Radio Direction Finding and Superresolution (2nd Edition)
Enlarged and revised second edition. Modern direction finders, capable of measuring elevation angles as well as azimuth angles on the components of multiray wavefields, have become powerful tools for research in ionospheric physics and HF radio propagation. The complexity of the problem of resolving closelyspaced rays requires the combined use of wide aperture antenna arrays, multichannel receiving systems and sophisticated digital processing techniques.
Inspec keywords: ionospheric techniques; interference (wave); antenna arrays; radio directionfinding
Other keywords: DF plots; tworay wave fields; circular arrays; bearing accuracy; superresolution; ionospheric modes; zeroaperture bearings; wavefield models; wave interference; ray paths; radio direction finding; wavefront analysis; interferometers; ionospheric tilts; directive array patterns; site errors; instrumental errors
Subjects: Antenna arrays; Instrumentation and techniques for aeronomy, space physics, and cosmic rays; Atmospheric, ionospheric and magnetospheric techniques and equipment; Radionavigation and direction finding
 Book DOI: 10.1049/PBEW033E
 Chapter DOI: 10.1049/PBEW033E
 ISBN: 9780863412387
 eISBN: 9781849193894
 Page count: 386
 Format: PDF

Front Matter
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1 Introduction
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The aim in this chapter is to review briefly the physical principles involved in radiodirection finding in the HF band. We shall consider only direction finding from fixed landbased stations, on signals propagated via the ionosphere. The abbreviation DF will be used here, as in most published literature, for both direction finder and direction finding; the correct choice should always be obvious from the context. The ideal direction finder would be capable of working over a wide frequency band, 360° of azimuth and 90° of elevation, of dealing with all forms of modulation, and of giving accurate and reliable bearings, even on comparatively brief transmissions. A signal such as a distress call from a ship may be brief and signal parameters such as frequency may not be optimum at all receiving points. The transmissions on which bearings are to be taken in normal applications of civil and military DF must therefore be regarded as noncooperative.

2 Ionospheric modes
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This chapter discusses ionospheric modes. Both singlehop and multiplehop echoes from the ionosphere are frequently seen in records made with verticalincidence sounders (ionosondes) radiating short pulses. It is convenient to regard the region of the ionosphere in which reflection occurs as a wellmarked layer; that is to say, the D, E, and Fregions of ionospheric physics tend to be visualised as discrete layers in raytracing applications. The surface of the earth and the reflecting layer act very much like a pair of parallel mirrors in optics, and the separate returns correspond to multiple images of the transmitter.

3 Wavefield models
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The aim in this chapter is to explore the basic properties of wavefields produced by a small number of rays. A mathematical approach is used to derive the basic equations defining the fields and the interpretation of the equations is then illustrated by means of computerproduced plots. Some of the results to be presented can be found in published DF literature, but the author knows of no book or paper where the theory is developed systematically. The main emphasis will be on the properties of fields in the ground plane, z = 0.

4 Zeroaperture bearings in tworay wavefields
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In this chapter we shall consider the 'bearing' defined by the normal to a curve of constant phase (CP) at a particular point in a tworay wavefield. It must be emphasised that 'bearing' is thus being given a very special meaning; it is a geometrical property of the wavefield. In effect, we consider the bearing indicated by a phasesensitive DF as its aperture shrinks to zero. Because this limiting case is an abstraction, we can continue to postpone any consideration of the behaviour of particular types of instrument; we are still dealing with the inherent properties of the field. Nevertheless, a number of general results can be derived that provide a useful insight into more practical situations.

5 Directive array patterns
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The aim of this chapter is to summarise those aspects of antenna array theory that are relevant to the design of DF systems. The reader is assumed to be familiar with the more basic standard properties of directive arrays. In the orthodox DF instrument, some or all of the antenna elements are connected together to form an array, the radiation pattern of which possesses desirable directional properties; in general terms, directivity is clearly an essential property of an array to be used for directional measurements.

6 Instrumental and site errors
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Although this book is not primarily concerned with DF hardware, it is convenient at this point to summarise the physical sources of measurement errors arising from instrumental and site effects. Reradiating objects in the vicinity of the receiving site are of particular interest here, because the rays from such objects add to the complexity of the wavefield to be resolved. The sources of error in DF can be broadly divided into four categories: (a) Instrumental errors, which can cause incorrect measurements in even the simplest situation, namely a single plane wavefront arriving at the DF array, (b) Site errors, i.e. distortions and deviations of the arriving wavefronts caused by features in the neighbourhood of the receiving site such as reradiating conductors, ground contours and gradients of ground conductivity, (c) Waveinterference errors, when the arrival of two or more rays produces swings in the indicated bearing. (d) Propagation errors, i.e. changes in ray direction produced by ionospheric tilts or magnetoionic effects.

7 An introduction to resolution techniques
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We shall assume from now on that a DF system is available that is capable of measuring correctly, in singleray conditions, at least the azimuth angle of arrival of the ray. In effect, instrumental errors and site errors will be regarded as negligible, although it is not essential to assume an absence of reradiating objects near the DF; some of the discrete rays in multiray problems could arrive from such objects. Between these two extremes of NADF and very wide aperture DF (VWADF) lie WADFs with array dimensions of the order of a few λ; the physical dimensions fall generally in the range 50 to 500 m. It is with such systems that we shall be concerned, and the underlying question is whether their classical resolving power can be enhanced by intelligent analysis of the timevarying wavefield produced by the complex fading of two or more rays.

8 Wave interference effects for circular arrays
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We now come to the question of how much can be deduced about the structure of a wavefield from an analysis of wave interference effects. The observations might consist of the fluctuations of the indicated bearing of a WADF as a function of time (this Chapter) or a sequence of phase difference measurements on various pairs of elements in an interferometer (Chapter 9). Not surprisingly, the amount of information that can be derived with an orthodox DF, wbich provides only one indicated bearing at any one moment, proves to be very limited. In the numerical simulations for this type of problem, therefore, attention will be confined to tworay wave fields and the basic question is whether the ray azimuths can be found. In order to distinguish between the zeroaperture bearing defined in Chapter 4 and the indicated bearing from a DF of finite aperture, we shall refer to the former as the tangent bearing; it is the direction normal to the tangent to a curve of constant phase at a specified point in the horizontal plane and is a property of the wavefield. A phasemeasuring DF of finite aperture, centred on the same point, would produce an approximation to the same bearing but would average in some way (to be investigated below) over a finite segment of the curve. Thus, the 'segment bearing' for a finite aperture approaches the tangent bearing as the aperture approaches zero.

9 Wave interference effects for interferometers
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We shall discuss here some of the problems involved in using an interferometer to make measurements on a CW signal in multiray conditions. Performance now depends on factors such as the exact sequence of operations, the speed with which it is carried out, the number of times it is repeated to obtain a time average, and the methods used to resolve ambiguities and to combine observations from separate arms. The optimum combination of processor and array configuration is by no means clear, and a full discussion of all possible permutations is beyond the scope of this book. All that will be attempted here is an introduction to the types of calculations that are needed in the interpretation of observations and the analysis of accuracy.

10 Wavefront analysis: the concept
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Signal processing in DF has, until recently, been almost entirely of an analogue nature. The spinning goniometer usually used in a Wullenweber system is an example: the inputs to the goniometer are the signals from a selected arc of elements and the output is fed through a singlechannel receiver which determines the narrow frequency band to be examined. The development of multichannel HF receivers in recent years has opened up the possibility of using instead some form of digital processing; the amplitudes and phases of the element signals at a selected frequency can be measured, recorded digitally and processed in a computer program. The experimental and mathematical techniques are referred to collectively as wavefront analysis (WFA). The aim is to resolve the wavefield into its component rays and to print out the ray parameters. Direct analytic or iterative attacks on the equations relating the ray parameters to the measured signals become very unattractive when more than a small number of rays are present. However, a general method of solution, applicable to almost any form of antenna array, can be used with fading signals (Gething, 1971). In this chapter some simple examples of the technique will be given for idealised problems involving neither noise nor measurement errors, in order to explain the principles involved.

11 Wavefront analysis using imperfect data
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Some of the problems that arise in the analysis of imperfect WFA data will now be briefly discussed. When the input data to a computer program are contaminated by noise and measurement errors it is no longer reasonable to expect a unique and correct answer from a minimum number of frames. Other potential sources of error include departures of the behaviour of the real wavefield from the assumed model, and differences between the real and computed patterns of the array; clearly, a measuring instrument of known properties is required if accurate measurements are to be made. Techniques for the statistical analysis of samples of imperfect data can, of course, be tested against simulated problems to which known amounts of noise, measurement errors, angle fluctuations etc. are added in a series of controlled numerical experiments. The ability of an array to resolve a given wavefield must eventually break down when the signal/noise ratio becomes sufficiently unfavourable. In a problem involving (say) two strong rays and one weak ray, it is likely to be the weak ray that is the first to be lost from the solution as the noise level increases. In these circumstances, a partiallycorrect solution might be obtained.

12 Superresolution algorithms
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The word superresolution is now widely used in place of the term wavefront analysis and the phrase 'the resolution of multicomponent wave fields'. It is used to describe techniques of signal processing that can be applied in many different fields such as radar, sonar, radio astronomy, seismology and spectral analysis, as well as radio direction finding. Some hundreds of papers on superresolution algorithms designed to determine the angular spectrum of the arriving rays have appeared in the literature over the last ten years, many based on concepts described in earlier chapters. In this chapter we describe the main categories of algorithms and their performance against simulated problems. Further developments and refinements of the processing techniques are discussed in Chapter 13. The aim in both chapters is to describe and interpret the important results rather than to reproduce mathematical detail from the research papers.

13 Additional tools for superresolution
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This chapter discusses the additional tools for superresolution. Partial coherence between two ionosphericallypropagated rays tends to increase the sampling time required to achieve the same resolution performance as with two incoherent rays.

14 Ray paths
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Suppose that the DOAs of several rays from the target transmitter have been determined. Suppose further that the state of the ionosphere in the general area between the transmitter and the DF is known in some detail. Then it is possible in principle to reverse the ray directions at the point of reception and retrace the ray paths back to the transmitter, using a suitable computer program to find the paths in the ionosphere. Ideally, all ray paths would ultimately be found to intersect at one point on the earth's surface, possibly after multiple hops. Not only would the DOAs at the DF site be correctly explained; the relative time delays between different paths would also be correctly predicted. That is to say, the versions of a given signal element received over various paths should trace back to a common timeorigin as well as a common spaceorigin. In practice, of course, exact intersections in time and space are not obtained because of measurement errors and of incomplete or imperfect knowledge about the state of the ionosphere. However, the objective of ray retracing procedures is to make the best possible use of the available DOAs and ionospheric data. Note that an estimate of the position of the transmitter is now obtained from a single DF site; in effect, a range is determined from elevation angles and ionospheric data, while a direction is determined from the measured azimuth angle or angles. The aim in this chapter is to summarise those parts of raytracing theory that are relevant to DF. For further details, textbooks such as those by Budden (1985) and Kelso (1964) should be consulted.

15 The effects of ionospheric tilts
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When the properties of an ionospheric layer vary with latitude and longitude, we can speak of gradients in the ionospheric parameters. The layer can be described, for brevity, as tilted, but it must be borne in mind that a thick layer of variable properties does not necessarily behave in the same way as a tilted mirror. With this proviso, we now consider the effects of tilts on DF measurements, possible ways in which tilt corrections might be applied and possible ways of acquiring the necessary information about the ionosphere. The effective tilt of an ionospheric layer of finite thickness can vary with the depth of penetration of a ray into the layer and is therefore dependent on the frequency and elevation angle of the ray, as well as the gradients of various ionospheric parameters.

16 Bearing accuracy and DF plots
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Azimuth measurements taken on transmitters of known position, i.e. check targets, can be compared with the 'ideal' answer, namely the true g.c. bearing, so that accuracy statistics can be compiled for various purposes described in Section 16.1.2 below. The same procedure cannot be used for measurements of elevation angles, where the 'true' answer is generally unknown. We shall therefore confine our attention in this chapter to azimuth measurements, which will now be referred to for brevity as bearings. It should be noted that bearing errors can, and do, arise from propagation effects as well as from instrumental effects, and that we are not directly concerned here with the physical causes of instrumental errors, which were discussed in Chapter 6.

17 Conclusions
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The aim in this brief final chapter is to summarise the major conclusions. Until recently, the output of a DF system was usually thought of as a single number, the indicated bearing of the target transmitter. Traditionally, the wavefield, no matter how complex, was interpreted in terms of a oneray field, with the ray arriving from a constant azimuth. The need to achieve mode resolution with WADF's of reasonable aperture (a few wavelengths) implies that the benefits of timesampling as well as spacesampling must be incorporated in the processing scheme.

Appendix 1: Circular WADF: parameters used in pattern calculations
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This appendix gives the parameters used in pattern calculations for circular wide aperture direction finders (WADF). It list these under the headings of: Array dimensions and Goniometer, and lists some simplifying assumptions.

Appendix 2: The theory of multiplicative processing
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In this appendix we consider the response of a particular multiplicative array to an incident field produced by one or two targets. The array can be scanned electronically or mechanically relative to the targets. The treat ment here is based on a paper by Shaw and Davies (1964).

Appendix 3: Vertical array of loops: parameters used in pattern calculations
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This appendix gives the parameters used in pattern calculations for vertical array of loops.

Appendix 4: An example of masquerading
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A specific example will now be given, in which two distinct twomode toy problems, A and B, give almost identical signals in the eightloop vertical array. Data are shown for three frames, but the method of deriving the ray parameters for B from those for A is so general that the masquerading can in fact continue indefinitely. The frequency is 6 MHz. Continued manipulation of equations of this kind leads to the conviction that any twomode problem involving angles below 152° (at least) at 6 MHz can be masqueraded by some other problem involving two or more modes. One mode can masquerade as two, but in this case the amplitude fluctuations of the two modes would show a strong correlation. Masquerading occurs chiefly at low frequencies and lowelevation angles and is a manifestation of inadequate resolving power. On this physical interpretation, it cannot be avoided by alterations in the number and dispositions of the elements within a given aperture; it can arise with irregular, as well as regular, spacings.

Appendix 5: Covariance matrix of sensor outputs
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The covariance matrix of sensor outputs are discussed. Matrix notation, matrix pencil, projection, data model, noise model, asymptotic form, and Toeplitz matrix are also discussed.

Appendix 6: Magnetoionic deviations calculated with the Jones threedimensional raytracing program
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This appendix discusses: ionospheric model; earth's magnetic field; and sign convention.

Appendix 7: Path integrals for parabolic layers
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This appendix discusses the equations of path integrals for parabolic layers.

Appendix 8: The effective tilt of a parabolic layer
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For simplicity, earthcurvature effects are neglected throughout this appendix. The appropriate integrals for a flat earth and horizontally stratified ionosphere are shown.

Appendix 9: Standard deviations of the best point estimate
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We suppose that a Best Point Estimate (BPE) has been calculated from n bearing lines by means of the standard Stansfield theory. The position of the BPE relative to an origin at the (unknown) true position of the transmitter is (X, Y), where X = (G/A), Y = (F/B).

Back Matter
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