Korr 1

chapter_02.tex

@@ -2,9 +2,9 @@
 The following subchapters shall supply the reader with the theoretical foundation of digital signal processing to better understand the following implementation of \ac{ANR} on a low-power signal processor.\\ \\
 The chapter begins with the description of signals, the problem of them interfering and the basics of digital signal processing in general, covering fundamental topics like signal representation, transfer functions and filters.\\
 Filters are used in various functional designs, therefore a short explanation into the concepts of Finite Impulse Response- and Infinite Impulse Response filters is indispensable.\\
-At this point an introduction into adaptive noise reduction follows, including a short overview of the most important steps in history, the general concept of \ac{ANR}, its design possibilities and its optimization possibilities in regard of error calculation.\\
+At this point an introduction into \ac{ANR} follows, including a short overview of the most important steps in history, the general concept of \ac{ANR}, its design possibilities and its optimization possibilities in regard of error calculation.\\
 With this knowledge covered, a realistic signal flow diagram of an implanted \ac{CI} system with corresponding transfer functions is designed, essential to implement \ac{ANR} on a low-power digital signal processor.\\
-At the end of chapter two, high-level Python simulations shall function as a practical demonstration of the recently presented theoretical background.\\ \\
+At the end of chapter two, high-level Python simulations shall function as a feasibility demonstration of the recently presented theoretical background.\\ \\
 Throughout this thesis, sampled signals are denoted in lowercase with square brackets (e.g. {x[n]}) to distinguish them from time-continuous signals (e.g. {x(t)}). Vectors are notated in lowercase bold font, whereas matrix are notated in uppercase bold font. Scalars are notated in normal lowercase font.\\ 
 \subsection{Signals and signal interference}
 A signal is a physical parameter (e.g. pressure, voltage) changing its value over time. Whereas in nature, a signal is always analog, meaning continuous in both time and amplitude, a digital signal is represented in a discrete form, being sampled at specific time intervals and quantized to finite amplitude levels.\\ \\
@@ -15,7 +15,7 @@ The term "signal interference" describes the overlapping of unwanted signals or
     \caption{Noisy signal containing different frequencies and cleaned signal. \cite{source_dsp_ch1}}
     \label{fig:fig_interference}
 \end{figure}
-\noindent In cochlear implant systems, speech signals must be reconstructed with high spectral precision to ensure intelligibility for the user. As signal interference can cause considerable degradation to the quality of said final audio signal, the objective of this thesis shall be the improvement of implant technology in regard of adaptive noise reduction.
+\noindent In cochlear implant systems, speech signals must be reconstructed with high spectral precision to ensure intelligibility for the user. As signal interference can cause considerable degradation to the quality of said final audio signal, the objective of this thesis is the improvement of implant technology in regard of adaptive noise reduction.
 \subsection{Fundamentals of digital signal processing}
 Digital signal processing describes the manipulation of digital signals on a \ac{DSP} through mathematical approaches. Analog signals have to be digitalized before being able to be handled by a \ac{DSP}. 
 \subsubsection{Signal conversion and representation}
@@ -64,7 +64,7 @@ During the description of transfer functions, the term ``filter'' was used but n
     \label{fig:fig_lowpass}
 \end{figure}
 \subsection{Filter designs}
-Before we continue with the introduction to the actual topic of this thesis, adaptive noise reduction, two very essential filter designs need further explanation - the Finite Impulse Response- and Infinite Impulse Response filter. 
+Before we continue with the introduction to the actual topic of this thesis, \ac{ANR}, two very essential filter designs need further explanation - the Finite Impulse Response- and Infinite Impulse Response filter. 
 \subsubsection{Finite Impulse Response filters}
 A \ac{FIR} filter, commonly referred to as a ``Feedforward Filter'' is defined through the property, that it uses only input values and not feedback from output samples to determine its filtering behavior - therefore, if the input signal is reduced to zero, the response of a \ac{FIR} filter reaches zero after a finite number of samples.\\ \\
 Equation \ref{equation_fir} specifies the input-output relationship of a \ac{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
@@ -102,7 +102,7 @@ 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.\\ \\
-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.\\ \\
+In the 1930s, the first real concept of \ac{ANC} 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]
     \centering
@@ -110,11 +110,11 @@ In the 1930s, the first real concept of active noise cancellation was proposed b
     \caption{Reconstruction of Lawrence Fogel´s patent in 1960. \cite{source_patent}}
     \label{fig:fig_patent}
 \end{figure}
-\noindent In contrary to the static filters in the beginning of the century, the active noise cancellation of Lueg and Widrow was far more advanced than just reducing a signal by a specific frequency portion like with the use of static filters, yet this technique still has their limitations as it is designed only to work within to a certain environment.\\ \\
-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.
+\noindent In contrary to the static filters in the beginning of the century, the \ac{ANC} approach of Lueg and Widrow was far more advanced than just reducing a signal by a specific frequency portion like with the use of static filters, yet this technique still has their limitations as it is designed only to work within to a certain environment.\\ \\
+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 \ac{ANR} particularly suitable for hearing devices, where environmental noise characteristics vary constantly.\\ \\
-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.\\ \\
+Static filters, like low- and high-pass filters, as described in the previous subchapter, 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 — \ac{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
@@ -122,7 +122,7 @@ Although active noise cancellation and adaptive noise reduction share obvious si
     \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 primary sensor (top) aims to receive the desired signal and outputs the corrupted signal $d[n]$, which consists out of the desired signal $s[n]$ and the corruption noise signal $n[n]$ after passing the transfer function of the sensor, whereas the secondary signal sensor aims to receive (ideally) only the noise signal and outputs the reference noise signal $x[n]$ after also passing the respective transfer function, which then feeds the adaptive filter. We assume at this point, that the corruption noise signal is uncorrelated to the desired 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 desired 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 \ac{CI} system in chapter 2.6.
+\noindent Figure \ref{fig:fig_anr} shows the basic concept of an adaptive filter design, represented through a feedback filter application. The primary sensor (top) aims to receive the desired signal and outputs the corrupted signal $d[n]$, which consists out of the desired signal $s[n]$ and the corruption noise signal $n[n]$ after passing the transfer function of the sensor, whereas the secondary signal sensor aims to receive (ideally) only the noise signal and outputs the reference noise signal $x[n]$ after also passing the respective transfer function, which then feeds the adaptive filter. We assume at this point, that the corruption noise signal is uncorrelated to the desired 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 desired signal $š[n]$. 
 \subsubsection{Fully adaptive vs. hybrid filter design}
 The basic \ac{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 (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. 
@@ -179,7 +179,7 @@ The Wiener filter aims to adjust its coefficients to generate a filter output, w
 If we square the error signal and calculate the expected value, we receive the Mean Squared Error $J$, mentioned in the previous chapter, which is the metric the Wiener filter aims to minimize by adjusting its coefficients $w$.
 \begin{equation}
 \label{equation_j}
- J = E(e[n]^2) = E(d^2[n])-2wE(d[n]x[n])+w^2E(x^2[n]) = MSE 
+ J = E(e[n]^2) = E(d^2[n])-2wE(d[n]x[n])+w^2E(x^2[n]) = \text{MSE}
 \end{equation}
 The terms contained in Equation \ref{equation_j} can be further be defined as:
 \begin{itemize}
@@ -255,9 +255,9 @@ The result of Equation \ref{equation_j_lms_final} can now be inserted into Equat
 \label{equation_lms}
  w[n+1] = w[n] - 2\mu e[n]x[n]
 \end{equation}
-The \ac{LMS} algorithm therefore updates the filter coefficients $w[n]$ after every sample by adding a correction term, which is calculated by the error signal $e[n]$ and the reference noise signal $x[n]$, scaled by the constant step size $\mu$. By iteratively applying the \ac{LMS} algorithm, the filter coefficients converge towards the optimal values that minimize the mean squared error between the desired signal and the filter output. When a predefined acceptable error level is reached, the adaptation process can be stopped to save computing power.\\ \\
+The \ac{LMS} algorithm therefore updates the filter coefficients $w[n]$ after every sample by adding a correction term, which is calculated by the error signal $e[n]$ and the reference noise signal $x[n]$, scaled by the constant step size $\mu$. By iteratively applying the \ac{LMS} algorithm, the filter coefficients converge towards the optimal values that minimize the mean squared error between the desired signal and the filter output.
  \subsection{Signal flow diagram of an implanted cochlear implant system}
- Now equipped with the necessary theoretical background about signal processing, adaptive noise reduction and the \ac{LMS} algorithm, a realistic signal flow diagram with the relevant transfer functions of an implanted cochlear implant system can be designed, which will serve as the basis for the implementation of \ac{ANR} on a low-power digital signal processor.
+ Now equipped with the necessary theoretical background about signal processing, \ac{ANR} and the \ac{LMS} algorithm, a realistic signal flow diagram with the relevant transfer functions of an implanted cochlear implant system can be designed, which will serve as the basis for the implementation of \ac{ANR} on a low-power \ac{DSP}
  \begin{figure}[H]
     \centering
     \includegraphics[width=1.1\linewidth]{Bilder/fig_anr_implant.jpg}
@@ -289,7 +289,7 @@ x[n] = v[n] * (E_nB)
 \end{equation}
 where $v[n]$ is the noise signal at its source.\\ \\
 Another possible signal interaction could be the leakage of the desired signal into the secondary sensor, leading to the partial removal of the desired signal from the output signal. This case is not illustrated in Figure \ref{fig:fig_anr_implant} as it won't be further evaluated in this thesis, but shall be mentioned for the sake of completeness.\\ \\ 
-At this point, the theoretical background and the fundamentals of adaptive noise reduction have been adequately 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 the \ac{ANR} algorithm under different circumstances to evaluate their performance in regard of noise reduction quality before the actual implementation on a low-power digital signal processor is conducted. 
+At this point, the theoretical background and the fundamentals of adaptive noise reduction have been adequately 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 the \ac{ANR} algorithm under different circumstances to evaluate their performance in regard of noise reduction quality before the actual implementation on a low-power \ac{DSP} is conducted.