Rechtschreibkorrektur -> Check Christoph

chapter_02.tex

@@ -101,7 +101,7 @@ A higher number of needed coefficients implies, that the filter itself needs mor
 
 \subsection{Introduction to Adaptive Noise Reduction}
 \subsubsection{History}
-The necessity for the use of electric filters arose the first time in the beginnings of the 20th century with the development of the quite young fields of tele- and radio-communication. At his time, engineers used static filters like low- or highpass filters to improve transmission quality - this fundamental techniques allowed limiting the frequency spectrum, by cutting out certain frequencies like high-pitched noises or humming. From this time on, the development of new filter designs accelerated, for example with the soon-to-be developed LC-filter by Otto Zobel, an American scientist working at the telecommunication company AT\&T. Until then, the used filters were static, meaning they didn't change their behavior over time.\\ \\
+The necessity for the use of electric filters arose the first time in the beginnings of the 20th century with the development of the quite young fields of tele- and radio-communication. At his time, engineers used static filters, like low- or highpass filters, to improve transmission quality - this fundamental techniques allowed limiting the frequency spectrum, by cutting out certain frequencies like high-pitched noises or humming. From this time on, the development of new filter designs accelerated, for example with the soon-to-be developed LC-filter by Otto Zobel, an American scientist working at the telecommunication company AT\&T. Until then, the used filters were static, meaning they didn't change their behavior over time.\\ \\
 In the 1930s, the first real concept of active noise cancellation was proposed by the German Physician Paul Lueg. Lueg patented the idea of two speakers emitting antiphase signals which cancel each other out. Though his patent was granted in 1936, back at the time, there was no technical possibility detect and process audio signals in a way, to make his noise cancellation actually work in a technical environment.\\ \\
 20 years after Lueg's patent, Lawrence Fogel patented a practical concept of noise cancellation, intended for noise suppression in aviation - this time, the technical circumstances of the 1950s enabled the development of an aviation headset, lowering the overall noise experienced by pilots in the cockpit of a helicopter or an airplane by emitting a 180 degree phase shifted signal of the recorded background noise of the cockpit into the pilots' headset. (see Figure \ref{fig:fig_patent}).  
 \begin{figure}[H]
@@ -114,18 +114,18 @@ In the 1930s, the first real concept of active noise cancellation was proposed b
 With the rapid advancement of digital signal processing technologies, noise cancellation techniques  evolved from static, hardware-based filters and physical soundwave cancellation towards more sophisticated approaches. In the then 1970s, the concept of digital adaptive filtering arose, allowing digital filters to adjust their parameters in real-time based on the characteristics of the incoming signal and noise. This marked a significant leap forward, as it enabled systems to deal with dynamic and unpredictable noise environments - the concept of adaptive noise reduction was born.
 \subsubsection{The concept of adaptive filtering}
 Adaptive noise reduction describes an advanced filtering method based on an error-metric and represents a significant advancement over these earlier methods by allowing the filter parameters to continuously adapt to the changing acoustic environment in real-time. This adaptability makes ANR particularly suitable for hearing devices, where environmental noise characteristics vary constantly.\\ \\
-Static filters low- and high-pass filters as described in the previous chapter feature coefficients that remain constant over time. They are designed for known, predictable noise conditions (e.g., removing a steady 50 Hz hum). While these filters are efficient and easy to implement, they fail to function when noise characteristics change dynamically.\\ \\
-Although active noise cancellation and adaptive noise reduction share obvious similarities, they differ fundamentally in their application and signal structure. While active noise cancellation aims to physically cancel noise in the acoustic domain — typically before, or at the time, the signal reaches the ear — ANR operates within the signal processing chain, attempting to extract the digital noisy component from the desired digital signal. In cochlear implant systems, the latter is more practical because the acoustic waveform is converted into electrical stimulation signals; thus, signal-domain filtering is the only feasible approach.
+Static filters, like low- and high-pass filters, as described in the previous chapter feature coefficients that remain constant over time. They are designed for known, predictable noise conditions (e.g., removing a steady 50 Hz hum). While these filters are efficient and easy to implement, they fail to function when noise characteristics change dynamically.\\ \\
+Although active noise cancellation and adaptive noise reduction share obvious similarities, they differ fundamentally in their application and signal structure. While active noise cancellation aims to physically cancel noise in the acoustic domain — typically before, or at the time, the signal reaches the ear — ANR operates within the signal processing chain, attempting to extract the noisy component from the digital signal. In cochlear implant systems, the latter is more practical because the acoustic waveform is converted into electrical stimulation signals; thus, signal-domain filtering is the only feasible approach.
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.8\linewidth]{Bilder/fig_anr.jpg}
     \caption{The basic idea of an adaptive filter design for noise reduction.}
     \label{fig:fig_anr}
 \end{figure}
-\noindent Figure \ref{fig:fig_anr} shows the basic concept of an adaptive filter design, represented through a feedback filter application. The target signal sensor (top) aims to receive the target signal and outputs the corrupted target signal $d[n]$, which consists out of the recorded target signal $s[n]$ and the corruption noise signal $n[n]$, whereas the noise signal sensor aims to receive (ideally) only the noise signal and outputs the recorded reference noise signal $x[n]$, which then feeds the adaptive filter. We assume at this point, that the corruption-noise signal is uncorrelated to the recorded target signal, and therefore separable from it. In addition, we assume, that the corruption noise signal is correlated to the reference noise signal, as it originates from the same source, but takes a different signal path. \\ \\ The adaptive filter removes a certain, noise-related, frequency part of the input signal and re-evaluates the output through its feedback design. The filter parameters are then adjusted and applied to the next sample to minimize the observed error $e[n]$, which also represents the approximated target signal $š[n]$. In reality, a signal contamination of the two sensors has to be expected, which will be illustrated in a more realistic signal flow diagram of an implanted CI system.
+\noindent Figure \ref{fig:fig_anr} shows the basic concept of an adaptive filter design, represented through a feedback filter application. The target signal sensor (top) aims to receive the target signal and outputs the corrupted target signal $d[n]$, which consists out of the recorded target signal $s[n]$ and the corruption noise signal $n[n]$, whereas the noise signal sensor aims to receive (ideally) only the noise signal and outputs the recorded reference noise signal $x[n]$, which then feeds the adaptive filter. We assume at this point, that the corruption-noise signal is uncorrelated to the recorded target signal, and therefore separable from it. In addition, we assume, that the corruption noise signal is correlated to the reference noise signal, as it originates from the same source, but takes a different signal path. \\ \\ The adaptive filter removes a certain, noise-related, frequency part of the input signal and re-evaluates the output through its feedback design. The filter parameters are then adjusted and applied to the next sample to minimize the observed error $e[n]$, which also represents the approximated target signal $š[n]$. In reality, a signal contamination of the two sensors has to be expected, which will be illustrated in a more realistic signal flow diagram of an implanted CI system in chapter 2.6.
 \subsubsection{Fully adaptive vs. hybrid filter design}
 The basic ANR concept illustrated in Figure \ref{fig:fig_anr} can be understood as a fully adaptive variant. A fully adaptive filter design works with a fixed number of coefficients of which everyone is updated after every sample processing. Even if this approach features the best performance in noise reduction, it also requires a relatively high amount of computing power, as every coefficient has to be re-calculated after every evaluation step.\\ \\
-To reduce the required computing power, a hybrid static/adaptive filter design can be taken into consideration instead. In this approach, the initial fully adaptive filter is split into a fixed and an adaptive part - the static filter removes a certain, known, or estimated, frequency portion of the noise signal, whereas the adaptive part only has to adapt to the remaining, unforecastable, noise parts. This approach reduces the number of coefficients required to be adapted, therefore lowering the required computing power. 
+To reduce the required computing power, a hybrid static/adaptive filter design can be taken into consideration instead (refer to Figure \ref{fig:fig_anr_hybrid}). In this approach, the initial fully adaptive filter is split into a fixed and an adaptive part - the static filter removes a certain, known, or estimated, frequency portion of the noise signal, whereas the adaptive part only has to adapt to the remaining, unforecastable, noise parts. This approach reduces the number of coefficients required to be adapted, therefore lowering the required computing power. 
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.8\linewidth]{Bilder/fig_anr_hybrid.jpg}
@@ -144,7 +144,7 @@ Adaptive filters rely on an error metric to self-reliantely evaluate their perfo
 The error signal $e[n]$, already illustrated in Figure \ref{fig:fig_anr} and \ref{fig:fig_anr_hybrid},  is calculated as the difference between the corrupted target signal $d[n]$ and the output signal of the filter $y[n]$.
 As we will see in the following chapters, a real world application of an adaptive filter system poses several challenges, which have to be taken into consideration when designing the filter. These challenges include:
 \begin{itemize}
-\item The error signal $e[n]$ is not a perfect representation of the recorded target signal $s[n]$ present in the corrupted 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 The error signal $e[n]$ is not a perfect representation of the recorded target signal $s[n]$ present in the corrupted 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 Although, 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 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 be 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 $š[n]$.
@@ -159,15 +159,15 @@ The minimization of the error signal $e[n]$ can be achieved by applying differen
 \end{itemize}
 As computational 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 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. \\ \\
+\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 of gradient descent have to be introduced. \\ \\
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.7\linewidth]{Bilder/fig_wien.jpg}
     \caption{Simple implementation of a Wiener filter.}
     \label{fig:fig_wien}
 \end{figure}
-\noindent The Wiener filter, the base of many adaptive filter designs, is a statistical filter used to minimize the Mean Square Error between a target signal and the output of a linear filter. The output $y[n]$ of the Wiener filter is the sum of the weighted input samples, where the weights are represented by the filter coefficients.
+\noindent The Wiener filter, the base of many adaptive filter designs, is a statistical filter used to minimize the Mean Squares Error between a target signal and the output of a linear filter. The output $y[n]$ of the Wiener filter is the sum of the weighted input samples, where the weights are represented by the filter coefficients.
 \begin{equation}
 \label{equation_wien}
  y[n] = w_0x[n] + w_1x[n-1] + ... + w_Mx[n-M] = \sum_{k=0}^{M} w_kx[n-k]
@@ -265,7 +265,7 @@ The LMS algorithm therefore updates the filter coefficients $w[n]$ after every s
     \caption{Realistic implant design.}
     \label{fig:fig_anr_implant}
 \end{figure}
-\noindent Figure \ref{fig:fig_anr_hybrid} showed us the basic concept of an ANR implementation, without a detailed description how the corrupted target signal $d[n]$ and the reference noise signal $x[n]$ are formed. Figure \ref{fig:fig_anr_implant} now shows a more complete and realistic signal flow diagram of an implanted cochlear implant system, with two signal sensors and an adaptive noise reduction circuit afterwards. The target signal sensor receives the target signal and the noise signal over their respective transfer functions and outputs the corrupted target signal $d[n]$, which consists out of the recorded target signal $s[n]$ and the recorded corruption noise signal $n[n]$, whereas the noise signal sensor aims to receive (ideally) only the noise signal $v[n]$ over its transfer function and outputs the reference noise signal $x[n]$, which then feeds the adaptive filter.\\ \\
+\noindent Figure \ref{fig:fig_anr_hybrid} showed us the basic concept of an ANR implementation, without a detailed description how the corrupted target signal $d[n]$ and the reference noise signal $x[n]$ are formed. Figure \ref{fig:fig_anr_implant} now shows a more complete and realistic signal flow diagram of an implanted cochlear implant system, with two signal sensors and an adaptive noise reduction circuit afterwards. The target signal sensor receives the target- and noise signal over their respective transfer functions and outputs the corrupted target signal $d[n]$, which consists out of the recorded target signal $s[n]$ and the recorded corruption noise signal $n[n]$, whereas the noise signal sensor aims to receive (ideally) only the noise signal $v[n]$ over its transfer function and outputs the reference noise signal $x[n]$, which then feeds the adaptive filter.\\ \\
 Additionally, now the relevant transfer functions of the overall system are illustrated in Figure \ref{fig:fig_anr_implant}. The transfer functions $D_n$, $F_n$, and $C_n$ describe the path from the signal sources to the chassis of the cochlear implant, where the sensors are located. As the sources and the relative location of the user to the sources can vary, these transfer functions are time-variant and unknown. From the chassis, there are two options for continuing the signal path - either directly to the microphone membranes of the respective sensors, represented through the transfer function $G$, or through mechanical vibrations of the implant's chassis, represented through the transfer functions $A$ and $B$. As the mechanical properties of the implanted cochlear systems are fixed, these transfer functions do not change over time, so they can be seen as time-invariant and known.\\ \\   
 The corrupted target signal $d[n]$ can therefore be mathematically described as: 
 \begin{equation}