Theory mit fractional delay erweitert

2 files changed, 56 insertions, 6 deletions

diff --git a/doc/thesis/chapters/implementation.tex b/doc/thesis/chapters/implementation.tex
index b2e5d72..e366881 100644
--- a/doc/thesis/chapters/implementation.tex
+++ b/doc/thesis/chapters/implementation.tex
@@ -78,11 +78,10 @@ Here its possible to add some AWGN noise in the channel line. Different paramete
 
 \subsection{Fading}
 %TO DO: übersetzen 
-Für das veranschaulichen des Fading effekts wurde ein eigener Block kreaiert und in den Channel implementiert. Dieser Block basiert auf einem FIR Filter. Es kann mit direcktem Pfad oder ohne dargestellt werden ( Line of Side ). Mit Hilfe dieses Filters wird die Verspätung der nebenpfaden dargestellt. Es ist möglich beliebig viele dieser Pfade mit unterschiedlicher stärke zu simulieren.
+Für die statische implementation und veranschaulichen des Fading effekts wurde ein eigener Block kreaiert und in den Channel implementiert. Dieser Block basiert auf einem FIR Filter. Es kann mit direcktem Pfad oder ohne dargestellt werden (Line of Side). Mit Hilfe dieses Filters wird die Verspätung der nebenpfaden dargestellt. Es ist möglich beliebig viele dieser Pfade mit unterschiedlicher stärke zu simulieren. Dieser Block wurde zusätzlich mit der methode in \ref{sec:fractional-delay} beschriben implementiert um nichtganzahlige delay werte zu erlauben.
 
 % Bild einfügen 
-\subsubsection{Fractional Delay}
-Problem Werte nur auf dem Sample übermitelt und keine dazwischen.
+
 
 
 
diff --git a/doc/thesis/chapters/theory.tex b/doc/thesis/chapters/theory.tex
index 64d412b..e3c0117 100644
--- a/doc/thesis/chapters/theory.tex
+++ b/doc/thesis/chapters/theory.tex
@@ -223,7 +223,7 @@ Equation \eqref{eqn:multipath-frequency-response} shows that the frequency respo
 	}
 \end{figure}
 
-\subsection{Discrete-time model}
+\subsection{Discrete-time model}\label{sec:Discrete-time-model}
 
 Since in practice signal processing is done digitally, it is meaningful to discuss the properties of a discrete-time model. To keep the complexity of the model manageable some assumptions are necessary, thus the sent discrete signal \(s(n)\)\footnote{This is an abuse of notation. The argument \(n\) is used to mean the \(n\)-th digital sample of \(s\), whereas \(s(t)\) is used for the analog waveform.} is assumed to have a finite single sided bandwidth \(W\). This implies that in the time-domain signal is a series of sinc-shaped pulses each shifted from the previous by a time interval \(T = 1 / (2W)\) (Nyquist rate):
 \begin{equation}
@@ -272,11 +272,62 @@ f = np.logspace(5, 8, num=320)
 multipath = tap(.8, 500e-9, f) + tap(.4, 300e-9, f)
 \end{lstlisting}
 
-\subsection{Difficulties caused by discrete time}
+\subsection{Fractional Delay}\label{sec:fractional-delay}
+% TO Do quelle:  http://users.spa.aalto.fi/vpv/publications/vesan_vaitos/ch3_pt1_fir.pdf
+
+\begin{figure}
+	\centering
+	\begin{subfigure}{.45\linewidth}
+		\includegraphics[width=\linewidth]{./figures/screenshots/Fractional_delay_6}
+		\caption{sinc function shifted by the delay = 6.0, with the sample points}
+	\end{subfigure}
+	\hskip 5mm
+	\begin{subfigure}{.45\linewidth}
+		\includegraphics[width=\linewidth]{./figures/screenshots/Fractional_delay_637}
+		\caption{sinc function shifted by the delay = 6.37, with the sample points}
+	\end{subfigure}
+	\label{fig:fractional-delay-sinc-plot}	
+\end{figure}
+
+As in \ref{sec:Discrete-time-model} mentioned a FIR filter can be used to cheat a discrete time model of multiparty fading. But with a FIR filter, the delays are set at the sample rate, so the delays are integer. When the delays are noninteger a approximation had to be done. 
+
+In the example shown in \figref{fig:fractional-delay-sinc-plot}. For a integer delays in the sinc function all sample values are zero except the one by the delayed sample, which is the amplitude value, here one. When the delay is a fractional number all samples are non-zero. In theory this filter is notrealizable because its noncasual and the impulse respond is infinity long. This problem can't be solve by adding them because of the imaginary part.
+
+To desing a noninteger digital delay FIR Filter a least square integral error design approximation could be chosen.  
+
+\begin{equation} \label{eqn:transfer-function-FIR}
+	H(z)=\sum_{n=0}^{N} h(n) z^{-n}
+\end{equation}
+
+The transfare function is given in \eqref{eqn:transfer-function-FIR}, where \(N\) is the order of the filter given in integer coefficients. To be mention for the approximation is that the error decreases with a higher filter order.
+
+The error function between the ideal frequency respond an the approximation should be minimized, for the best possible approximation. 
+  
+\begin{equation} \label{eqn:error-function}
+		E\left(e^{j \omega}\right)=H\left(e^{j \omega}\right)-H_{\mathrm{id}}\left(e^{j \omega}\right)
+\end{equation}
+
+The impulse respond of such least squared fractional delay filter in \eqref{eqn:impuls-respond}. Only positive values are used to make the sinc-function casual. 
+
+\begin{equation} \label{eqn:impuls-respond}
+	h(n)= \begin{cases}\operatorname{sinc}(n-D), & 0 \leq n \leq N \\ 0, & \text { otherwise }\end{cases}
+\end{equation}
+
+To simplify the calculation, the assumption was made that the filter order is an odd number. With this assumption the exact order for the filter can be found out with \eqref{eqn:filter-order} and the integer delay \(D_{\text {int }}\).
+\begin{equation} \label{eqn:filter-order}
+	N = 2 D_{\text {int }} + 1
+\end{equation}
+
+
+The first non-zero sample can be find out with the help of the index M in \eqref{eqn: M first non-zero sample}.With the help of this index it can also be said whether the FIR filter is causal or not. For \(M \geq 0\) casual and if  \(M < 0\) noncasual, an so notrealizable. With the assumption that \(N\) is an odd number \(M \) should always be \( 0\) else something went wrong.
+
+\begin{equation}\label{eqn: M first non-zero sample}
+	M = \lfloor D\rfloor-\frac{N-1}{2} \quad \text { for odd } N
+\end{equation}
 
 
+%% TO DO : Mention windowing or not ?
 
-\skelpar{Not sampling at peaks of sincs.}
 \skelpar{Discrete frequency response. Discuss bins, etc.}
 
 \subsection{Statistical model}