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\section{Theoretical Background}
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.\\
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 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}.
@@ -15,7 +15,7 @@ Digital signal processing describes the manipulation of digital signals on a ded
\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 distortions of the 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, 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
@@ -41,7 +41,7 @@ When we discuss signals in a mathematical way, we need to explain the term ``tra
\label{fig:fig_transfer}
\end{figure}
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.\\ \\
During the description of transfer functions, the term ``filter'' was used but not yet defined. A filter can be understood as a component in signal processing, designed to modify or extract specific parts of a signal by selectively allowing certain frequency ranges to pass while attenuating others. Filters can be static, meaning they always extract the same portion of a signal, or adaptive, meaning they change their filtering-behavior over time according to their environment. Examples for static filter include low-pass-, high-pass-, band-pass- and band-stop filters, each tailored to isolate or remove particular frequency content (refer to Figure \ref{fig:fig_lowpass}).
During the description of transfer functions, the term ``filter'' was used but not yet defined. A filter can be understood as a component in signal processing, designed to modify or extract specific parts of a signal by selectively allowing certain frequency ranges to pass while attenuating others. Filters can be static, meaning they always extract the same portion of a signal, or adaptive, meaning they change their filtering behavior over time according to their environment. Examples for static filter include low-pass-, high-pass-, band-pass- and band-stop filters, each tailored to isolate or remove particular frequency content (refer to Figure \ref{fig:fig_lowpass}).
\begin{figure}[H]
\centering
\includegraphics[width=0.8\linewidth]{Bilder/fig_lowpass.jpg}
@@ -50,9 +50,9 @@ During the description of transfer functions, the term ``filter'' was used but n
\end{figure}
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, adaptive noise reduction, two very essential filter designs need further explanation - the Finite Impulse Response- and Infinite Impulse Response filter.
\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. \\ \\
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 always 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
\begin{equation}
\label{equation_fir}
@@ -83,28 +83,28 @@ Figure \ref{fig:fig_iir} visualizes a simple IIR filter with one feedforward coe
\subsection{Introduction to Adaptive Nose 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 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}).
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]
\centering
\includegraphics[width=0.8\linewidth]{Bilder/fig_patent.jpg}
\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}
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.\\ \\
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 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}
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.\\ \\
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 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.
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.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\linewidth]{Bilder/fig_anr.jpg}
\caption{Most simple variant of ANR}
\caption{Basic variant of }
\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.
Figure \ref{fig:fig_anr} shows the basic concept of an adaptive filter design, represented through a feedback filter application. The signal sensor aims to recieve the input signal, which consists out of the target signal and the noise signal, whereas the noise sensor aims to recieve (ideally) only the noise signal, which then feeds the adaptive filter. 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. 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.
\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 \\ \\