\section{Theory}
\label{sec:calibcant:theory}
-% TODO: deprecated in favor of sec:cantilever-calib:intro
-
-Rather than computing the variance of $x(t)$ directly, we attempt to
-filter out noise by fitting the power spectral density (\PSD)%
-\nomenclature[PSDa]{$\PSD$}{Power spectral density in angular
- frequency space}\index{PSD@\PSD}\nomenclature{$\omega$}{Angular
- frequency (radians per second)} of $x(t)$ to the theoretically
-predicted \PSD\ for a damped harmonic oscillator (\cref{eq:model-psd})
-\begin{align}
- \ddt{x} + \beta\dt{x} + \omega_0^2 x &= \frac{F_\text{thermal}}{m} \\
- \PSD(x, \omega) &= \frac{G_1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;,
-\end{align}
-\index{Damped harmonic oscillator}
-where $G_1\equiv G_0/m^2$, $\omega_0$, and $\beta$ are used as the
-fitting parameters (see \cref{eq:model-psd}).%
-\index{$\beta$}\index{$\gamma$} The variance of $x(t)$ is then given
-by \cref{eq:DHO-var}
-\begin{equation}
- \avg{x(t)^2} = \frac{\pi G_1}{2\beta\omega_0^2} \;,
-\end{equation}
-which we can plug into the equipartition theorem
-(\cref{eq:equipart}) yielding
-\begin{align}
- \kappa = \frac{2 \beta \omega_0^2 k_BT}{\pi G_1} \;.
-\end{align}
-From \cref{eq:GO}, we find the expected value of $G_1$ to be
+Our cantilever can be approximated as a damped harmonic
+oscillator\index{damped harmonic oscillator}
\begin{equation}
- G_1 \equiv G_0/m^2 = \frac{2}{\pi m} k_BT \beta \;. \label{eq:Gone}
-\end{equation}
-
-
-\section{Theoretical power spectral density for a damped harmonic oscillator}
-\label{sec:setup}
-
-Our cantilever can be approximated as a damped harmonic oscillator
-\begin{equation}
- m\ddt{x} + \gamma \dt{x} + \kappa x = F(t) \;, \label{eq:DHO}
+ m\ddt{x} + \gamma \dt{x} + \kappa x = F(t) \;,
+ \label{eq:DHO}
% DHO for Damped Harmonic Oscillator
\end{equation}
-where $x$ is the displacement from equilibrium,
- $m$ is the effective mass,
- $\gamma$ is the effective drag coefficient,
- $\kappa$ is the spring constant, and
- $F(t)$ is the external driving force.
+where $x$ is the displacement from equilibrium\index{$x$},
+ $m$ is the effective mass\index{$m$},
+ $\gamma$ is the effective drag coefficient\index{$\gamma$},
+ $\kappa$ is the spring constant\index{$\kappa$}, and
+ $F(t)$ is the external driving force\index{$F(t)$}.
During the non-contact phase of calibration,
$F(t)$ comes from random thermal noise.
+
\nomenclature{$\beta$}{Damped harmonic oscillator drag-acceleration
- coefficient $\beta \equiv \gamma/m$}\index{$\beta$}%
+ coefficient $\beta \equiv \gamma/m$}
\nomenclature{$\gamma$}{Damped harmonic oscillator drag coefficient
- $F_\text{drag} = \gamma\dt{x}$}\index{$\gamma$}%
-\index{damped harmonic oscillator}%
+ $F_\text{drag} = \gamma\dt{x}$}
\nomenclature{$\dt{s}$}{First derivative of the time-series $s(t)$
- with respect to time. $\dt{s} = \deriv{t}{s}$}%
+ with respect to time. $\dt{s} = \deriv{t}{s}$}
\nomenclature{$\ddt{s}$}{Second derivative of the time-series $s(t)$
- with respect to time. $\ddt{s} = \nderiv{2}{t}{s}$}%
+ with respect to time. $\ddt{s} = \nderiv{2}{t}{s}$}
-In the following analysis, we use the unitary, angular frequency Fourier transform normalization
+In the following analysis, we use the unitary, angular frequency
+Fourier transform normalization
\begin{equation}
\Four{x(t)} \equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}}\;.
\end{equation}
\nomenclature{\Four{s(t)}}{Fourier transform of the time-series
- $s(t)$. $s(f) = \Four{s(t)}$}\index{Fourier transform}
+ $s(t)$.
+ $s(f) = \Four{s(t)}
+ \equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{s(t) e^{-i \omega t}}$
+ }\index{Fourier transform}
We also use the following theorems (proved elsewhere):
\begin{align}
%\begin{equation}
% x*y \equiv \iInfInf{\tau}{x(t-\tau)y(\tau)}.
%\end{equation}
-As a corollary to Parseval's theorem, we note that the one sided power spectral density per unit time (\PSD) defined by
+As a corollary to Parseval's theorem, we note that the one sided power
+spectral density per unit time (\PSD) defined by
\begin{align}
- \PSD(x, \omega) &\equiv \normLimT 2 \left| x(\omega) \right|^2
+ \PSD(x, \omega) &\equiv \normLimT 2 \abs{x(\omega)}^2
&\text{\citep{PSD}} \label{eq:psd-def}
\end{align}
\index{PSD@\PSD}
= \iOInf{\omega}{\PSD(x,\omega)} \;, \label{eq:parseval-var}
\end{align}
where $t_T$ is the total time over which data has been aquired.
+%
+\nomenclature[PSDo]{$\PSD$}{Power spectral density in angular
+ frequency space
+ \begin{equation}
+ \PSD(g, w) \equiv \normLimT 2 \magSq{ \Four{g(t)}(\omega) }
+ \end{equation}}
+\nomenclature{$\omega$}{Angular frequency (radians per second)}
+\nomenclature{$\abs{z}$}{Absolute value (or magnitude) of $z$. For
+ complex $z$, $\abs{z}\equiv\sqrt{z\conj{z}}$.}
We also use the Wiener-Khinchin theorem,
which relates the two sided power spectral density $S_{xx}(\omega)$
and $\conj{x}$ represents the complex conjugate of $x$.
\nomenclature{$\conj{z}$}{Complex conjugate of $z$}
-
-\subsection{Fitting deflection voltage directly}
-
-In order to keep our errors in measuring $\sigma_p$ seperate from
-other errors in measuring $\avg{x(t)^2}$, we can fit the voltage
-spectrum before converting to distance.
-\begin{align}
- \ddt{V_p}/\sigma_p + \beta\dt{V_p}/\sigma_p + \omega_0^2 V_p/\sigma_p
- &= F_\text{thermal} \\
- \ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p
- &= \sigma_p\frac{F_\text{thermal}}{m} \\
- \ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p
- &= \frac{F_\text{thermal}}{m_p} \\
- \PSD(V_p, \omega) &= \frac{G_{1p}}
- { (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 } \\
- \avg{V_p(t)^2} &= \frac{\pi G_{1p}}{2\beta\omega_0^2}
- = \frac{\pi \sigma_p^2 G_{1}}{2\beta\omega_0^2}
- = \sigma_p^2 \avg{x(t)^2} \;,
-\end{align}
-where $m_p\equiv m/\sigma_p$, $G_{1p}\equiv G_0/m_p^2=\sigma_p^2 G_1$.
-Plugging into the equipartition theorem yeilds
-\begin{align}
- \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
- = \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}} \;.
-\end{align}
-
-From \cref{eq:Gone}, we find the expected value of $G_{1p}$ to be
-\begin{equation}
- G_{1p} \equiv \sigma_p^2 G_1 = \frac{2}{\pi m} \sigma_p^2 k_BT \beta \;.
- \label{eq:Gone-p}
-\end{equation}
-
-
-\subsection{Fitting deflection voltage in frequency space}
-
-Note: the math in this section depends on some definitions from
-section \cref{sec:setup}.
-
-As yet another alternative, you could fit in frequency
-$f\equiv\omega/2\pi$ instead of angular frequency $\omega$. But we
-must be careful with normalization. Comparing the angular frequency
-and normal frequency unitary Fourier transforms
-\begin{align}
- \Four{x(t)}(\omega)
- &\equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}} \\
- \Fourf{x(t)}(f) &\equiv \iInfInf{t}{x(t) e^{-2\pi i f t}}
- = \iInfInf{t}{x(t) e^{-i \omega t}}
- = \sqrt{2\pi}\cdot\Four{x(t)}(\omega=2\pi f) \;,
-\end{align}
-from which we can translate the \PSD
-\begin{align}
- \PSD(x, \omega) &\equiv \normLimT 2 \magSq{ \Four{x(t)}(\omega) } \\
- \begin{split}
- \PSD_f(x, f) &\equiv \normLimT 2 \magSq{ \Fourf{x(t)}(f) }
- = 2\pi \cdot \normLimT 2 \magSq{ \Four{x(t)}(\omega=2\pi f) } \\
- &= 2\pi \PSD(x, \omega=2\pi f) \;.
- \end{split}
-\end{align}
-\nomenclature{$t$}{Time (seconds)}
-\index{PSD@\PSD!in frequency space}
-The variance of the function $x(t)$ is then given by plugging into
-\cref{eq:parseval-var} (our corollary to Parseval's theorem)
-\begin{align}
- \avg{x(t)^2} &= \iOInf{\omega}{\PSD(x,\omega)}
- = \iOInf{f}{\frac{1}{2\pi}\PSD_f(x,f)2\pi\cdot}
- = \iOInf{f}{\PSD_f(x,f)} \;.
-\end{align}
-Therefore
-\begin{align}
- \begin{split}
- \PSD_f(V_p, f) &= 2\pi\PSD(V_p,\omega)
- = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
- = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
- &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
- = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2}
- \end{split} \\
- \avg{V_p(t)^2} &= \frac{\pi G_{1f}}{2\beta_f f_0^2} \;.
-% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
-% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
-\end{align}
-where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
-$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
-\begin{align}
- \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
- = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;.
-\end{align}
-
-From \cref{eq:Gone}, we expect $G_{1f}$ to be
-\begin{equation}
- G_{1f} = \frac{G_{1p}}{8\pi^3}
- = \frac{\sigma_p^2 G_1}{8\pi^3}
- = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
- = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
- \label{eq:Gone-f}
-\end{equation}
-
-
-% TODO: re-integrate the following
-
-% \begin{split}
-% \PSD_f(V_p, f) =
-% 2\pi\PSD(V_p,\omega)
-% = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
-% = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
-% &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
-% \end{split} \\
-
-% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
-% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
-
-%where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
-%$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
-
-%From \cref{eq:Gone}, we expect $G_{1f}$ to be
-%\begin{equation}
-% G_{1f} = \frac{G_{1p}}{8\pi^3}
-% = \frac{\sigma_p^2 G_1}{8\pi^3}
-% = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
-% = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
-% \label{eq:Gone-f}
-% \end{equation}
-
\subsection{Highly damped case}
+\label{sec:calibcant:ODHO}
-For highly damped systems, the inertial term becomes insignificant
- ($m \rightarrow 0$).
-This model is commonly used for optically trapped beads\citep{TODO}.
-Because it is simpler and solutions are more easily available%
-\citep{grossman05,TODO},
-it will server to outline the general approach before we dive into the
-general case.
+For highly damped systems, the inertial term in \cref{eq:DHO} becomes
+insignificant ($m \rightarrow 0$). This model is commonly used for
+optically trapped beads\citep{TODO}. Because it is simpler and
+solutions are more easily available\citep{grossman05,TODO}, it will
+serve to outline the general approach before we dive into the general
+case.
Fourier transforming \cref{eq:DHO} with $m=0$ and applying
\cref{eq:four-deriv} we have
% ODHO stands for very Over Damped Harmonic oscillator
\begin{align}
(i \gamma \omega + \kappa) x(\omega) &= F(\omega) \label{eq:ODHO-freq} \\
- |x(\omega)|^2 &= \frac{|F(\omega)|^2}{\kappa^2 + \gamma^2 \omega^2} \;.
+ \abs{x(\omega)}^2 &= \frac{\abs{F(\omega)}^2}
+ {\kappa^2 + \gamma^2 \omega^2} \;.
\label{eq:ODHO-xmag}
\end{align}
\index{Damped harmonic oscillator!extremely overdamped}
\index{PSD@\PSD}
Because thermal noise is white (not autocorrelated + Wiener-Khinchin
-Theorem), we can denote the one sided thermal power spectral density
-per unit time by
+Theorem), we can write the one sided thermal power spectral density
+per unit time as
\begin{equation}
\PSD(F, \omega) = G_0
= \normLimT 2 \magSq{F(\omega)} \;. \label{eq:GOdef} % label O != zero
\end{equation}
-Plugging \cref{eq:GOdef} into \cref{eq:ODHO-psd} we have
+Plugging \cref{eq:GOdef} into \cref{eq:ODHO-psd-F} we have
\begin{equation}
\PSD(x, \omega) = \frac{G_0}{\kappa^2 + \gamma^2\omega^2} \;.
+ \label{eq:ODHO-psd-GO}
\end{equation}
This is the formula we would use to fit our measured \PSD, but let us
go a bit farther to find the expected \PSD\ and thermal noise given
-$m$, $\gamma$ and $\kappa$.
+$\gamma$ and $\kappa$.
Integrating over positive $\omega$ to find the total power per unit
time yields
= \iOInf{\omega}{\frac{G_0}{\kappa^2 + \gamma^2\omega^2}}
= \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{\kappa^2 + z^2}}
= \frac{G_0 \pi}{2 \gamma \kappa} \;,
+ \label{eq:ODHO-psd-int}
\end{align}
-where the integral is solved in \cref{sec:integrals}.
+where the integral is solved in \cref{sec:integrals:highly-damped}.
Plugging into our corollary to Parseval's theorem (\cref{eq:parseval-var}),
\begin{equation}
Plugging \cref{eq:ODHO-var} into \cref{eq:equipart} we have
\begin{align}
\kappa \frac{G_0 \pi}{2 \gamma \kappa} &= k_BT \\
- G_0 &= \frac{2 \gamma k_BT}{\pi} \;.
+ G_0 &= \frac{2 \gamma k_BT}{\pi} \;. \label{eq:ODHO-GO}
\end{align}
-So we expect $x(t)$ to have a power spectral density per unit time given by
+Combining \cref{eq:ODHO-psd-GO,eq:ODHO-GO}, we expect $x(t)$ to have a
+power spectral density per unit time given by
\begin{equation}
- \PSD(x, \omega) = \frac{2}{\pi}
+ \PSD(x, \omega) = \frac{2}{\pi}
\cdot
\frac{\gamma k_BT}{\kappa^2 + \gamma^2\omega^2} \;.
\label{eq:ODHO-psd}
\index{PSD@\PSD}
\subsection{General form}
+\label{sec:calibcant:SHO}
The procedure here is exactly the same as the previous section. The
integral normalizing $G_0$, however, becomes a little more
complicated.
-Fourier transforming \cref{eq:DHO} and applying \cref{eq:four-deriv} we have
+Fourier transforming \cref{eq:DHO} and applying \cref{eq:four-deriv}
+we have
\begin{align}
(-m\omega^2 + i \gamma \omega + \kappa) x(\omega) &= F(\omega)
\label{eq:DHO-freq} \\
(\omega_0^2-\omega^2 + i \beta \omega) x(\omega) &= \frac{F(\omega)}{m} \\
- |x(\omega)|^2 &= \frac{|F(\omega)|^2/m^2}
+ \abs{x(\omega)}^2 &= \frac{\abs{F(\omega)}^2/m^2}
{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;,
\label{eq:DHO-xmag}
\end{align}
-where $\omega_0 \equiv \sqrt{\kappa/m}$ is the resonant angular
-frequency and $\beta \equiv \gamma / m$ is the drag-aceleration
-coefficient.
-\index{Damped harmonic oscillator}\index{beta}\index{gamma}
-\nomenclature{$\omega_0$}{Resonant angular frequency (radians per second)}
-\index{$\omega_0$}
-
-We compute the \PSD\ by plugging \cref{eq:DHO-xmag} into \cref{eq:psd-def}
+where $\omega_0 \equiv \sqrt{\kappa/m}$\index{$\omega_0$} is the
+resonant angular frequency and $\beta \equiv \gamma / m$ is the
+drag-acceleration coefficient.\index{Damped harmonic
+ oscillator}\index{$\gamma$}\index{$\kappa$}\index{$\beta$}
+
+\nomenclature{$\omega_0$}{Resonant angular frequency (radians per
+ second)}
+
+We compute the \PSD\ by plugging \cref{eq:DHO-xmag} into
+\cref{eq:psd-def}
\begin{equation}
\PSD(x, \omega)
- = \normLimT \frac{2 |F(\omega)|^2/m^2}
+ = \normLimT \frac{2 \abs{F(\omega)}^2/m^2}
{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;.
\label{eq:DHO-psd-F}
\end{equation}
\index{PSD@\PSD}
Plugging \cref{eq:GOdef} into \cref{eq:DHO-psd-F} we have
-\begin{equation}
- \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;.
- \label{eq:model-psd}
-\end{equation}
-Integrating over positive $\omega$ to find the total power per unit time yields
+\begin{align}
+ \PSD(x, \omega) &= \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;,
+ \label{eq:model-psd} \\
+ &= \frac{G_1}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2} \;,
+ \label{eq:model-psd-Gone}
+\end{align}
+where $G_1\equiv G_0/m^2$ consolidates the unknown fitting parameters
+without loss of generality.
+
+Integrating over positive $\omega$ to find the total power per unit
+time yields
\begin{align}
\iOInf{\omega}{\PSD(x, \omega)}
&= \frac{G_0}{2m^2}
= \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2}
= \frac{G_0 \pi}{2m^2\beta\omega_0^2}
= \frac{G_0 \pi}{2m^2\beta \frac{\kappa}{m}} \\
- &= \frac{G_0 \pi}{2m \beta \kappa} \;.
+ &= \frac{G_0 \pi}{2m \beta \kappa} \;,
+ \label{eq:DHO-psd-int}
\end{align}
-The integration is detailed in \cref{sec:integrals}.
+where the integration is solved in \cref{sec:integrals:general}.
By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have
-\begin{equation}
- \avg{x(t)^2} = \frac{G_0 \pi}{2m^2\beta\omega_0^2} \;. \label{eq:DHO-var}
-\end{equation}
+\begin{align}
+ \avg{x(t)^2}
+ &= \frac{G_0 \pi}{2m^2\beta\omega_0^2} \;, \label{eq:DHO-var} \\
+ &= \frac{G_1 \pi}{\beta\omega_0^2} \;, \label{eq:DHO-var-Gone}
+\end{align}
Plugging \cref{eq:DHO-var} into the equipartition theorem
(\cref{eq:equipart}) we have
\begin{align}
- \kappa \frac{G_0 \pi}{2m \beta \kappa} &= k_BT \\
- G_0 &= \frac{2}{\pi} k_BT m \beta \;. \label{eq:GO}
+ \kappa \frac{G_0 \pi}{2m \beta \kappa} &= k_BT \, \\
+ G_0 &= \frac{2}{\pi} k_BT m \beta \;, \label{eq:GO} \\
+ G_1 &\equiv \frac{G_0}{m^2} = \frac{2}{\pi m} k_BT \beta \;. \label{eq:Gone}
\end{align}
-So we expect $x(t)$ to have a power spectral density per unit time given by
+Combining \cref{eq:model-psd,eq:GO}, we expect $x(t)$ to have a power
+spectral density per unit time given by
\begin{equation}
\PSD(x, \omega) = \frac{2 k_BT \beta}
{ \pi m \p[{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}] }\;.
\end{equation}
\index{PSD@\PSD}
-As expected, the general form \cref{eq:DHO-psd} reduces to the
-extremely overdamped form \cref{eq:ODHO-psd}. Plugging in for
-$\beta\equiv\gamma/m$ and $\omega_0\equiv\sqrt{\kappa/m}$,
+As expected, we can recover the extremely overdamped form
+\cref{eq:ODHO-psd} from the general form \cref{eq:DHO-psd}. Plugging
+in for $\beta\equiv\gamma/m$ and $\omega_0\equiv\sqrt{\kappa/m}$,
\begin{align}
\lim_{m\rightarrow 0} \PSD(x, \omega)
&= \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
- { \pi m^2 \p[{(\kappa/m-\omega^2)^2 + \gamma^2/m^2\omega^2}] }
+ { \pi m^2 \p[{\p({\frac{\kappa}{m}-\omega^2})^2 + \frac{\gamma^2}{m^2}\omega^2}] }
= \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
{ \pi \p[{(\kappa-m\omega^2)^2 + \gamma^2\omega^2}] } \\
&= \frac{2}{\pi}
\cdot
\frac{\gamma k_BT}{\kappa^2 + \gamma^2\omega^2} \;.
\end{align}
+
+\subsection{Fitting deflection voltage directly}
+\label{sec:calibcant:voltage}
+
+In order to keep our errors in measuring $\sigma_p$ separate from
+other errors in measuring $\avg{x(t)^2}$, we can fit the voltage
+spectrum before converting to distance. Plugging \cref{eq:x-from-Vp}
+into \cref{eq:DHO},
+\begin{align}
+ \frac{\ddt{V_p}}{\sigma_p} + \beta\frac{\dt{V_p}}{\sigma_p}
+ + \omega_0^2 \frac{V_p}{\sigma_p}
+ &= F(t) \\
+ \ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p
+ &= \sigma_p\frac{F(t)}{m} \\
+ \ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p
+ &= \frac{F(t)}{m_p} \;,
+\end{align}
+where $m_p\equiv m/\sigma_p$. This has the same form as
+\cref{eq:DHO}, which can be rearranged to:
+\begin{align}
+ \ddt{x} + \frac{\gamma}{m} \dt{x} + \frac{\kappa}{m} x &= \frac{F(t)}{m} \\
+ \ddt{x} + \beta \dt{x} + \omega_0^2 x &= \frac{F(t)}{m} \;,
+\end{align}
+so the \PSD\ of $V_p(t)$ will be the same as the \PSD\ of $x(t)$,
+after the replacements $x\rightarrow V_p(t)$ and $m\rightarrow m_p$.
+Making these replacements in \cref{eq:model-psd-Gone,eq:DHO-var-Gone},
+we have
+\begin{align}
+ \PSD(V_p, \omega) &= \frac{G_{1p}}
+ { (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 } \\
+ \avg{V_p(t)^2} &= \frac{\pi G_{1p}}{2\beta\omega_0^2}
+ = \frac{\pi \sigma_p^2 G_{1}}{2\beta\omega_0^2}
+ = \sigma_p^2 \avg{x(t)^2} \;,
+\end{align}
+where $G_{1p}\equiv G_0/m_p^2=\sigma_p^2 G_1$.
+Plugging into the equipartition theorem yeilds
+\begin{align}
+ \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
+ = \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}} \;.
+\end{align}
+
+From \cref{eq:Gone}, we find the expected value of $G_{1p}$ to be
+\begin{equation}
+ G_{1p} \equiv \sigma_p^2 G_1 = \frac{2}{\pi m} \sigma_p^2 k_BT \beta \;.
+ \label{eq:Gone-p}
+\end{equation}
+
+\subsection{Fitting deflection voltage in frequency space}
+\label{sec:calibcant:frequency}
+
+As another alternative, you could fit in frequency
+$f\equiv\omega/2\pi$ instead of angular frequency $\omega$. The
+analysis will be the same, but we must be careful with normalization.
+Comparing the angular frequency and normal frequency unitary Fourier
+transforms
+\begin{align}
+ \Four{x(t)}(\omega)
+ &\equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}} \\
+ \Fourf{x(t)}(f) &\equiv \iInfInf{t}{x(t) e^{-2\pi i f t}}
+ = \iInfInf{t}{x(t) e^{-i \omega t}}
+ = \sqrt{2\pi}\cdot\Four{x(t)}(\omega=2\pi f) \;,
+\end{align}
+from which we can translate the \PSD
+\begin{align}
+ \PSD(x, \omega) &\equiv \normLimT 2 \magSq{ \Four{x(t)}(\omega) } \\
+ \begin{split}
+ \PSD_f(x, f) &\equiv \normLimT 2 \magSq{ \Fourf{x(t)}(f) }
+ = 2\pi \cdot \normLimT 2 \magSq{ \Four{x(t)}(\omega=2\pi f) } \\
+ &= 2\pi \PSD(x, \omega=2\pi f) \;.
+ \end{split}
+\end{align}
+\nomenclature{$t$}{Time (seconds)}
+\index{PSD@\PSD!in frequency space}
+The variance of the function $x(t)$ is then given by plugging into
+\cref{eq:parseval-var} (our corollary to Parseval's theorem)
+\begin{align}
+ \avg{x(t)^2} &= \iOInf{\omega}{\PSD(x,\omega)}
+ = \iOInf{f}{\frac{1}{2\pi}\PSD_f(x,f)2\pi\cdot}
+ = \iOInf{f}{\PSD_f(x,f)} \;.
+\end{align}
+Therefore
+\begin{align}
+ \begin{split}
+ \PSD_f(V_p, f) &= 2\pi\PSD(V_p,\omega)
+ = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
+ = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
+ &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
+ = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2}
+ \end{split} \\
+ \avg{V_p(t)^2} &= \frac{\pi G_{1f}}{2\beta_f f_0^2} \;.
+% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
+% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
+\end{align}
+where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
+$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
+\begin{align}
+ \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
+ = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;.
+\end{align}
+
+From \cref{eq:Gone-p}, we expect $G_{1f}$ to be
+\begin{equation}
+ G_{1f} = \frac{G_{1p}}{8\pi^3}
+ = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
+ = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
+ \label{eq:Gone-f}
+\end{equation}
+
+
+% TODO: re-integrate the following
+
+% \begin{split}
+% \PSD_f(V_p, f) =
+% 2\pi\PSD(V_p,\omega)
+% = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
+% = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
+% &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
+% \end{split} \\
+
+% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
+% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
+
+%where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
+%$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
+
+%From \cref{eq:Gone}, we expect $G_{1f}$ to be
+%\begin{equation}
+% G_{1f} = \frac{G_{1p}}{8\pi^3}
+% = \frac{\sigma_p^2 G_1}{8\pi^3}
+% = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
+% = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
+% \label{eq:Gone-f}
+% \end{equation}
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