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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
@@ -15,17 +15,17 @@ 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
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}
\label{equation1}
x[n] = [x[n-N+1],x[n-N+2],...,x[n-1],x[n]]
\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,