XMM-Newton Science Analysis System


colsmooth (colsmooth-1.9.1) [xmmsas_20230412_1735-21.0.0]

Introduction.

The task convolves a specified column of a fits file by a user-specified convolving function. This can be used to smooth the values in the column. Specifically, for input vector $x$ defined from $i=1$ to $i=N$ and convolver $c$ defined from $-M$ to $M$, colsmooth performs the following transformation:

  $$y_{i} = \frac{\sum_{j=-M}^{M} F_{i,j}}{\sum_{j=-M}^{M} G_{i,j}}$$ (1)

where

  $$F_{i,j} = \left\{ \begin{array}{ll} c_{j} \ x_{i-j} & \textrm{if $i-j \in [1,N]$}\\ 0 & \textrm{else}\\ \end{array}\right.$$ (2)

and the weight term $G_{i,j}$ is given by

$$G_{i,j} = \left\{ \begin{array}{ll} c_{j} & \textrm{if $i-j \in [1,N]$}\\ 0 & \textrm{else}\\ \end{array}\right.$$

These formulae automatically normalise the convolver $c$. If this is not desired (indicated by setting the parameter normalise to `no'), the $y$ values are multiplied by $\sum_{j=-M}^{M} c_{j}$ in addition to the above.

The user may also specify an uncertainty column. The values of this column, which we denote by $u$, are transformed as follows:

  $$v_{i} = \frac{\sqrt{\sum_{j=-M}^{M} H_{i,j}}}{\sum_{j=-M}^{M} G_{i,j}}$$ (3)

The vector $v$ here represents the transformed uncertainty values. The quantity $H$ is given by

$$H_{i,j} = \left\{ \begin{array}{ll} c_{j}^{2} \ u_{i-j}^{2} & \textrm{if $i-j \in [1,N]$}\\ 0 & \textrm{else}\\ \end{array}\right.$$

and $G$ is as given above. The $v$ values can be `unnormalised' in the same way as $y$ above.

It is probably as well to remind users that the benefits of smoothing are almost entirely cosmetic. The fundamental amount of information in the column cannot of course be increased by such processing. In fact use of a low-pass filter may be deceptive: the amount of noise may not have decreased very much although the graph looks much better because of the suppression of the more visible high-frequency end of the noise spectrum.

This task is not XMM-specific.