Signal and image frequency axes in the rest frame

The discussion of Section [*] is still valid for both the signal and image frequency axes. Eq. [*] can easily be rewritten

  $\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig}}^{\ensuremath{\mathrm{rest...
...th{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},
$ (20)
and
  $\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima}}^{\ensuremath{\mathrm{rest...
...th{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},
$ (21)
with
  $\displaystyle \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensurem...
...m{\ensuremath{\mathrm{sys}}}}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{c}}.
$ (22)
\ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}} is simply the tuning frequency entered by the user. However, \ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{rest}}}} must be computed. The easiest way is to define the side band separation as
  $\displaystyle \ensuremath{SB_{\ensuremath{\mathrm{sep}}}^{\ensuremath{\mathrm{o...
...math{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{obs}}}} \right] }.
$ (23)
It is straightfoward to show that it is constant and equal to
  $\displaystyle \ensuremath{SB_{\ensuremath{\mathrm{sep}}}^{\ensuremath{\mathrm{o...
...= 2\,\ensuremath{f_\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}}.
$ (24)
In the rest (source) frame, the following relations hold
  $\displaystyle \frac{\ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{...
... = 1+\ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}{}.
$ (25)
The side band separation in the rest frame is defined as
  $\displaystyle \ensuremath{SB_{\ensuremath{\mathrm{sep}}}^{\ensuremath{\mathrm{r...
...ath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{rest}}}} \right] }.
$ (26)
It is then straightforward to show that
  $\displaystyle \ensuremath{SB_{\ensuremath{\mathrm{sep}}}^{\ensuremath{\mathrm{o...
...suremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}} \right] },
$ (27)
and
  $\displaystyle
\ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\math...
...uremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}} \right] }}.
$ (28)