diff --git a/Bilder/Thumbs.db b/Bilder/Thumbs.db index 1d9f26a..ebad2ed 100644 Binary files a/Bilder/Thumbs.db and b/Bilder/Thumbs.db differ diff --git a/Bilder/fig_fir.jpg b/Bilder/fig_fir.jpg index 43144fd..5007301 100644 Binary files a/Bilder/fig_fir.jpg and b/Bilder/fig_fir.jpg differ diff --git a/Bilder/fig_iir.jpg b/Bilder/fig_iir.jpg index 8add645..dbb2e62 100644 Binary files a/Bilder/fig_iir.jpg and b/Bilder/fig_iir.jpg differ diff --git a/chapter_02.tex b/chapter_02.tex index 5cb1502..407c718 100644 --- a/chapter_02.tex +++ b/chapter_02.tex @@ -52,27 +52,31 @@ Examples for an adaptive filter is the Least-Mean-Square-Algorithm used for adap \subsection{Filter designs} Before we continue with the introduction to the actual topic of this thesis, ANR, two very essential filter designs need further explanation - the Finite Impulse Response- (FIR) and Infinite Impulse Response-filters (IIR). \subsubsection{Finite Impulse Response-filters} -A Finite Impulse Respone filter, commonly referred to as a ''Feedforward Filter'' is defined through the property, that it uses only present and past input values and not feedback from output samples - therefore the response of a FIR-filter reaches zero after a finite number of samples. Due to the fact, that there is no feedback, a FIR-filter offers unconditional stability, meaning that the filter response converges, no matter how the coefficients are set. A disadvantage to the FIR-desgin is the relatively slow frequency response compared to its IIR counterpart. +A Finite Impulse Respone filter, commonly referred to as a ''Feedforward Filter'' is defined through the property, that it uses only present and past input values and not feedback from output samples - therefore the response of a FIR-filter reaches zero after a finite number of samples. Due to the fact, that there is no feedback, a FIR-filter offers unconditional stability, meaning that the filter response converges, no matter how the coefficients are set. A disadvantage to the FIR-desgin is the relatively slow frequency response compared to its IIR counterpart. \\ \\ +Equation \ref{equation_fir} specifies the input-output relationship of a FIR-filter - $x[n]$ is the input sample, $y[n]$ is output sample, and $b_0$ to $b_M$ the filter coefficients and M the length of the filter \begin{equation} \label{equation_fir} - y[n] = \sum_{k=1}^{M} b_kx[n-k] + y[n] = \sum_{k=0}^{M} b_kx[n-k] = b_0x[n] + b_1x[n-1] + ... + b_Mx[n-M] \end{equation} +Figure \ref{fig:fig_fir} visualizes a simple FIR-filter with two coefficients - the first sample is multiplied with the operator $b_0$ whereas the following sample $b1$ is multiplied with the operator $b_1$ before added back together. The Operator $Z^{-1}$ represents a delay operator. \begin{figure}[H] \centering \includegraphics[width=0.8\linewidth]{Bilder/fig_fir.jpg} - \caption{FIR-filter} + \caption{FIR-filter example with two feedforward operators} \label{fig:fig_fir} \end{figure} \subsubsection{Infinite Impulse Response-filters} -A Ininite Impulse Respone filter, commonly referred to as a ''Feedback Filter'' does, in contrary to its FIR-counterpart, use past output samples in addition to current and past input samples - therefore the response of an IIR-filter theoretically continues indefinitely. This recursive nature allows IIR filter to achieve a sharp frequency response with significantly fewer coefficients than an equivalent FIR filter but it also opens up the possibility, that the filter-response diverges, depending on the set coefficients. +An Ininite Impulse Respone filter, commonly referred to as a ''Feedback Filter'' can be seen as an extension of the FIR-filter. In contrary to it's counterpart, it also uses past output samples in addition to current and past input samples - therefore the response of an IIR-filter theoretically continues indefinitely. This recursive nature allows IIR filter to achieve a sharp frequency response with significantly fewer coefficients than an equivalent FIR-filter, but it also opens up the possibility, that the filter response diverges, depending on the set coefficients.\\ \\ +Equation \ref{equation_iir} specifies the input-output relationship of a IIR-filter. In addition to \ref{equation_fir} there is now a second term included, where $a_0$ to $a_N$ are the feedback coefficients with its own length N. \begin{equation} \label{equation_iir} - y[n] = \sum_{k=1}^{M} b_kx[n-k] - \sum_{k=1}^{N} a_ky[n-k] + y[n] = \sum_{k=0}^{M} b_kx[n-k] - \sum_{k=0}^{N} a_ky[n-k] = b_0x[n] + ... + b_Mx[n-M] - a_0y[n] - ... - a_Ny[n-N] \end{equation} +Figure \ref{fig:fig_iir} visualizes a simple IIR-filter with one feedforward coefficient and one feedback coefficient. The first sample passes through the adder after it was multiplied with $b_0$. After that, it is passed back after being multiplied with $a_0$ and is added two the second sample, also multplied with $b_0$. \begin{figure}[H] \centering \includegraphics[width=0.8\linewidth]{Bilder/fig_iir.jpg} - \caption{IIR-filter} + \caption{IIR-filter example with one feedforward operator and one feedback operator} \label{fig:fig_iir} \end{figure} diff --git a/drawio/FIR.drawio b/drawio/FIR.drawio index 1ee190a..b4eda65 100644 --- a/drawio/FIR.drawio +++ b/drawio/FIR.drawio @@ -1,6 +1,6 @@ - + - + @@ -25,7 +25,7 @@ - + @@ -36,7 +36,7 @@ - + @@ -45,7 +45,7 @@ - + @@ -54,7 +54,7 @@ - + @@ -83,10 +83,10 @@ - + - + diff --git a/drawio/IIR.drawio b/drawio/IIR.drawio index bef0c41..8e4a6df 100644 --- a/drawio/IIR.drawio +++ b/drawio/IIR.drawio @@ -1,4 +1,4 @@ - + @@ -58,30 +58,33 @@ - + - + - + - + - + - + - + + + +