second), whereas \(\omega\) is for digital frequencies (in radians / sample).
+
+\paragraph{Transformations} Recall the three important operations for the analysis of analog signals.
+\begin{flalign*}
+  \textit{Fourier Transform} &&
+  X(\Omega) &= \int_\mathbb{R} x(t) e^{j\Omega t} \,dt \\
+  %
+  \textit{Inverse Fourier Transform} &&
+  x(t) &= \int_\mathbb{R} X(\Omega) e^{j\Omega t} \,\frac{d\Omega}{2\pi} \\
+  %
+  \textit{Laplace Transform} &&
+  X(s) &= \int_\mathbb{R} x(t) e^{-st} \,dt
+\end{flalign*}
+The Laplace transform reduces to the Fourier transform under the substitution \(s = j\Omega\).
+
+\paragraph{Linear Systems}
+Recall that superposition holds.
+Thus the system is characterized completely by the impulse response function \(h(t)\).
+The output in the time domain \(y(t)\) is given by the convolution product
+\[
+  y(t) = h(t) * x(t) = \int_\mathbb{R} h(t - t') x(t') \,dt',
+\]
+and in the frequency domain \(Y(\Omega) = H(\Omega) X(\Omega)\), where \(H(\Omega)\) is the Fourier transform of \(h(t)\).
+
+% Analog signals:
+% TODO: FT of eigenfunctions e^{j\Omega_k t\}
+
 \section{Sampling and reconstruction}
 
+Sampling theorem: \(f_s = 2 f_\text{max}\) is called Nyquist rate. In other words you need at least 2 samples/cycle to reconstruct a signal.
+%% TODO: ideal sampler
+Nyquist intervals are bounded by Nyquist frequencies, i.e. \(\left[-f_s / 2, f_2 / 2\right]\)
+
+Alias frequency \(f_a = f \pmod f_s\).
+
+Anti-aliasing: analog LP prefilter cutoff \@ \(f_s/2\)
+
+Processing: Upper limit on sampling frequency given by processing time \(T_\text{proc}\). Thus \(2f_\text{max} \leq f_s \leq f_\text{proc}\).
+
+
+
+
 \end{document}
diff --git a/tex/docmacros.sty b/tex/docmacros.sty
new file mode 100644
index 0000000..41c7673
--- /dev/null
+++ b/tex/docmacros.sty
@@ -0,0 +1,15 @@
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{tex/docmacros}[2021/09/27 v0.1 Document Macros]
+
+\RequirePackage{amsmath}
+\RequirePackage{amsthm}
+
+% Probability
+\DeclareMathOperator{\expectation}{E}
+\newcommand{\E}[1]{\expectation \left\{#1\right\}}
+
+\DeclareMathOperator{\variance}{Var}
+\newcommand{\Var}[1]{\variance \left\{#1\right\}}
+
+\DeclareMathOperator{\probability}{Pr}
+\renewcommand{\Pr}[1]{\probability \left\{#1\right\}}
-- 
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