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\section{Overview}  This document presents a parametric model to help design an Interferometric Array providing a total cost estimate for the resulting solution.  In particular, this document Array. It  describes the relationship of design parameters in section 2 with performance objectives in section 3. In particular, this document provides an explanation of the design parameters and objectives the array might have. These explanations enable the reader to both assess completeness of the model, and accuracy of the mathematical relationships as well.  Section 4 provides the python code used to generate data in the format required for visual analysis of array design options performance. The python code is consistent with parameters and objectives selection in sections 2 and 3, and the mathematical relationships between them.  \section{Variables}  This section aims to include all relevant design parameters that might influence the cost of the array. selected performance objectives in section 4.  \subsection{Antenna Aspects}  \subsubsection{Collecting Area}  \subsubsection{Temperature}  \subsubsection{Efficiency}  An overall antenna efficiency measure is 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}  SEFD = {\frac{T_{sys}}{\frac{\eta_a A}{2k_B}}}  \end{equation}  in units of Janskys 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}).  \subsection{Pad Aspects}  \subsubsection{Quantity}  \subsubsection{Position} 
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\section{Objectives}  This section aims to include array performance objectives that might be influenced by design choices.  \subsection{Minimize Brightness Sensitivity Limit}  An overall antenna performance measure is 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}  SEFD = {\frac{T_{sys}}{\frac{\eta_a A}{2k_B}}}  \end{equation}  in units of Janskys 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 N apertures with the same $SEFD$, observing the same bandwidth $\Delta\nu$, during the same integration time $t_{int}$, 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{(N(N-1) \Delta \nu t_{int}}}}   \end{equation}  in units of Janskys per synthesized beam area with $\eta_s$ most important factor being correlator efficiency $\eta_c = \frac{\text{correlator sensitivity}}{\text{sesitivity of a perfect analog correlator having the same } t_{int}}$ (see \cite{sensitivity}).  \subsection{Operations Costs}  \subsubsection{Components reliability}  \subsubsection{Maintenance complexity}  \subsubsection{Calibration Software Costs}  \subsubsection{Calibration Hardware Costs}  \subsubsection{Power Consumption Cost}  \subsubsection{Re-configuration Systems Operation Cost}  \subsection{Up-front Costs}  \subsubsection{Cost of Antennas Construction}  A commonly used rule of thumb for the cost of an antenna is that it is proportional to $D^{\alpha}$, where $\alpha \approx 2.7$ for values of $D$ from a few meters to tens of meters (see \cite{moran}). For $N$ antennas of diameter $D$ meters with accuracy $\frac{\lambda}{16}$, where $\lambda$ is in millimeters we could use \cite{mmadesign} as an upper limit for Antenna construction cost. 
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\text{Correlator cost} = 2N^2 + 112N +1360  \end{equation}  \subsubsection{Cost of Re-configuration Systems Construction}  \subsection{Operations Costs}  \subsubsection{Components reliability}  \subsubsection{Maintenance complexity}  \subsubsection{Calibration Software Costs}  \subsubsection{Calibration Hardware Costs}  \subsubsection{Power Consumption Cost}  \subsubsection{Re-configuration Systems Operation Cost}  \section{Mathematical Formulation}  Thus if we are using 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}}  & & f(x) = {\frac{2k_B}{\eta_s \eta_a}}{\frac{T_{sys}}{\pi x^2 \sqrt{2 \Delta \nu \tau_{acc}}}}\\  & \text{subject to}  & & X_{ij} = M_{ij}, \; (i,j) \in \Omega, \\  &&& X \succeq 0.  \end{aligned}  \end{equation*}  \section{Array Performance Data Generation - Python Implementation}  \section{Visualization Tool Notes}  \section{Conversation notes}