2022.03.25 — discrete fourier transform and lockin detection
here we continue our discussion, started earlier this week, of our mathematical analysis of lockin detection.
we focus in this post on how to apply equation 9 of that post when we have an autocorrelation $s(t)$ defined only at discrete time separations
$t = 0, \Delta t, 2 \Delta t, \ldots, N \Delta t$, i.e. some discrete series of values $s_0, s_1, \ldots, s_N$.
suppose we take a series of $2N+1$ measurements of some voltage $v_1,\ldots,v_{2N+1}$ spaced by a fixed time interval $\Delta t$.
what follows will hold for a general series of measurements, but ultimately we will be interested in the case where
\begin{equation}
\label{ac}
\left(v_1,v_2,\ldots,v_{N+1},\ldots,v_{2N},v_{2N +1}\right) = \left(s_{N},s_{N-1},\ldots,s_{0},\ldots,s_{N-1},s_{N}\right)
\end{equation}
anyways, for any series $v_n$ of length $2N + 1$ we can define a function $v(t)$ of a continuous time variable $t$
\begin{equation}
v(t) = \sum_{m=-N}^{+N} \hat{v}_m e^{-i 2 \pi \frac{m}{(2N+1)\Delta t} t}
\label{idft}
\end{equation}
where
\begin{equation}
\hat{v}_m = \frac{1}{2N + 1}\sum_{n=1}^{2N+1} v_n e^{i 2 \pi \frac{nm}{2N +1}}
\label{dft}
\end{equation}
so that the function $v(t)$ evaluated at a time $t = n\Delta t$ is equal to the nth measurement $v_n$:
\begin{equation*}
\begin{split}
v(n \Delta t)
& =
\sum_m \hat{v}_m e^{-i 2 \pi \frac{m}{(2N+1)\Delta t} n \Delta t}
\\
& =
\sum_m \hat{v}_m e^{-i 2 \pi \frac{mn}{(2N+1)}}
\\
& =
\sum_m \sum_{n'} \frac{1}{2N +1} v_{n'} e^{i 2 \pi \frac{n'm}{2N +1}} e^{-i 2 \pi \frac{mn}{(2N+1)}}
\\
& =
\sum_m \sum_{n'} \frac{1}{2N +1} v_{n'} e^{i 2 \pi \frac{\left(n'-n\right)m}{2N +1}}
\\
& =
\sum_{n'} \frac{1}{2N +1} v_{n'} \left(\sum_{m=1}^{2N+1} e^{i 2 \pi \frac{\left(n'-n\right)m}{2N +1}}\right)
\\
(!) & =
\sum_{n'} \frac{1}{2N +1} v_{n'} \left( \left(2N + 1\right)\delta_{nn'}\right)
\\
& =
v_n
\end{split}
\end{equation*}
for an explanation of the step marked (!), see the footnote of https://chris.reillybrothers.net/blog/20190521.html.
in the line before the (!) we change the limits for the $m$ sum from those defined in equation \ref{idft} by exploiting the property $\hat{v}_{m + \left(2N + 1\right)} = \hat{v}_m$.
what is the fourier transform $\hat{v}(\omega)$ of $v(t)$?
\begin{equation}
\begin{split}
\hat{v}(\omega)
& =
\int_{-\infty}^{+\infty} d t v(t) e^{i \omega t}
\\
& =
\int_{-\infty}^{+\infty} d t
\left(
\sum_m \hat{v}_m e^{-i 2 \pi \frac{m}{(2N+1)\Delta t} t}
\right)
e^{i \omega t}
\\
& =
\sum_m
\hat{v}_m
\int_{-\infty}^{+\infty} d t
e^{
i \left(
\omega - 2 \pi \frac{m}{(2N+1)\Delta t}
\right)
t
}
\\
& =
\sum_m
\hat{v}_m
\left(
2 \pi \delta \left(
\omega - \omega_m
\right)
\right)
\\
& =
2 \pi
\sum_m
\hat{v}_m
\delta \left(
\omega - \omega_m
\right)
\end{split}
\label{ft}
\end{equation}
where
\begin{equation}
\omega_m = 2 \pi \frac{m}{(2N+1)\Delta t} = m \Delta \omega
\end{equation}
so we find that the fourier transform of $v(t)$ is a comb of delta functions from $\omega_{-N}$ to $\omega_{+N}$ spaced by $\Delta \omega$.
letting $v_n$ take the values as indicated in equation \ref{ac},
the fourier transform $\hat{v}(\omega)$ becomes the power spectral density $P_s(\omega)$,
so that, according to equation 9 from 2022-03-21,
the root-mean-squared deviation $\delta y$ of the lockin output from a noise input with autocorrelation $s(t)$,
estimated with a time series $s_n$ at time separations $\Delta t$, can be written as
\begin{equation}
\label{spiky}
\begin{split}
\delta y
& = \sqrt{
\frac{1}{4 \pi}
\left( P_s * P_g \right)(\omega)
}
\\
& = \sqrt{
\frac{1}{4 \pi}
\left(
2 \pi \sum_{m}' \hat{v}_m
\right)
}
\\
& = \sqrt{
\frac{1}{2}
\sum_{m}' \hat{v}_m
}
\\
\end{split}
\end{equation}
where $\sum_{m}'$ denotes a restricted sum over the values $m$ such that $|\omega_m - \omega_o| < \Omega$,
where $\omega_o$ is the lockin modulation frequency and $\Omega = \frac{1}{\tau}$,
where $\tau$ is the lockin time constant.
we can also replace the delta functions $\delta(\omega - \omega_m)$ by rectangles of unit area and width $\Delta \omega$ centered about $\omega_m$,
thus obtaining a "filled out" power spectral density $\tilde{P}_s(\omega)$ where
\begin{equation}
\label{filledout}
\tilde{P}_s(\omega) = 2 \pi \frac{\hat{v}_m}{\Delta \omega}
\end{equation}
where $m$ minimizes the distance $|\omega - \omega_m|$.
this form of the power spectral density has more visual appeal, since we can obtain $\delta y$ by:
-
integrating $\tilde{P}_s(\omega)$
(which should be a relatively smooth function to the extent that $\Delta \omega$ is small)
from $\omega_o - \Omega$ to $\omega_o + \Omega$, then
-
dividing the result by $4 \pi$, and finally
-
taking the square root
for completeness and convience we include the formula for estimating $\delta y$ in terms of frequency.
the "spiky" power spectral density $\bar{P}_s(f)$, in units of V2 per Hz,
is identical in form to the associated radial frequency ($\omega$) expression (see equation \ref{ft}):
\begin{equation}
\bar{P}_s(f)
= 2 \pi \sum_m \hat{v}_m \delta \left(
f - f_m
\right)
\end{equation}
where $f_m = m \Delta f$ and $\Delta f = \frac{\Delta \omega}{2 \pi} = \frac{1}{(2N + 1) \Delta t}$.
the associated expression for $\delta y$ is also identical to the $\omega$ case:
\begin{equation}
\delta y = \sqrt{
\frac{1}{2}
\sum_{m}' \hat{v}_m
}
\end{equation}
where now the sum extends over values $m$ where $|f_m - f_o| < f_{\tau}$,
where $f_o = \frac{\omega_o}{2 \pi}$
and $f_{\tau} = \frac{\Omega}{2 \pi}$.
for the "filled out" power spectral density $\tilde{\bar{P}}_s(f)$ we need only make the substitution
$\Delta \omega \to \Delta f$, i.e.
\begin{equation}
\tilde{\bar{P}}_s(\omega) = 2 \pi \frac{\hat{v}_m}{\Delta f}
\end{equation}
the procedure for obtaining $\delta y$ from $\tilde{\bar{P}}_s(f)$ is identical to that of $\tilde{P}_s(f)$,
though our new integration limits now run from $f_o - f_{\tau}$ to $f_o + f_{\tau}$.