deletions | additions       diff --git a/subsubsection_Survey_Speed_label_sec__.tex b/subsubsection_Survey_Speed_label_sec__.tex  index 4f169c3..0914caa 100644  --- a/subsubsection_Survey_Speed_label_sec__.tex  +++ b/subsubsection_Survey_Speed_label_sec__.tex 
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\bar{\Omega}_B = { \sum_k w_k \Omega_{Bk} \over \sum_k w_k }  \end{equation}  where for uniform weights $w_k=$const.\ and uniform frequency coverage over the band we can approximate by the integral  \begin{eqnarray} \begin{equation}  \bar{\Omega}_B&  =&  {1\over \nu_{max}-\nu_{min}}\,\int_{\nu_{min}}^{\nu_{max}} d\nu\,\Omega_B(\nu)  \qquad d\nu\,\Omega_B(\nu).  \end{equation}  Assuming the beam FWHM scales inversely by frequency, then  \bar{\Omega}_B = {\nu^2_0 \over \nu_{min}\,\nu_{max}}\,\Omega_B(\nu_0)  \qquad\qquad  \Omega_B(\nu) = \Omega_B(\nu_0)\,\left( {\nu_0 \over \nu}\right)^2 \\  & = & {\nu^2_0 \over \nu_{min}\,\nu_{max}}\,\Omega_B(\nu_0)  \end{eqnarray}  assuming \end{equation}  or equivalently $\bar{\Omega}_B =\Omega_B(\bar{\nu})$ where $\bar{\nu}=\sqrt{\nu_{min}\,\nu_{max}}$ is  the beam FWHM scales inversely by geometric mean  frequency. For full S-band $\nu_{min}=$2GHz, $\nu_{max}=$4GHz, and $\nu_{0}=$3GHz then \begin{equation}\label{eq:ombeff}  \bar{\Omega}_B= {9\over8}\,\Omega_B(\nu_0=3{\rm GHz}) \qquad \qquad\qquad  \bar{\theta}_P = \sqrt{9\over8}\,\theta_P(3{\rm GHz}) \end{equation} (this holds for any 2:1 band relative to the mid-band FWHM).