Synopsis

#include <lal/Resample.h>

One of the crucial problems in searching for constant-frequency astrophysical signals is removing the effects of Doppler modulation due to the Earth's motion. This is normally accomplished by constructing a canonical time coordinate $\tau$ of an inertial frame (i.e. the barycentred time), and decimating/resampling the data at fixed intervals in $\tau$. The reconstructed $\tau$ depends on the direction to the source relative to the Earth's motion; in addition, slow intrinsic parameterized modulations in the source frequency can also be corrected by this coordinate transformation.

Most of the routines in this module assume that $\tau$ can be piecewise expanded as a Taylor series in $t$. That is, one defines a set of fitting regions $T_i=[t_{\mathrm{bound}(i-1)}, t_{\mathrm{bound}(i)}]$, and a set of fitting points $t_{(i)}\in T_i$. In each region one then writes:

$$\tau(t) = \sum_{k=0} \frac{1}{k!}c_{k(i)}(t-t_{(i)})^k \; .$$ (9.1)

Since one is normally interested in tracking the difference $\tau(t)-t$, one can also write the expansion as:

$$\tau(t)-t = \sum_{k=0} a_{k(i)}(t-t_{(i)})^k \; ,$$ (9.2)

where

$$a_{0(i)} = c_{0(i)}-t_{(i)} \; ,$$

$$a_{1(i)} = c_{1(i)}-1 \; ,$$

$$a_{k(i)} = c_{k(i)}/k! \; , \; k\geq2 \; .$$

These are the polynomial coefficients normally assumed in the modules under this header.

The procedure for resampling according to $\tau$ is normally combined with decimating the time series. That is, one takes a time series sampled at constant intervals $\Delta t$ in $t$, and samples it at constant intervals $d\Delta t$ in $\tau$, where the decimation factor $d$ is normally taken to be an integer $\geq1$. When $\tau$ and $t$ are drifting out of phase relatively slowly, this means that most of the time every $d^\mathrm{th}$ sample in the original time series becomes the next sample in the decimated time series. However, when $\tau$ and $t$ drift out of synch by an amount $\pm\Delta t$, one can force the decimated time series to track $\tau$ (rather than $t$) by sampling the $d\pm1^\mathrm{th}$ next datum (rather than the $d^\mathrm{th}$). If the drift is sufficiently rapid or $d$ is sufficiently large, one may be forced to choose the point $d\pm2$, $d\pm3$, etc.; the size of this adjustment is called the correction shift. The number of (resampled) time intervals between one correction point and the next is called the correction interval.

Unless otherwise specified, all time variables and parameters in the functions under this header can be assumed to measure the detector time coordinate $t$. Canonical times are specified by giving the difference $\tau-t$.