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chapter_02.tex
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@@ -159,55 +159,104 @@ The minimization of the error signal $e[n]$ can by achieved by applying differen
\end{itemize}
As computaional efficiency is a key requirement for the implementation of real-time ANR on a low-power digital signal processor, the Least Mean Squares algorithm is chosen for the minimization of the error signal and therefore will be further explained in the following subchapter.
\subsubsection{Use of Least Mean Squares algorithm in adaptive filtering}
Before the Least Mean Squares algorithm can be explained in detail, the Wiener filter and the concept of gradient descent have to be introduced.
\subsubsection{The Wiener filter and 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. \\ \\
\begin{figure}[H]
\centering
\includegraphics[width=0.7\linewidth]{Bilder/fig_wien.jpg}
\caption{Simple implementation of a Wien 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 square 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 Mean Square Error between a target 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]
\end{equation}
The Wiener filter aims to adjust it´s coefficients to generate a filter output, which resembles the corruption-noise $n[n]$ contained in the target signal $d[n]$ as close as possible. After the filter output is substracted from the target signal, we recvieve the error signal $e[n]$, which represents the cleaned signal $š[n]$ after the noise-component has been removed.
The Wiener filter aims to adjust it´s coefficients to generate a filter output, which resembles the corruption noise signal $n[n]$ contained in the target signal $d[n]$ as close as possible. After the filter output is substracted from the target signal, we recvieve the error signal $e[n]$, which represents the cleaned signal $š[n]$ after the noise component has been removed. For better unsderstanding, a simple Wiener filter with one coefficient shall be illustrated in the following mathematical approach, before the generalization to an n-dimensional filter is made.
\begin{equation}
\label{equation_wien_error}
e[n] = d[n] - y[n] = d[n] - wx[n]
\end{equation}
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 it´s coefficients $w$.
\begin{equation}
\label{equation_wien_error}
\label{equation_j}
J = E(e[n]^2) = E(d^2[n])-2wE(d[n]x[n])+w^2E(x^2[n]) = MSE
\end{equation}
The termns contained in Equation \ref{equation_wien_error} can be further be defined as:
The terms contained in Equation \ref{equation_j} can be further be defined as:
\begin{itemize}
\item $\sigma^2$ = $E(d^2[n])$: The expected value of the squared corrupted target signal - a constant term independent of the filter coefficients $w$.
\item P = $E(d[n]x[n])$: The cross-correlation between the corrupted target signal and the noise reference signal - a measure of how similar these two signals are.
\item R = $E(x^2[n])$: The auto-correlation of the noise reference signal - a measure of the signal's spectral power.
\item R = $E(x^2[n])$: The auto-correlation (or serial-correlation) of the noise reference signal - a measure of the similarity of a signal with it´s delayed copy and therefore of the signal's spectral power.
\end{itemize}
For a large number of samples, Equation {\ref{equation_wien_error}} can therefore be further simplified and written as:
Equation {\ref{equation_j}} can therefore be further simplified and written as:
\begin{equation}
\label{equation_wien_error_final}
\label{equation_j_simple}
J = \sigma^2 - 2wP + w^2R
\end{equation}
As every part of Equation \ref{equation_wien_error_final} beside $w^2$ is constant, the MSE is a quadratic function of the filter coefficients $w$, offering a calculatable minimum. To find this minimum, we can calculate the derivative of $J$ with respect to $w$ and set it to zero:
As every part of Equation \ref{equation_j_simple} beside $w^2$ is constant, $j$ is a quadratic function of the filter coefficients $w$, offering a calculatable minimum. To find this minimum, the derivative of $J$ with respect to $w$ can be calculated and set to zero:
\begin{equation}
\label{equation_gradient_j}
\label{equation_j_gradient}
\frac{dJ}{dw} = -2P + 2wR = 0
\end{equation}
Solving Equation \ref{equation_gradient_j} for $w$ delivers the equation to calculate the optimal coefficients for the Wiener filter::
Solving Equation \ref{equation_j_gradient} for $w$ delivers the equation to calculate the optimal coefficients for the Wiener filter:
\begin{equation}
\label{equation_wien_optimal}
w_{opt} = \frac{P}{R}
\label{equation_w_optimal}
w_{opt} = {P}R^{-1}
\end{equation}
To find the optimal set of coefficients $w$ minimizing the Mean Squared Error $J$, we can apply the concept of gradient descent. Gradient descent is an iterative optimization algorithm used to find the minimum of a function by moving in the direction of the steepest descent, which is determined by the negative gradient of the function. In our case, we want to minimize the MSE $J$ by adjusting the filter coefficients $w$. The update rule for the coefficients using gradient descent can be expressed as:
\begin{figure}[H]
\centering
\includegraphics[width=0.7\linewidth]{Bilder/fig_w_opt.jpg}
\caption{Minimum of the Mean Square Error J located at the optimcal coefficient w* \cite{source_dsp_ch9}}
\label{fig:fig_mse}
\end{figure}
\noindent If the Wiener filter now consists not out of one coefficient, but out of several coefficients, Equation \ref{equation_wien} can be written in a matrix form as
\begin{equation}
\label{equation_wien_matrix}
y[n] = \sum_{k=0}^{M} w_kx[n-k] = \textbf{W}^T\textbf{X}[n]
\end{equation}
where \textbf{X} is the input signal matrix and \textbf{W} the filter coefficient matrix.
\begin{align}
\label{equation_input_vector}
\textbf{X}[n] = [x[n],x[n-1],...,x[n-M]]^T \\
\label{equation_coefficient_vector}
\textbf{W}[n] = [w_0,w_1,...,w_M]^T
\end{align}
Equation \ref{equation_j} can therefore also be rewritten in matrix form to:
\begin{equation}
\label{equation_j_matrix}
J = \sigma^2 - 2\textbf{W}^TP + \textbf{W}^TR\textbf{W}
\end{equation}
After settings the derivative of Equation \ref{equation_j_matrix} to zero and solving for $W$, we receive the optimal filter coefficient matrix:
\begin{equation}
\label{equation_w_optimal_matrix}
\textbf{W}_{opt} = PR^{-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 algortihm aims to to minimize the MSE $J$ 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 \nabla J(w(n))
w(n+1) = w(n) - \mu \frac{dJ}{dw}
\end{equation}
where $\mu$ is the constant step size determining the rate of convergence. Figure \ref{fig:fig_w_opt} visualizes the concept of stepwise minimization of the MSE $J$ using gradient descent. After the derivative of $J$ with respect to $w$ r4aches zero, the optimal coefficients $w_{opt}$ are found and the coefficients are no longer updated.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{Bilder/fig_gradient.jpg}
\caption{Vizualization of the steepest decent alorithm used on the Mean Squared Error. \cite{source_dsp_ch9}}
\label{fig:fig_w_opt}
\end{figure}
\subsubsection{The Least Mean Squares algorithm}
The given approach of the steepest decent algorithm in the subchapter above still involves the calculation of the derivative of the MSE $\frac{dJ}{dw}$, which is also a compuational expensive operation to calulate, as it requires knowledge of the statistical properties of the input signals (cross-correlation P and auto-correlation R). Therefore, in energy critical real-time applications, like the implementation of ANR on a low-power DSP, a sample-based aproxmation in form of a Least Mean Squares (LMS) algorithm is used instead. The LMS algorithm approximates the gradient of the MSE by using the instantaneous estimates of the cross-correlation and auto-correlation. To achieve this, we remove the statistical expectation out of the MSE $J$ and take the derivative to obtain a samplewise approximate of $\frac{dJ}{dw[n]}$.
\begin{align}
\label{equation_j_lms}
J = e[n]^2 = (d[n]-wx[n])^2 \\
\label{equation_j_lms_final}
\frac{dJ}{dw[n]} = 2(d[n]-w[n]x[n])\frac{d(d[n])-w[n]x[n]}{dw[n]} = -2e[n]x[n]
\end{align}
The result of Equation \ref{equation_j_lms_final} can now be inserted into Equation \ref{equation_gradient} to receive the LMS update rule for the filter coefficients:
\begin{equation}
\label{equation_lms}
w[n+1] = w[n] + 2\mu e[n]x[n]
\end{equation}
The LMS algorithm therefore updates the filter coefficients $w[n]$ after every sample by adding a correction term, which is 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 LMS algorithm, the filter coefficients converge towards the optimal values that minimize the mean squared error between the target signal and the filter output. When a predefined acceptable error level is reached, the adaptation process can be stopped to save computing power.\\ \\
\subsection{Signal flow diagram of an implanted cochlear implant system}
\subsection{Derivation of the system’s transfer function based on the problem setup}
\subsection{Example applications and high-level simulations using Python}

13
literature.bib
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@@ -23,21 +23,21 @@
@misc{source_dsp1,
author = {Li Tan, Jean Jiang},
title = {Digital Signal Processing Fundamentals and Applications 2nd Ed},
title = {Digital Signal Processing Fundamentals and Applications 3rd Ed},
howpublished = {Elsevier Inc.},
year = {2013},
note = {ISBN: 978-0-12-415893-1}
}
@misc{source_dsp_ch1,
author = {Li Tan, Jean Jiang},
title = {Digital Signal Processing Fundamentals and Applications 2nd Ed},
title = {Digital Signal Processing Fundamentals and Applications 3rd Ed},
howpublished = {Elsevier Inc.},
year = {2013},
note = {Chapter 1}
}
@misc{source_dsp_ch2,
author = {Li Tan, Jean Jiang},
title = {Digital Signal Processing Fundamentals and Applications 2nd Ed},
title = {Digital Signal Processing Fundamentals and Applications 3rd Ed},
howpublished = {Elsevier Inc.},
year = {2013},
note = {Chapter 2}
@@ -49,3 +49,10 @@
year = {1960},
note = {Pat. Nr. 2966549}
}
@misc{source_dsp_ch9,
author = {Li Tan, Jean Jiang},
title = {Digital Signal Processing Fundamentals and Applications 3rd Ed},
howpublished = {Elsevier Inc.},
year = {2013},
note = {Chapter 9}
}