\documentclass[12pt]{article}
\title{Derivation of matched filter response}
\author{NGL}
\begin{document}
\maketitle

\section{Introduction: matched filter design}

See also the submitted article.

\subsection{Continuous case}

Let us denote the stationary noise time dependence as $w(t)$. Then the noise
spectrum is by Wiener-Khinchine relation
\[
S(\omega)=\frac{1}{2\pi}\int K(t) e^{-i\omega t} dt
\]
where $K(t')=\langle w(t_1) w(t_1+t)\rangle$ is the correlation
function, independent of $t_1$ because of stationarity.

Normalization is such that
\[
\langle w^2(t)\rangle=\int_{-\infty}^{+\infty} S(\omega)\,d\omega
\]
Usually the function in use is $S_f(f)$, where $f=\omega/(2\pi)$.
$S_f(f)$ is defined only for $f>0$ and is normalized so that
\[
\langle w^2(t)\rangle=\int_{0}^{+\infty} S_f(f)\,df
\]
It is related to $S(\omega)$ by $S(f)=4\pi S(\omega)$.

Denote the signal shape $F(t)$, and its Fourier transform
\[
\tilde{F}(\omega)=\int_{-\infty}^{+\infty}F(t)e^{-i\omega t}dt
\]
The inverse Fourier transform is
\[
F(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\tilde{F}(\omega)
e^{i\omega t}d\omega
\]


Let us consider signals going through a (linear) filter with {\em
response function} $H(t)$. It is defined as the output of the filter
when the input is the delta function $x(t)=\delta(t)$.  If the input
is $x(t)$, then the output is The signal after filter is given by
\[
y(t)=\int_{-\infty}^{+\infty}H(t')x(t-t')dt'
\]
The {\em transfer function} $\tilde{H}(\omega)$ is defined as the
Fourier transform of $H(t)$. If the noise spectrum at input is
$S(\omega)$, then the noise spectrum at output is
\[
S_y(\omega)=|\tilde{H}(\omega)|^2 S_x(\omega)
\]

The matched filter transfer function is given by
\[
\tilde{H}(\omega)=\frac{\tilde{F}(\omega)}{S(\omega)}
\]
Thus, the noise spectrum on the output of the matched filter is
\[
S_{mf}(\omega)=\frac{|\tilde{F}(\omega)|^2}{S(\omega)}
\]

\subsection{Discrete case}
Analogous. See the paper.

\section{Derivation for AUTEC}
A generic bipolar pulse signal is
\[
F(t)=-\frac{1}{\tau}te^{-\frac{t^2}{2\tau^2}+\frac{1}{2}}
\]
The various factors are chosen so that the maximum and minimum are at
$t=\pm \tau$ and also $p(\mp \tau)=\pm 1$.

The spectrum of this signal is
\[
\tilde{F}(\omega)=i\omega\tau^2\sqrt{2\pi}e^{-\frac{\omega^2\tau^2}{2}+
\frac{1}{2}}
\]

The noise has Knudsen spectrum
\[
S(f)=S_0\left(\frac{f}{f_0}\right)^{-\alpha}
\]
where $\alpha\approx 1.7$. For convenience in analytic derivation,
take $\alpha=2$. In terms of $\omega$ we have
\[
S(\omega)=\frac{S_0}{4\pi}\left(\frac{\omega_0}{\omega}\right)^2
\]
where $\omega_0=2\pi f_0$.

The matched filter transfer function is
\[
\tilde{H}(\omega)=\frac{\tilde{F}(\omega)}{S(\omega)}=
\frac{i\omega^3\tau^2 4\pi\sqrt{2\pi}}{S_0\omega_0^2}
e^{-\frac{\omega^2\tau^2}{2}+\frac{1}{2}}
\]
The response function is
\[
H(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}
\frac{i\omega^3\tau^2 4\pi\sqrt{2\pi}}{S_0\omega_0^2}
e^{-\frac{\omega^2\tau^2}{2}+\frac{1}{2}+i\omega t} d\omega=
\frac{4\pi}{S_0\omega^2}
\frac{e^{-\frac{t^2}{2\tau^2}+\frac{1}{2}}}{\tau^2}\left(
\frac{t^3}{\tau^3}-3\frac{t}{\tau}
\right)
\]

\end{document}
