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The subject of this book can be broadly described as the principles of radio interferometry applied to the measurement of natural radio signals from cosmic sources. The uses of such measurements lie mainly within the domains of astrophysics, astrometry, and geodesy. As an introduction, we consider in this chapter the applications of the technique, some basic terms and concepts, and the historical development of the instruments and their uses.The fundamental concept of this book is that the image, or intensity distribution, of a source has a Fourier transform that is the two-point correlation function of the electric field, whose components can be directly measured by an interferometer. This Fourier transform is normally called the fringe visibility function, which in general is a complex quantity. The basic formulation of this principle is called the van Cittert–Zernike theorem(see Chap. ), derived in the 1930s in the context of optics but not widely appreciated by radio astronomers until the publication of the well-known textbook by Born and Wolf (). The techniques of radio interferometry developed from those of the Michelson stellar interferometer without specific knowledge of the van Cittert–Zernike theorem. Many of the principles of interferometry have counterparts in the field of X-ray crystallography (seeBeevers and Lipson ).

In this chapter, we provide a simplified analysis of interferometry and introduce several important concepts. We first consider an interferometer in one dimension and discuss the effect of finite bandwidth and show how the interferometer response can be interpreted as a convolution. We extend the analysis to two dimensions and discuss circumstances in which three-dimensional imaging can be undertaken. This chapter is intended to provide a broad introduction to the principles of synthesis imaging to facilitate the understanding of more detailed development in later chapters. A brief introduction to the theory of Fourier transforms is given in Appendix .

In this chapter, we introduce the full two-dimensional analysis of the interferometer response, without small-angle assumptions, and then investigate the small-field approximations that simplify the transformation from the measured visibility to the intensity distribution. There is a discussion of the relationship between the cross-correlation of the received signals and the cross power spectrum, which results from the Wiener–Khinchin relation and is fundamental to spectral line interferometry. An analysis of the basic response of the receiving system is also given. The appendix considers some approaches to the representation of noiselike signals, including the analytic signal, and truncation of the range of integration.

In this chapter, we start to examine some of the practical aspects of interferometry. These include baselines, antenna mounts and beam shapes, and the response to polarized radiation, all of which involve geometric considerations and coordinate systems. The discussion is concentrated on Earth-based arrays with tracking antennas, which illustrate the principles involved, although the same principles apply to other systems such as those that include one or more antennas in Earth orbit.

This chapter opens with a brief review of some basic considerations of antennas. The main part of the chapter is concerned with the configurations of antennas in interferometers and synthesis arrays. It is convenient to classify array designs as follows:

Examples of these types of arrays are described, and their spatial transfer functions (i.e., spatial sensitivities) are compared. Other concerns include the size and number of antennas needed in an array. Also discussed is the technique of forming images from direct Fourier transformation of the electric field on an aperture.

This chapter is concerned with the response of the receiving system that accepts the signals from the antennas, amplifies and filters them, and measures the cross-correlations for the various antenna pairs. We show how the basic parameters of the system affect the output. Some of the effects were introduced in earlier chapters and are here presented in a more detailed development that leads to consideration of system design in Chaps.  and . At some point in the processing chain between the antenna and the correlator output, the form of the signals is changed from an analog voltage to a digital format, and the resulting data are thereafter processed by computer-type hardware. This does not affect the mathematical analysis of the processing and is not considered in this chapter. However, the digitization introduces a component of quantization noise, which is analyzed in Chap. 

In this chapter, we consider certain aspects of the design of the interferometric system in more detail. This discussion primarily involves parts of the system where the signals are in analog form. The trend in technology has been to convert signals as early as possible in the signal chain, following the antennas, into digital form to facilitate data handling, avoid low-level distortions, and generally take advantage of the rapid progress in the development of digital equipment and computers. Three key items are discussed: (1) low noise amplification of signals at the antenna output to minimize the effect of additive noise, (2) phase-stable transmission systems that allow the transfer of reference timing and phase signals from the central communications hub of the instrument to the antennas, and (3) the synchronous phase switching systems needed to eliminate spurious responses in the correlator output. The analysis here leads to specification of tolerances on system parameters that are necessary to achieve the goals of sensitivity and accuracy.

The use of digital rather than analog instrumentation offers important practical advantages in the transmission of signals over long baselines, the implementation of compensating time delays, and the measurement of cross-correlation of signals. In digital delay circuits, the accuracy of the delay depends on the accuracy of the timing pulses in the system, and long delays accurate to tens of picoseconds are more easily achieved digitally than by using analog delay lines. Furthermore, there is no distortion of the signal by the digital units other than the calculable effects of quantization. In contrast, with an analog system, it is difficult to keep the shape of the frequency response within tolerances while delay elements are switched into and out of the signal channels. Correlators with wide dynamic range are readily implemented digitally, including those with multichannel output, as needed for spectral line observations. Analog multichannel correlators employ filter banks to divide the signal passband into many narrow channels. Such filters, when subject to temperature variations, can be a source of phase instability. Finally, except at the highest bit rates (frequencies), digital circuits need less adjustment than analog ones and are better suited to replication in large numbers for large arrays.

In 1967, a new technique of interferometry was developed in which the receiving elements were separated by such a large distance that it was expedient to operate them independently with no real-time communication link. This was accomplished by recording the data on magnetic tape for later cross-correlation at a central processing station. The technique was called very-long-baseline interferometry (VLBI), a term recalling the earlier long-baseline interferometers at Jodrell Bank Observatory, in which the elements were connected by microwave links that had reached 127 km in length. The principles involved in VLBI are fundamentally the same as those involved in interferometers with connected elements. The tape recorder and its successor, disk storage, can be considered as an IF delay line of limited capacity with an unusually long propagation time, weeks instead of microseconds. The use of tape and disk recording media is motivated entirely by economics and places substantial limitations on the system. Satellite links have been demonstrated (Yen et al. 1977), but their high cost discourages their use.

This chapter is concerned with the calibration and Fourier transformation of visibility data, mainly as applied to Earth-rotation synthesis. Methods for the evaluation of the visibility measurements on a rectangular grid of points, necessary for the use of the discrete Fourier transform as implemented with the fast Fourier transform (FFT) algorithm, are discussed. Phase and amplitude closure conditions, which are valuable calibration tools, are also described. Analysis of the causes of certain types of image defects is given. Special consideration is given for certain observing modes, such as spectral line, and conversion of frequency to velocity is described. In addition, methods of extracting astronomical information directly from visibility data by model fitting are described. These techniques are important even with arrays having excellent (, ) coverage. Some methods of calculating Fourier transforms before the advent of the FFT are discussed in Appendix 10.3.