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(17) |
Finally from eqs. 12 and 19, the resampled channel weight is4:
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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 is resampled onto a spectrum
with the same resolution but with a shifted value
at reference channel (fig. 2). In such a case, the
channel weight
we can deduce, and its associated
,
are:
1. |
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2. |
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3. |
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This should be kept in mind when averaging e.g. two and
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
are equal!
One should also take care that eq. 21 assumes
uncorrelated input channels. Resampling a spectrum which
was already resampled (e.g.
) introduces
a correlation between more contiguous channels. In particular
this equation should not be used to compute the associated
weights.
The weight at eq. 21 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 . 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: