next up previous contents
Next: Signal and image frequency Up: Heterodyne receivers and signal/image Previous: USB and LSB frequency   Contents

Signal and image frequency axes in the observatory frame

The RF tuning frequency, \ensuremath{f_\ensuremath{\mathrm{RFtuned}}^{\ensuremath{\mathrm{obs}}}}, can be choosen in either band. This band is called the signal band. The other band is then called the image band. The associated RF frequencies are written \ensuremath{f_\ensuremath{\mathrm{sig}}^{\ensuremath{\mathrm{obs}}}} and \ensuremath{f_\ensuremath{\mathrm{ima}}^{\ensuremath{\mathrm{obs}}}}. We obtain

\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{sig}}^{\ensuremath{\mathr...
..._\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}},
\end{displaymath} (55)

and
\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{ima}}^{\ensuremath{\mathr...
..._\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}}.
\end{displaymath} (56)

The sign of the product $\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\,\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}$ is arbitrary. Indeed, it only defines how the brightnesses are ordered in memory, i.e., whether the frequency increases or decreases when \ensuremath{i_{\ensuremath{\mathrm{}}}} increases. It thus is simpler to assume that this product is always positive. This implies that the signal frequency increases and the image frequency decreases when \ensuremath{i_{\ensuremath{\mathrm{}}}} increases. In other words, we adopt the convention that the signal brightnesses are ordered by increasing frequency when \ensuremath{i_{\ensuremath{\mathrm{}}}} increases, i.e., the product $\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\,\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}$ is always positive.

As the absolute value of \ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}} is 1, we will greatly simplify the equations by replacing $\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\,\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}$ with \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}} with the convention that $\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}} > 0$ in Eqs 55 and 56. We can then rewrite them as

\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{sig}}^{\ensuremath{\mathr...
..._\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}},
\end{displaymath} (57)

and
\begin{displaymath}
\ensuremath{f_\ensuremath{\mathrm{ima}}^{\ensuremath{\mathr...
..._\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}}.
\end{displaymath} (58)

with
\begin{displaymath}
\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}} > 0.
\end{displaymath} (59)


next up previous contents
Next: Signal and image frequency Up: Heterodyne receivers and signal/image Previous: USB and LSB frequency   Contents
Gildas manager 2015-03-01