Channel weight of a resampled spectrum

With the assumption of independent channel intensities, the weight $w_{R'}(i)$ of this resampled channel can be derived from its variance:

$$\ensuremath{\mathrm{var}}(T_{R'}(i)) = \ensuremath{\mathrm{var}} \left( \sum_{j=j_{\ensuremath{\mathrm{min}}}}^{j_{\ensuremath{\mathrm{max}}}} \alpha_R(j) \times T_R(j) \right)$$ (14)

$$= \sum_{ j=j_{\ensuremath{\mathrm{min}}}}^{j_{\ensuremath{\mathrm{max}}}} \alpha_R(j)^2 \times \ensuremath{\mathrm{var}} \left( T_R(j) \right)$$ (15)

$$= \frac{1}{\beta^2} \sum_{j=j_{\ensuremath{\mathrm{min}}}}^{j_{\ensuremath{\mathrm{max}}}} f_R(j)^2 \, w_R(j)$$ (16)

where eq.  is derived from the general variance property:

$$\ensuremath{\mathrm{var}} (aX+bY) = a^2 \ensuremath{\mathrm{var}}(X) + b^2 \ensuremath{\mathrm{var}}(Y)$$ (17)

where $a$ and $b$ are numerical constants and when $X$ and $Y$ are independent random variables.

Finally from eqs.  and , the resampled channel weight is⁴:

Figure: Resampling of $R$ spectrum into ${R'}$, when the resolution is preserved but the reference channel is shifted by $0.0$ (1), $0.25$ (2) and $0.5$ (3) channel.

This relation has a non-intuitive effect on the resampled spectrum: its weight is, in the general case, different from the original one. Let's assume that the spectrum $R$ is resampled onto a spectrum $R'$ with the same resolution but with a shifted value $x_{\ensuremath{\mathrm{val}}}$ at reference channel (fig. ). In such a case, the channel weight $w_{R'}$ we can deduce, and its associated $\sigma_{R'}$, are:

1.	$x_{\ensuremath{\mathrm{val}}} \rightarrow x_{\ensuremath{\mathrm{val}}}+0.00$:	$w_{R'} = w_R$,	$\sigma_{R'} = \sigma_R$
2.	$x_{\ensuremath{\mathrm{val}}} \rightarrow x_{\ensuremath{\mathrm{val}}}+0.25$:	$w_{R'} = 1.6 \times w_R$,	$\sigma_{R'} = \sigma_R / \sqrt{1.6}$
3.	$x_{\ensuremath{\mathrm{val}}} \rightarrow x_{\ensuremath{\mathrm{val}}}+0.50$:	$w_{R'} = 2 \times w_R$,	$\sigma_{R'} = \sigma_R / \sqrt{2}$

assuming all the $R$ channels have the same weight (i.e. same $\sigma_R(j)$). The extra factor $w_{R'}/w_{R}$ affects either the SIGMA and the TIME weightings. These examples show that, depending on the desired resampling, the weight of the recomputed spectrum may be different. In the two latter cases, a correlation has been introduced between contiguous channels. From the physical point of view, this can be explained from the fact that one resampled channel contains more information than one original channel, e.g. in case number 3 it has an equal contribution of the two original ones: the noise is reduced by a factor $\sqrt{2}$.

This should be kept in mind when averaging e.g. two $R$ and $T$ spectra with same resolution but shifted X-axis. If a SIGMA weighting is invoked, the average won't be found at the mean distance of the two spectra even if their $\sigma$ are equal!

One should also take care that eq.  assumes uncorrelated input channels. Resampling a spectrum which was already resampled (e.g. $R' \rightarrow R''$) introduces a correlation between more contiguous channels. In particular this equation should not be used to compute the associated weights.

The weight at eq.  apply to the TIME and SIGMA weighting, where the computations above have a physical meaning and one can expect consistant values for integration time, channel width and $\sigma$. For the EQUAL weighting, user expects the two input spectra to have the same weight whatever their abscissa axis definition: this means that a re-EQUALization of the channels must come after the resampling. The ad hoc weighting is in this case:

where $w_R(j)$ is either $1.0$ (see section ) for a new input spectrum, or any (possibly not constant) value for the reentrant sum. This preserves a correct ponderation of the reentrant sum in front of a new input spectrum.