Demian Arancibia

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OVERVIEW 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 § [sec:var] and a set of equations for research and cost objectives in § [sec:obj]. A spreadsheet that uses these design parameters and produces a CSV file for analysis of the emerging Pareto Front is introduced in § [sec:spreadsheet]. This output enables the of Multiple Objective Visual Analytics (MOVA) for complex engineered systems as proposed in . PARAMETERS This section presents selected design parameters that influence selected objectives in § [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. Antenna Parameters Antenna Collecting Area We will use A in this document as each array element collecting area (thus we could also write π ⋅ D², with D being the dish diameter). Antenna Efficiency We will use ηa in this document as the antenna efficiency with \eta_a = } \cdot } \cdot } \cdot } as defined in . Antenna Quantity We will use N in this document as the number of array elements. Antenna Pad Parameters 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. 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. Reciever System Parameters Number of Receivers per Array Element We will use R as the number of frequency bands, being Ri the different frequency bands. If the array bandwidth is λmax − λmin, it is useful for our analysis to use wavelength λ = λmin. 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. Receivers Efficiency Signal Processing and Transmision Parameters 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) Correlator Aspects 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. Efficiency We will use ηc as correlator efficiency in this document, with \eta_c(t_{int}) = }{ t_{int}} as defined in . OBJECTIVES This section aims to include array performance objectives that might be influenced by design variables in § [sec:var]. Fourier Plane Coverage As derived in , the antenna diameter determines its beam size $ \approx {D}$. If the plane area ${\lambda}$ is divided in cells of size ${\lambda}$ then N_{occ} \leqslant \pi ({D})^2 Point Source Sensitivity An overall measure of performance is the System Equivalent Flux Density, SEFD, defined in as the flux density of a source that would deliver the same amount of power: SEFD = {}{{2k_B}}} in units of Janskys where Tsys is the system temperature including contributions from receiver noise, feed losses, spillover, atmospheric emission, galactic background and cosmic background, and kB = 1.380 × 10−23 Joule K−1 is the Boltzmann constant. According to , if we assume N apertures with the same SEFD, observing the same bandwidth Δν, during the same integration time tint, then weak-source limit in the sensitivity of a synthesis image of a single polarization is \Delta I_m = {{\eta_s }}{{}}} in units of Janskys per synthesized beam area, with ηs most important factor being correlator efficiency ηc. Surface Brightness Sensitivity Operations Costs Components reliability Maintenance complexity Calibration Software Costs Calibration Hardware Costs Power Consumption Cost Re-configuration Systems Operation Cost Up-front Costs Cost of Antennas Construction According to , a commonly used rule of thumb for the cost of an antenna is that it is proportional to Dα, where α ≈ 2.7 for values of D from a few meters to tens of meters. For N antennas of diameter D meters with accuracy ${16}$, where λ is in millimeters we could use as an upper limit for Antenna construction cost. = {10})^{2.7}}{(\lambda^{0.7})} + 500 in K$. Cost of Front-end system For M frequency bands, each 30% wide, and dual polarization we could use as an upper limit for Front-End System Cost: = 45MN + 200M in K$. Cost of LO system We could use as an upper limit for LO System Cost: = 80N+100 in K$. IF Transmission Cost We could use as an upper limit for IF Transmission Cost: = 8BN + 30N + 400 in K$. Correlator Cost We could use Correlator Cost as an upper limit: = 2N^2 + 112N +1360 in K$. Cost of Re-configuration Systems Construction DATA FOR VISUAL ANALYTICS - SPREADSHEET IMPLEMENTATION This section presents a spreadsheet that produces data in the right format for performing visual analytics, consistent with variables in § [sec:var] and objectives in § [sec:obj]. VISUALIZATION TOOL NOTES CONVERSATION NOTES Engineering cost vs. Calibration cost Tricky because you can compensate antenna quality with software. So the equations must capture this trade off.