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In numerical simulations of planetary dynamos there is an abrupt transition in the dynamics of both the velocity and magnetic fields at a ‘local’ Rossby number of 0.1. For smaller Rossby numbers there are helical columnar structures aligned with the rotation axis, which efficiently maintain a dipolar field. However, when the thermal forcing is increased, these columns break down and the field becomes multi-polar. Similarly, in rotating turbulence experiments and simulations there is a sharp transition at a Rossby number of
${\sim}0.4$
. Again, helical axial columnar structures are found for lower Rossby numbers, and there is strong evidence that these columns are created by inertial waves, at least on short time scales. We perform direct numerical simulations of the flow induced by a layer of buoyant anomalies subject to strong rotation, inspired by the equatorially biased heat flux in convective planetary dynamos. We assess the role of inertial waves in generating columnar structures. At high rotation rates (or weak forcing) we find columnar flow structures that segregate helicity either side of the buoyant layer, whose axial length scale increases linearly, as predicted by the theory of low-frequency inertial waves. As the rotation rate is weakened and the magnitude of the buoyant perturbations is increased, we identify a portion of the flow which is more strongly three-dimensional. We show that the flow in this region is turbulent, and has a Rossby number above a critical value
$Ro^{crit}\sim 0.4$
, consistent with previous findings in rotating turbulence. We suggest that the discrepancy between the transition value found here (and in rotating turbulence experiments), and that seen in the numerical dynamos (
$Ro^{crit}\sim 0.1$
), is a result of a different choice of the length scale used to define the local
$Ro$
. We show that when a proxy for the flow length scale perpendicular to the rotation axis is used in this definition, the numerical dynamo transition lies at
$Ro^{crit}\sim 0.5$
. Based on this we hypothesise that inertial waves, continually launched by buoyant anomalies, sustain the columnar structures in dynamo simulations, and that the transition documented in these simulations is due to the inability of inertial waves to propagate for
$Ro>Ro^{crit}$
.
In this study, monolingual (English) and bilingual (English/Spanish, English/Urdu) five- and six-year-old children completed a grammaticality judgment test in order to assess their awareness of the grammaticality of two types of syntactic constructions in English: word order and gender representation. All children were better at detecting grammatically correct and incorrect word order constructions than gender constructions, regardless of language group. In fact, bilingualism per se did not impact the results as much as receptive vocabulary range. For example, children with the highest receptive vocabulary scores were more accurate in detecting incorrect word order constructions (i.e., word order violations, semantic anomalies) and incorrect gender agreement than children in the lower receptive vocabulary ranges. However, no differences were found between the ranges for ambiguous gender constructions. These results highlight the importance of receptive vocabulary ability on syntactic awareness performance, regardless of language group.
Phased arrays date back to the very earliest days of radio. The German physicist Karl Ferdinand Braun constructed a three element, switchable array in 1909 to enhance radio transmission in one direction. Early phased arrays achieved beam steering through applying a progressive phase to each element of a one- or two-dimensional array; the concept may be found in almost every book on antenna theory, e.g. [1]. The contemporary usage extends to include control of both the amplitude and phase (or time-delay) excitations of each radiating element in a multiantenna system [2].
While the analytical tools covered in this book are applicable to phased array antennas for all applications, the concepts and examples in the book are organized around the design and optimization of high-sensitivity radio frequency and microwave receivers. Radio astronomy is an especially challenging application of this technology, and will feature strongly in this book. Although parabolic dishes have dominated antenna technology since the early 1960s, to the point where dishes have become largely synonymous with radio telescopes in the popular imagination,1 many early discoveries in radio astronomy were made using phased arrays [3]. The same is true for the large dishes (often over 30 m in diameter) used by telecommunication ground stations and for deep space tracking in the same timeframe; but again, phased arrays were far from forgotten, playing an important role in the first Approach and Landing System (ALS) and post- WWII early warning systems.
Parabolic dishes have probably reached the apogee of their design in recent years, and since they are fundamentally large mechanical systems, their cost is dominated by the cost of materials and labour – neither of which is likely to change dramatically in the foreseeable future. In the radio astronomy community, the currently accepted guideline is that the cost of a dish scales since the area only increases as, building ever-larger steerable dishes is clearly not a viable method for increasing sensitivity, which is directly proportional to collecting area. Additionally, steerable dishes in particular involve moving parts, bringing significant maintenance requirements. Phased arrays, on the other hand, are fundamentally electronic systems, whose cost is increasingly dominated by processing. Moore's Law provides the prospect of continuing – and dramatic – reductions in processing costs.
Computational electromagnetics (CEM), the numerical solution of Maxwell's equations, is now a well-established field [1]–[3]. Growing steadily from pioneering work in the 1960s, a range of commercial CEM tools is now available.Well known general purpose software packages in the field are FEKO (Altair), HFSS (Ansys), and Microwave Studio (CST); specifically for reflector analysis, GRASP (TICRA) is a leading code. There are also many public domain codes available, with the venerable NEC-2 code among the most popular. From the perspective of the underlying algorithm, the most widely used methods are the method of moments (MoM), finite element method (FEM) and the finite difference time domain method (FDTD). Many textbooks on computational electromagnetics are available; a basic introduction to FDTD, MoM, and FEM can be found in [4].
Large, broadband aperture and phased array antenna systems can have characteristic dimensions beyond the size that commercial modeling tools can accommodate within a reasonable computation time or memory requirement. Array antenna systems exhibit multiscale features and include both dielectric and metal materials, and elements in the array are strongly coupled. Simulations over a wide range of frequencies and beam scan angles must be considered, and for PAFs the model must include a large reflector. In particular for PAFs, edge effects are important, so analytical infinite array approximations have limited value and full wave techniques are required. Given the computationally intensive nature of the analysis, optimization of astronomical antenna systems by evaluating the entire structure in full detail is challenging.
For radio astronomy applications in particular, designers try to minimize the inclusion of significant volumes of dielectric material in the phased array antennas, to avoid the losses which dielectrics usually bring with them. As a result, the MoM is usually the most efficient technique, as it handles highly (or perfectly) conducting antennas in a very efficient formulation, and using surface or volume equivalence, dielectric materials can be included in the analysis. Since many commercial software packages are built around FEM and FDTD, it is also common for these methods to be used in the design of antenna systems for high sensitivity receiver applications.
The single antenna receiver reviewed in Chapter 2 can be fully characterized using classical antenna terms, such as directivity, gain, radiation efficiency, noise figure, noise temperature, and sensitivity. For an active array receiver, only the first and last of these figures of merit (directivity and sensitivity), are well-defined concepts. The pattern directivity is well defined and in principle can be measured for any receiver (see Sec. 5.2), and sensitivity can be determined from the SNR achieved with a signal of interest of known intensity. The other antenna terms require special treatment of the effects of antenna losses and amplifier stages and other nonreciprocal components in the active array system. In this chapter, we use the analytical framework developed in Chapters 4 and 5 to define efficiencies, antenna gain, system noise, and other figures of merit in a way that properly accounts for antenna losses and active, nonreciprocal components in the active antenna receiver system.
Specialized figures of merit have been widely used in the phased array antenna community for many decades. For transmitting arrays, the power radiated by one element in the array accounting for active impedance mismatch is measured by the element efficiency of Hannan [1] for transmitting arrays. To characterize the performance of a receiving array, the array signal processing community employs the SNR gain of the array relative to a single sensor, or the array gain, which is well defined and can easily be applied to any receiver system, no matter how complicated. All that is required is an identifiable output voltage or power and a measure of the signal and noise contributions at the output, so that the SNR at the output of the receiver system can be quantified. The multiple input multiple output (MIMO) communications community has developed other figures of merit that focus on measuring the performance of MIMO antenna arrays in terms of antenna diversity and channel capacity.
For high sensitivity applications, where optimizing the system noise performance is particularly critical to achieving stringent design goals, fine characterization of the efficiency and noisiness of critical components and subsystems has required the development of additional figures of merit for active array receivers. The most recent revision of the IEEE Standard for Definitions of Terms for Antennas [2] includes new terms for active antenna arrays that have resulted from research in the field of high sensitivity phased array receivers for radio astronomy.
The classical approach to array antenna analysis is to represent the field radiated by the array as a product of the radiation pattern of one element and an array factor that includes the locations and excitations of each element in the array. This approach has been used for many decades to design phased arrays for many applications, from radar and terrestrial communications to satellite systems and radio telescopes.
For high performance receiving arrays, the approximate factorization of the array radiation pattern into an element pattern and array factor may not be accurate enough to use when designing for stringent performance requirements. Mutual coupling causes element radiation patterns to differ across the array, and this must be taken into account in the design process from the beginning. For this reason, we will only briefly consider the classical array factor technique in this chapter, and move instead to a more sophisticated analysis method based on overlap integrals and network theory.
The array factor method certainly has enduring value. The array factor provides an intuitive way to analyze, understand, visualize, and design steered beams and array radiation pattern. It can be taught in a simple way to students of array theory, yet is useful in designing highly sophisticated multiantenna systems. Optimal designs, such as the Chebyshev array, can be readily treated within this framework.
For the high sensitivity applications of interest in this book, the array factor method is only useful for rough designs, and a more advanced approach is needed. The overlap integral and network theory approach comes at a price. It relies on numerical approximations or full wave simulations of embedded element patterns, and is not amenable to pencil and paper treatments. The analysis methodology used in this book is intended from the ground up to be used with numerical methods and optimization tools. As a middle ground between the array factor method and full wave numerical modeling, we will also develop the lossless, resonant, minimum scattering approximation.
The standard approach to antenna theory is to treat transmitting antennas first, followed by the receiving case. After reviewing the classical array factor method and other basic concepts of simple array antennas, we will develop the overlap integral and network theory first for a transmitting array.
The treatment of single antennas in Chapter 2 introduced key concepts for receivers, including gain, directivity, effective area, impedance matching, noise figure, equivalent noise temperature, signal to noise ratio, and sensitivity. The treatment was then extended in Chapter 4 to arrays of transmitting elements. We now turn our attention to the main focus of the book, modeling high-sensitivity receiving antenna arrays.
For transmitters, the distribution of the signal power radiated by the antenna system is the primary consideration. Noise radiated by a transmitter is usually of secondary importance. For receivers, both the signal and noise response of the system are important, and the ultimate figure of merit for the performance of the system is the ratio of received signal of interest to the system noise (SNR). For this reason, the treatment of receiving arrays is more complicated than transmitters, as system noise must be brought into the analysis.We will begin by extending the network theory treatment developed in Chapter 4 to receiving arrays, and then bring in the noise modeling concepts that were introduced in Chapter 2.
Receiving Array Network Model
As a model for a receiving array, we will consider a basic narrowband active receive array architecture consisting of antenna elements terminated by low noise amplifiers, followed by receiver chains and a beamforming network which applies a complex gain constant (magnitude scaling and phase shift) to the signal from each element and sums the weighted signals to form a single scalar output, as in Fig. 5.1.
The canonical block diagram shown in Fig. 5.1 can represent many different types of antenna array receiver systems used in a variety of applications, and there are many variations on this basic architecture. Signals may be combined with an analog network, or the beamforming can be done using sampling followed by digital signal processing. For broadband applications, beamforming can be done with a time delay network, but more commonly the signal is processed digitally in frequency subbands using a narrowband beamformer architecture. Digital signal processing systems can form many simultaneous beams, so that the beamformer block is repeated in parallel for each formed beam.
In common with many other fields of electrical and electronic engineering, real-time digital signal processing (DSP) has revolutionized the processing of datastreams from phased arrays. As will be appreciated from the discussions in this book, modern radio telescopes such as LOFAR and MWA include beamformed stations (comprising individual elements) which are then correlated with each other to form the interferometric array; similarly, the ASKAP and APERTIF designs have PAF feeds on dishes, which again are correlated. In this chapter, we consider several threads of real-time DSP for correlation, beamforming and frequency channelizing.
We start by addressing real-time DSP for interferometers; here, the major cost is at the correlator stage, and this is the first topic considered, following the presentation in [1] for the theoretical background. On the one hand, many of the operations required are relatively simple – multiply and accumulate (integrate) – but there is also the requirement to perform very rapid Fourier transforms. Of course, the fast Fourier transform (FFT) is the key tool here. Aperture arrays and phased array feeds typically require Fourier transforms and correlators for array calibration and for observation-mode array signal processing, so these computational blocks have applications to all of the types of arrays considered in this book.
We then consider beamforming. Beams can be computed in real time, or if accumulated beam power estimates per channel rather than beamformer voltage time series sample outputs are sufficient for a given observation, then the array outputs can be correlated, followed by post-correlation beamforming. The major computational load for real time beamforming is the formation of weighted sums of array outputs. Array calibration with a correlator is computationally intensive, but the calculation of the beamformer weights themselves is generally negligible from a loading perspective. In most current applications, beamformer weights are computed infrequently, and this is done offline, not in real time. As the array response changes due to instrumental electronic drift, structural deformations, or ionospheric effects, beamformer weights may need to be updated at periods ranging from weeks at L band to minutes at mmwave frequencies. In the RFI mitigation scenario, fast relative motion between the SoI and interfering sources implies the need for rapid recalculation of weights (on the order of 1 to 500 ms).
The general disciplines of calibrating phased arrays, constructing beam weighting coefficients, and performing computations on the output signals from a phased array system, all belong to broad field known as array signal processing. Basic topics from array signal processing, as well as advanced topics such as radio frequency interference mitigation, are surveyed in this chapter. For phased arrays used in demanding applications like radio astronomy, a priori array calibration methods generally are insufficiently accurate. In practice, array calibration generally involves measured signal responses or correlations with bright sources, so calibration is included here in the same treatment as beamforming and signal processing.
Beamforming
In the context of antenna array receivers, beamforming is the process of linearly weighting and combining signals from array elements in order to form a desired spatial response pattern. An example of response pattern is shown in Fig. 10.1. Beamforming can be viewed as spatial filtering where the discrete-in-space samples of the propagating wavefront (i.e., the outputs of the distinct antenna elements of the array, each in a different position) are used as inputs to a linear filter. Typically the beamformer is designed by adjusting the element weights to achieve higher gain, directivity, sensitivity, and signal to noise ratio than is possible from any single array antenna element. In the process, the high gain field of view is narrowed to a relatively small directional region know as the beam main lobe, or simply the beam. The lower response peaks outside this main lobe region are undesirable artifacts of the beamforming process and are pattern sidelobes.
Beams may be steered to a desired direction by inserting time delays in the signal path of each element to compensate for differential propagation delays across the array for wavefronts arriving from that direction. These time aligned signals sum coherently in the beamformer and so higher gain is achieved in the desired steering direction. For narrowband signals this steering time delay may be replaced by simply multiplying element signal streams by the equivalent complex phase shift e−jωkτi where ωk is the narrowband subband center radian frequency for the kth channel and τi is the alignment time delay correction for the ith element (this is equivalent to the phase shift derived in Sec. 4.1).
In this chapter, we turn from the theory of arrays and array feeds and analysis methods to the design of phased array antennas. As in Chapter 10, the application focus is primarily on radio astronomy, since these are often the systems with the most challenging specifications – due to the extremely weak signals involved. As indicated in Chapter 1, phased arrays in radio astronomy are used in two quite distinct configurations: aperture arrays (AAs), and phased array feeds (PAFs). Recapping briefly, AAs have a direct view of the sky, whereas PAFs lie in the focal plane of a reflector. The design considerations for AAs and PAFs diverge substantially.
AAs tend to involve very large numbers of elements (the EMBRACE system to be discussed contained around 20 000 elements, and SKA1-LOW is planned to have more than 100 000), whereas PAFs usually involve tens or hundreds of elements. For AAs, there are important system considerations, including whether a sparse or dense configuration should be adopted, and additionally, whether the topology of the array should be regular or random (irregular). These choices are usually driven by the desired application(s) – which in radio astronomy are the science cases – and of course cost. In particular for AAs operating at the lower radio astronomy frequencies (typically the VHF radio band), the radio sky is so noisy that these systems are sky noise dominated, permitting different design optimizations. For some of these designs, infinite array analysis offers useful design information. Furthermore, for a typical radio telescope, the clustering of elements into stations, and the overall layout of these stations for interferometric imaging (the topic of Chapter 11) are also important design issues, requiring optimization.
PAFs generally operate in a different milieu in radio astronomy, typically in the UHF bands (and higher). Here, very high performance and ultra-low noise characteristics are crucial, the number of elements is limited, edge effects of finite arrays are very important, and the full theory developed earlier in this book must be brought to bear on their design. Additionally, the interaction with the reflector must also be included. Many of these systems are retro-fitted to existing dishes, so the interferometric imaging considerations have already been addressed in the original design, but there are newbuilds with PAFs – most notably ASKAP – where this must also be taken into account.
Many excellent books on phased array antennas are already in print. The theoretical and mathematical content commonly found in books on this topic has stabilized in the last decade or two, but the past ten years has seen a dramatic departure from classical analytical tools in the methodologies used to design and optimize phased arrays. Our goal is to gather in one place recent advances in the mathematical framework for phased array analysis and create a book for which the theoretical treatment reflects the state-ofthe- art in the academic literature and is equal to the task of designing antenna arrays for applications with demanding performance requirements.
One of the themes of this book is the design of phased arrays based on computer simulation. The last few decades have seen enormous progress in computational electromagnetics (CEM), which has revolutionized our ability to analyze antennas rigorously. The framework developed in this book can be used either with analytical approximations for the antenna response, or with computational electromagnetics tools for more accurate results.
Another theme is the dedicated analysis of active receiving arrays using network theory and the signal correlation matrix formulation. Most antenna textbooks focus almost exclusively on the antenna as a transmitting system, and then mention that the reciprocity theorem extends this analysis to receiving systems. Modern phased array systems involve active front ends, with low noise amplifiers in close proximity to the antenna or even integrated into the antenna feed. The system in this case is nonreciprocal, and some conventional antenna concepts no longer apply. Many systems use digital beamforming, which provides additional degrees of freedom when compared to classical analog beamformers. These systems require a new vocabulary and new figures of merit to properly describe them; where possible, we reconcile these with traditional antenna terms.
The authors have been closely involved in modern developments in phased arrays, computational electromagnetics, and array signal processing for applications in challenging fields such as radio astronomy for many years. This text represents the first synthesis of these modern design methods as a book; this is complemented by a review of classical methods such as the array factor approximation.
Up to now, the focus of this book has been on phased arrays, or beamformers. For radio astronomy, another type of array, namely interferometers or correlator arrays or synthesis arrays are very important. For mapping applications (such as sky surveys), interferometers gather information more efficiently than phased arrays [1, Sec. 5.3], which is why many of the existing and proposed array-based astronomical instruments described earlier in this book are in fact hierarchical arrays. Each dish may have a PAF array feed or each station may consist of an aperture array, and the dishes and stations are in turn part of a sparse synthesis array which may extend over very long distances – including into space in VLBI.
The key difference between a phased array and a correlator array is the method by which beam-scanning is accomplished. In the former, a phased array points the narrow beam formed in a particular direction, which then must be scanned over the field of interest, whereas at any instant, a correlator array responds simultaneously to the whole field of view of the individual antennas (i.e., dishes or stations), also known as the primary beam. (Here, we assume that the phased array forms only one beam simultaneously.) Interferometry accomplishes this by Fourier transforming the cross-correlation products of each receiver's voltages with every other. The complex exponentials in the Fourier integral can be seen as simultaneously weighting these correlations with appropriate phase variations, thus effectively scanning the beam with one pointing. Modern correlation arrays also usually compute the self-products, which also permit classical beamforming to be accomplished.
Modern sparse synthesis array calibration relies on regular on-the-fly measurements of known calibrator sources, and also uses a method known as self-calibration; work is presently in progress on methods suitable for even higher dynamic range and incorporating more subtle instrumentation effects. In this chapter, we review these methods, observing that calibration and imaging for compact arrays and these sparse synthesis arrays are somewhat different procedures. The basics of sparse synthesis array calibration and interferometric imaging have been known for decades, but new systems, in particular aperture arrays, have thrown up new challenges in this field.
Having discussed the basic theory and approaches to phased array receiver modeling, as well as figures of merit and system characterization, we now turn to specific details of the design and fabrication of the front end aperture itself. A wide variety of element types and configurations have been explored. Chief examples include the wideband sinuous element of the early explorations of PAFs at NRAO [1], dipole elements for PAFs [2], [3] as well as aperture arrays [4], tapered slot antennas or Vivaldi elements [5], [6], derivatives of the TSA element such as the egg-crate array, the checkerboard array [7]–[9] and other current-sheet implementations, horn elements [10], and of course the ubiquitous microstrip patch antenna [11] and the patch excited cup antenna designed by RUAG Space [12]. Many of these designs were surveyed in Chapter 1. In this chapter, we review considerations on selecting an appropriate element type, methods for design optimization, and fabrication issues. Receiver electronics is also discussed, with a specific focus on low noise amplifiers. Signal transport is also briefly addressed. The chapter concludes with an overview of downconversion and sampling, as well as the analog filters which these processes require.
Frequency and Bandwidth
Perhaps the most fundamental criteria in selecting an element type for a phased array system are the operating frequency and bandwidth. Frequency of operation might be considered the initial point in selecting an element, but because a given antenna type can be scaled, within fabrication limitations, to resonate or operate over a wide range of frequencies, bandwidth is in some ways the more critical driver. Element types can be divided most simply into narrowband, resonant antennas (dipole, patch) and wideband antennas (sinuous antenna, Vivaldi, checkerboard, and others). The division between narrowband and wideband is of course not a precise cutoff, but 10% to 20% relative bandwidth (i.e., bandwidth divided by the design center frequency), might be considered the upper limit of narrowband antenna types, and antennas with wider relative bandwidth would be considered to be wideband or ultrawideband. Bandwidth limitations are discussed further in Sec. 9.4.