Update slides and typo in theory

2 files changed, 102 insertions, 73 deletions

diff --git a/doc/slides/slides.tex b/doc/slides/slides.tex
index ae1e81b..e355794 100644
--- a/doc/slides/slides.tex
+++ b/doc/slides/slides.tex
@@ -1,5 +1,6 @@
 % !TeX program = xelatex
 % !TeX encoding = utf8
+% !TeX root = slides.tex
 \documentclass[xetex, onlymath, handout]{beamer}
 \usefonttheme{serif}
 \usetheme{hsr}
@@ -7,6 +8,8 @@
 % use lmodern for math
 \usepackage{lmodern}
 
+\usepackage{tex/docmacros}
+
 %% Pretty figures
 \usepackage{circuitikz}  % Electric diagrams
 \usepackage{pgfplots}    % Pretty plots
@@ -59,100 +62,88 @@
 
 \section{Multipath Fading}
 
-\begin{frame}{Multipath Fading sketch}
+\begin{frame}{Multipath fading}
 	\begin{figure}
 		\centering
 		\input{figures/tikz/multipath-sketch}
 	\end{figure}
-	\begin{equation} \label{eqn:multipath-impulse-response}
-		h(\tau, t) = \sum_k c_k(t) \delta(\tau - \tau_k(t)),
-	\end{equation}
+  \vspace{\baselineskip}
+  \[
+    r(t) = \sum_k c_k s(t - \tau_k).
+  \]
+\end{frame}
+
+\begin{frame}[fragile]{Impulse reponse of a multipath fading channel}
+  \begin{figure}
+    \centering
+    \input{figures/tikz/multipath-impulse-response}
+  \end{figure}
+  \[
+		h(\tau, t) = \sum_k c_k(t) \delta(\tau - \tau_k(t))
+  \]
 \end{frame}
 
-\begin{frame}{Spectrum of a multipath fading channel}
+\begin{frame}[fragile]{Spectrum of a multipath fading channel}
 	\begin{figure}
 		\centering
 		\resizebox{\linewidth}{!}{
 			\input{figures/tikz/multipath-frequency-response-plots}
-			% \skelfig[width = .8 \linewidth, height = 3cm]{}
 		}
 	\end{figure}
-	\begin{equation} 
-		H(f, t) = \int_\mathbb{R} \sum_k c_k(t) \delta(\tau - \tau_k(t)) e^{-2\pi jf\tau} \, d\tau
-		= \sum_k c_k(t) e^{-2\pi jf \tau_k(t)}.
-	\end{equation}
 \end{frame}
 
-
-
 \subsection{Discrete-time model}
 
-\begin{frame}{Discrete-time model}
+\begin{frame}[fragile]{Discrete-time and FIR}
 	\begin{figure}
 		\centering
-		\input{figures/tikz/tapped-delay-line}
+    \resizebox{\linewidth}{!}{
+      \input{figures/tikz/tapped-delay-line}
+    }
 	\end{figure}
-	\begin{equation} 
-		h_l(m) = \sum_k c_k(mT) \sinc(l - \tau(mT)/T)
-	\end{equation}
+  \vspace{\baselineskip}
+  \[
+    h_l(m) = \sum_k c_k(mT) \sinc\left(l - \frac{\tau_k(mT)}{T}\right)
+  \]
 \end{frame}
 
-
 \subsection{Statistical model}
 
 \begin{frame}[fragile]{Statistical model}
   \begin{columns}
-    \begin{column}{.5\linewidth}
-      \begin{itemize}
-        \item Raileigh distribution (NLOS)
-        \item Rician distribution (LOS) 
-     \end{itemize}
+    \begin{column}{.4\linewidth}
+      Assuming WSSUS
+      \[
+        \theta_k \sim \mathcal{U}(0, 2\pi)
+      \]
+      The NLOS Fading
+      \[
+        f = \lim_{N \to \infty} \frac{1}{\sqrt{N}} \sum_{k=1}^N e^{j\theta_k}
+      \]
+      \[
+        f \sim \mathrm{ Rayleigh}
+      \]
+      if there is a LOS
+      \[
+        f \sim \mathrm{ Rice}(K)
+      \]
     \end{column}
-    \begin{column}{.5\linewidth}
-  	  \begin{figure}
-  		\centering
-  		\resizebox{!}{4cm}{%
-  			\input{figures/tikz/ring-of-scattering-objects}
-  		}
-  	 \end{figure}
+    \begin{column}{.6\linewidth}
+      \begin{figure}
+        \centering
+        \resizebox{\linewidth}{!}{%
+          \input{figures/tikz/ring-of-scattering-objects}
+          }
+      \end{figure}
     \end{column}
   \end{columns}
 \end{frame}
 
-
-
-
-
 \section{Implementation}
 
 %TODO: Mabe picture Hardware, Bicture GR.
 
-\begin{frame}{Tools}
-  \begin{columns}
-	\begin{column}{.5\linewidth}
-		\begin{itemize}
-			\item Software Stack
-				\begin{itemize}
-					\item GNU Radio
-					\item Dear PyGUI
-				\end{itemize}
-			\item Hardware
-			\begin{itemize}
-				\item USRP B210
-			\end{itemize}
-		\end{itemize}
-	\end{column}
-	\begin{column}{.5\linewidth}
-		\begin{figure}
-		\centering
-		\includegraphics[frame, width = \linewidth]{figures/screenshots/gui_screenshot}
-		\end{figure}
-	\end{column}
-\end{columns}
-\end{frame}
-
-
-\begin{frame}{Blockdiagram}
+\begin{frame}{Block Diagram}
 	\begin{figure}
 		\centering
 		\resizebox{.9\linewidth}{!}{
@@ -162,26 +153,65 @@
 	\end{figure}
 \end{frame}
 
+\subsection{Transmitter and Receiver Chains}
 
+\begin{frame}{Transmitter}
+  \begin{figure}
+    \centering
+    \includegraphics[width=\linewidth]{figures/picture/PC210002}
+  \end{figure}
+\end{frame}
 
-\subsection{Transmitter and Receiver chain}
-
-\begin{frame}{Transmitter chain}
-	
+\begin{frame}{Framed data packets}
+  \begin{figure}
+    \centering
+    \resizebox{\linewidth}{!}{
+      \input{figures/tikz/packet-frame}
+    }
+  \end{figure}
+  \begin{itemize}
+    \item Very short payload
+    \item \(k\)-Byte preamble is a Barker code \texttt{0x1f35} for Sync
+    \item Should be replaced with CAZAC
+  \end{itemize}
 \end{frame}
 
-\begin{frame}{Receiver chain}
-	
+\begin{frame}{Receiver}
+  \begin{figure}
+    \centering
+    \includegraphics[width=\linewidth]{figures/picture/PC210011}
+  \end{figure}
 \end{frame}
 
+
+% \begin{frame}{Tools}
+%   \begin{columns}
+% 	\begin{column}{.5\linewidth}
+% 		\begin{itemize}
+% 			\item Software Stack
+% 				\begin{itemize}
+% 					\item GNU Radio
+% 					\item Dear PyGUI
+% 				\end{itemize}
+% 			\item Hardware
+% 			\begin{itemize}
+% 				\item USRP B210
+% 			\end{itemize}
+% 		\end{itemize}
+% 	\end{column}
+% 	\begin{column}{.5\linewidth}
+% 		\begin{figure}
+% 		\centering
+% 		\includegraphics[width = \linewidth]{figures/screenshots/gui_screenshot}
+% 		\end{figure}
+% 	\end{column}
+% \end{columns}
+% \end{frame}
+
 \subsection{Channel model}
 
 \begin{frame}{Discrete-time model}
-	\begin{figure}
-		\centering
-		\input{figures/tikz/qpsk-simulations-static}
-	\end{figure}
-	the 1 tap model the fading tap was \(0.2\delta(n - 0.25)\), and for the 4 tap model uses \(0.2 \delta(n - 0.25) + 0.08 \delta(n - 3.25) + 0.5 \delta(n - 4) + 0.4 \delta(n - 6.3)\). In both cases the delays are given in samples.
+
 \end{frame}
 
 \begin{frame}{Statistical model}
@@ -198,7 +228,6 @@
 
 %%Tools
 
-
 \end{document}
 
 % vim:et:ts=2:sw=2:wrap:nolinebreak:
diff --git a/doc/thesis/chapters/theory.tex b/doc/thesis/chapters/theory.tex
index 87c5867..50759bb 100644
--- a/doc/thesis/chapters/theory.tex
+++ b/doc/thesis/chapters/theory.tex
@@ -261,7 +261,7 @@ Finally, the substitution \(l = m - n\) eliminates the sender's sample counter \
 \end{equation}
 This result is very similar to the continuous time model described by \eqref{eqn:multipath-impulse-response} in the sense that each received digital sample is a sent sample convolved with a different discrete channel response (because of time variance). To see how the discrete CIR
 \begin{equation} \label{eqn:discrete-multipath-impulse-response}
-	h_l(m) = \sum_k c_k(mT) \sinc\left(l - \frac{\tau(mT)}{T}\right)
+	h_l(m) = \sum_k c_k(mT) \sinc\left(l - \frac{\tau_k(mT)}{T}\right)
 \end{equation}
 is different from \eqref{eqn:multipath-impulse-response} consider again the plot of \(h(\tau,t)\) in \figref{fig:multipath-impulse-response}. The plot of \(h_l(m)\) would have discrete axes with \(m\) replacing \(t\) and \(l\) instead of \(\tau\). Because of the finite bandwidth in the \(l\) axis instead of Dirac deltas there would be superposed sinc functions.