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chapter_02.tex
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@@ -146,20 +146,20 @@ As we will see in the following chapters, a real world application of an adaptiv
\begin{itemize}
\item The error signal $e[n]$ is not a perfect representation of the recorded target signal $s[n]$ present in the corrputed target signal $d[n]$, as the adaptive filter can only approximate the noise signal based on its current coefficients, which in general do not represent the optimal solution at that given time.
\item Altough, the corruption noise signal $n[n]$ and the reference noise signal $x[n]$ are correlated, they are not identical, as they take different signal paths from the noise source to their respective sensors. This discrepancy can lead to imperfect noise reduction, as the adaptive filter has to estimate the relationship between these two signals.
\item The recorded target signal $s[n]$ is not directly available, as it is only available combined with the corruption noise signal $n[n]$ in the form of $d[n]$ while there is no reference available. Therefore, the error signal $e[n]$, respectively $š[n]$, of the adaptive filter serves as an approximation of the clean speech signal and is used as an indirect measure of the filter's performance, guiding the adaptation process by its own stepwise minimization.
\item The reference noise signal $x[n]$ fed into the adaptive filter could also contaminated with parts of the target signal. If this circumstance occurs, it can lead to undesired effects if not handled properly.
\item The recorded target signal $s[n]$ is not directly available, as it is only available combined with the corruption noise signal $n[n]$ in the form of $d[n]$ while there is no reference available. Therefore, the error signal $e[n]$, respectively $š[n]$, of the adaptive filter serves as an approximation of the clean target signal and is used as an indirect measure of the filter's performance, guiding the adaptation process by its own stepwise minimization.
\item The reference noise signal $x[n]$ fed into the adaptive filter could also contaminated with parts of the target signal. If this circumstance occurs is not handled properly, it could lead to the undesired removal of parts of the target signal from the output signal $š[n]$.
\end{itemize}
The goal of the adaptive filter is therefore to minimize this error signal over time, thereby improving the quality of the output signal by reducing it by it´s noise-component.\\
The minimization of the error signal $e[n]$ can by achieved by applying different error metrics used to evaluate the performance of an adaptive filter, including:
\begin{itemize}
\item Mean Squared Error (MSE): This metric calculates the averaged square of the error between the expected value and the observed value over a predefined period. It is sensitive to large errors and is commonly used in adaptive filtering applications.
\item Least Mean Squares (LMS): This metric focuses on minimizing the mean squared error by adjusting the filter coefficients iteratively based on the error signal by applying the gradient descent method. It is computationally efficient and widely used in real-time applications.
\item Normalized Least Mean Squares (NLMS): An extension of the LMS algorithm that normalizes the step size based on the power of the input signal, improving convergence speed and stability.
\item Recursive Least Squares (RLS): This metric aims to minimize the weighted sum of squared errors, providing faster convergence than LMS but at the cost of higher computational complexity.
\item Normalized Least Mean Squares (NLMS): An extension of the LMS algorithm that normalizes the step size based on the input signal, improving convergence speed.
\item Recursive Least Squares (RLS): This metric aims to minimize the weighted sum of squared errors, providing faster convergence than the LMS metric but at the cost of higher computational effort.
\end{itemize}
As computaional efficiency is a key requirement for the implementation of real-time ANR on a low-power digital signal processor, the Least Mean Squares algorithm is chosen for the minimization of the error signal and therefore will be further explained in the following subchapter.
As computaional efficiency is a key requirement for the implementation of real-time ANR on a low-power DSP, the Least Mean Squares algorithm is chosen for the minimization of the error signal and therefore will be further explained in the following subchapter.
\subsubsection{The Wiener filter and Gradient Descent}
\subsubsection{The Wiener filter and the concept of Gradient Descent}
Before the Least Mean Squares algorithm can be explained in detail, the Wiener filter and the concept Gradient Descent have to be introduced. \\ \\
\begin{figure}[H]
\centering
@@ -279,7 +279,7 @@ The noise reference signal $x[n]$ can be mathematically described as:
x[n] = v[n] * (C_nB)
\end{equation}
where $v[n]$ is the noise signal at its source and $x[n]$ is the recorded reference noise signal after passing the transfer functions.\\ \\
Another possible signal interaction could be the leakage of the target signal into the noise signal sensor, leading to undesired effects. This case is not illustrated in Figure \ref{fig:fig_anr_implant} as it wont be further evaluated in this thesis, but shall be mentioned for the sake of completeness at this point.\\ \\
Another possible signal interaction could be the leakage of the target signal into the noise signal sensor, leading to the partly removal of the target signal from the output signal. This case is not illustrated in Figure \ref{fig:fig_anr_implant} as it wont be further evaluated in this thesis, but shall be mentioned for the sake of completeness.\\ \\
At this point, the thereotical background and the fundamentals of adaptive noise reduction have been adequatly introduced and explained as necessary for the understanding of the following chapters of this thesis. The next chapter will now focus on practical high level simulations of different filter concepts and LMS algorithm variations to evaluate their performance in regard of noise reduction quality before the actual implementation on a low-power digital signal processor is conducted.