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| -rw-r--r-- | doc/thesis/chapters/implementation.tex | 12 | ||||
| -rw-r--r-- | doc/thesis/chapters/theory.tex | 21 |
2 files changed, 22 insertions, 11 deletions
diff --git a/doc/thesis/chapters/implementation.tex b/doc/thesis/chapters/implementation.tex index e0e357b..51f6646 100644 --- a/doc/thesis/chapters/implementation.tex +++ b/doc/thesis/chapters/implementation.tex @@ -91,7 +91,8 @@ What is here referred to as envelope detector has the purpose of synchronizing t \paragraph{Polyphase Clock Sync} %% To Do : nochmals anschauen ob dieese erklärung verständlich ist und richtig interpretiert wurde. -With the the polyphase clock sync the symbols can be synchronized by preforming a time synchronization with the help of multiple filterbanks. For that the derivation of the filtered signal should be minimized whish turns to a better SNR. This works with the help of two filterbanks, one of them contains the filters of the signal adapted to the pulse shaping with several phases. The other contains its derivative. So in the time domain it has a sinc shape, for the output Signal the sinc peak should be on a sample, with the fact that sinc(0) = 1 and sinc(0)' = 0 an error signal can be generated which tells how far away from the peak it is. This error Signal should be zero this is possible with the help of a loop second order whish constants the number of the filterbank and the rate. This rate is generated because of the clock difference between the transmitter and reviver to synchronies the receiver the filter goes through the phases. For the output one sample per symbol is enough. +With the the polyphase clock sync the symbols can be synchronized by preforming a time synchronization with the help of multiple filterbanks. For that the derivation of the filtered signal should be minimized whish turns to a better SNR. +%This works with the help of two filterbanks, one of them contains the filters of the signal adapted to the pulse shaping with several phases. The other contains its derivative. So in the time domain it has a sinc shape, for the output Signal the sinc peak should be on a sample, with the fact that sinc(0) = 1 and sinc(0)' = 0 an error signal can be generated which tells how far away from the peak it is. This error Signal should be zero this is possible with the help of a loop second order whish constants the number of the filterbank and the rate. This rate is generated because of the clock difference between the transmitter and reviver to synchronies the receiver the filter goes through the phases. For the output one sample per symbol is enough. \paragraph{Equalizer} @@ -250,11 +251,16 @@ Here its possible to add some AWGN noise in the channel line. Different paramete % %For the a first simulation with some fading the 16QAM simulation model has been extended with a FIR-Filter in the Chanel. The results of this simulation are shown in \figref{fig:simul16QAM} and \figref{fig:simul16QAM_1} as the blue Signal. -\subsection{Fading} +\subsection{Fading with Discrete-time model} %TO DO: übersetzen -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. + +Für die statische gemäss \ref{sec:Discrete-time-model} 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 +\subsection{Fading with Statistical model} + +\ref{statistical_model} + \subsection{Measurements} diff --git a/doc/thesis/chapters/theory.tex b/doc/thesis/chapters/theory.tex index b01a31f..12ebb67 100644 --- a/doc/thesis/chapters/theory.tex +++ b/doc/thesis/chapters/theory.tex @@ -288,42 +288,47 @@ In the example shown in \figref{fig:fractional-delay-sinc-plot}. For a integer d To desing a noninteger digital delay FIR Filter a least square integral error design approximation could be chosen. +Where the transfare function is given as: + \begin{equation} \label{eqn:transfer-function-FIR} H(z)=\sum_{n=0}^{N} h(n) z^{-n} \end{equation} +The Order of the filter is given with the help of \(N\), this value only contains integer coefficients. +To be mention for the approximation is that the error decreases with a higher filter order. -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. +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. +The impulse respond of such a 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 \(D_{\text {int }}\). +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 the help of 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. - +The first non-zero sample can be find out with the help of the index M: \begin{equation}\label{eqn: M first non-zero sample} M = \lfloor D\rfloor-\frac{N-1}{2} \quad \text { for odd } N \end{equation} +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. + + %% TO DO : Mention windowing or not ? \skelpar{Discrete frequency response. Discuss bins, etc.} -\subsection{Statistical model} +\subsection{Statistical model} \label{statistical_model} Because as mentioned earlier it is difficult to estimate the time-dependent parameters of \(h_l(m)\) in many cases it is easier to model the components of the CIR as stochastic processes, thus greatly reducing the number of parameters. This is especially effective for channels that are constantly changing, because by the central limit theorem the cumulative effect of many small changes tends to a normal statistical distribution. @@ -334,7 +339,7 @@ Recall that \(h_l(m)\) is a function of time because \(c_k\) and \(\tau_k\) chan for some parameter \(\sigma\). Loosely speaking, the distribution needs to be ``circular'' because \(h_l\) is a complex number, which is two dimensional, it can however be understood as \(\Re{h_l} \sim \mathcal{N}(0, \sigma^2)\) and \(\Im{h_l} \sim \mathcal{N}(\mu, \sigma^2)\), i.e. having each component being normally distributed. - +% TO DO : Picture of gaussian distribution \begin{subequations} \begin{align} R_{l} (k) &= \E{h_l(m) h_l^*(m+k)}, \\ |