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-\begin{document}

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-{\Huge{delay space stepping strategy}}

-

-Blaise Thompson \hfill last modified \mydate

-

-\dotfill

-

-Linear stepping is more expensive than it needs to be.

-

-Want to capture the dynamic range of the data as quickly as possible.

-

-Typically have exponential decay dynamics (perhaps multi-exponential)\dots we can capitalize on this. We want to take high resolution data at early delays and low resolution data at late delays.

-

-Of course, we don't want to throw away any information we would otherwise be entitled to.

-

-Conceptually we want to 'linearize' the data, so that each subsequent delay step accounts for the same change in signal.

-

-Signal goes exponentially...

-

-\begin{eqnarray}

-S &=& \me^{-\frac{t}{\tau}} \\

-\log{(S)} &=& -\frac{t}{\tau} \\

-t &=& -\tau\log{(S)}

-\end{eqnarray}

-

-So to step linearly in $t$, my step size has to go as $-\tau\log{(S)}$.

-

-We want to go linearly in signal, meaning that we want to divide $S$ into even sections. If $S$ goes from 0 to 1 and we choose to acquire $N$ points,

-

-\begin{eqnarray}

-t_n &=& -\tau\log{\left(\frac{n}{N}\right)}.

-\end{eqnarray}

-

-Note that $t_n$ starts at long times and approaches zero delay. So the first $t_1$ is the smallest signal and $t_N$ is the largest.

-

-Now we can start to consider realistic cases, like where $\tau$ is not quite known and where some other longer dynamics persist (manifested as a static offset). Since these values are not separable in a general system, I'll keep $S$ normalized between 0 and 1.

-

-\begin{eqnarray}

-S &=& (1-c)\me^{-\frac{t}{\tau_{\mathrm{actual}}}} + c \\

-S_n &=& (1-c)\me^{-\frac{-\tau_{\mathrm{step}}\log{\left(\frac{n}{N}\right)}}{\tau_{\mathrm{actual}}}} + c \\

-S_n &=& (1-c)\me^{-\frac{\tau_{\mathrm{step}}}{\tau_{\mathrm{actual}}} \log{\left(\frac{N}{n}\right)}} + c \\

-S_n &=& (1-c)\left(\frac{N}{n}\right)^{-\frac{\tau_{\mathrm{step}}}{\tau_{\mathrm{actual}}}} + c \\

-S_n &=& (1-c)\left(\frac{n}{N}\right)^{\frac{\tau_{\mathrm{step}}}{\tau_{\mathrm{actual}}}} + c

-\end{eqnarray}

-

-

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-\begin{figure}[!htb]

-	\centering

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-	\caption{}

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-

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