commit 466fd9b5e85c1daae1be12b34431d4dc56e1d816
Author: jatinchowdhury18 <[email protected]>
Date: Wed, 6 Feb 2019 00:53:13 -0800
Initial commit
Diffstat:
14 files changed, 2161 insertions(+), 0 deletions(-)
diff --git a/.gitignore b/.gitignore
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+# References folder
+Refs
+
+# Tex artifacts
+*.synctex.gz
+*.aux
+*.bbl
+*.blg
+420_paper.log
+*.out
diff --git a/Music_420_Project_Proposal.pdf b/Music_420_Project_Proposal.pdf
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diff --git a/Notes/Continuous Time Considerations.md b/Notes/Continuous Time Considerations.md
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+---
+author: Jatin Chowdhury
+title: Continuous Time Equations for Analog Tape Modeling
+date: 2/1/2019
+---
+
+# Record head
+
+For an instantaneous current I, the magnetic field output of the record head is given as a function of distance along the tape 'x', and depth into the tape 'y' (Karlqvist medium field approximation) [Bertram, page 60]:
+
+$$ H_x(x,y) = \frac{1}{\pi} H_0 \Big(\tan^{-1} \Big(\frac{(g/2) + x}{y} \Big) + \tan^{-1} \Big(\frac{(g/2) - x}{y} \Big) \Big) $$
+$$ H_y(x,y) = \frac{1}{2 \pi} H_0 \ln \Big(\frac{((g/2) - x)^2 + y^2}{((g/2) + x)^2 + y^2} \Big) $$
+
+where $g$ = head gap, and $H_0$ = deep gap field, given by:
+
+$$ H_0 = \frac{NIE}{g} $$
+
+where $N$ = number of turns coils of wire around the head, and $E$ = head efficiency given by:
+
+$$ E = \frac{1}{1 + \frac{l A_g}{\mu_r g} \int_{core} \frac {d \vec{l}}{A(l)}} $$
+
+where $A_g$ is the gap area, $\mu_r$ is the core permeability relative to free space ($\mu_0$), $g$ if the gap width, and $A(l)$ is the cross-sectional area of the core as a function of length.
+
+# Tape Magnetisation
+
+## Deadzone
+
+For low current, the field is insufficient to create a change in magnetisation. For high current the field saturates. The effective field magnetising the tape $H_h$ can be described as follows:
+
+$$ H_h = \begin{cases} 0 & H \le S^* H_c \\
+ H & H > S^* H_c
+\end{cases} $$
+
+where $S^*$ = hysteresis loop squareness, and $H_c$ = coercivity.
+
+## Hysteresis
+
+The magnetostatic field recorded to magnetic tape can be described using a hysteresis loop. A circuit simulation of a hysteresis loop by Martin Holters and Udo Zolzer, using the Jiles-Atherton magnetisation model can be found at http://dafx16.vutbr.cz/dafxpapers/08-DAFx-16_paper_10-PN.pdf. They use the following differential equation to describe magnetisation 'M' as a function of magnetic field 'H':
+
+$$ \frac{dM}{dH} = \frac{(1-c) \delta_M (M_{an} - M)}{(1-c) \delta k - \alpha (M_{an} - M)} + c \frac{dM_{an}}{dH} $$
+
+where $M_{an}$ is the anisotropic magnetisation given by:
+
+$$ M_{an} = M_s L \Big( \frac{H + \alpha M}{a} \Big) $$
+
+where $M_s$ is the magnetisation saturation, and $L$ is the Langevin function.
+
+# Playback head
+
+## Ideal playback voltage
+
+The ideal playback voltage as a function of tape magnetisation is given by [Bertram, page 121]:
+
+$$ V(x) = NWEv \mu_0 \int_{-\infty}^{\infty} dx' \int_{-\delta/2}^{\delta/2} dy' \vec{h}(x' + x, y') \cdot \frac{\vec{M}(x', y')}{dx} $$
+
+where $N$ = number of turns of wire, $W$ = with of the playhead, $E$ = playhead efficiency, $v$ = tape speed. Note that $V(x) = V(vt)$ for constant $v$. $\vec{h}(x, y)$ is defined as:
+
+$$ \vec{h} (x, y) \equiv \frac{\vec{H} (x, y)}{NIE} $$
+
+where $\vec{H} (x, y)$ is the same as for the record head.
+
+## Loss effects
+
+There are several frequency-dependent loss effects associated with playback, described as follows [Kadis, page 126]:
+
+$$ V(x) = V_0(x) [e^{-kd}] \Big[\frac{1 - e^{-k \delta}}{k \delta} \Big] \Big[\frac{\sin (kg /2)}{kg/2} \Big] $$
+
+where $k$ = wave number.
+
+### Spacing Loss
+
+$$ g_s = e^{-kd} $$
+
+where $d$ is the distance between the tape and the playhead.
+
+### Gap Loss
+
+$$ g_g = \frac{\sin (kg /2)}{kg/2} $$
+
+where $g$ is the gap with of the play head.
+
+### Thickness Loss
+
+$$ g_t = \frac{1 - e^{-k \delta}}{k \delta} $$
+
+where $\delta$ is the thickness of the tape.
+
+# Conclusion
+If each of these components is digitized, a digital physical model of the analog tape machine can be constructed.
diff --git a/Notes/Continuous Time Considerations.pdf b/Notes/Continuous Time Considerations.pdf
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diff --git a/Notes/Digitizing Hysteresis.md b/Notes/Digitizing Hysteresis.md
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+---
+author: Jatin Chowdhury
+title: Continuous Time Equations for Analog Tape Modeling
+date: 2/1/2019
+---
+
+## Hysteresis
+
+The magnetostatic field recorded to magnetic tape can be described using a hysteresis loop. A circuit simulation of a hysteresis loop by Martin Holters and Udo Zolzer, using the Jiles-Atherton magnetisation model can be found at http://dafx16.vutbr.cz/dafxpapers/08-DAFx-16_paper_10-PN.pdf. They use the following differential equation to describe magnetisation 'M' as a function of magnetic field 'H':
+
+$$ \frac{dM}{dH} = \frac{(1-c) \delta_M (M_{an} - M)}{(1-c) \delta k - \alpha (M_{an} - M)} + c \frac{dM_{an}}{dH} $$
+
+where $M_{an}$ is the anisotropic magnetisation given by:
+
+$$ M_{an} = M_s L \Big( \frac{H + \alpha M}{a} \Big) $$
+
+where $M_s$ is the magnetisation saturation, and $L$ is the Langevin function:
+
+$$ L(x) = \coth (x) - \frac{1}{x} $$
+$$ L'(x) = \frac{1}{x^2} - \coth^2(x) + 1 $$
+$$ L''(x) = 2 \coth(x) \cdot (\coth^2(x) - 1) - \frac{2}{x^3} $$
+
+Let $Q(t) = \frac{H + \alpha M}{a}$
+
+Differentiating, we get:
+
+$$ \frac{dM}{dt} = \frac{(1-c) \delta_M (M_sL(Q) - M)}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{dH}{dt} + c \frac{M_s}{a} \Big(\frac{dH}{dt} + \alpha \frac{dM}{dt} \Big) L'(Q) $$
+
+(wrong see paper)
+$$
+\frac{d^2 M}{dt^2} = \frac{(1-c) \delta_M (M_sL(Q) - M)}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{d^2H}{dt^2} +
+\frac{(1-c) \delta_M (M_sL'(Q) - \dot{M})}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{dH}{dt} +
+\frac{(1-c) \delta_M (M_sL(Q) - M)(-\alpha (M_sL'(Q) - \dot{M}))}{((1-c) \delta k - \alpha (M_sL(Q) - M))^2} \frac{dH}{dt} +
+c \frac{M_s}{a} \Big(\frac{dH}{dt} c \frac{M_s}{a} (\frac{dH}{dt} + \alpha \frac{dM}{dt}) L''(Q) + \frac{d^2H}{dt^2} L'(Q) + \alpha \frac{dM}{dt} c \frac{M_s}{a} (\frac{dH}{dt} + \alpha \frac{dM}{dt}) L''(Q) + \alpha \frac{d^2M}{dt^2} L'(Q) \Big)
+$$
+
+Simplified (wrong see paper):
+$$ \frac{d^2 M}{dt^2} = \frac{
+\frac{(1-c) \delta_M (M_sL(Q) - M)}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{d^2H}{dt^2} +
+\frac{(1-c) \delta_M (M_sL'(Q) - \dot{M})}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{dH}{dt} +
+\frac{(1-c) \delta_M (M_sL(Q) - M)(-\alpha (M_sL'(Q) - \dot{M}))}{((1-c) \delta k - \alpha (M_sL(Q) - M))^2} \frac{dH}{dt} +
+c \frac{M_s}{a} \Big(\frac{d^2H}{dt^2} L'(Q) + c \frac{M_s}{a} L''(Q) (\dot{H} + \alpha \dot{M})^2 \Big)}
+{1 - c \alpha \frac{M_s}{a} L'(Q)}
+$$
+
+
+$$ \frac{dM}{dt} = \frac{\frac{(1-c) \delta_M (M_sL(Q) - M)}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{dH}{dt} + c \frac{M_s}{a} \frac{dH}{dt} L'(Q)}{1 - c \alpha \frac{M_s}{a} L'(Q)} = f(t, M, \vec{u}) $$
+
+where $\vec{u} = \begin{bmatrix}
+H \\
+\dot{H} \\
+\ddot{H}
+\end{bmatrix}$
+
+Using trapezoidal rule:
+
+$$ \dot{\hat{H}}(n) = 2 \frac{\hat{H}(n) - \hat{H}(n-1)}{T} - \dot{\hat{H}}(n-1) $$
+
+and similar for $\ddot{\hat{H}}$. Now, using the semi-implicit trapezoidal rule [Yeh]:
+
+$$ \hat{M}(n) = \hat{M}(n-1) + \frac{T}{2} \frac{f[n, \hat{M}(n-1), \vec{u}(n)] + f[n-1, \hat{M}(n-1), \vec{u}(n-1)]}{1 - \frac{T}{2}\ddot{\hat{M}}(n-1)} $$
diff --git a/Notes/Digitizing Hysteresis.pdf b/Notes/Digitizing Hysteresis.pdf
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diff --git a/Paper/420_paper.pdf b/Paper/420_paper.pdf
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diff --git a/Paper/420_paper.tex b/Paper/420_paper.tex
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+% !TeX program = pdflatex
+% !BIB program = bibtex
+% Template LaTeX file for DAFx-19 papers
+%
+% To generate the correct references using BibTeX, run
+% latex, bibtex, latex, latex
+% modified...
+% - from DAFx-00 to DAFx-02 by Florian Keiler, 2002-07-08
+% - from DAFx-02 to DAFx-03 by Gianpaolo Evangelista
+% - from DAFx-05 to DAFx-06 by Vincent Verfaille, 2006-02-05
+% - from DAFx-06 to DAFx-07 by Vincent Verfaille, 2007-01-05
+% and Sylvain Marchand, 2007-01-31
+% - from DAFx-07 to DAFx-08 by Henri Penttinen, 2007-12-12
+% and Jyri Pakarinen 2008-01-28
+% - from DAFx-08 to DAFx-09 by Giorgio Prandi, Fabio Antonacci 2008-10-03
+% - from DAFx-09 to DAFx-10 by Hannes Pomberger 2010-02-01
+% - from DAFx-10 to DAFx-12 by Jez Wells 2011
+% - from DAFx-12 to DAFx-14 by Sascha Disch 2013
+% - from DAFx-15 to DAFx-16 by Pavel Rajmic 2015
+% - from DAFx-16 to DAFx-17 by Brian Hamilton 2016
+% - from DAFx-18 to DAFx-19 by Dave Moffat 2019
+%
+% Template with hyper-references (links) active after conversion to pdf
+% (with the distiller) or if compiled with pdflatex.
+%
+% 20060205: added package 'hypcap' to correct hyperlinks to figures and tables
+% use of \papertitle and \paperauthorA, etc for same title in PDF and Metadata
+%
+% 1) Please compile using latex or pdflatex.
+% 2) If using pdflatex, you need your figures in a file format other than eps! e.g. png or jpg is working
+% 3) Please use "paperftitle" and "pdfauthor" definitions below
+
+%------------------------------------------------------------------------------------------
+% ! ! ! ! ! ! ! ! ! ! ! ! user defined variables ! ! ! ! ! ! ! ! ! ! ! ! ! !
+% Please use these commands to define title and author(s) of the paper:
+\def\papertitle{Real-time Physical Model for Sony TC-260 Tape Machine}
+\def\paperauthorA{Jatin Chowdhury}
+
+% Authors' affiliations have to be set below
+
+%------------------------------------------------------------------------------------------
+\documentclass[twoside,a4paper]{article}
+\usepackage{dafx_19}
+\usepackage{amsmath,amssymb,amsfonts,amsthm}
+\usepackage{euscript}
+\usepackage[latin1]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{ifpdf}
+
+\usepackage[english]{babel}
+\usepackage{caption}
+\usepackage{subfig} % or can use subcaption package
+\usepackage{color}
+
+\setcounter{page}{1}
+\ninept
+
+\usepackage{times}
+% Saves a lot of ouptut space in PDF... after conversion with the distiller
+% Delete if you cannot get PS fonts working on your system.
+
+% pdf-tex settings: detect automatically if run by latex or pdflatex
+\newif\ifpdf
+\ifx\pdfoutput\relax
+\else
+ \ifcase\pdfoutput
+ \pdffalse
+ \else
+ \pdftrue
+\fi
+
+\ifpdf % compiling with pdflatex
+ \usepackage[pdftex,
+ pdftitle={\papertitle},
+ pdfauthor={\paperauthorA},
+ colorlinks=false, % links are activated as colror boxes instead of color text
+ bookmarksnumbered, % use section numbers with bookmarks
+ pdfstartview=XYZ % start with zoom=100% instead of full screen; especially useful if working with a big screen :-)
+ ]{hyperref}
+ \pdfcompresslevel=9
+ \usepackage[pdftex]{graphicx}
+ \usepackage[figure,table]{hypcap}
+\else % compiling with latex
+ \usepackage[dvips]{epsfig,graphicx}
+ \usepackage[dvips,
+ colorlinks=false, % no color links
+ bookmarksnumbered, % use section numbers with bookmarks
+ pdfstartview=XYZ % start with zoom=100% instead of full screen
+ ]{hyperref}
+ % hyperrefs are active in the pdf file after conversion
+ \usepackage[figure,table]{hypcap}
+\fi
+\usepackage{cleveref}
+
+\title{\papertitle}
+
+\affiliation{
+\paperauthorA \,}
+{\href{http://ccrma.stanford.edu}{Center for Computer Research in Music and Acoustics} \\ Stanford University \\ Palo Alto, CA \\ {\tt \href{mailto:[email protected]}{[email protected]}}}
+
+\begin{document}
+% more pdf-tex settings:
+\ifpdf % used graphic file format for pdflatex
+ \DeclareGraphicsExtensions{.png,.jpg,.pdf}
+\else % used graphic file format for latex
+ \DeclareGraphicsExtensions{.eps}
+\fi
+
+\maketitle
+
+\section{Abstract}
+For decades, analog magnetic tape recording was the most popular
+method for recording music, but has been replaced over the past 30 years first by
+DAT tape, and then by DAWs \cite{Kadis}. Despite being replaced
+by higher quality technology,
+many have sought to recreate a "tape" sound through digital effects, despite
+the distortion, tape "hiss", and other oddities analog tape produced.
+To quote Brian Eno, "Whatever you now find weird, ugly, uncomfortable, and nasty
+about a new medium will surely become its signature. CD distortion, the jitteriness
+of digital video, the crap sound of 8-bit - all of these will be cherished
+and emulated as soon as they can be avoided." \cite{Eno}. The following describes
+a physical model of an analog tape machine, starting from basic physical
+principles.
+
+\section{Continuous Time System}
+Audio recorded to and played back from a tape machine can be thought of as going
+through three distinct processors: the record head, tape magnetisation, and the play
+head.
+
+\subsection{The Record Head}
+For an instantaneous input current $I(t)$, the magnetic field output of the record
+head is given as a function of distance along the tape 'x', and depth into
+the tape 'y' (Karlqvist medium field approximation) \cite{1994tmr..book.....B}.
+
+\begin{multline}
+ H_x(x,y) = \frac{1}{\pi} H_0 \Big(\tan^{-1} \Big(\frac{(g/2) + x}{y} \Big) + \\
+ \tan^{-1} \Big(\frac{(g/2) - x}{y} \Big) \Big)
+ \label{eq1}
+\end{multline}
+\begin{equation}
+ H_y(x,y) = \frac{1}{2 \pi} H_0 \ln \Big(\frac{((g/2) - x)^2 + y^2}{((g/2) + x)^2 + y^2} \Big)
+ \label{eq2}
+\end{equation}
+%
+where $g$ = head gap, and $H_0$ = deep gap field, given by:
+
+\begin{equation}
+ H_0 = \frac{NEI}{g}
+\end{equation}
+%
+where $N$ = number of turns coils of wire around the head, and $E$ = head
+efficiency given by:
+
+\begin{equation}
+ E = \frac{1}{1 + \frac{l A_g}{\mu_r g} \int_{core} \frac {d \vec{l}}{A(l)}}
+\end{equation}
+%
+where $A_g$ is the gap area, $\mu_r$ is the core permeability relative to
+free space ($\mu_0$), $g$ if the gap width, and $A(l)$ is the cross-sectional
+area of the core as a function of length.
+
+\subsection{Tape Magnetisation}
+The magnetic field being recorded to tape can be described using
+a hysteresis loop, as follows \cite{1994tmr..book.....B}:
+
+\begin{equation}
+ \vec{M}(x,y) = F_{Loop}(\vec{H}(x,y))
+\end{equation}
+%
+where $F_{Loop}$ is a generalized hysteresis function.
+\newline\newline
+Using the Jiles-Atherton magnetisation model, the following
+differential equation describes magnetisation 'M' as a function
+of magnetic field 'H' \cite{Hysteresis}:
+
+\begin{equation}
+ \frac{dM}{dH} = \frac{(1-c) \delta_M (M_{an} - M)}{(1-c) \delta k - \alpha (M_{an} - M)} + c \frac{dM_{an}}{dH}
+ \label{eq5}
+\end{equation}
+%
+where $c$ is the ration of normal and anhysteric initial susceptibilities,
+$k$ is a measure of the width of the hysteresis
+loop, $\alpha$ is a mean field parameter, representing inter-domain
+coupling, and
+
+\begin{equation}
+ \delta = \begin{cases}
+ 1 & \text{if H is increasing} \\
+ -1 & \text{if H is decreasing}
+ \end{cases}
+\end{equation}
+\begin{equation}
+ \delta_M = \begin{cases}
+ 1 & \text{if $\delta$ and $M_{an} - M$ have the same sign} \\
+ 0 & \text{otherwise}
+ \end{cases}
+\end{equation}
+%
+$M_{an}$ is the anisotropic magnetisation given by:
+
+\begin{equation}
+ M_{an} = M_s L \Big( \frac{H + \alpha M}{a} \Big)
+\end{equation}
+%
+where $M_s$ is the magnetisation saturation, $a$ characterizes the shape
+of the anhysteric magnetisation and $L$ is the Langevin function:
+
+\begin{equation}
+ L(x) = \coth (x) - \frac{1}{x}
+\end{equation}
+
+\subsection{Play Head}
+\subsubsection{Ideal Playback Voltage}
+The ideal playback voltage as a function of tape magnetisation is given by
+\cite{1994tmr..book.....B}:
+
+\begin{equation}
+ V(x) = NWEv \mu_0 \int_{-\infty}^{\infty} dx' \int_{-\delta/2}^{\delta/2} dy' \vec{h}(x' + x, y') \cdot \frac{\vec{M}(x', y')}{dx}
+ \label{eq11}
+\end{equation}
+%
+where $N$ = number of turns of wire, $W$ = width of the playhead, $E$ = playhead
+efficiency, $v$ = tape speed, and $\mu_0$ is the permeability of free space.
+Note that $V(x) = V(vt)$ for constant $v$. $\vec{h}(x, y)$ is defined as:
+
+\begin{equation}
+ \vec{h} (x, y) \equiv \frac{\vec{H} (x, y)}{NIE}
+ \label{eq12}
+\end{equation}
+%
+where $\vec{H} (x, y)$ can be calculated by \cref{eq1,eq2}.
+
+\subsubsection{Loss Effects}
+There are several frequency-dependent loss effects associated with playback,
+described as follows \cite{Kadis}:
+
+\begin{equation}
+ V(x) = V_0(x) [e^{-kd}] \Big[\frac{1 - e^{-k \delta}}{k \delta} \Big] \Big[\frac{\sin (kg /2)}{kg/2} \Big]
+\end{equation}
+%
+where $k$ = wave number, $d$ is the distance between the tape and the playhead,
+$g$ is the gap width of the play head, and $\delta$ is the thickness of the tape.
+
+\section{Digitizing the System}
+\subsection{Record Head}
+For simplicity, let us assume,
+\begin{equation}
+ \vec{H}(x,y,t) = \vec{H}(0,0,t)
+\end{equation}
+%
+In this case $H_y \equiv 0$, and $H_x \equiv H_0$. Thus,
+\begin{equation}
+ H(t) = \frac{NEI(t)}{g}
+ \label{eq15}
+\end{equation}
+%
+or,
+\begin{equation}
+ \hat{H}(n) = \frac{NE\hat{I}(n)}{g}
+\end{equation}
+
+\subsection{Hysteresis}
+Beginning from \cref{eq5}, we can find the derivative of $M$ w.r.t. time,
+as in \cite{Hysteresis}:
+\begin{multline}
+ \frac{dM}{dt} = \frac{(1-c) \delta_M (M_sL(Q) - M)}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{dH}{dt} + \\
+ c \frac{M_s}{a} \Big(\frac{dH}{dt} + \alpha \frac{dM}{dt} \Big) L'(Q)
+\end{multline}
+%
+where $Q = \frac{H + \alpha M}{a}$, and
+
+\begin{equation}
+ L'(x) = \frac{1}{x^2} - \coth^2(x) + 1
+\end{equation}
+%
+Taking the second derivative, we find:
+\begin{multline}
+ \frac{d^2 M}{dt^2} = \frac{(1-c) \delta_M (M_sL(Q) - M)}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{d^2H}{dt^2}\\
+ + \frac{(1-c) \delta_M (c \frac{M_s}{a} (\frac{dH}{dt} + \alpha \frac{dM}{dt}) L'(Q) - \dot{M})}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{dH}{dt}\\
+ + \frac{(1-c) \delta_M (M_sL(Q) - M)(-\alpha (c \frac{M_s}{a} (\frac{dH}{dt} + \alpha \frac{dM}{dt}) L'(Q) - \dot{M}))}{((1-c) \delta k - \alpha (M_sL(Q) - M))^2} \frac{dH}{dt}\\
+ + c \frac{M_s}{a} \Big(\frac{dH}{dt} c \frac{M_s}{a} (\frac{dH}{dt} + \alpha \frac{dM}{dt}) L''(Q) + \frac{d^2H}{dt^2} L'(Q) + \alpha \frac{dM}{dt} c \frac{M_s}{a} (\frac{dH}{dt} + \alpha \frac{dM}{dt}) L''(Q) + \alpha \frac{d^2M}{dt^2} L'(Q) \Big)
+\end{multline}
+%
+where
+\begin{equation}
+ L''(x) = 2 \coth(x) \cdot (\coth^2(x) - 1) - \frac{2}{x^3}
+\end{equation}
+%
+Solving for $\frac{dM}{dt}$ and $\frac{d^2M}{dt^2}$, we get:
+\begin{equation}
+ \frac{dM}{dt} = \frac{\frac{(1-c) \delta_M (M_sL(Q) - M)}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{dH}{dt} + c \frac{M_s}{a} \frac{dH}{dt} L'(Q)}{1 - c \alpha \frac{M_s}{a} L'(Q)} = f(t, M, \vec{u})
+\end{equation}
+\begin{multline}
+ \frac{d^2 M}{dt^2} = \frac{1}{1 - c \alpha \frac{M_s}{a} L'(Q)} \Bigg[
+\frac{(1-c) \delta_M (M_sL(Q) - M)}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{d^2H}{dt^2}\\
++ \frac{(1-c) \delta_M (c \frac{M_s}{a} (\frac{dH}{dt} + \alpha \frac{dM}{dt}) L'(Q) - \dot{M})}{(1-c) \delta k - \alpha (M_sL(Q) - M)} \frac{dH}{dt}\\
++ \frac{(1-c) \delta_M (M_sL(Q) - M)(-\alpha (c \frac{M_s}{a} (\frac{dH}{dt} + \alpha \frac{dM}{dt}) L'(Q) - \dot{M}))}{((1-c) \delta k - \alpha (M_sL(Q) - M))^2} \frac{dH}{dt}\\
++ c \frac{M_s}{a} \Big(\frac{d^2H}{dt^2} L'(Q) + c \frac{M_s}{a} L''(Q) (\dot{H} + \alpha \dot{M})^2 \Big) \Bigg]
+\end{multline}
+%
+where $\vec{u} = \begin{bmatrix}
+ H \\
+ \dot{H}
+ \end{bmatrix}$.
+\newline\newline
+Using the trapezoidal rule:
+
+\begin{equation}
+ \dot{\hat{H}}(n) = 2 \frac{\hat{H}(n) - \hat{H}(n-1)}{T} - \dot{\hat{H}}(n-1)
+\end{equation}
+\begin{equation}
+ \ddot{\hat{H}}(n) = 2 \frac{\dot{\hat{H}}(n) - \dot{\hat{H}}(n-1)}{T} - \ddot{\hat{H}}(n-1)
+\end{equation}
+%
+Then, using the semi-implicit trapezoidal rule for non-linear differential
+equations \cite{Yeh}.
+\begin{multline}
+ \hat{M}(n) = \hat{M}(n-1)\\
+ + \frac{T}{2} \frac{f[n, \hat{M}(n-1), \hat{\vec{u}}(n)] + f[n-1, \hat{M}(n-1), \hat{\vec{u}}(n-1)]}{1 - \frac{T}{2}\ddot{\hat{M}}(n-1)}
+\end{multline}
+
+\subsection{Play Head}
+By combining \cref{eq11} with \cref{eq12,eq15}, we get:
+\begin{equation}
+ V(t) = NWEv \mu_0 g M(t)
+\end{equation}
+%
+or,
+\begin{equation}
+ \hat{V}(n) = NWEv \mu_0 g \hat{M}(t)
+\end{equation}
+%
+
+%\newpage
+\nocite{*}
+\bibliographystyle{IEEEbib}
+\bibliography{references}
+
+\end{document}
diff --git a/Paper/IEEEbib.bst b/Paper/IEEEbib.bst
@@ -0,0 +1,1021 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IEEE.bst %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Bibliography Syle file for articles according to IEEE instructions
+% [email protected] <22-JUN-93>
+% modified from unsrt.bib. Contributions by Richard H. Roy
+
+ENTRY
+ { address
+ author
+ booktitle
+ chapter
+ edition
+ editor
+ howpublished
+ institution
+ journal
+ key
+ month
+ note
+ number
+ organization
+ pages
+ publisher
+ school
+ series
+ title
+ type
+ volume
+ year
+ }
+ {}
+ { label }
+
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+
+FUNCTION {init.state.consts}
+{ #0 'before.all :=
+ #1 'mid.sentence :=
+ #2 'after.sentence :=
+ #3 'after.block :=
+}
+
+STRINGS { s t }
+
+FUNCTION {output.nonnull}
+{ 's :=
+ output.state mid.sentence =
+ { ", " * write$ }
+ { output.state after.block =
+% next line commented out by rhr and changed to write comma
+% { add.period$ write$
+ { ", " * write$
+ newline$
+ "\newblock " write$
+ }
+ { output.state before.all =
+ 'write$
+ { add.period$ " " * write$ }
+ if$
+ }
+ if$
+ mid.sentence 'output.state :=
+ }
+ if$
+ s
+}
+
+FUNCTION {output}
+{ duplicate$ empty$
+ 'pop$
+ 'output.nonnull
+ if$
+}
+
+FUNCTION {output.check}
+{ 't :=
+ duplicate$ empty$
+ { pop$ "empty " t * " in " * cite$ * warning$ }
+ 'output.nonnull
+ if$
+}
+
+FUNCTION {output.bibitem}
+{ newline$
+ "\bibitem{" write$
+ cite$ write$
+ "}" write$
+ newline$
+ ""
+ before.all 'output.state :=
+}
+
+FUNCTION {fin.entry}
+{ add.period$
+ write$
+ newline$
+}
+
+% 5/24/89 rhr
+% modified fin.entry function - prints note field after body of entry
+%FUNCTION {fin.entry}
+%{ add.period$
+% note empty$
+% 'write$
+% { "\par\bgroup\parindent=0em " * annote * "\par\egroup " * write$
+% }
+% if$
+% newline$
+%}
+
+FUNCTION {new.block}
+{ output.state before.all =
+ 'skip$
+ { after.block 'output.state := }
+ if$
+}
+
+% new block without terminating last block with a comma
+FUNCTION {new.ncblock}
+{
+ write$
+ newline$
+ "\newblock "
+ before.all 'output.state :=
+}
+
+FUNCTION {new.nccont}
+{
+ write$
+ " "
+ before.all 'output.state :=
+}
+
+FUNCTION {new.sentence}
+{ output.state after.block =
+ 'skip$
+ { output.state before.all =
+ 'skip$
+ { after.sentence 'output.state := }
+ if$
+ }
+ if$
+}
+
+FUNCTION {not}
+{ { #0 }
+ { #1 }
+ if$
+}
+
+FUNCTION {and}
+{ 'skip$
+ { pop$ #0 }
+ if$
+}
+
+FUNCTION {or}
+{ { pop$ #1 }
+ 'skip$
+ if$
+}
+
+FUNCTION {new.block.checka}
+{ empty$
+ 'skip$
+ 'new.block
+ if$
+}
+
+FUNCTION {new.block.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.block
+ if$
+}
+
+FUNCTION {new.sentence.checka}
+{ empty$
+ 'skip$
+ 'new.sentence
+ if$
+}
+
+FUNCTION {new.sentence.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.sentence
+ if$
+}
+
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+ { pop$ "" }
+ 'skip$
+ if$
+}
+
+FUNCTION {emphasize}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "{\em " swap$ * "}" * }
+ if$
+}
+
+FUNCTION {boldface}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "{\bf " swap$ * "}" * }
+ if$
+}
+
+%FUNCTION {boldface}
+%{ 's swap$ :=
+% s "" =
+% { "" }
+% { "{\bf " s * "}" * }
+% if$
+%}
+%
+INTEGERS { nameptr namesleft numnames }
+
+FUNCTION {format.names}
+{ 's :=
+ #1 'nameptr :=
+ s num.names$ 'numnames :=
+ numnames 'namesleft :=
+ { namesleft #0 > }
+ { s nameptr "{ff~}{vv~}{ll}{, jj}" format.name$ 't :=
+ nameptr #1 >
+ { namesleft #1 >
+ { ", " * t * }
+ { numnames #2 >
+ { "," * }
+ 'skip$
+ if$
+ t "others" =
+ { " et~al." * }
+ { " and " * t * }
+ if$
+ }
+ if$
+ }
+ 't
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+}
+
+FUNCTION {format.authors}
+{ author empty$
+ { "" }
+ { author format.names }
+ if$
+}
+
+FUNCTION {format.editors}
+{ editor empty$
+ { "" }
+ { editor format.names
+ editor num.names$ #1 >
+ { ", Eds." * }
+ { ", Ed." * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.title}
+{ title empty$
+ { "" }
+ { "``" title "t" change.case$ * }
+ if$
+}
+
+FUNCTION {n.dashify}
+{ 't :=
+ ""
+ { t empty$ not }
+ { t #1 #1 substring$ "-" =
+ { t #1 #2 substring$ "--" = not
+ { "--" *
+ t #2 global.max$ substring$ 't :=
+ }
+ { { t #1 #1 substring$ "-" = }
+ { "-" *
+ t #2 global.max$ substring$ 't :=
+ }
+ while$
+ }
+ if$
+ }
+ { t #1 #1 substring$ *
+ t #2 global.max$ substring$ 't :=
+ }
+ if$
+ }
+ while$
+}
+
+FUNCTION {format.date}
+{ year empty$
+ { month empty$
+ { "" }
+ { "there's a month but no year in " cite$ * warning$
+ month
+ }
+ if$
+ }
+ { month empty$
+ 'year
+ { month " " * year * }
+ if$
+ }
+ if$
+}
+
+% FUNCTION {format.date}
+% { year empty$
+% 'year
+% { " " year * }
+% if$
+% }
+
+FUNCTION {format.btitle}
+{ title emphasize
+}
+
+FUNCTION {tie.or.space.connect}
+{ duplicate$ text.length$ #3 <
+ { "~" }
+ { " " }
+ if$
+ swap$ * *
+}
+
+FUNCTION {either.or.check}
+{ empty$
+ 'pop$
+ { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+ if$
+}
+
+FUNCTION {format.bvolume}
+{ volume empty$
+ { "" }
+ { "vol." volume tie.or.space.connect
+ series empty$
+ 'skip$
+ { " of " * series emphasize * }
+ if$
+ "volume and number" number either.or.check
+ }
+ if$
+}
+
+FUNCTION {format.number.series}
+{ volume empty$
+ { number empty$
+ { series field.or.null }
+ { output.state mid.sentence =
+ { "number" }
+ { "Number" }
+ if$
+ number tie.or.space.connect
+ series empty$
+ { "there's a number but no series in " cite$ * warning$ }
+ { " in " * series * }
+ if$
+ }
+ if$
+ }
+ { "" }
+ if$
+}
+
+FUNCTION {format.edition}
+{ edition empty$
+ { "" }
+ { output.state mid.sentence =
+ { edition "l" change.case$ " edition" * }
+ { edition "t" change.case$ " edition" * }
+ if$
+ }
+ if$
+}
+
+INTEGERS { multiresult }
+
+FUNCTION {multi.page.check}
+{ 't :=
+ #0 'multiresult :=
+ { multiresult not
+ t empty$ not
+ and
+ }
+ { t #1 #1 substring$
+ duplicate$ "-" =
+ swap$ duplicate$ "," =
+ swap$ "+" =
+ or or
+ { #1 'multiresult := }
+ { t #2 global.max$ substring$ 't := }
+ if$
+ }
+ while$
+ multiresult
+}
+
+FUNCTION {format.pages}
+{ pages empty$
+ { "" }
+ { pages multi.page.check
+ { "pp." pages n.dashify tie.or.space.connect }
+ { "p." pages tie.or.space.connect }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.vol.num.pages}
+{
+volume empty$
+ {"" }
+ {"vol. " volume *}
+if$
+number empty$
+ 'skip$
+ {", no. " number * *}
+if$
+pages empty$
+ 'skip$
+ { duplicate$ empty$
+ { pop$ format.pages }
+ { ", pp. " * pages n.dashify * }
+ if$
+ }
+if$
+}
+
+%FUNCTION {format.vol.num.pages}
+%%boldface added 3/17/87 rhr
+%{ volume field.or.null boldface
+% number empty$
+% 'skip$
+% { "(" number * ")" * *
+% volume empty$
+% { "there's a number but no volume in " cite$ * warning$ }
+% 'skip$
+% if$
+% }
+% if$
+% pages empty$
+% 'skip$
+% { duplicate$ empty$
+% { pop$ format.pages }
+% { ":" * pages n.dashify * }
+% if$
+% }
+% if$
+%}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+ 'format.pages
+ { type empty$
+ { "chapter" }
+ { type "l" change.case$ }
+ if$
+ chapter tie.or.space.connect
+ pages empty$
+ 'skip$
+ { ", " * format.pages * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.in.ed.booktitle}
+{ booktitle empty$
+ { "" }
+ { editor empty$
+ { "in " booktitle emphasize * }
+ { "in " booktitle emphasize * ", " * format.editors * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {empty.misc.check}
+{ author empty$ title empty$ howpublished empty$
+ month empty$ year empty$ note empty$
+ and and and and and
+ { "all relevant fields are empty in " cite$ * warning$ }
+ 'skip$
+ if$
+}
+
+FUNCTION {format.thesis.type}
+{ type empty$
+ 'skip$
+ { pop$
+ type "t" change.case$
+ }
+ if$
+}
+
+FUNCTION {format.tr.number}
+{ type empty$
+ { "Tech. {R}ep." }
+ 'type
+ if$
+ number empty$
+ { "t" change.case$ }
+ { number tie.or.space.connect }
+ if$
+}
+
+FUNCTION {format.article.crossref}
+{ key empty$
+ { journal empty$
+ { "need key or journal for " cite$ * " to crossref " * crossref *
+ warning$
+ ""
+ }
+ { "In {\em " journal * "\/}" * }
+ if$
+ }
+ { "In " key * }
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {format.crossref.editor}
+{ editor #1 "{vv~}{ll}" format.name$
+ editor num.names$ duplicate$
+ #2 >
+ { pop$ " et~al." * }
+ { #2 <
+ 'skip$
+ { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+ { " et~al." * }
+ { " and " * editor #2 "{vv~}{ll}" format.name$ * }
+ if$
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.book.crossref}
+{ volume empty$
+ { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+ "In "
+ }
+ { "vol." volume tie.or.space.connect
+ " of " *
+ }
+ if$
+ editor empty$
+ editor field.or.null author field.or.null =
+ or
+ { key empty$
+ { series empty$
+ { "need editor, key, or series for " cite$ * " to crossref " *
+ crossref * warning$
+ "" *
+ }
+ { "{\em " * series * "\/}" * }
+ if$
+ }
+ { key * }
+ if$
+ }
+ { format.crossref.editor * }
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {format.incoll.inproc.crossref}
+{ editor empty$
+ editor field.or.null author field.or.null =
+ or
+ { key empty$
+ { booktitle empty$
+ { "need editor, key, or booktitle for " cite$ * " to crossref " *
+ crossref * warning$
+ ""
+ }
+ { "In {\em " booktitle * "\/}" * }
+ if$
+ }
+ { "In " key * }
+ if$
+ }
+ { "In " format.crossref.editor * }
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {article}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title ",''" * "title" output.check
+ new.ncblock
+ crossref missing$
+ { journal emphasize "journal" output.check
+ format.vol.num.pages output
+ format.date "year" output.check
+ }
+ { format.article.crossref output.nonnull
+ format.pages output
+ }
+ if$
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {book}
+{ output.bibitem
+ author empty$
+ { format.editors "author and editor" output.check }
+ { format.authors output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ new.block
+ format.btitle "title" output.check
+ crossref missing$
+ { format.bvolume output
+ new.block
+ format.number.series output
+ new.sentence
+ publisher "publisher" output.check
+ address output
+ }
+ { new.block
+ format.book.crossref output.nonnull
+ }
+ if$
+ format.edition output
+ format.date "year" output.check
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {booklet}
+{ output.bibitem
+ format.authors output
+ new.block
+ format.title ",''" * "title" output.check
+ new.nccont
+ howpublished address new.block.checkb
+ howpublished output
+ address output
+ format.date output
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {inbook}
+{ output.bibitem
+ author empty$
+ { format.editors "author and editor" output.check }
+ { format.authors output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ new.block
+ format.btitle "title" output.check
+ crossref missing$
+ { format.bvolume output
+ format.chapter.pages "chapter and pages" output.check
+ new.block
+ format.number.series output
+ new.sentence
+ publisher "publisher" output.check
+ address output
+ }
+ { format.chapter.pages "chapter and pages" output.check
+ new.block
+ format.book.crossref output.nonnull
+ }
+ if$
+ format.edition output
+ format.date "year" output.check
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title ",''" * "title" output.check
+ new.ncblock
+ crossref missing$
+ { format.in.ed.booktitle "booktitle" output.check
+ format.bvolume output
+ format.number.series output
+ format.chapter.pages output
+ new.sentence
+ publisher "publisher" output.check
+ address output
+ format.edition output
+ format.date "year" output.check
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ format.chapter.pages output
+ }
+ if$
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {inproceedings}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title ",''" * "title" output.check
+ new.ncblock
+ crossref missing$
+ { format.in.ed.booktitle "booktitle" output.check
+ address empty$
+ { organization publisher new.sentence.checkb
+ organization output
+ format.date "year" output.check
+ }
+ { address output.nonnull
+ format.date "year" output.check
+ organization output
+ }
+ if$
+ format.bvolume output
+ format.number.series output
+ format.pages output
+ publisher output
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ format.pages output
+ }
+ if$
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {conference} { inproceedings }
+
+FUNCTION {manual}
+{ output.bibitem
+ author empty$
+ { organization empty$
+ 'skip$
+ { organization output.nonnull
+ address output
+ }
+ if$
+ }
+ { format.authors output.nonnull }
+ if$
+ new.block
+ format.btitle "title" output.check
+ author empty$
+ { organization empty$
+ { address new.block.checka
+ address output
+ }
+ 'skip$
+ if$
+ }
+ { organization address new.block.checkb
+ organization output
+ address output
+ }
+ if$
+ format.edition output
+ format.date output
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title ",''" * "title" output.check
+ new.ncblock
+ "M.S. thesis" format.thesis.type output.nonnull
+ school "school" output.check
+ address output
+ format.date "year" output.check
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {misc}
+{ output.bibitem
+ format.authors output
+ title howpublished new.block.checkb
+ format.title ",''" * output
+ new.nccont
+ howpublished new.block.checka
+ howpublished output
+ format.date output
+ new.block
+ note output
+ fin.entry
+ empty.misc.check
+}
+
+FUNCTION {phdthesis}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.btitle "title" output.check
+ new.block
+ "Ph.D. thesis" format.thesis.type output.nonnull
+ school "school" output.check
+ address output
+ format.date "year" output.check
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+ editor empty$
+ { organization output }
+ { format.editors output.nonnull }
+ if$
+ new.block
+ format.btitle "title" output.check
+ format.bvolume output
+ format.number.series output
+ address empty$
+ { editor empty$
+ { publisher new.sentence.checka }
+ { organization publisher new.sentence.checkb
+ organization output
+ }
+ if$
+ publisher output
+ format.date "year" output.check
+ }
+ { address output.nonnull
+ format.date "year" output.check
+ new.sentence
+ editor empty$
+ 'skip$
+ { organization output }
+ if$
+ publisher output
+ }
+ if$
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title ",''" * "title" output.check
+ new.ncblock
+ format.tr.number output.nonnull
+ institution "institution" output.check
+ address output
+ format.date "year" output.check
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title ",''" * "title" output.check
+ new.ncblock
+ note "note" output.check
+ format.date output
+ fin.entry
+}
+
+FUNCTION {default.type} { misc }
+
+MACRO {jan} {"Jan."}
+
+MACRO {feb} {"Feb."}
+
+MACRO {mar} {"Mar."}
+
+MACRO {apr} {"Apr."}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"June"}
+
+MACRO {jul} {"July"}
+
+MACRO {aug} {"Aug."}
+
+MACRO {sep} {"Sept."}
+
+MACRO {oct} {"Oct."}
+
+MACRO {nov} {"Nov."}
+
+MACRO {dec} {"Dec."}
+
+MACRO {acmcs} {"ACM Computing Surveys"}
+
+MACRO {acta} {"Acta Informatica"}
+
+MACRO {cacm} {"Communications of the ACM"}
+
+MACRO {ibmjrd} {"IBM Journal of Research and Development"}
+
+MACRO {ibmsj} {"IBM Systems Journal"}
+
+MACRO {ieeese} {"IEEE Transactions on Software Engineering"}
+
+MACRO {ieeetc} {"IEEE Transactions on Computers"}
+
+MACRO {ieeetcad}
+ {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"}
+
+MACRO {ipl} {"Information Processing Letters"}
+
+MACRO {jacm} {"Journal of the ACM"}
+
+MACRO {jcss} {"Journal of Computer and System Sciences"}
+
+MACRO {scp} {"Science of Computer Programming"}
+
+MACRO {sicomp} {"SIAM Journal on Computing"}
+
+MACRO {tocs} {"ACM Transactions on Computer Systems"}
+
+MACRO {tods} {"ACM Transactions on Database Systems"}
+
+MACRO {tog} {"ACM Transactions on Graphics"}
+
+MACRO {toms} {"ACM Transactions on Mathematical Software"}
+
+MACRO {toois} {"ACM Transactions on Office Information Systems"}
+
+MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"}
+
+MACRO {tcs} {"Theoretical Computer Science"}
+
+READ
+
+STRINGS { longest.label }
+
+INTEGERS { number.label longest.label.width }
+
+FUNCTION {initialize.longest.label}
+{ "" 'longest.label :=
+ #1 'number.label :=
+ #0 'longest.label.width :=
+}
+
+FUNCTION {longest.label.pass}
+{ number.label int.to.str$ 'label :=
+ number.label #1 + 'number.label :=
+ label width$ longest.label.width >
+ { label 'longest.label :=
+ label width$ 'longest.label.width :=
+ }
+ 'skip$
+ if$
+}
+
+EXECUTE {initialize.longest.label}
+
+ITERATE {longest.label.pass}
+
+FUNCTION {begin.bib}
+{ preamble$ empty$
+ 'skip$
+ { preamble$ write$ newline$ }
+ if$
+ "\begin{thebibliography}{" longest.label * "}" * write$ newline$
+}
+
+EXECUTE {begin.bib}
+
+EXECUTE {init.state.consts}
+
+ITERATE {call.type$}
+
+FUNCTION {end.bib}
+{ newline$
+ "\end{thebibliography}" write$ newline$
+}
+
+EXECUTE {end.bib}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%% End of IEEE.bst %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
diff --git a/Paper/dafx_19.sty b/Paper/dafx_19.sty
@@ -0,0 +1,432 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% File: dafx_19.sty (LaTeX Document style option "dafx_19")
+%
+% Usage: \documentclass{article}
+% \usepackage{dafx_19}
+%
+% Or for LaTeX 2.09:
+% Usage: \documentstyle[...,dafx_19,...]{article}
+%
+% Purpose:
+%
+% Style file for Conference on Digital Audio Effects (DAFx-19).
+% Modified from waspaa99.sty
+% Features:
+% - correct page size (175mm x 226mm)
+% - use times-roman font as default font (roman font) in order to achieve
+% best quality on paper and screen
+% - twocolumn format
+% - boldfaced, numbered, and centered section headings
+% - correct subsection and subsubsection headings
+% - use \title{xx} for title, will automatically be typeset uppercase (not need for the user to do so)
+%
+% NEW NEW NEW
+%
+% - use \affiliation{author1, author2, ... and authorN}{address} for author(s) <typeset in italics>
+% with a single affiliation (address)
+% - use \affiliationS{author}{address1}{address2} for author(s) <typeset in italics>
+% with double affiliation (address)
+% - use \twoaffiliations{author1}{address1}{author2}{address2}
+% for two (or more) authors with two separate addresses
+% - use \threeaffiliations{author1}{address1}{author2}{address2}{author3}{address3}
+% for three (or more) authors with three separate addresses
+% - use \fouraffiliations{author1}{address1}{author2}{address2}{author3}{address3}{author4}{address4}
+% for four (or more) authors with four separate addresses
+% - note: no need for \name \address \author nor \date
+%
+% - optional: can use \thanks{xx} within \affiliation, \affiliations, etc.
+% asterisk is not printed after name nor in footnote
+% - optional: can use \sthanks{xx} after each name within \name or
+% \twoauthors if different thanks for each author,
+% footnote symbol will appear for each name and footnote
+% - optional: use \ninept to typeset text in 9 pt; default is 10pt.
+%
+% Example of use for one or more authors at a common address and
+% common support. For distinct support acknowledgments,
+% use \sthanks{xx} after each name.
+%
+% \documentstyle[dafx_19]{article}
+% \title{Title of the paper}
+% \affiliation{George P. Burdell and John Q. Professor
+% \thanks{This work was supported by...}}
+% {Common address, department \\
+% City, etc \\
+% optional e-mail address}
+%
+% \begin{document}
+% OPTIONAL --> \ninept <-- OPTIONAL, for nine pt only
+% \maketitle
+% \begin{abstract}
+% This is the abstract for my paper.
+% \end{abstract}
+% .
+% Insert text of paper
+% .
+% \end{document}
+%
+% Example of use for authors at two distinct addresses with only
+% one support acknowledgment. For distinct support acknowledgments,
+% use \sthanks{xx} after each name.
+%
+% \documentstyle[dafx_19]{article}
+% \title{Title of the paper}
+% \twoaffiliations{John Doe
+% \thanks{This work was supported by...}}
+% {Doe's address, department \\
+% City, etc \\
+% optional e-mail address}
+% {Judy Smith}
+% {Smith's address, department \\
+% City, etc \\
+% optional e-mail address}
+%
+% \begin{document}
+% OPTIONAL --> \ninept <-- OPTIONAL, for nine pt only
+% \maketitle
+% \begin{abstract}
+% This is the abstract for my paper.
+% \end{abstract}
+% .
+% Insert text of paper
+% .
+% \end{document}
+%
+%
+% PLEASE REPORT ANY BUGS
+%
+% Author: Stephen Martucci -- [email protected]
+% Modifications: Jyri Huopaniemi -- [email protected]
+% the Dafx crew -- [email protected]
+% The DAFx04 crew -- [email protected]
+% The DAFx06 crew -- [email protected]
+% The DAFx07 crew -- [email protected]
+% The DAFx08 crew -- [email protected]
+% The DAFx09 crew -- [email protected]
+% The DAFx10 crew -- [email protected]
+% The DAFx12 crew -- [email protected]
+% The DAFx13 crew -- [email protected]
+% The DAFx14 crew -- [email protected]
+% The DAFx16 crew -- [email protected]
+% The DAFx17 crew -- [email protected]
+% The DAFx19 crew -- [email protected]
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%change default font to times-roman
+\renewcommand{\sfdefault}{phv}
+\renewcommand{\rmdefault}{ptm}
+\renewcommand{\ttdefault}{pcr}
+
+\oddsidemargin -0.31in
+\evensidemargin -0.31in
+
+\topmargin 0truept
+\headheight 12truept
+\footskip 0truept
+\textheight 226truemm
+\textwidth 175truemm
+\voffset -14truept
+\headsep 18truept
+
+\twocolumn
+\columnsep 8truemm
+\pagestyle{empty}
+\def\ninept{\def\baselinestretch{0.95}\let\normalsize\small\normalsize}
+
+\def\maketitle{\par
+ \begingroup
+ \def\thefootnote{}
+ \def\@makefnmark{\hbox to 6pt{$^{\@thefnmark}$\hss}}
+ \if@twocolumn
+ \twocolumn[\@maketitle]
+ \else \newpage
+ \global\@topnum\z@ \@maketitle \fi\@thanks
+ \endgroup
+ \setcounter{footnote}{0}
+ \let\maketitle\relax
+ \let\@maketitle\relax
+ \gdef\thefootnote{\arabic{footnote}}\gdef\@@savethanks{}%
+ \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax}
+
+\def\@maketitle{\newpage
+ \null
+ \vskip 1em
+ \begin{center}
+ {\large \bf \@title \par} \vskip 2em
+ {%
+ %\large
+ %\ninept
+ \lineskip .5em
+ \begin{tabular}[t]{c}\@name \\ \@address
+ \end{tabular}\par}
+ \end{center}
+ \par
+ \vskip 1.4em}
+
+\def\title#1{\gdef\@title{\MakeUppercase{#1}}}
+\gdef\@title{\uppercase{title of paper}}
+\gdef\@name{{\em Name of author}\\}
+\gdef\@address{Address - Line 1 \\
+ Address - Line 2 \\
+ Address - Line 3}
+
+\let\@@savethanks\thanks
+\def\thanks#1{\gdef\thefootnote{}\@@savethanks{#1}}
+\def\sthanks#1{\gdef\thefootnote{\fnsymbol{footnote}}\@@savethanks{#1}}
+
+\def\affiliationSpace{2mm}
+
+\def\affiliation#1#2{\gdef\@address{}
+ \gdef\@name{
+ \begin{minipage}[t]{\linewidth}
+ \centering
+ {\em #1}\\[\affiliationSpace]
+ #2\relax
+ \end{minipage}
+ }
+}
+
+\def\affiliations#1#2#3{\gdef\@address{}
+ \gdef\@name{\begin{tabular}{@{}c@{}}
+ {\em #1} \\ \vspace*{\affiliationSpace}\\
+ \begin{tabular}{@{}c@{}}
+ #2\relax
+ \end{tabular}\hskip 0.3in
+ \begin{tabular}{@{}c@{}}
+ #3\relax
+ \end{tabular}
+ \end{tabular}
+ }
+}
+
+\def\twoaffiliations#1#2#3#4{\gdef\@address{}
+ \gdef\@name{
+ \begin{minipage}[t]{.5\linewidth}
+ \centering
+ {\em #1}\\[\affiliationSpace]
+ #2\relax
+ \end{minipage}%
+ \begin{minipage}[t]{.5\linewidth}
+ \centering
+ {\em #3}\\[\affiliationSpace]
+ #4\relax
+ \end{minipage}%
+ }
+}
+
+\def\threeaffiliations#1#2#3#4#5#6{\gdef\@address{}
+ \gdef\@name{
+ \begin{minipage}[t]{.333\linewidth}
+ \centering
+ {\em #1}\\[\affiliationSpace]
+ #2\relax
+ \end{minipage}%
+ \hfill%
+ \begin{minipage}[t]{.333\linewidth}
+ \centering
+ {\em #3}\\[\affiliationSpace]
+ #4\relax
+ \end{minipage}%
+ \hfill%
+ \begin{minipage}[t]{.333\linewidth}
+ \centering
+ {\em #5}\\[\affiliationSpace]
+ #6\relax
+ \end{minipage}%
+ }
+}
+
+\def\fouraffiliations#1#2#3#4#5#6#7#8{\gdef\@address{}
+ \gdef\@name{
+ \begin{minipage}[t]{.5\linewidth}
+ \centering
+ {\em #1}\\[\affiliationSpace]
+ #2\relax
+ \end{minipage}%
+ \begin{minipage}[t]{.5\linewidth}
+ \centering
+ {\em #3}\\[\affiliationSpace]
+ #4\relax
+ \end{minipage}%
+ \\[22mm]%
+ \begin{minipage}[t]{.5\linewidth}
+ \centering
+ {\em #5}\\[\affiliationSpace]
+ #6\relax
+ \end{minipage}%
+ \begin{minipage}[t]{.5\linewidth}
+ \centering
+ {\em #7}\\[\affiliationSpace]
+ #8\relax
+ \end{minipage}%
+ }
+}
+
+\def\@sect#1#2#3#4#5#6[#7]#8{
+ \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname.\hskip 0.6em}
+ \begingroup \ifnum #2=1\bf\centering
+ {\interlinepenalty \@M
+ \@svsec\uppercase{#8}\par}\else\ifnum #2=2\bf
+ \noindent{\interlinepenalty \@M \@svsec #8\par}\else\it
+ \@hangfrom{\hskip\parindent}{\interlinepenalty \@M
+ \@svsec #8\par}\fi\fi\endgroup
+ \csname #1mark\endcsname{#7}\addcontentsline
+ {toc}{#1}{\protect\numberline{\csname the#1\endcsname} #7}
+ \@tempskipa #5\relax
+ \@xsect{\@tempskipa}}
+
+\def\abstract{\begin{center}
+{\bf ABSTRACT\vspace{-.5em}\vspace{0pt}}
+\end{center}}
+\def\endabstract{\par}
+
+\def\copyrightnotice#1{\gdef\@copyrightnotice{#1}}\let\@copyrightnotice\relax
+\def\toappear#1{\gdef\@toappear{#1}}\let\@toappear\relax
+
+%%% Strings for the preprint titlepage footer:
+
+\newif\if@preprint\@preprintfalse
+\def\ds@preprint{\@preprinttrue}
+\def\ps@preprint{\let\@mkboth\@gobbletwo\def\@oddhead{}
+ \def\@oddfoot{\rlap{\@toappear}\hfil\thepage\hfil
+ \llap{\@copyrightnotice}\gdef\@toappear{}\gdef\@copyrightnotice{}}}
+
+\if@preprint\ps@preprint
+\else\ps@empty\flushbottom\fi
+
+\def\thebibliography#1{\section{References}\list
+ {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
+ \advance\leftmargin\labelsep
+ \usecounter{enumi}}
+ \def\newblock{\hskip .11em plus .33em minus .07em}
+ \sloppy\clubpenalty4000\widowpenalty4000
+ \sfcode`\.=1000\relax}
+\let\endthebibliography=\endlist
+
+\flushbottom
+%%%%% fancyheading included
+\def\lhead{\@ifnextchar[{\@xlhead}{\@ylhead}}
+\def\@xlhead[#1]#2{\gdef\@elhead{#1}\gdef\@olhead{#2}}
+\def\@ylhead#1{\gdef\@elhead{#1}\gdef\@olhead{#1}}
+
+\def\chead{\@ifnextchar[{\@xchead}{\@ychead}}
+\def\@xchead[#1]#2{\gdef\@echead{#1}\gdef\@ochead{#2}}
+\def\@ychead#1{\gdef\@echead{#1}\gdef\@ochead{#1}}
+
+\def\rhead{\@ifnextchar[{\@xrhead}{\@yrhead}}
+\def\@xrhead[#1]#2{\gdef\@erhead{#1}\gdef\@orhead{#2}}
+\def\@yrhead#1{\gdef\@erhead{#1}\gdef\@orhead{#1}}
+
+\def\lfoot{\@ifnextchar[{\@xlfoot}{\@ylfoot}}
+\def\@xlfoot[#1]#2{\gdef\@elfoot{#1}\gdef\@olfoot{#2}}
+\def\@ylfoot#1{\gdef\@elfoot{#1}\gdef\@olfoot{#1}}
+
+\def\cfoot{\@ifnextchar[{\@xcfoot}{\@ycfoot}}
+\def\@xcfoot[#1]#2{\gdef\@ecfoot{#1}\gdef\@ocfoot{#2}}
+\def\@ycfoot#1{\gdef\@ecfoot{#1}\gdef\@ocfoot{#1}}
+
+\def\rfoot{\@ifnextchar[{\@xrfoot}{\@yrfoot}}
+\def\@xrfoot[#1]#2{\gdef\@erfoot{#1}\gdef\@orfoot{#2}}
+\def\@yrfoot#1{\gdef\@erfoot{#1}\gdef\@orfoot{#1}}
+
+\newdimen\headrulewidth
+\newdimen\footrulewidth
+\newdimen\plainheadrulewidth
+\newdimen\plainfootrulewidth
+\newdimen\headwidth
+\newif\if@fancyplain
+\def\fancyplain#1#2{\if@fancyplain#1\else#2\fi}
+
+% Command to reset various things in the headers:
+% a.o. single spacing (taken from setspace.sty)
+% and the catcode of ^^M (so that epsf files in the header work if a
+% verbatim crosses a page boundary)
+\def\fancy@reset{\restorecr
+ \def\baselinestretch{1}%
+ \ifx\undefined\@newbaseline% NFSS not present; 2.09 or 2e
+ \ifx\@currsize\normalsize\@normalsize\else\@currsize\fi%
+ \else% NFSS (2.09) present
+ \@newbaseline%
+ \fi}
+
+% Initialization of the head and foot text.
+
+\headrulewidth 0.4pt
+\footrulewidth\z@
+\plainheadrulewidth\z@
+\plainfootrulewidth\z@
+
+\lhead[\fancyplain{}{\sl\rightmark}]{\fancyplain{}{\sl\leftmark}}
+% i.e. empty on ``plain'' pages, \rightmark on even, \leftmark on odd pages
+\chead{}
+%\rhead[\fancyplain{}{\sl\leftmark}]{\fancyplain{}{\sl\rightmark}}
+% i.e. empty on ``plain'' pages, \leftmark on even, \rightmark on odd pages
+\lfoot{}
+\cfoot{}
+\rfoot{}
+
+% Put together a header or footer given the left, center and
+% right text, fillers at left and right and a rule.
+% The \lap commands put the text into an hbox of zero size,
+% so overlapping text does not generate an errormessage.
+
+\def\@fancyhead#1#2#3#4#5{#1\hbox to\headwidth{\fancy@reset\vbox{\hbox
+{\rlap{\parbox[b]{\headwidth}{\raggedright#2\strut}}\hfill
+\parbox[b]{\headwidth}{\centering#3\strut}\hfill
+\llap{\parbox[b]{\headwidth}{\raggedleft#4\strut}}}\headrule}}#5}
+
+
+\def\@fancyfoot#1#2#3#4#5{#1\hbox to\headwidth{\fancy@reset\vbox{\footrule
+\hbox{\rlap{\parbox[t]{\headwidth}{\raggedright#2\strut}}\hfill
+\parbox[t]{\headwidth}{\centering#3\strut}\hfill
+\llap{\parbox[t]{\headwidth}{\raggedleft#4\strut}}}}}#5}
+
+\def\headrule{{\if@fancyplain\let\headrulewidth\plainheadrulewidth\fi
+\hrule\@height\headrulewidth\@width\headwidth \vskip-\headrulewidth}}
+
+\def\footrule{{\if@fancyplain\let\footrulewidth\plainfootrulewidth\fi
+\vskip-0.3\normalbaselineskip\vskip-\footrulewidth
+\hrule\@width\headwidth\@height\footrulewidth\vskip0.3\normalbaselineskip}}
+
+\def\ps@fancy{%
+\@ifundefined{@chapapp}{\let\@chapapp\chaptername}{}%for amsbook
+\@ifundefined{chapter}{\def\sectionmark##1{\markboth
+{\uppercase{\ifnum \c@secnumdepth>\z@
+ \thesection\hskip 1em\relax \fi ##1}}{}}%
+\def\subsectionmark##1{\markright {\ifnum \c@secnumdepth >\@ne
+ \thesubsection\hskip 1em\relax \fi ##1}}}%
+{\def\chaptermark##1{\markboth {\uppercase{\ifnum \c@secnumdepth>\m@ne
+ \@chapapp\ \thechapter. \ \fi ##1}}{}}
+\def\sectionmark##1{\markright{\uppercase{\ifnum \c@secnumdepth >\z@
+ \thesection. \ \fi ##1}}}}%
+\csname ps@headings\endcsname % use \ps@headings defaults if they exist
+\ps@@fancy
+\gdef\ps@fancy{\@fancyplainfalse\ps@@fancy}%
+\headwidth\textwidth}
+\def\ps@fancyplain{\ps@fancy \let\ps@plain\ps@plain@fancy}
+\def\ps@plain@fancy{\@fancyplaintrue\ps@@fancy}
+\def\ps@@fancy{%
+\def\@mkboth{\protect\markboth}%
+\def\@oddhead{\@fancyhead\@lodd\@olhead\@ochead\@orhead\@rodd}%
+\def\@oddfoot{\@fancyfoot\@lodd\@olfoot\@ocfoot\@orfoot\@rodd}%
+\def\@evenhead{\@fancyhead\@rodd\@elhead\@echead\@erhead\@lodd}%
+\def\@evenfoot{\@fancyfoot\@rodd\@elfoot\@ecfoot\@erfoot\@lodd}%
+}
+\def\@lodd{\if@reversemargin\hss\else\relax\fi}
+\def\@rodd{\if@reversemargin\relax\else\hss\fi}
+
+\let\latex@makecol\@makecol
+\def\@makecol{\let\topfloat\@toplist\let\botfloat\@botlist\latex@makecol}
+\def\iftopfloat#1#2{\ifx\topfloat\empty #2\else #1\fi}
+\def\ifbotfloat#1#2{\ifx\botfloat\empty #2\else #1\fi}
+\def\iffloatpage#1#2{\if@fcolmade #1\else #2\fi}
+
+\pagestyle{fancy}
+\renewcommand{\headrulewidth}{0pt}
+\renewcommand{\footrulewidth}{-5mm}
+\lhead{\em{\small{}}}%Proceedings of the 22$^{\text{\itshape nd}}$ International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2--6, 2019}}}
+\rhead{}
+\lfoot{}
+\rfoot{}
+\cfoot{{\vskip 11mm}{\small DAFX-\thepage}}{}
+
diff --git a/Paper/references.bib b/Paper/references.bib
@@ -0,0 +1,80 @@
+@book{Zolzer:2011:DDA:2028616,
+ author = {Zolzer, Udo},
+ title = {DAFX: Digital Audio Effects},
+ year = {2011},
+ isbn = {0470665998, 9780470665992},
+ edition = {2nd},
+ publisher = {Wiley Publishing},
+}
+
+@Misc{Carl,
+ title={Cassette Tape Physical Modeling and Analysis},
+ author={Richard Carl, Daniel Fine, Benjamin Schmitz},
+ school={University of Rochester, Department of Audio and Music Engineering},
+ howpublished={\url{http://www2.ece.rochester.edu/~zduan/teaching/ece472/projects/2018/RichardCarl_DanielFine_BenjaminSchmitz_Cassette_FinalReport.pdf}}
+}
+
+@Misc{Howell,
+ title={A Comparison of Analog vs Digital Tape Echo},
+ author={Benjamin Howell},
+ school={Birmingham City University},
+ howpublished={\url{https://www.academia.edu/10437785/Benijah_HowellBSc_-_A_Comparison_of_analog_VS_Digital_Tape_Echo}}
+}
+
+@BOOK{1994tmr..book.....B,
+ author = {{Bertram}, H.~N.},
+ title = "{Theory of Magnetic Recording}",
+booktitle = {Theory of Magnetic Recording, by H.~Neal Bertram, pp.~372.~ISBN 0521445124.~Cambridge, UK: Cambridge University Press, April 1994.},
+ year = 1994,
+ month = apr,
+ pages = {372},
+ adsurl = {http://adsabs.harvard.edu/abs/1994tmr..book.....B},
+ adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+@book{Camras:1987:MRH:27189,
+ author = {Camras, Marvin},
+ title = {Magnetic Recording Handbook},
+ year = {1987},
+ isbn = {0442262620},
+ publisher = {Van Nostrand Reinhold Co.},
+ address = {New York, NY, USA},
+}
+
+@book {Kadis,
+ title = {The Science of Sound Recording},
+ year = {2012},
+ pages = {256},
+ publisher = {Focal Press},
+ organization = {Focal Press},
+ edition = {1},
+ address = {Waltham, MA},
+ issn = {978-0-240-82154-2},
+ url = {http://www.focalpress.com/books/audio/the_science_of_sound_recording.aspx},
+ author = {Jay Kadis}
+}
+
+@book {Eno,
+ title = {A Year with Swollen Appendices},
+ year = {1996},
+ pages = {424},
+ publisher = {Faber and Faber},
+ issn = {0-571-17995-9},
+ author = {Brian Eno}
+ }
+
+@inproceedings{Hysteresis,
+ author = {Holters, Martin and Zölzer, Udo},
+ year = {2016},
+ month = {09},
+ pages = {},
+ title = {Circuit Simulation with Inductors and Transformers Based on the Jiles-Atherton Model of Magnetization}
+}
+
+@phdthesis{Yeh,
+ author = {D.T. Yeh},
+ title = {Digital Implementation of Musical Distortion Circuits by Analysis and Simulation},
+ school = {Stanford Univeristy},
+ year = 2009,
+ month = 6
+}
diff --git a/Simulations/Sim1 M_H.png b/Simulations/Sim1 M_H.png
Binary files differ.
diff --git a/Simulations/Sim1 M_t.png b/Simulations/Sim1 M_t.png
Binary files differ.
diff --git a/Simulations/hystersis.py b/Simulations/hystersis.py
@@ -0,0 +1,129 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+T = 48000
+M_s = 1.6e6
+a = 1.1e3
+alpha = 1.6e-3
+k = 4.0e2
+c = 1.7e-1
+
+# Langevin function
+def L (x):
+ if (abs (x) > 10 ** -4):
+ return (1 / np.tanh (x)) - (1/x)
+ else:
+ return (x / 3)
+
+# Langevin derivative
+def L_d (x):
+ if (abs(x) > 10 ** -4):
+ return (1 / x ** 2) - (1 / np.tanh (x)) ** 2 + 1
+ else:
+ return (1 / 3)
+
+# Langevin 2nd derivative
+def L_d2 (x):
+ if (abs(x) > 10 ** -3):
+ return 2 * (1 / np.tanh (x)) * ((1 / np.tanh (x)) ** 2 - 1) - (2 / x ** 3)
+ else:
+ return -(2 / 15) * x
+
+# trapezoidal rule derivative
+def deriv (x_n, x_n1, xDeriv_n1):
+ return (2 / T) * (x_n - x_n1) - xDeriv_n1
+
+# dM/dt or "non-linear function"
+def f (M, H, H_d):
+ Q = (H + alpha * M) / a
+ M_diff = M_s * L (Q) - M
+ delta = 1 if H_d > 0 else -1
+ delta_M = 1 if np.sign (delta) == np.sign (M_diff) else 0
+ L_prime = L_d (Q)
+
+ denominator = 1 - c * alpha * (M_s / a) * L_prime
+
+ t1_num = (1 - c) * delta_M * M_diff
+ t1_den = (1 - c) * delta * k - alpha * M_diff
+ t1 = (t1_num / t1_den) * H_d
+
+ t2 = c * (M_s / a) * H_d * L_prime
+
+ return (t1 + t2) / denominator
+
+# d^2M/dt^2
+def M_d2 (M, M_d, H, H_d, H_d2):
+ Q = (H + alpha * M) / a
+ M_diff = M_s * L (Q) - M
+ delta = 1 if H_d > 0 else -1
+ delta_M = 1 if np.sign (delta) == np.sign (M_diff) else 0
+ L_prime = L_d (Q)
+
+ denominator = 1 - c * alpha * (M_s / a) * L_prime
+
+ t1_num = (1 - c) * delta_M * M_diff
+ t1_den = (1 - c) * delta * k - alpha * M_diff
+ t1 = (t1_num / t1_den) * H_d2
+
+ M_diff2 = c * (M_s / a) * (H_d + alpha * M_d) * L_d (Q) - M_d
+ t2_num = (1 - c) * delta_M * M_diff2
+ t2 = (t2_num / t1_den) * H_d
+
+ t3_num = (1 - c) * delta_M * M_diff * (-alpha) * M_diff2
+ t3 = (t3_num / t1_den**2) * H_d
+
+ L_2prime = L_d2 (Q)
+ t4 = c * (M_s / a) * (H_d2 * L_prime + c * (M_s / a) * L_2prime * (H_d + alpha * M_d)**2)
+
+ return (t1 + t2 + t3 + t4) / denominator
+
+def M_n (M_n1, f_n, f_n1, M_d2_n1):
+ return M_n1 + (T / 2) * (f_n + f_n1) / (1 - (T/2) * M_d2_n1)
+
+t = np.linspace (0, 10, T * 50)
+H_in = np.concatenate ((1000 * np.sin (2 * np.pi * 2 * t[0:T*5]), 900 * np.sin (2 * np.pi * 2 * t[T*5:T*10]), \
+ 800 * np.sin (2 * np.pi * 2 * t[T*10:T*15]), 750 * np.sin (2 * np.pi * 2 * t[T*15:T*20]), \
+ 700 * np.sin (2 * np.pi * 2 * t[T*20:T*25]), 650 * np.sin (2 * np.pi * 2 * t[T*25:T*30]), \
+ 600 * np.sin (2 * np.pi * 2 * t[T*30:T*35]), 550 * np.sin (2 * np.pi * 2 * t[T*35:T*40]),
+ 500 * np.sin (2 * np.pi * 2 * t[T*40:T*45]), 400 * np.sin (2 * np.pi * 2 * t[T*45:T*50])))
+# plt.plot (t, H_in)
+# plt.show()
+
+M_out = np.zeros (T * 50)
+M_n1 = 0
+H_n1 = 0
+H_d_n1 = 0
+H_d2_n1 = 0
+
+n = 0
+percent = 0
+for H in H_in:
+ H_d = deriv (H, H_n1, H_d_n1)
+ H_d2 = deriv (H_d, H_d_n1, H_d2_n1)
+
+ f_n = f (M_n1, H, H_d)
+ f_n1 = f (M_n1, H_n1, H_d_n1)
+ M_d2_n1 = M_d2 (M_n1, f_n1, H_n1, H_d_n1, H_d2_n1)
+
+ M = M_n (M_n1, f_n, f_n1, M_d2_n1)
+
+ M_n1 = M
+ H_n1 = H
+ H_d_n1 = H_d
+ H_d2_n1 - H_d2
+
+ M_out[n] = (M)
+ n += 1
+
+ curPercent = (int) (n / (T * 50) * 100)
+ if (curPercent > percent):
+ percent = curPercent
+ print (str (percent) + "% completed")
+
+MH = plt.figure (1)
+plt.plot (H_in, M_out)
+#MH.show()
+
+Mt = plt.figure (2)
+plt.plot (t, M_out)
+plt.show()
