Review notes from lecture 1

1 files changed, 92 insertions, 4 deletions

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-%% TODO: publish to CTAN
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+\documentclass[margin=small]{tex/hsrzf}
 
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 % Packages
 
-%% TODO: publish to CTAN
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+\usepackage{tex/docmacros}
+
+%% Font configuration
+\usepackage{fontspec}
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+\setmainfont[Ligatures = TeX]{TeX Gyre Pagella}
 
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@@ -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}