Principle

Changing the source systemic velocity is a change of source reference frame. The signal and image frequency scales in the measurement frame must stay constant under this transform. However they change in the source reference frame. Indeed

  $\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{meas}}}...
...}}}) \, \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{rest}}}},
$ (54)
with
  $\displaystyle \ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{me...
...\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{meas}}}}}{\ensuremath{c}}.
$ (55)
Hence
  $\displaystyle \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{rest...
...}}{1+\ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}}.
$ (56)