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\title{Interferometric Array Multi-Objective Visual Analytics}
\author{Demian Arancibia}
\affil{Affiliation not available}
\date{\today}
\maketitle
\section{Overview}\label{sec:intro}
This document presents a parametric model to help design an Interferometric Array. It focuses in the value vs. cost trade-off inherent to many of its architecture definitions. This is a Multiple Objective problem. This document describes design parameters to consider in \S~\ref{sec:var} and a set of equations for research and cost objectives in \S~\ref{sec:obj}.
A spreadsheet that uses these design parameters and produces a CSV file for analysis of the emerging Pareto Front is introduced in \S~\ref{sec:spreadsheet}. This output enables the of Multiple Objective Visual Analytics (MOVA) for complex engineered systems as proposed in \cite{mova}.
\section{Parameters}\label{sec:var}
This section presents selected design parameters that influence selected objectives in \S~\ref{sec:obj}. We will select design parameters that are specification agnostic. As an example of this, the parameters will be relevant to multiple antenna specifications, including offset Gregorian and symmetric Cassegrain.
\subsection{Antenna Parameters}
\subsubsection{Antenna Collecting Area}
We will use $A$ in this document as each array element collecting area (thus we could also write $\pi \cdot D^2$, with $D$ being the dish diameter).
\subsubsection{Antenna Efficiency}
We will use $\eta_a$ in this document as the antenna efficiency with
\begin{equation}\label{eq:antenna_efficiency}
\eta_a = \eta_{\text{surface eff.}} \cdot \eta_{\text{aperture blockage}} \cdot \eta_{\text{feed spillover eff.}} \cdot \eta_{\text{illumination taper eff.}}
\end{equation} as defined in \cite{antenna}.
\subsubsection{Antenna Quantity}
We will use $N$ in this document as the number of array elements.
\subsection{Antenna Pad Parameters}
\subsubsection{Pad Quantity}
We will use $P$ in this document as the number of pad built for the array. In case re-configuration of the array is envisioned, there might be a bigger number of pads ready for aperture connection to the system.
\subsubsection{Pad Position}
We will use the geographic latitudes and longitudes to establish pad location in this document. We will calculate the length of the possible baselines using pad positions. We will also calculate length and complexity of the roads, fiber and power networks needed using pad positions. We will use $B$ as the maximum array element separation in any single configuration.
\subsection{Reciever System Parameters}
\subsubsection{Number of Receivers per Array Element}
We will use $R$ as the number of frequency bands, being $R_i$ the different frequency bands. If the array bandwidth is $\lambda_{max} - \lambda_{min}$, it is useful for our analysis to use wavelength $\lambda = \lambda_{min}$.
Notes: high bandwidth ration: up to 7 might be practical, but could compromise Ae/Tsys. High absolute bandwidth is challenging for digitalization. up to 20GHz might be practical.
\subsubsection{Receivers Efficiency}
\subsubsection{Signal Processing and Transmision Parameters}
\subsubsubsection{Downconversion Scheme}
Notes: directly at RF (no reference), single sideband down conversion (LO and timing reference), double sideband (IQ) down conversion (two LO, two references, LO tunable.
\subsubsubsection{Instantaneus bandwidth}
\subsubsubsection{Quantization}
Bits per sample (dynamic range)
\subsection{Correlator Aspects}
\subsubsection{Position}
We will geographic latitude and longitude to establish correlator location in this document. We will calculate fiber, power and road network aspects based in this information.
\subsubsection{Efficiency}
We will use $\eta_c$ as correlator efficiency in this document, with
\begin{equation}\label{eq:correlator_efficiency}
\eta_c(t_{int}) = \frac{\text{correlator sensitivity}}{\text{sesitivity of a perfect analog correlator having the same } t_{int}}
\end{equation}
as defined in \cite{sensitivity}.
\section{Objectives}\label{sec:obj}
This section aims to include array performance objectives that might be influenced by design variables in \S~\ref{sec:var}.
\subsection{Fourier Plane Coverage}
As derived in \cite{design}, the antenna diameter determines its beam size $\theta_{ant} \approx \frac{\lambda}{D}$. If the plane area $\frac{B}{\lambda}$ is divided in cells of size $\frac{D}{\lambda}$ then
\begin{equation}\label{eq:fourier}
N_{occ} \leqslant \pi (\frac{B}{D})^2
\end{equation}
\subsection{Point Source Sensitivity}
An overall measure of performance is the System Equivalent Flux Density, $SEFD$, defined in \cite{sensitivity} as the flux density of a source that would deliver the same amount of power:
\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, and $k_B = 1.380 \times 10^{-23}$ Joule $K^{-1}$ is the Boltzmann constant.
According to \cite{sensitivity}, if we assume N apertures with the same $SEFD$, observing the same bandwidth $\Delta\nu$, during the same integration time $t_{int}$, then 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 $\eta_c$.
\subsection{Surface Brightness 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}
According to \cite{moran}, 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. 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.
\begin{equation}\label{eq:antenna_cost}
\text{Antenna Cost} = \frac{890N(\frac{D}{10})^{2.7}}{(\lambda^{0.7})} + 500
\end{equation}
in $K\$$.
\subsubsection{Cost of Front-end system}
For $M$ frequency bands, each 30\% wide, and dual polarization we could use \cite{mmadesign} as an upper limit for Front-End System Cost:
\begin{equation}\label{eq:fe_cost}
\text{Front-End System Cost} = 45MN + 200M
\end{equation}
in $K\$$.
\subsubsection{Cost of LO system}
We could use \cite{mmadesign} as an upper limit for LO System Cost:
\begin{equation}\label{eq:lo_cost}
\text{LO System Cost} = 80N+100
\end{equation}
in $K\$$.
\subsubsection{IF Transmission Cost}
We could use \cite{mmadesign} as an upper limit for IF Transmission Cost:
\begin{equation}\label{eq:IF_Tx_cost}
\text{IF Transmission Cost} = 8BN + 30N + 400
\end{equation}
in $K\$$.
\subsubsection{Correlator Cost}
We could use \cite{mmadesign} Correlator Cost as an upper limit:
\begin{equation}\label{eq:correlator}
\text{Correlator cost} = 2N^2 + 112N +1360
\end{equation}
in $K\$$.
\subsubsection{Cost of Re-configuration Systems Construction}
\section{Data for visual analytics - Spreadsheet implementation}\label{sec:spreadsheet}
This section presents a spreadsheet that produces data in the right format for performing visual analytics, consistent with variables in \S~\ref{sec:var} and objectives in \S~\ref{sec:obj}.
\section{Visualization Tool Notes}
\section{Conversation notes}
\subsection{Engineering cost vs. Calibration cost}
Tricky because you can compensate antenna quality with software. So the equations must capture this trade off.
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