Import standard modules:
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import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from IPython.display import HTML
HTML('../style/course.css') #apply general CSS
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HTML('../style/code_toggle.html')
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When thinking about engineering and the development of an instrument from a practical point of view the key concept to get across is that any measurement is a loss in information and is noisy. In the case of astronomy, there is the potential to measure infinite frequency bandwidth, frequency resolution, time bandwidth, and time resolution across a $4\pi$ area of the sky. In reality, the instruments that we can build have limits on all these parameters, so we must be selective about which information we retain based on the scientific goals and engineering limitations when an instrument is build and how it is used.
EB:RW: above: When thinking in terms of engineering, and more specifically the development of an instrument from a practical point of view, the key thin to keep in mind is that *any measurement is a loss in information - and is noisy*. For astronomy, we would *want* to measure over an infinitely large bandwidth in both frequency and time, at an infinitely fine resolution in frequency and time, and across a $4\pi$ area of the sky. In *practice*, the instruments we **can** build have limits on all these parameters, and so we must choose what information we retain and what information we sacrifice... - based on our scientific goals, current engineering limitations, and budget when an instrument is conceived - taking into account how we intend to build it and how we intend to use it.
In the case of radio interferometry the instrument is the collection of receiving elements (be they dishes or dipole antennas or any number of exotic telescopes) which make up the array and the electronic chains used to detect, filter, and amplify the incoming signals. Each stage and component of the instrument has an intrinsic effect on the true sky signal. Much of the process of being able to produce a scientifically useful result from an interferometric array is being able to understand and correcting for (up to some accuracy) these intrinsic effects.
EB:RW: above: In radio interferometry, an *instrument* is what we call a specific *array* of receiving elements (be they dishes, dipole antennas, or any number of exotic telescopes) along with the associated electronic chains - whose purpose is to detect, filter, and amplify the incoming signals. Each stage and component of the instrument has an intrinsic effect on the true sky signal. Much of the work of producing a scientifically useful result from an interferometric array relies on understanding these intrinsic effects, and correctinging our final signal for them (up to some accuracy).
A mathematical framework for describing an instrumental effect on a polarized electromagnetic wave, called Jones Calculus, have been well developed. Jones notation has been further used to formulate the Radio Interferometric Measurement Equation (RIME) which is a generic equation which describes the transform of the original electromagnetic signal to the measured signal. The RIME is deceptively simple but fully encapsulates the effects of the radio interferometer system along with any intervening effects such as the ionosphere, atmosphere, intersteller medium, etc.
EB:RW: above: There exists a well-developped mathematical framework for describing an instrumental effect on a polarized electromagnetic wave: Jones Calculus. The notation from this formalism has been extended to formulate the *Radio Interferometric Measurement Equation (RIME)*, a generic equation which describes the transformation of the original electromagnetic signal into the final measured signal. The RIME is deceptively simple, but fully encapsulates the effects of the instrument (i.e. the full radio interferometer system), along with any physical effects affecting the signal as it propagates from an astronomical source to the instrument (e.g. as the ionosphere, atmosphere, intersteller medium, etc).
This chapter will begin with the mathematical description of Jones Calculus and the RIME before moving on to explain the dominant instrumental effects seen in interferometric arrays. These effects are seperated into two types: direction-independent effects (DIEs) which affects signals from al direction of the sky in the same way and direction-dependent effects (DDEs) which vary based on the sky position of the where a signal originated. For example, the gain of a source due to it's position relative to the direction a dish is pointed is a direction-dependent effect. If the dish is moved the source will either increase or descrease in gain based on the beam pattern of the dish. And, the variations due to the electronic stability of the system is a direction-independent effect because the effect happens after the signal has been received.
EB:RW: above: This chapter will begin with a mathematical description of Jones Calculus and the RIME, before moving on to typical instrumental effects affecting interferometric arrays. These effects are seperated into two types: *direction-independent effects (DIEs)*, which affect signals in the same way regardless of sky direction, and *direction-dependent effects (DDEs)*, which vary depending on the sky position of a signal's source.
EB:RW: above: An example of a DIE is the signal variations due to the system's electronic stability, because the effect occurs after the signal has been received - and thus affects signals from every direction equally. An example of a DDE is the gain of a source due to its position relative to a dish's pointing direction: if the dish moves, the source's gain will change as a function of the dish's *beam pattern*.
As we will see in this chapter, the measurement of the sky will be affected by the design and stability of the system electronics, power pattern of the receiving elements, the type of mounts and pointing of dishes, the intervening mediums such as the atmosphere, and radio frequency interference (RFI) due to humans, terrestrial sources such as lightning, or the Sun.
EB:RW: above: As we will see in this chapter, an instrument's measurement of the sky will be affected by the design and stability of its electronics ($\S$ 7.3 ➞, $\S$ 7.4 ➞), the power pattern of its receiving elements ($\S$ 7.5 ➞), its type of mounts and dish pointing ($\S$ 7.5 ➞), intervening media such as the atmosphere ($\S$ 7.7 ➞), and radio frequency interference (RFI) ($\S$ 7.8 ➞), which are caused by various factors such as human activity, terrestrial sources such as lightning, or the Sun. </span
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Next: 7.1 Jones Notation