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\section{Design Variables}  This section aims to include all relevant design parameters that might influence the selected performance objectives.  \subsection{Antenna Aspects}  \subsubsection{Antenna Effective Collecting \subsubsection{Collecting  Area} \subsubsection{Antenna Temperature}  \subsubsection{Antenna Pointing Accuracy} \subsubsection{Temperature}  \subsubsection{Efficiency}  \subsection{Pad Aspects}  \subsubsection{Quantity}  \subsubsection{Position}  \subsection{Receiver Aspects}  \subsubsection{Bandwidth}  \subsubsection{Reciever Temperature}  \subsubsection{Calibration Cost} \subsubsection{Temperature}  \subsubsection{Efficiency}  \subsection{Correlator Aspects}  \subsubsection{Position}  \subsubsection{Efficiency}  \section{Performance Objectives}  This section aims to includeall the  array performance objectives that might be influenced by design choices. \subsection{Minimize Brightness Sensitivity Limit}  It is often instructive to express the system temperature in terms of the System Equivalent Flux Density, $SEFD$, defined as the flux density of a source that would deliver the same amount of power (see \cite{sensitivity2}):   \begin{equation}\label{eq:system_equivalent_flux_density} 
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\end{equation}  where $T_{sys}$ is the system temperature including contributions from receiver noise, feed losses, spillover, atmospheric emission, galactic background and cosmic background, $k_B = 1.380 \times 10^{-23}$ Joule $K^{-1}$ is the Boltzmann constant, $A$ is the antenna collecting area (thus we could also write $\pi \cdot D^2$), and $\eta_a$ is the antenna efficiency with $\eta_a = \eta_{\text{surface efficiency}} \cdot \eta_{\text{aperture blockage}} \cdot \eta_{\text{feed spillover efficiency}} \cdot \eta_{\text{illumination taper efficiency}}$ (see \cite{antenna}).  If we assume each aperture has N apertures with  the same $SEFD$, observes obserbing  the same bandwidth $\Delta\nu$, during the same correlator accumulation time $\tau_{acc}$, then the weak-source limit in the sensitivity of a synthesis image of a single polarization is \begin{equation}\label{eq:sens}  \Delta I_m = {\frac{1}{\eta_s }}{\frac{SEFD}{\sqrt{2 }}{\frac{SEFD}{\sqrt{(N(N-1)  \Delta \nu \tau_{acc}}}} \end{equation}  in units of Janskys per synthesized beam area.   with $\eta_s$ most important factor being correlator efficiency $\eta_c = \frac{\text{correlator sensitivity}}{\text{sesitivity of a perfect analog correlator having the same } \tau_{acc}}$ (see \cite{sensitivity}).  Hence the  Thus if we are using x, vector $x = {\text{antenna diameter}, \text{antenna efficiency}, \}$,  the antenna diameter, as the optimization variable the problem we would like to solve is: \begin{equation*}  \begin{aligned}  & \underset{x}{\text{minimize}} 
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