From 971fd096ce8e869f4d70ecb18dec84ff907da57c Mon Sep 17 00:00:00 2001
From: Naoki Pross <np@0hm.ch>
Date: Mon, 4 Oct 2021 19:15:20 +0200
Subject: Continue sampling

---
 DigSig1.tex | 76 ++++++++++++++++++++++++++++++++++++++++++++-----------------
 1 file changed, 55 insertions(+), 21 deletions(-)

(limited to 'DigSig1.tex')

diff --git a/DigSig1.tex b/DigSig1.tex
index 2ca6207..10379ae 100644
--- a/DigSig1.tex
+++ b/DigSig1.tex
@@ -8,13 +8,19 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % Packages
 
-\usepackage{tex/hsrstud}
-\usepackage{tex/docmacros}
-
 %% Font configuration
 \usepackage{fontspec}
-% \usepackage{gfsbaskerville}
-\setmainfont[Ligatures = TeX]{TeX Gyre Pagella}
+\usepackage{fouriernc}
+
+%% Own packages
+% \usepackage{tex/hsrstud}
+\usepackage{tex/docmacros}
+
+%% Mathematics
+\usepackage{amssymb}
+
+%% Frames
+\usepackage{framed}
 
 %% Language configuration
 \usepackage{polyglossia}
@@ -37,7 +43,7 @@
 \authoremail{naoki.pross@ost.ch}
 \author{\textsl{Naoki Pross} -- \texttt{\theauthoremail}}
 
-\title{\texttt{\themodule} Lecture Notes}
+\title{Digital Signal Processing Lecture Notes}
 \date{\thesemester}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -66,7 +72,7 @@ The \emph{distribution function} of a RV is a function \(F_x : \mathbb{R} \to [0
 \[
   F_x(\alpha) = \Pr{x \leq \alpha}.
 \]
-The probability density function (PDF) is
+The \emph{probability density function} (PDF) of a RV is
 \[
   f_x(\alpha) = \frac{dF_x}{d\alpha}.
 \]
@@ -90,7 +96,7 @@ for example
 \end{align*}
 The \emph{variance} of a RV is
 \[
-  \sigma^2 = \Var{x} = \E{(x - \E{x})^2} = \E{x^2} - \E{x}^2,
+  \Var{x} = \sigma^2 = \E{(x - \E{x})^2} = \E{x^2} - \E{x}^2,
 \]
 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 \(\E{x^n}\).
@@ -99,9 +105,10 @@ The variance is sometimes also called the \emph{second moment} of a RV, the \emp
 
 \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).
+In this document we will use the notation \(\Omega = 2\pi f\) for physical analog frequencies (in radians / second), and \(\omega\) for digital frequencies (in radians / sample).
 
-\paragraph{Transformations} Recall the three important operations for the analysis of analog signals.
+\subsection{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 \\
@@ -114,7 +121,7 @@ The variance is sometimes also called the \emph{second moment} of a RV, the \emp
 \end{flalign*}
 The Laplace transform reduces to the Fourier transform under the substitution \(s = j\Omega\).
 
-\paragraph{Linear Systems}
+\subsection{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
@@ -128,16 +135,43 @@ and in the frequency domain \(Y(\Omega) = H(\Omega) X(\Omega)\), where \(H(\Omeg
 
 \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 \(2f_\text{max} \leq f_s \leq f_\text{proc}\).
-
+To sample a signal \(x(t)\) it means to measure (take) the value at a periodic interval every \(T\) seconds. \(T\) is thus called the \emph{sample interval} and \(f_s =1/T\) is the \emph{sampling frequency}. We will introduce the notation
+\[
+  x[n] = x(nT)
+\]
+to indicate that a signal is a set of discrete samples.
+
+\subsection{Sampling theorem}
+
+To represent a signal \(x(t)\) by its samples \(x[n]\) two conditions must be met:
+\begin{enumerate}
+  \item \(x(t)\) must be \emph{bandlimited}, i.e. there must be a frequency \(f_\text{max}\) after which the spectrum of \(x(t)\) is always zero.
+  \item The sampling rate \(f_s\) must be chosen so that
+    \[
+      f_s \geq 2 f_\text{max}.
+    \]
+\end{enumerate}
+In other words you need at least 2 samples / period to reconstruct a signal.
+When \(f_s = 2 f_\text{max}\), the edge case, the sampling rate is called \emph{Nyquist rate}. 
+The interval \(\left[-f_s / 2, f_2 / 2\right]\), and its multiples are called \emph{Nyquist intervals}, as they are bounded by the Nyquist frequencies.
+It would be good to have an arbitrarily high sampling frequency but in reality there is upper limit given by processing time \(T_\text{proc}\). Thus \(2f_\text{max} \leq f_s \leq f_\text{proc}\).
+
+\subsection{Discrete-Time Fourier Transform}
+
+Mathematically speaking, to sample a signal is equivalent multiplying a function with the \emph{impulse train distribution}\footnote{Sometimes it is also called the Dirac comb.}
+\[
+  \Comb_T (t) = \sum_{n = -\infty}^{\infty} \delta(t - nT),
+\]
+so \(x[n] = \Comb_T(t)\, x(t)\).
+Interestingly the impulse train is periodic, and has thus a Fourier series with all coefficients equal to \(1/T\).
+So the Fourier transform of a comb is also a comb, i.e.
+\[
+  \Comb_T(t) \leftrightarrow \Comb_{1/T}(\Omega).
+\]
+ 
+\subsection{Spectrum replication and aliasing}
+% Alias frequency \(f_a = f \pmod{f_s}\).
+% Anti-aliasing: analog LP prefilter cutoff \@ \(f_s/2\)
 
 
 
-- 
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