% vim: set ts=2 sw=2 noet spell:

\chapter{Theory}

\begin{figure}
	\centering
	\input{figures/tikz/overview}
	\caption{
		Block diagram for a general wireless communication system with annotated signal names. Frequency domain representations of signals use the uppercase symbol of their respective time domain name.
		\label{fig:notation}
	}
\end{figure}

In this section we will briefly give the mathematical background required by the modulation schemes used in the project. The notation used is summarised in figure \ref{fig:notation}. For conciseness encoding schemes and (digital) signal processing calculations are left out and discussed later. Thus for this section \(m_e = m\).

\section{Quadrature amplitude modulation}

Quadrature amplitude modulation is a family of modern digital modulation methods, that use an analog carrier signal. The simple yet effective idea behind QAM is to encode extra information into an orthogonal carrier signal, thus increasing the number of bits sent per unit of time.

\begin{figure}
	\centering
	\input{figures/tikz/qpks-constellation}
	\caption{
		% TODO: write caption
		\label{fig:qpks-constellation}
	}
\end{figure}

\subsection{Phase Shift Keying (PSK)}

PSK is a popular modulation type for data transmission\cite{Meyer2011}. With a bipolar binary signal, the amplitude remains constant and only the phase will be changed with phase jumps of 180 degrees, which can be seen as a multiplication of the carrier signal with $\pm$ 1. That is alow known as binary phase shift keying.

\begin{figure}
	% TODO: Better Image
	% https://sites.google.com/site/billmahroukelec675/bipolar-phase-shift-keying
	\includegraphics[width=5cm]{./image/BPSK2.png}
\end{figure}

\subsection{Quadrature Phase Shift Keying (QPSK)}

Two bits are modulated at ones with the same bandwidth as a 2-PSK so more informations are transmitted at the same time. \cite{Meyer2011}
%TODO: Image Signal Raum 
Most times there is noise and the points on the constellation diagram become a surface. 
If the surfaces overlap there will be a problem with decoding. 

\section{Fading}

\subsection{Geometric Model}


\subsection{Statistical Model}

%% TODO: write about advantage of statistical model instead of geometric

\paragraph{Continuous time model}

Continuous time small scale fading channel response.

time varying channel impulse response:
\begin{equation}
	h(t, \tau) = \sum_k c_k (t) \delta(\tau - \tau_k(t))
\end{equation}

received signal \(y = h * x\), i.e. convolution with channel model. 

\subsection{Time discretization of the model}

%% TODO: explain why

Assume \(x\) is a time discrete signal with and bandwidth \(W\), thus the pulse is sinc shaped
\begin{equation}
	x(t) = \sum_n x[n] \sinc(t/T - n)
\end{equation}
Ideal sampling at rate \(2W\) of \(y\) gives
