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\section{Introduction}
\subsection{Motivation}
According to the World Health Organization (WHO), around 1.6 billion people over 14 years worldwide suffer from any kind of hearing loss. Included in this 1.6 billion people, around 430 million suffer from disabling hearing loss (up to deafness), requiring rehabilitation. In the case of disabling hearing loss, the possibility of using an Implant System has revolutionized auditory rehabilitation by restoring partial hearing. Despite steady progress in implant technology over the past decades, the system still faces its limitations. Complex auditory environments, like static noises overlain by a person speaking, can still propose a considerable challenge for users of auditory implants compared to people with a healthy hearing. \\ \\
According to the World Health Organization (WHO), around 1.6 billion people over 14 years worldwide suffer from any kind of hearing loss. Included in this 1.6 billion people, around 430 million suffer from disabling hearing loss (up to deafness), requiring rehabilitation. In the case of disabling hearing loss, the possibility of using an implant system solution has revolutionized auditory rehabilitation by restoring partial hearing. Despite steady progress in implant technology over the past decades, the system still faces its limitations. Complex auditory environments, like static noises overlain by a person speaking, can still propose a considerable challenge for users of auditory implants compared to people with a healthy hearing. \\ \\
Therefore, the improvement of implant performance in regard to the suppression of disturbance noises is therefore a crucial step in the development of more user-friendly implant solutions which provide users with more natural sound perception and greater listening comfort.
\\ \\
By addressing these challenges, this work aims to contribute to the next generation of cochlear implant technology, ultimately enhancing the auditory experience and quality of life for people with severe hearing impairments.
\subsection{Introduction to Cochlear Implant Systems}
A Cochlear Implant (CI) System is a specialized form of hearing aid, used to restore partly or complete deafness. In contrary to standard hearing aids, CI's do not just amplify the audio signal received by the ear, but stimulate the auditory nerve itself directly through electric pulses.\\ \\
Usually, a CI System consists out of an external processor (``audio processor'') receiving the ambient audio signal, processing it, and then transmitting it inductively via a transmission coil through the skin to the cochlear implant itself, implanted on the patient's skull (see Figure \ref{fig:fig_synchrony}). The CI stimulates the auditory nerves inside the cochlear through charge pulses, thus enabling the patient to hear the received audio signal as sound.\\
\subsection{Introduction to cochlear implant systems}
A cochlear implant (CI) System is a specialized form of hearing aid, used to restore partly or complete deafness. In contrary to standard hearing aids, CI's do not just amplify the audio signal received by the ear, but stimulate the auditory nerve itself directly through electric pulses.\\ \\
Usually, a CI system consists out of an external processor with a microphone (``audio processor'') receiving the ambient audio signal, processing it, and then transmitting it inductively via a transmission coil through the skin to the cochlear implant itself, implanted on the patient's skull (see Figure \ref{fig:fig_synchrony}). The CI stimulates the auditory nerves inside the cochlear through charge pulses, thus enabling the patient to hear the received audio signal as sound.\\
\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{Bilder/fig_synchrony.png}
\caption{Sketch of a MED-EL Synchrony Cochlear Implant with a Sonnet 3 Audio Processor \cite{source_synchrony}}
\label{fig:fig_synchrony}
\end{figure}
The pulse transmission to the cochlear is realized through a silicone electrode with embedded metal contacts. Said electrode is inserted into the cochlear through a drilled hole in the bone, where, depending on the insertion-depth, different contact-areas stimulate different parts of the frequency-spectrum of the hearing sense. The smaller end of the electrode-array inserted deep into the cochlear stimulate low-frequencies, whereas the larger beginning of the array stimulates high-frequencies. (see figure \ref{fig:fig_electrode}).
The pulse transmission to the cochlear is realized through a silicone electrode with embedded metal contacts. Said electrode is inserted into the cochlear through a drilled hole in the bone, where, depending on the insertion depth, different contact areas stimulate different parts of the frequency spectrum of the hearing sense. The smaller end of the electrode array inserted deep into the cochlear stimulates low frequencies, whereas the larger part at the beginning of the array stimulates high frequencies. (see Figure \ref{fig:fig_electrode}).
\begin{figure}[H]
\centering
\includegraphics[width=0.8\linewidth]{Bilder/fig_electrode.jpg}
\caption{Visualization of a MED-EL electrode inserted into a human cochlear. \cite{source_electrode}}
\label{fig:fig_electrode}
\end{figure}
As for any head worn hearing aid, the audio processor of a CI system does not only pick up the desired ambient audio signal, but also any sort of interference noises from different sources. This circumstance leads to a decrease in the quality of the final audio signal. Reducing this interference noise through Adaptive Noise Reduction (ANR), implemented on a low-power Digital Signal Processor (DSP), which can be powered within the electrical limitations of a CI system, is the topic of this master's thesis.
As for any head worn hearing aid, the audio processor of a CI system does not only pick up the desired ambient audio signal, but also any sort of interference noises from different sources. This circumstance leads to a decrease in the quality of the final audio signal for the user. Reducing this interference noise through adaptive noise reduction, implemented on a low-power digital signal processor, which can be powered within the electrical limitations of a CI system, is the topic of this master's thesis.
\subsection{The problem of signal interference in audio processing}
A signal is a physical parameter (e.g. pressure, voltage) changing its value over time. The term "Signal Interference" describes the overlapping of two or more signals resulting in a new signal. \\ \\A simple example of a desirable signal interference would be the sound generated by playing several strings of a guitar. Hitting one string results in a pure sine wave of a designated frequency (depending on which note is played), perceptible as sound. Hitting a chord (consisting of several strings), the separate sine waves of the strings combine to a new signal through the process of signal interference - in this case a desired, harmonic sound. (see Figure \ref{fig:fig_interference})
A signal is a physical parameter (e.g. pressure, voltage) changing its value over time. The term "signal interference" describes the overlapping of two or more signals resulting in a new signal. \\ \\A simple example of a desirable signal interference would be the sound generated by playing several strings of a guitar. Hitting one string results in a pure sine wave of a designated frequency (depending on which note is played), perceptible as sound. Hitting a chord (consisting of several strings), the separate sine waves of the strings combine to a new signal through the process of signal interference - in this case a desired, harmonic sound. (see Figure \ref{fig:fig_interference})
\begin{figure}[H]
\centering
\includegraphics[width=0.8\linewidth]{Bilder/fig_interference.png}
\caption{Signal interference of three separate tones resulting in an E-Minor chord.}
\label{fig:fig_interference}
\end{figure}
In technical environments signal interference is also common when electromagnetic and acoustic noise coexist. Such conditions can cause electromagnetic coupling or broadband acoustic noise that degrades microphone input and digital transmission
Therefor, in auditory applications, signal interference can cause a considerable degradation to the quality of the final signal, posing an additional challenge to aurally impaired people using an implant solution for rehabilitation.
Thus, the objective of this thesis shall be the improvement of implant technology in regard of noise reduction.
\subsection{Implementation of Adaptive Noise Reduction in Cochlear Implant Systems}
The above problem statement of signal interference shows its significance in the improvement of CI systems. For persons with a healthy hearing sense, the addition of noise to an observed signal may just mean a decrease in hearing comfort, whereas for aurally impaired people it can make the difference in the basic understanding of information. As everyday environments present fluctuating background noise - from static crowd chatter to sudden sounds of different characteristics — that can severely degrade speech perception, the ability to suppress noise is a crucial benefit for users of cochlear implant systems. \\ \\
Adaptive Noise Reduction (ANR) (also commonly referred as Adaptive Noise Cancellation (ANC)), is an advanced signal-processing technique that adjusts the parameters of a digital filter to suppress unwanted noise from a signal while preserving the desired target-signal. In contrary to static filters (like a high- or low-pass filter), ANR uses real-time feedback to adjust said digital filter to adapt to the current circumstances.\\ \\
Adaptive noise reduction (ANR) (also commonly referred as adaptive noise cancellation (ANC)), is an advanced signal-processing technique that adjusts the parameters of a digital filter to suppress unwanted noise from a signal while preserving the desired target signal. In contrary to static filters (like a high- or low-pass filter), ANR uses real-time feedback to adjust said digital filter to adapt to the current circumstances.\\ \\
The challenge in the implementation of ANR in CI systems lies in the limited capacities. As the CI system is powered by a small battery located in the audio processor, energy efficiency is crucial for a possible solution of the described problem of noise interference. Any approach to a reduction of interference noise must be highly optimized with regard to computing power and implemented on dedicated low-power hardware, being able to be powered within the limitations of a CI system.\\ \\
The main solution concept of this thesis is the optimization of the adaptive filter of the ANR algorithm in combination with the used low-power hardware. Its goal is, to deliver the best possible result in interference noise reduction while still being able to be powered by the limited resources of a CI system. Different variants, like the fully adaptive filter, the hybrid static/adaptive filter and different optimization approaches of the latter one are low-level simulated on the dedicated DSP. Especially, the different optimization strategies of the hybrid static/adaptive filter algorithm shall be evaluated and compared in regard of their required computing power, and therefore, their required power consumption. Depending on the kind of interference noise, the frequency and the intensity, a promising optimization approach is the reduction of adaptation steps per sample while still maintaining an adequate quality of the filtered audio signal.\\ \\
The main solution concept of this thesis is the optimization of the adaptive filter of the ANR algorithm in combination with the used low-power hardware. Its goal is, to deliver the best possible result in interference noise reduction while still being able to be powered by the limited resources of a CI system. Different variants, like the fully adaptive filter, the hybrid static/adaptive filter and different optimization approaches of the latter one are low-level simulated on the dedicated digital signal processor. Especially, the different optimization strategies of the hybrid static/adaptive filter algorithm shall be evaluated and compared in regard of their required computing power, and therefore, their required power consumption. Depending on the kind of interference noise, the frequency and the intensity, a promising optimization approach is the reduction of adaptation steps per sample while still maintaining an adequate quality of the filtered audio signal.\\ \\
Due to the fact, that the CI system is powered by a battery with a relatively small capacity, the firmware is required to work with the least power possible. Therefore, optimization in regard to a minimization of needed processor clocks is aimed for.

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chapter_02.tex
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\section{Theoretical Background}
The following subchapters shall equip the reader with the theoretical foundations of digital signal processing to better understand the following implementation of ANR on a low-power signal processor.\\ \\
We will begin with the fundamentals of digital signal processing in general, covering topics like signals, transfer-functions and filters.\\
To fully understand ANR, a short deep-dive into the concepts of Finite Impulse Response- and Infinite Impulse Response filters is indispensable.\\
From this point we will continue into the history and the mathematical concepts of ANR, its real-time feedback possibilities and its use of the Least Mean Square (LMS) Algorithm.\\
With this knowledge covered, we will design a realistic signal flow diagram and the corresponding transfer functions, of an implanted CI system essential to implement a functioning ANR on a low-power DSP.\\
The following subchapters shall supply the reader with the theoretical foundation of digital signal processing to better understand the following implementation of ANR on a low-power signal processor.\\ \\
The chapter begins with the basics of digital signal processing in general, covering fundamental topics like signals, transfer functions and filters.\\
Filters are used in various functional designs, therefore a short deep-dive into the concepts of Finite Impulse Response- and Infinite Impulse Response filters is indispensable.\\
At this point an introduction into ANR follows, including a short overview of the most important steps in history, the general concept of ANR, its design possibilities and its use of the Least-Mean-Square algorithm.\\
With this knowledge covered, a realistic signal flow diagram of an implanted CI system with corresponding transfer functions is designed, essential to implement 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.\\ \\
Chapter 2 is relying on the textbook ``Digital Signal Processing Fundamentals and Applications 2nd Ed'' by Tan and Jiang \cite{source_dsp1}.
\subsection{Fundamentals of Digital Signal Processing}
Digital Signal Processing (DSP) describes the manipulation of an analog signal trough mathematical approaches after it has been recorded and converted into a digital form. Nearly every part of the modern daily live, be it communication via cellphones, X-Ray imaging or picture editing, is affected by DSP.
\subsection{Fundamentals of digital signal processing}
Digital signal processing describes the manipulation of digital signals on a dedicated processor (often called ``digital signal processor (DSP)'') trough mathematical approaches. Analog signals have to be digitalized before being able to be handled by a DSP. Nearly every part of the modern daily live, be it communication via cellphones, X-Ray imaging or picture editing, is affected by signal processing.
\subsubsection{Signals}
\begin{figure}[H]
\centering
@@ -15,8 +15,8 @@ Digital Signal Processing (DSP) describes the manipulation of an analog signal t
\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 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 distortions of the signal.\\ \\
Throughout this thesis, sampled signals are denoted in lowercase with square brackets (e.g. {x[n]}) to distinguish them from continuous-time signals
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 distortions of the signal.\\ \\
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)}).\\
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
\begin{equation}
@@ -25,7 +25,7 @@ The discrete digital signal can be viewed as a sequence of finite samples with
\end{equation}
where x[n] is the current sample and x[n-1] is the preceding sample.
\subsubsection{Time domain vs. frequency domain}
A signal (either analog or digital) can be displayed and analyzed in two ways: the time spectrum and the frequency spectrum. The time spectrum shows the amplitude of the signal over time - like the sine waves from Figure \ref{fig:fig_interference}. If a fast Fourier transformation (FFT) is applied to the signal in the time spectrum, we receive the same signal in the frequency spectrum, now showing the frequencies present in the signal (refer to Figure \ref{fig:fig_fft}).\\ \\
A signal (either analog or digital) can be displayed and analyzed in two ways: the time spectrum and the frequency spectrum. The time spectrum shows the amplitude of the signal over time - like the sine waves from Figure \ref{fig:fig_interference}. If a Fast Fourier Transformation (FFT) is applied to the signal in the time spectrum, we receive the same signal in the frequency spectrum, now showing the frequencies present in the signal (refer to Figure \ref{fig:fig_fft}).\\ \\
\begin{figure}[H]
\centering
\includegraphics[width=0.8\linewidth]{Bilder/fig_fft.jpg}
@@ -48,9 +48,9 @@ During the description of transfer functions, the term ``filter'' was used but n
\caption{Behavior of a low-pass-filter. \cite{source_dsp_ch2}}
\label{fig:fig_lowpass}
\end{figure}
Examples for an adaptive filter is the Least-Mean-Square-Algorithm used for adaptive noise reduction, which will be introduced in the following chapters.
Examples for an adaptive filter is a digital filter adapted by the Least-Mean-Square algorithm used for adaptive noise reduction, which will be introduced in the following chapters.
\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- and Infinite Impulse Response-filters.
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- and Infinite Impulse Response filters.
\subsubsection{Finite Impulse Response filters}
A Finite Impulse Response (FIR) 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 design is the relatively slow frequency response compared to its Infinite Impulse Response 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
@@ -82,7 +82,8 @@ Figure \ref{fig:fig_iir} visualizes a simple IIR filter with one feedforward coe
\subsection{Introduction to Adaptive Nose Reduction}
\subsubsection{History}
In the beginnings of the 20th century, filter techniques were limited to the use of static filters like low- or highpass filters. The fundamental techniques allow limiting the frequency spectrum, by cutting out certain frequency like high-pitched noises. 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.\\ \\
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 and 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 the 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
@@ -90,11 +91,21 @@ In the beginnings of the 20th century, filter techniques were limited to the use
\caption{Patent of a device for lowering ambient noise to improve intelligence by Lawrence Fogel in 1960 \cite{source_patent}}
\label{fig:fig_patent}
\end{figure}
The final step to real adaptive noise cancellation was made with the introduction of the fundamental Least-Mean-Square (LMS) algorithm in 1960 by Widrow and Hoff, which will be discussed in a later chapter in detail.
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 and time-invariant signals.\\ \\
With the introduction of the fundamental Least-Mean-Square (LMS) algorithm in 1960 by Widrow and Hoff, the last necessary step was made to revolutionize the field of signal filtering. With this mathematical approach it was possible, to leave the area of static filters and active noise cancellation and move to a far more sophisticated signal processing technique - adaptive noise reduction.
\subsubsection{The concept of adaptive filtering}
As already mentioned in the introduction, environmental noise severely degrades cochlear implant user's speech understanding and listening comfort. The traditional concept of static noise reduction, such as fixed filters, are not a feasible solution due to dynamic acoustic conditions where the type, intensity, and spectral composition of noise can change rapidly. Adaptive Noise Reduction addresses this problem by using adaptive filters that can automatically adjust their parameters in real time, continuously optimizing the system's response to changing environments.\\ \\
The practical concepts from the previous chapters were based analog noise suppression, were a microphone measures the noise and a fixed circuit generates the antiphase signal - this means, the system only works in a specified environment with time-invariant disturbing noise and there is no real adaptiveness to it. The concept of adaptive filtering on the other hand is based on the idea, that a digital filter is learning in real-time through a feedback system what frequencies to filter and what no to filter. \\ \\
Figure XXX shows the basic concept of an adaptive filter design, represented through a combination of a feedforward- and feedback filter application.
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 (for example, removing a steady 50 Hz hum or high-frequency hiss). 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 desired signal component from a noisy 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{Most simple variant of ANR}
\label{fig:fig_anr}
\end{figure}
Figure \ref{fig:fig_anr} shows the basic concept of an adaptive filter design, represented through a combination of a feedforward- and feedback filter application.
\subsubsection{Static vs. hybrid filter design}
\subsubsection{Introduction to the Least Mean Square algorithm}
Allowing an automatic adaption of the filter coefficients depending on the surrounding by stepwise minimization of the squared error \\ \\
\subsection{Signal flow diagram showing the origin of the useful signal,

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