Spectrometers in radio-astronomy

Typical radio-astronomy spectrometers (e.g., correlators or Fourier transform spectrometers) deliver the brightness along a given line of sight at regularly spaced frequencies in the observatory frame. The spectrum can be represented as a set of brightness temperatures $T(\ensuremath{i_{\ensuremath{\mathrm{}}}})$, where the frequency axis in the observatory frame is defined as

  $\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}...
...ath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}},
$ (1)
where \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}( \ensuremath{i_{\ensuremath{\mathrm{}}}}) is the frequency at the channel \ensuremath{i_{\ensuremath{\mathrm{}}}}, \ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{obs}}}} the frequency at the reference channel \ensuremath{i_{\ensuremath{\mathrm{0}}}}, and \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}} the channel spacing.