Abgabe

chapter_02.tex

@@ -4,14 +4,14 @@ The chapter begins with the description of signals, the problem of them interfer
 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 \ac{ANR} follows, including a short overview of the most important steps in history, the general concept of \ac{ANR} and its design- and 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.\\ \\
-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.
+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.\\ \\ Chapter 2, especially the mathematical foundation, is oriented on the textbook ``Digital Signal Processing Fundamentals and Applications 3rd Ed'' by Tan and Jiang \cite{source_dsp1}.
 \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.\\ \\
-The term ``signal interference'' describes the overlapping of unwanted signals or noise with the desired signal, degrading the overall quality and intelligibility of the processed information. A simple example of signal interference is shown in Figure \ref{fig:fig_interference} - the noisy signal (top) consists out of several signals of different frequencies, representing both the desired signal and unwanted noise. The cleaned signal (bottom) shows the signal after unwanted frequencies has been cut off by a filter.\\ \\
+The term ``signal interference'' describes the overlapping of unwanted signals or noise with the desired signal, degrading the overall quality and intelligibility of the processed information. A simple example of signal interference is shown in Figure \ref{fig:fig_interference} - the noisy signal (top) consists out of several signals of different frequencies, representing both the desired signal and unwanted noise. The cleaned signal (bottom) shows the signal after unwanted frequencies have been cut off by a filter.\\ \\
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.8\linewidth]{Bilder/fig_interference.jpg}
-    \caption{Noisy signal containing different frequencies and cleaned signal. \cite{source_dsp_ch1}}
+    \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 is the improvement of implant technology in regard of adaptive noise reduction.
@@ -21,15 +21,15 @@ Digital signal processing describes the manipulation of digital signals on a \ac
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.8\linewidth]{Bilder/fig_dsp.jpg}
-    \caption{Block diagram of processing an analog input signal to an analog output signal with digital signal processing in between. \cite{source_dsp_ch1}}
+    \caption{Block diagram of processing an analog input signal to an analog output signal with digital signal processing in between \cite{source_dsp_ch1}}
     \label{fig:fig_dsp}
 \end{figure}
-Before digital signal processing can be applied to an analog signal like human voice, several steps are required beforehand. An analog signal, continuous in both time and amplitude, is passed through an initial filter, which limits the frequency bandwidth. An analog-digital converter then samples and quantities the signal into a digital form, now discrete in time and amplitude. This digital signal can now be processed, before (possibly) being converted to an analog signal again (refer to Figure \ref{fig:fig_dsp}). The sampling rate defines, in how many samples per second are taken from the analog signal - a higher sample rate delivers a more accurate digital representation of the signal but also uses more resources. According to the Nyquist–Shannon sampling theorem, the sample rate must be at least twice the highest frequency component present in the signal to avoid aliasing of the signal (refer to Figure \ref{fig:fig_nyquist}). Aliasing describes the phenomenon, that high frequency parts of a signal are wrongly interpreted, if the sampling rate of the analog signal is too low. The digitalized signal then contains low frequencies, which don't occur in the original signal.
+Before digital signal processing can be applied to an analog signal like human voice, several steps are required beforehand. An analog signal, continuous in both time and amplitude, is passed through an initial filter, which limits the frequency bandwidth. An analog-digital converter then samples and quantities the signal into a digital form, now discrete in time and amplitude. This digital signal can now be processed, before (possibly) being converted to an analog signal again (refer to Figure \ref{fig:fig_dsp}). The sampling rate defines, how many samples per second are taken from an analog signal - a higher sample rate delivers a more accurate digital representation of the signal but also uses more resources. According to the Nyquist–Shannon sampling theorem, the sample rate must be at least twice the highest frequency component present in the signal to avoid aliasing of the signal (refer to Figure \ref{fig:fig_nyquist}). Aliasing describes the phenomenon, that high frequency parts of a signal are wrongly interpreted, if the sampling rate of the analog signal is too low. The digitalized signal then contains low frequencies, which don't occur in the original signal.
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.8\linewidth]{Bilder/fig_nyquist.jpg}
     \caption{Adequate (top) and inadequate 
-    (bottom) sampling frequency of a signal. \cite{source_dsp_ch1}}
+    (bottom) sampling frequency of a signal \cite{source_dsp_ch1}}
     \label{fig:fig_nyquist}
 \end{figure}
 \noindent The discrete digital signal can be viewed as a sequence of finite samples with its amplitude being a discrete value, like a 16- or 32-bit integer. A signal vector of the length N, containing N samples, is therefore notated as
@@ -43,7 +43,7 @@ A signal (either analog or digital) can be displayed and analyzed in two ways: t
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.8\linewidth]{Bilder/fig_fft.jpg}
-    \caption{Sampled digital signal in the time spectrum and in the frequency spectrum. \cite{source_dsp_ch1}}
+    \caption{Sampled digital signal in the time spectrum and in the frequency spectrum \cite{source_dsp_ch1}}
     \label{fig:fig_fft}
 \end{figure}
 \subsubsection{Transfer functions and filters}
@@ -51,7 +51,7 @@ When we discuss signals in a mathematical way, we need to explain the term ``tra
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.8\linewidth]{Bilder/fig_transfer.jpg}
-    \caption{Simple representation of a transfer function taking a noisy input signal and delivering a clean output signal. \cite{source_dsp_ch1}}
+    \caption{Simple representation of a transfer function taking a noisy input signal and delivering a clean output signal \cite{source_dsp_ch1}}
     \label{fig:fig_transfer}
 \end{figure}
 \noindent In digital signal processing, especially in the design of a noise reduction algorithm, transfer functions are essential for modeling and analyzing filters, amplifiers, and the pathway of the signal itself. By understanding a system’s transfer function, one can predict how sound signals are altered and therefore how filter parameters can be adapted to deliver the desired output signal.\\ \\
@@ -76,7 +76,7 @@ As there are three operators present in the filter, three samples are needed bef
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.8\linewidth]{Bilder/fig_fir.jpg}
-    \caption{\ac{FIR} filter example with three feedforward operators.}
+    \caption{\ac{FIR} filter example with three feedforward operators}
     \label{fig:fig_fir}
 \end{figure}
 \subsubsection{Infinite Impulse Response filters}
@@ -90,7 +90,7 @@ Figure \ref{fig:fig_iir} visualizes a simple \ac{IIR} filter with two feedforwar
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.8\linewidth]{Bilder/fig_iir.jpg}
-    \caption{\ac{IIR} filter example with two feedforward operators and two feedback operators.}
+    \caption{\ac{IIR} filter example with two feedforward operators and two feedback operators}
     \label{fig:fig_iir}
 \end{figure}
 \subsubsection{FIR- vs. IIR-filters}
@@ -98,7 +98,7 @@ Due to the fact, that there is no feedback, a \ac{FIR} filter offers uncondition
 The recursive nature of a \ac{IIR} filter, in contrary, allows achieving a sharp frequency response with significantly fewer coefficients than an equivalent \ac{FIR} filter, but it also opens up the possibility, that the filter response diverges, depending on the set coefficients.\\ \\
 A higher number of needed coefficients implies, that the filter itself needs more time to complete its signal response, as the group delay is increased.
 
-\subsection{Introduction to Adaptive Noise Reduction}
+\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.\\ \\
@@ -106,7 +106,7 @@ In the 1930s, the first real concept of active noise cancellation was proposed b
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.7\linewidth]{Bilder/fig_patent.jpg}
-    \caption{Reconstruction of Lawrence Fogel´s patent in 1960. \cite{source_patent}}
+    \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 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.\\ \\
@@ -114,11 +114,11 @@ With the rapid advancement of digital signal processing technologies, noise canc
 \subsubsection{The concept of adaptive filtering}
 \ac{ANR} 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 subchapter, feature coefficients that remain constant over time. They are designed for known, predictable noise conditions (e.g. removing a steady 50 Hz hum originating from a power supply). 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.
+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 relevant 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.}
+    \caption{The basic concept 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 $š[n]$. 
@@ -128,7 +128,7 @@ To reduce the required computing power, a hybrid static/adaptive filter design c
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.8\linewidth]{Bilder/fig_anr_hybrid.jpg}
-    \caption{Hybrid adaptive filter design for noise reduction with a static part and an adaptive part.}
+    \caption{Hybrid adaptive filter design for noise reduction with a static part and an adaptive part}
     \label{fig:fig_anr_hybrid}
 \end{figure}
 \subsection{Adaptive optimization strategies}
@@ -151,21 +151,21 @@ The goal of the adaptive filter is therefore to minimize this error signal over
 The minimization of the error signal $e[n]$ can be achieved by applying different error metrics and algorithms used to evaluate the performance of an adaptive filter, including:
 \begin{itemize}
 \item \ac{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 \ac{LMS}: The \ac{LMS} is an algorithm, focused on minimizing the \ac{MSE} 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 \ac{LMS}: The \ac{LMS} algorithm, developed by Bernard Widrow in 1960, focuses on minimizing the \ac{MSE} 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 \ac{NLMS}: An extension of the \ac{LMS} algorithm that normalizes the step size based on the input signal, improving convergence speed.
 \item \ac{RLS}: This algorithm aims to minimize the weighted sum of squared errors, providing faster convergence than the \ac{LMS} algorithm but at the cost of higher computational effort.
 \end{itemize}
-As computational efficiency is a key requirement for the implementation of real-time \ac{ANR} on a low-power \ac{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.
+As computational efficiency is a key requirement for the implementation of real-time \ac{ANR} on a low-power \ac{DSP}, the \ac{LMS} 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 of gradient descent have to be introduced. \\ \\
+Before the \ac{LMS} 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.}
+    \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 Squares Error between a desired 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 \ac{MSE} between a desired 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]
@@ -229,7 +229,7 @@ After settings the derivative of Equation \ref{equation_j_matrix} to zero and so
 \label{equation_w_optimal_matrix}
  \textbf{w}_{opt} = \textbf{P}\textbf{R}^{-1}
 \end{equation}
-\noindent For a large filter, the numerical solution of Equation \ref{equation_w_optimal_matrix} can be computational expensive, as it involves the inversion of potential large matrix. Therefore, to find the optimal set of coefficients $w$, the concept of gradient descent, introduced by Widrow\&Stearns in 1985, can be applied. The gradient decent algorithm aims to minimize the MSE iteratively sample by sample, by adjusting the filter coefficients $w$ in small steps towards the direction of the steepest descent to find the optimal coefficients. The update rule for the coefficients using gradient descent can be expressed as
+\noindent For a large filter, the numerical solution of Equation \ref{equation_w_optimal_matrix} can be computational expensive, as it involves the inversion of potential large matrix. Therefore, to find the optimal set of coefficients $w$, the concept of gradient descent (introduced by Widrow\&Stearns in 1985 \cite{source_anr}) can be applied. The gradient decent algorithm aims to minimize the MSE iteratively sample by sample, by adjusting the filter coefficients $w$ in small steps towards the direction of the steepest descent to find the optimal coefficients. The update rule for the coefficients using gradient descent can be expressed as
 \begin{equation}  
 \label{equation_gradient}
  w(n+1) = w(n) - \mu \frac{dJ}{dw} 
@@ -238,7 +238,7 @@ where $\mu$ is the constant step size determining the rate of convergence. Figur
 \begin{figure}[H]
     \centering
     \includegraphics[width=0.9\linewidth]{Bilder/fig_gradient.jpg}
-    \caption{Visualization of the steepest decent algorithm used on the Mean Squared Error. \cite{source_dsp_ch9}}
+    \caption{Visualization of the steepest decent algorithm used on the Mean Squared Error \cite{source_dsp_ch9}}
     \label{fig:fig_w_opt}
 \end{figure}
 \subsubsection{The Least Mean Squares algorithm}
@@ -260,19 +260,19 @@ The \ac{LMS} algorithm therefore updates the filter coefficients $w[n]$ after ev
  \begin{figure}[H]
     \centering
     \includegraphics[width=1.1\linewidth]{Bilder/fig_anr_implant.jpg}
-    \caption{Filter design for an implanted cochlear implant system with two signal sensors.}
+    \caption{Filter design for an implanted cochlear implant system with two signal sensors}
     \label{fig:fig_anr_implant}
 \end{figure}
 \noindent Figure \ref{fig:fig_anr_hybrid} showed us the basic concept of an \ac{ANR} implementation, without a detailed description how the corrupted 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 following definitions of the involved signals shall help to better understand the involved signals and their interactions:
 \begin{itemize}
-\item Desired signal: The wanted signal, like human voice, which shall be restored.
-\item Noise signal: The unwanted signal, like background noise, which shall be reduced.
-\item Recorded desired signal: The desired signal after passing the transfer function to the primary sensor.
-\item Corruption noise signal: The noise signal after passing the transfer function to the primary sensor.
-\item Reference noise signal: The noise signal after passing the transfer function to the secondary sensor.
-\item Corrupted signal: The combination of the recorded desired signal and the corruption noise signal
-\item Filter output / Cleaned signal: The output signal of the \ac{ANR} algorithm, representing the desired signal after noise reduction. This signal also equals the error signal of the adaptive filter.
+\item Desired signal $t[n]$: The wanted signal, like human voice, which shall be restored.
+\item Noise signal $v[n]$: The unwanted signal, like background noise, which shall be reduced.
+\item Recorded desired signal $s[n]$: The desired signal after passing the transfer function to the primary sensor.
+\item Corruption noise signal $n[n]$: The noise signal after passing the transfer function to the primary sensor.
+\item Reference noise signal $x[n]$: The noise signal after passing the transfer function to the secondary sensor.
+\item Corrupted signal $d[n]$: The combination of the recorded desired signal and the corruption noise signal
+\item Filter output $e[n]$: The output signal of the \ac{ANR} algorithm, representing the desired signal after noise reduction. This signal also equals the error signal of the adaptive filter.
 \end{itemize}
 The primary sensor receives the desired- and noise signal over their respective transfer functions and outputs the corrupted signal $d[n]$, which consists out of the recorded desired signal $s[n]$ and the 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 $C_n$, $D_n$, and $E_n$ describe the path from the signal sources to the cochlear implant system. As the sources, the relative location of the user to the sources and the medium between them can vary, these transfer functions are time-variant and unknown. After the signals reached the implant systems, we establish the possibility, that the remaining path of the signals is mainly depended on the sensitivity curve of the respective sensors and therefore can be seen as time-invariant and known.\\ \\