\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 § \ref{sec:var} and a set of equations for research and cost objectives in § \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 § \ref{sec:spreadsheet}. This output enables the of Multiple Objective Visual Analytics (MOVA) for complex engineered systems as proposed in (Woodroof 2013).

\label{sec:var} This section presents selected design parameters that influence selected objectives in § \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.

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).

We will use \(\eta_a\) in this document as the antenna efficiency with \[\label{eq:antenna_efficiency} \eta_a = \eta_{\text{surface eff.}} \cdot \eta_{\text{aperture blockage}} \cdot \eta_{\text{feed spillover eff.}} \cdot \eta_{\text{illumination taper eff.}}\] as defined in (Napier 1999).

We will use \(N\) in this document as the number of array elements.

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.

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.

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.

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.

Bits per sample (dynamic range)

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.

We will use \(\eta_c\) as correlator efficiency in this document, with \[\label{eq:correlator_efficiency} \eta_c(t_{int}) = \frac{\text{correlator sensitivity}}{\text{sesitivity of a perfect analog correlator having the same } t_{int}}\] as defined in (Crane 1989).

\label{sec:obj} This section aims to include array performance objectives that might be influenced by design variables in § \ref{sec:var}.

As derived in (Hjellming 1989), 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 \[\label{eq:fourier} N_{occ} \leqslant \pi (\frac{B}{D})^2\]

An overall measure of performance is the System Equivalent Flux Density, \(SEFD\), defined in (Crane 1989) as the flux density of a source that would deliver the same amount of power: \[\label{eq:system_equivalent_flux_density} SEFD = {\frac{T_{sys}}{\frac{\eta_a A}{2k_B}}}\] 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 (Crane 1989), 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 \[\label{eq:sens} \Delta I_m = {\frac{1}{\eta_s }}{\frac{SEFD}{\sqrt{(N(N-1) \Delta \nu t_{int}}}}\] in units of Janskys per synthesized beam area, with \(\eta_s\) most important factor being correlator efficiency \(\eta_c\).

According to (Thomson 2001), 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 (Brown 1987) as an upper limit for Antenna construction cost. \[\label{eq:antenna_cost} \text{Antenna Cost} = \frac{890N(\frac{D}{10})^{2.7}}{(\lambda^{0.7})} + 500\] in \(K\$\).

For \(M\) frequency bands, each 30% wide, and dual polarization we could use (Brown 1987) as an upper limit for Front-End System Cost: \[\label{eq:fe_cost} \text{Front-End System Cost} = 45MN + 200M\] in \(K\$\).

We could use (Brown 1987) as an upper limit for LO System Cost: \[\label{eq:lo_cost} \text{LO System Cost} = 80N+100\] in \(K\$\).

We could use (Brown 1987) as an upper limit for IF Transmission Cost: \[\label{eq:IF_Tx_cost} \text{IF Transmission Cost} = 8BN + 30N + 400\] in \(K\$\).

We could use (Brown 1987) Correlator Cost as an upper limit: \[\label{eq:correlator} \text{Correlator cost} = 2N^2 + 112N +1360\] in \(K\$\).

\label{sec:spreadsheet} This section presents a spreadsheet that produces data in the right format for performing visual analytics, consistent with variables in § \ref{sec:var} and objectives in § \ref{sec:obj}.

Tricky because you can compensate antenna quality with software. So the equations must capture this trade off.

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