(8) |

(9) |

ensure that the weighted mean is the maximum likelihood estimator of the mean, under the assumption of independent (

Let's call the resampled version of such as it is aligned with the desired output spectrum . The intensity of at channel can be written (see fig. ):

where:- and are the channel range of input spectrum which are overlapped at least partially by the output channel number ,
- is the used fraction of the different input channels. It
quantifies their contribution in the selected range:

if the input channel is out of the range, if the output channel fully overlaps the input channel, otherwise (proportionally to the overlap).

(11) | |||

(12) |

The normalization factor ensures that the total integrated intensity is preserved during the resampling process. With these, can be written:

(13) |