\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} + k 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,
 $k$ is the spring constant, and
 $F(t)$ is the external driving force.
During the non-contact phase of calibration,
 $F(t)$ comes from random thermal noise.

In the following analysis, we use the unitary, angular frequency Fourier transform normalization
\begin{equation}
  \F{x(t)} \equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}}
\end{equation}

We also use the following theorems (proved elsewhere):
\begin{align}
  \cos\left(\frac{\theta}{2}\right) &= \pm\sqrt{\frac{1}{2}[1+\cos(\theta)]}
     &\text{\cite{cos_halfangle},} \label{eq.cos_halfangle} \\
  \F{\nderiv{n}{t}{x(t)}} &= (i \omega)^n x(\omega)
     &\text{\cite{four_deriv},} \label{eq.four_deriv} \\
%  \F{x*y} &= x(\omega) y(\omega),  \label{eq.four_conv}
%     & \text{and} \\
  \iInfInf{t}{\magSq{x(t)}} &= \iInfInf{\omega}{\magSq{x(w)}}
     &\text{(Parseval's)\cite{parseval}.} \label{eq.parseval}
\end{align}
%where $x*y$ denotes the convolution of $x$ and $y$,
%\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
\begin{align}
  \PSD(x, \omega) &\equiv \normLimT 2 \left| x(\omega) \right|^2 
     &\text{\cite{PSD}} \label{eq.psd_def}
\end{align}
relates to the variance by
\begin{align}
  \avg{x(t)^2}
     &= \iLimT{\magSq{x(t)}}
     = \normLimT \iInfInf{\omega}{\magSq{x(\omega)}}
     = \iOInf{\omega}{\PSD(x,\omega)}, \label{eq.parseval_var}
\end{align}
where $t_T$ is the total time over which data has been aquired.


We also use the Wiener-Khinchin theorem,
which relates the two sided power spectral density $S_{xx}(\omega)$
to the autocorrelation function $r_{xx}(t)$ via
\begin{align}
  S_{xx}(\omega) &= \F{ r_{xx}(t) }
       &\text{(Wiener-Khinchin)\cite{wiener_khinchin},} \label{eq.wiener_khinchin}
\end{align}
where $r_{xx}(t)$ is defined in terms of the expectation value
\begin{align}
  r_{xx}(t) &\equiv \avg{x(\tau)\conj{x}(\tau-t)}
       &\text{\cite{wikipedia_wiener_khinchin}}
\end{align}
and $\conj{x}$ represents the complex conjugate of $x$.
