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@@ -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}