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Radio is what makes it both powerful and frustrating, and radio, an old technology after all, is the key to understanding its idiosyncrasies. The attentive reader will notice, though, that the real backdrop for this book is the physics of the electromagnetic wave, rather than digital communication. That said, there remains a real need for a better set of diagnostic tools for wireless professionals, and for a more ambitious engineering reach into the physical layer, beyond simply making the transceiver work.

Multitudes of mobile devices crowd the physical spaces, as well as the available spectrum, opening up problems in security and logistics. Obviously, true security in the radio medium can be found only in good encryption — this is a lesson that goes as far back as World War II and Bletchley Park — but it does help to know where your interlocutors are. It is always difficult to quantify things that did not happen, but I am inclined to believe that a lot of wired mischief is deterred by the fact that your wired connection can be traced to your desk, at least within a typical workplace!

Knowing the location, or at least the general direction of the other party, is helpful with the logistics of radio communication as well. When radiolocation is combined with the ability to steer the transmission, radio communication becomes something in the nature of a focused personal conversation.

Other areas of ubiquitous wireless communication still present an untouched field for the application of radiolocation. Exposition in this book tends to follow the direction from the general and abstract toward the specific and concrete. It is centered around an engineering project, which hopefully anchors and focuses the discussion. To be more specific, the exposition centers on radiolocation by measuring signal amplitudes in multiple directions, and on an actual device that performs this radiolocation around the full circle of the horizon.

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The medium of radio is, of course, fundamentally analog, and radiolocation is really an exercise in quantitative measurement. For that reason, I have emphasized the quantitative and real-time aspects of the architecture and design. As the book progresses toward the full description of the implemented architecture, the discussion explores design alternatives, and seeks to justify the engineering choices that were made along the way.

Chapter 1 offers an overview of physical phenomena underlying radio communication: it describes the electromagnetic wave and its interactions with matter, leading to the highly important topic of antenna physics. This chapter will be of most use to those engineering denizens of the digital world who do not routinely venture below the data link layer.

Chapters 2 and 3 deal with the mathematical issues that are at the core of reconstructing the direction of the radio wave. Chapter 2 develops the direction-finding algorithm and investigates its numerical aspects; this chapter is fundamental to understanding the implementation of radiolocation, as we describe it in subsequent chapters.

Chapter 3 develops a broader method of radio imaging, including multi- Preface ix ple sources and radiolocation over the full sphere of directions. This chapter lays theoretical groundwork for future imaging architectures, and the reader may choose to omit it on first reading. Chapters 4 to 7 build the architecture of the Wi-Fi radiolocator, from the antenna to the complete radiolocation transceiver; these chapters form the engineering core of the book.

Chapter 8 aggregates various implementation details that are peripheral to the main topic, but are nevertheless essential for the device design. Chapter 9 looks forward, toward applying the lessons of this work to other, potentially more intricate wireless protocols. In following this book, the reader will benefit from some previous familiarity with the concepts in wireless communication, and from college-level physics and mathematics. The discussion of radiolocation is supplemented with topics in radio fundamentals, and the exposition is grounded in basic physics, but the reader will not be hampered by the lack of specialized background knowledge.

Where it seemed necessary, I have provided introductions to some well-established side topics, in the form of appendices. These appendices contain additional mathematical details, or give concept summaries, but they are not meant to be complete expositions. Throughout the book, I have also provided references to standard, and hopefully approachable, textbooks in the field. My greatest expectation of the reader may be that of familiarity with board-level electronics and embedded systems design.

Practical experience in these areas will be helpful in envisioning some of the implementation details, details that I left out in order not to drift too far away from the main topic. Attempting to cover that background would have resulted in an entirely different book! Bloomington, IN Danko Antolovic Acknowledgments Every book draws upon the contributions and help of many people and institutions.

I wish to acknowledge here Steven Wallace who, as director of the Advanced Network Management Lab at Indiana University, recognized the importance of radiolocation in wireless communication early on. Building the prototypes described in this book was greatly facilitated by the efforts of John Poehlman and the excellent staff of Electronic Instrument Services of the Chemistry Department, at Indiana University.

I appreciate the years of our collaboration and friendship, for knowledge is indeed advanced best through a free and generous exchange of ideas. Image Formation in Spherical Geometry. Ever since James Clerk Maxwell spelled out its fundamental equations in , its tenets have been verified and reverified by measurements, its formalism developed and made more elegant.

It is also a science with wide application, since, except for the force of gravity, the vast majority of phenomena with which we come into contact every day is electrical in nature. The differences between these phenomena stem entirely from different ways in which photons of different wavelength interact with matter. In conjunction with quantum-mechanical principles, electrical force underlies the structure of atoms and molecules, and therefore all of chemistry, crystallography, and molecular biology as well.

Solid and liquid state, all that we perceive as bulk or extension in space, is maintained by a quantum-mechanical balance of electromagnetic forces. The same electrical force provides a remarkably flexible and efficient method for the transport and distribution of energy, the electric grid. It is the basis of our communication and digital information technology, not to even speak of consumer electronics.

We cannot do justice to this vast field in this book, nor is that our purpose; excellent textbooks on the subject abound Jackson , Pozar , Russer , Born and Wolf In this introductory chapter we review topics in electrodynamics that form the foundations for understanding radio technology. We hope to convey to the reader some of the subtlety of the formalism of electrodynamics, although we relegate one or two of the more intricate mathematical topics to the Chap. We begin with that simple and remarkable phenomenon, the electromagnetic wave in empty space.

That is the classical pre-quantum view, which is perfectly adequate for our purposes. This disturbance is physically described as the presence of two vector fields, electric field E and magnetic field H, at any point in space. Otherwise, the small volume contains a source or a sink of the field, such as an electric charge.

We have presented divergence and curl in their integral forms, which are highly intuitive, but these are true differential operators in spatial coordinates. For further exposition of this topic, the reader should consult any standard textbook on electrodynamics or mathematical physics Jackson ; Byron and Fuller ; Morse and Feshbach Equations 1. Let us mention here that there is an asymmetry in the electrodynamics of the world, in the sense that there are no known magnetic charges.

In the presence of electric charges, 1. The formalism of electrodynamics would allow magnetic charges, and there are no known reasons for them not to exist — they have simply not been observed so far. They are related to the speed of light in the following way: c D. We can separate the fields into two equations, at the price of having equations of the second order: We first act upon 1.

It acts on scalar fields, as well as component-wise on vector fields, and yields scalars or vectors correspondingly. Physically, Laplacian of a field at a point in space is proportional to the difference between the average value of the field over an infinitesimal volume around the point, and the actual value of the field at that point see Sect. The fields change in time by accelerating toward their average values, but since their first time derivative is not zero at equilibrium, they overshoot and continue in an oscillatory motion.

From these equations we see that electromagnetic phenomena in empty space, away from the material anchors of electric charges and currents, must be wavelike disturbances propagating at the speed of light. This equation is also homogeneous and linear, which means that any linear combination of any solutions if solutions exist is also a solution. Naturally, this applies to 1.

Equation 1. So far, nothing in this discussion is specific to electrodynamics: for example, when applied to two dimensions, this formalism describes oscillations of an elastic membrane as well. Let us now introduce the electromagnetic fields in the form. E D E0 exp.

In analogy with the analysis for u in 1. We may ask, naively, whether the two fields must really have the same wave vector and the same frequency, and whether there are any restrictions on the vectors E0 and H0: Wave equations 1. We notice, first, that because of the differentiation properties of the exponential function, vector differential operations become algebraic operators for vectors of the type V D V0 exp. Suppose that there were separate electric and magnetic frequencies and wave vectors; it follows from 1. H H0 exp. The electromagnetic wave in vacuum is transversal in both of its components, and the components are mutually orthogonal as well see Fig.

Here, we invoke without proof the Poynting theorem, a general theorem about the energy flow in electromagnetic fields see Jackson We see that, for the plane wave, S points in the same direction as k: The wave carries energy in the direction of its propagation, as one would expect. Vectors k, E and H form a right-handed coordinate system 1.

As we already mentioned in passing, the structure of our everyday matter is determined by the electric force, and it is the richness of this structure that gives rise to the variety of interactions between the electromagnetic wave and material objects. These remarks are true for all of the electromagnetic radiation, but we concentrate here on the part of the spectrum that is commonly known as radio waves. On the largest terrestrial scales, radio waves encounter the surface of the sea and land, interact with the ionized upper atmosphere, and traverse large distances around the globe.

Gases interact very little with radio waves, but all condensed matter does, and even mist in the air has an effect over long distances. Coming closer to the human scale, radio waves are blocked by mountains, reflected by tall buildings, attenuated by walls, absorbed by foliage. On the size scale of an office, short-range microwave communication is affected by the presence of metallic structures inside walls, by furniture, and even by people passing by.

Since microwave radio in this cluttered setting is our focus of interest, we devote some attention in this chapter to understanding the two main types of materials encountered: dielectrics and conductors. When describing electrodynamics within material media, it is common to introduce two new fields: electric displacement D and magnetic induction B. They are related to the electric and magnetic field: D D "E 1. The equations contain quantities which describe aggregate properties of the matter and are not part of the electromagnetic theory proper.

These quantities must be determined by empirical observation, or derived from a theoretical description of the medium. Most materials fall into one of two broad classes: dielectrics, within which the macroscopic current J is identical to zero, and conductors, in which it is not. Dielectrics, crystalline or amorphous, consist of rigid covalent structures, in which every electron is confined to a small region, either an atom or a chemical bond; in the presence of an external field, an electron can depart from its average position only very little.

Conductors are characterized by freely moving charges. In the crystalline lattice of metals, a portion of the available electrons resides in quantum states that are distributed over the bulk of the solid. These electrons can move relatively freely from one end of the piece of metal to the other, giving rise to macroscopic currents. Likewise, conductive ionic solutions such as seawater contain moving charges in the form of charged molecular species ions , which again move relatively freely through the nonrigid structure of the liquid.

This classification is somewhat crude, because the distinction is a matter of degree in, for example, semimetals and semiconductors, but it nevertheless accounts well for many observed phenomena. Now, we examine the behavior of the electromagnetic wave within these two types of material Fowles For our purposes we can assume that the material is non-ferromagnetic, i. It is also convenient to express the electric effect of the medium as the polarization vector. Its complex interactions with the neighboring matter are simplified into a frictional force, proportional to the speed of its movement.

This is a very simplistic description of a dielectric, but it suffices to account qualitatively for the properties important in radio wave propagation. The strategy is to determine how the polarization P depends on E, and to solve the wave equation 1. Assuming the harmonic time behavior exp. This becomes a solution for 1.

This is a direct consequence of the frictional term in 1. We can express the wave number as the complex index of refraction, n D. At the resonant frequency of the oscillating electron, however, the attenuation index has a maximum, and the dielectric grows opaque.

It can be easily shown that the two quantities are related as "r D n2 1. The loss tangent is small, ca. As in Sect.

Second, for the steady state of constant or slowly changing fields and currents. Following the same steps as in Sect. They are roughly equal at the so-called plasma3 frequency: 3 Plasma is a generic term for a gaseous low-density assembly of charged particles. Free electrons in a metal are well described as a plasma, which is held in place by the overall positive charge anchored to the lattice of metal atoms.

We see from 1. Below the plasma frequency, conductors are opaque. We conclude this discussion on wave propagation in materials with a few words on reflection. Reflectance is defined as the fraction of energy bounced back from the material surface. For conductors below plasma frequency, how- 1. Conductors, therefore, reflect most of the radiated energy, and what enters the material is dissipated within the skin depth. In addition, plasma frequencies of metals lie almost without exception in the ultraviolet region ca.

Antennas straddle the boundary: They transfer the energy between moving charges currents and propagating electromagnetic waves. They differ from the usual lumped components and transmission lines, because their electric properties are not amenable to simplifying single-feature descriptions: They are neither pure capacitors, nor inductors, nor waveguides. Antennas come in a wide variety of designs, and describing them correctly usually requires invoking the formalism of electrodynamics at a fairly fundamental level. They are the least intuitive and most fascinating elements in the signal path of radio communication.

Directional antennas play a large role in selective use of the radio medium, such as long-distance point-to-point links, and in radiolocation. We review here the fundamentals of the electromagnetic theory of antennas, and devote some attention to the design principles used in the directional varieties. Generally speaking, the radiating antenna is a current of known shape in space and behavior in time, typically confined to a metallic conductor.

The current induces an electromagnetic wave, which propagates into infinite distance. Incidentally, the analysis is usually easier for a radiating antenna, but we see that the results are applicable to the receiving antenna as well Sect. To obtain the field around the antenna, one must solve 1. We outline the procedure below, but the reader is referred to standard texts for a detailed discussion Balanis , Elliott Assuming harmonic time dependence of currents and fields, and skipping numerous mathematical steps, we state the solution of the Equation 1. Magnetic and electric fields follow from 1.

Cumbersome appearance of the integral in 1. Such analysis is not our purpose here; instead, we use 1. This is the static or near-field region around the antenna. We do this by using two slightly inconsistent approximations. Taking curls of A according to 1.

This is the radiation region, or far-field region, of the antenna, characterized by the radial outflow of energy, and by spherical waves whose angular distribution is independent of the distance r. We shall look at the interference first. Let us calculate the current integral in 1. The bracketed term, called the array factor, expresses the phase differences, which are due to the spatial arrangement of the radiators. Phase delays between adjacent sources equal kd cos 1. This is the direction in which the radiated power [square of the expression in 1.

The maximum is in the direction perpendicular to the array, where the waves from all sources are in phase. The array is placed vertically in the center of the pattern and has the same geometry as that in Fig. Both consist of point sources, whose phase differences are due to the spatial arrangements, and which produce wave reinforcement in certain directions and cancellations in others see Fig. However, arrays of radio antennas need not be linear: Circular and other arrangements are also used.

This phase difference determines the direction of constructive wave interference, and is the basis of beam-steering techniques for antenna arrays. An interesting variation on the antenna array is the helical antenna. We refrain from the full mathematical treatment of the helix see Elliott , but, in a rough approximation, one can regard the corresponding current elements on each turn as the elements of an array see Fig.

The signal source drives a propagating current wave in the helix — we assume that the helix is long enough to radiate away all the power, and that there is no significant reflection of the current wave back from the far end. Waves emitted by the adjacent current elements then reinforce each other in the direction of the axis of the helix; a reflective ground plane is placed at the feed end, and the antenna exhibits a nicely formed main lobe in the end-fire direction.

If it did, its power flux on the surface of a large sphere in the far field would be strictly radial, and of constant magnitude on the whole surface. Because of Fig. This is topologically equivalent to combing the entire surface of a furry sphere: it cannot be done without creating swirls or breaks. But swirls and breaks are discontinuities in the field infinite spatial derivatives and are not physically possible; therefore, an isotropic source is not possible either. The paragon of all reflector antennas is of course the paraboloid dish, since we know from elementary geometry that all rays emanating from the focus of the parabola are reflected in the same direction.

Conversely, a plane wave traveling along the direction of the axis converges upon reflection on a single point, the focus. For the purposes of radiolocation, reflectors are more interesting as image forming receivers, and a paraboloid reflector is indeed analogous to a telescope, an instrument that maps the delocalized, space-filling plane wave of light into an illuminated dot in the focal plane.

Direction of the wave relative to the instrument can then be deduced from the position of that dot. Departing from geometrical optics, we can envision the reflection of the plane wave as re-emission of spherical waves from every point on the surface of the mirror. Constructive interference between these re-emissions forms a wave that converges on the focus. Because the wavelength is not infinitely small, relative to the size of the mirror, we expect to see a diffraction pattern in the focal plane, rather than a pure dot of illumination.

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In fact, diffraction patterns caused by apertures are well known Fowles , Born and Wolf , and the illumination pattern caused by the circular opening of radius R is plotted in Fig. The abscissa in this plot is the angle away from the axis, represented in the form kR sin. As we should expect, this pattern is similar to the beam pattern of the array antenna, since both have their origins in the same physics. This is the image size, or the angle which the paraboloid dish will sweep while detecting one point source, and is obviously a measure of the resolution of the antenna resolution is usually defined as the minimum separation between two still discernible sources, and is half of D.

We see that the resolution is better for larger dishes and for shorter wavelengths. To put things into perspective, let us compare a 1. The angular image size turns out to be ca. This type of antenna consists of an area of conductor, placed above a larger ground plane, and separated from it by a dielectric layer. It is obvious that patch antennas can be fabricated using the printed circuit board technology, and they are often integrated into the circuit boards of inexpensive wireless electronics.

Being flat, they are frequently used where space is at a premium, such as in avionics applications. Their main drawback is that they are relatively inefficient radiators, compared to other types of antennas, and that they have a narrow bandwidth around their resonant frequency. The first question is why patch antennas work at all. Two metal plates separated by a thin dielectric layer form a capacitor, and why should a capacitor radiate?

The answer is that the radiation from a patch antenna is due entirely to fringe effects, the noncontainment of the field at the edges of the patch. That explains the inherent inefficiency: Most of the electromagnetic energy is trapped in standing waves within the patch, and only a small fraction radiates out. Nevertheless, practical advantage of the compact design is considerable, and patch antennas enjoy great popularity in microwave communication. To explain how these antennas work, let us assume that the patch is rectangular, and therefore forms a rectangular resonant cavity with the ground plane see Fig.

Without going too deeply into the theory of resonators, we can assert on intuitive grounds that the main mode of oscillation the one with lowest frequency forms a standing wave along the largest dimension L, and that the electric field is directed from one conductor to the other. Figure 1. Fields in the shorter slots. Radiation from the longer slots cancels out. The rectangular patch is, in effect, a two-element array with a mirror plane, the geometry of its radiators held in place by the standing wave inside a resonant cavity.

The antenna has a broad lobe facing upward. Due to the boundary conditions, electric field forms a cosine wave, and M cancels out along the slots L. The magnetic field forms a sine wave in the L direction and is parallel to M Similarly, a circular patch can be described as a circular radiating slot.

It is particularly interesting that two oscillation modes of the circular patch, at the same frequency and spatially orthogonal to each other, can be excited with two antenna feeds that are out of phase by a quarter-period in time, thus creating circular polarization. Quantitative analysis of patch antennas requires description of the fringing effects either by semi-empirical formulas or by complete electromagnetic field simulations. We refer the interested reader to Balanis for a detailed discussion, including a semi-empirical analysis of the rectangular patch.

References Balanis, C. In this chapter, we will delve into our chosen method of amplitude-based radiolocation, but let us first draw some useful analogies with optical detectors and optical image formation. A radiolocation method that relies on the amplitude, or power measurements, must involve at least one directional detector.

This implies that we must measure the power in a number of directions in the vicinity of the putative maximum. In the approximation of geometrical optics, where the diffraction phenomena are neglected, formation of an image can be viewed as just such a many-directional measurement of amplitude. For every direction within its field of vision, a telescope, or even a pinhole camera, has a preferentially sensitive spot in its image plane. When a wave arrives from that direction, it causes an infinitely narrow maximum readout at the right place in the image plane.

We know that it is a maximum, because the adjacent directions are also measured, and yield zero intensity: what we see what the film or the CCD array sees is a bright spot, the image of a distant point source. As we have discussed in Sects. The wavelength is typically not small relative to the antenna size, and directional antennas have prominent wide lobes, due to wave diffraction. Since a simple one-feed radio antenna is intrinsically a single sensor, finite width of its sensing field is of some practical benefit for detecting the presence of a source in the first place.

Multiple measurements that are required to find a maximum are traditionally accomplished by swinging the antenna around until the strongest signal is received. We would, of course, prefer to do all the directional measurements at once, i. We know that the image will be broadened by diffraction, and that it must be detected by a spatial array of sensors, to establish the location of the maximum intensity. There are two plausible options for image formation. The first alternative is an aperture instrument, such as a reflector telescope; reflecting mirrors are widely used in rotating radar antennas, but we contemplate here forming an image with a stationary instrument.

Referring back to the example in Sect. At the focal distance of, let us say 1 m, that is a 19 cm image diameter. The image area will have to be covered with sensors whose sizes and mutual distances are small relative to the wavelength. This means that the sensors miniature antennas will have very low gain, and will have to be supplemented with low-noise amplifiers. Furthermore, aperture instruments tend to have a narrow vision field, and wideangle coverage is usually desired in radiolocation. It is difficult to construct wideangle aperture instruments, especially in the reflector design, where the image plane obstructs the aperture to begin with.

A radio lens could be used instead since the refractor design can have a wider field than the reflector, but radio lenses are bulky and expensive. The second alternative, the one that we pursue in our implementation, performs multiple simultaneous directional measurements with an array of directional sensors, and is somewhat akin to the compound eyes of insects. There is no optical image, in the usual sense of a converging front of mutually reinforcing waves; instead, the direction of the incoming wave is reconstructed in circuitry or in code from the slight differences in the way in which it affects the adjacent sensors.

Figure 2. The parallel rays, which stand for a plane wave, impinge upon a curved array of eyelets, causing the adjacent ones to produce simultaneous outputs, which either differ in intensity or are different discrete signals, as shown in the figure — the eyelets have some directional sensitivity. Non-overlapping sensors would lead to blind spots, and sensors without angular sensitivity could not discern the directions that lie between them, even if their fields overlapped.

This is the level of sensory complexity that we have implemented in radiolocation. Direction-sensitive outputs from adjacent sensors are combined to construct an image of the source of the wave some dragonflies come close to doing just that. Furthermore, unlike the reflectors and refractors, which invariably suffer distortions in directions away from their optical axis, accuracy of the compound eyes is uniform throughout the vision field.

Analogously, a suitable multi-element antenna can perform radiolocation equally accurately in all directions, without ever having to move. However, individual sensors of this compound antenna must be directional and, unlike the sensors in the image plane of the telescope, must therefore be comparable in size to the wavelength. This places a practical limit on the number of elements, and limits the density of the coverage of directions. Function F.

We now choose a particular value of the independent variable, we can call it , and we define a function F. In symbols: F. We apply this to our array of sensors, which we here assume to be in the form of a planar ring of directional antennas. Variable ' becomes the angle around the ring, with appropriate periodic conditions. The angle of the incoming wave is. Signal strengths, which the wave induces on the antennas, are equal to L. Because of 2. Such a configuration is shown 2. These conclusions are also valid for spatial arrangements of axially symmetrical lobes in arbitrary directions.

Radiolocation in Ubiquitous Wireless Communication

It is immediately obvious that this conclusion does not hold if the lobe shape is not axially symmetrical. Obviously, the angular sensitivity and overlap of the antennas are crucial: they yield a source direction that is determined, redundantly and robustly, by the signals of several all antennas that contribute points along the rotated lobe. We should point out, however, that we assumed in Sect.

In practice, the optimization must involve at least two variational parameters: the rotation angle and the scale. It is clear that the direction finding relies on numerical computation of some intensity. Since there are stringent time requirements that need to be observed, implementation of the algorithm will be helped by an understanding of its underlying mathematical features. In the next two sections, we investigate the effects that the error function and the lobe shape have on this computation.

This is the ideal limit, in which inputs from an infinitely dense array of sensors fully delineate a shape identical to that of an antenna lobe. Real inputs are somewhat sparse — that is why pattern matching has to be performed in the first place — but it is nevertheless instructive to investigate how the match of two identical lobes behaves under variations.

Let us establish some conventions and nomenclature. The original unvaried lobe shape is defined by a function L. The error function is the contribution to the error at each point along the lobe all d' integrals in this chapter are taken over the full circle.

In order for the numerics of iterative optimization to work well, the variational error should also be continuous, and have continuous first and second derivatives, at the minimum point. Our variational error will be calculated by numerical integration of a suitable error function, along all points on a lobe; furthermore, it will be recalculated for every iteration step. Obviously, it is critical that we select a simple and fast error function, even at the expense of some physical plausibility: wrong choice may result in an unacceptably slow calculation.

The intuitive first choice of error, the area between displaced lobes, is actually a poor error measure. It is a slowly changing function along the lobe, and its integral over the length of the lobe curve provides a variational error that is well balanced over all parts of the lobe. The problem with this error measure is that it involves calculating distances along directions perpendicular to the lobe curve, and therefore requires long divisions. By expanding L. Parabolic cross sections in the directions of the coordinate axes are described by the error curvatures. Integrals in 2. Comparing quantities of different dimensions is not physically meaningful, 32 2 Radiolocation with Multiple Directional Antennas but we are interested only in the accuracy of computational approximations, and the physical dimension of the error is not significant [see 2.

In order to perform the numerical analyses of this and the following sections, we use a convenient model lobe pattern, that of the linear array of point sources from Sect. The pattern of this antenna array is described by the analytical formulas 1. We can think of this as an array with a mirror behind it, and with somewhat unnaturally weak side lobes. We can readily adjust the width of the lobe by changing the array parameters.

This model pattern is in fact fairly similar to the pattern of the helical antenna used in our experimental prototype; this similarity should not surprise us too much because, as we have mentioned in Sect. These trends can only be more pronounced in narrower lobes. We will return to this point later. Apart from the expected increase in all variants 2. This difference is due to different powers of L0.

The curvature integrand, shown in Fig. Cartesian and angular approximations also improve for wider lobes. Fortunately, the radiolocation application has no requirements for the accuracy of the scaling, and the errors in the two parameters are not strongly linked. We will discuss this point in the next section.

Notice that the vertical scale is much finer than in Fig. The main variational parameter is of course the angle of rotation, but as we observed in Sect. We can, of course, normalize the inputs to a fixed maximum value, and such normalization yields a good initial value of the scaling parameter, but this is not strictly correct, except in the case of exact alignment of the source with one antenna direction. Hence, there is a need for simultaneous optimization of direction and scale.

This was driven by an engineering consideration: the measurement circuit AD, described in Sect.

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Rather than calibrating the individual parts, we allowed for the base variation. This variation increases the computing time, and could well be eliminated in favor of a one-time calibration. Such calibration incurs practical costs also, and the selection of one course of action over another will depend on practical considerations of a specific implementation. We should note here that the principles discussed so far do not depend on the choice of scale in which antenna data are expressed, and we switch freely between linear and decibel scales, as convenient. We always perform the minimization calculations in the linear scale, for an empirical reason: the logarithmic scale is biased in favor of lower magnitudes.

This makes it a good visualization tool, but it gives the secondary lobes and their inevitable inaccuracies too great a say in determining the bearing. In minimizing the error function, we utilized the widely used Nelder—Mead algorithm; it is one of many minimum-finding algorithms in multiple dimensions.

For N variational parameters, the algorithm calculates the values of the error function on the vertices of a polyhedron in the N C 1 dimensional space. On the basis of some reasonable heuristic assumptions about the topography of the continuous and differentiable error function, it changes the size and shape of the polyhedron iteratively, until it settles on a vanishingly small polyhedron around the minimum. For a good description of the Nelder—Mead algorithm, we refer the reader to Lagarias et al.

As we discussed in Sect.

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Let us now estimate how accurately we must minimize the variational error. We see from Figs. If the iterations terminate in a point close to the origin, as in the Case 1 in Fig. Diagonality of the Hess matrix in 2. The above discussion amounts to saying that for nonsymmetrical lobes, inaccurate scaling biases the optimization of the rotation, which is plausible.

There are limitations, however, to decreasing the lobe width. First, the more directional an antenna is, the larger it 38 2 Radiolocation with Multiple Directional Antennas must be, relative to the wavelength. For example, narrow-beam linear arrays must be long, utilizing the mutual reinforcement of many wave sources to restrict the spreading of radiation; this is reflected in 1. The same can be said of reflectors, helices and various multi-element Yagi-Uda see e. Second, narrow beam diminishes the overlap between antenna elements, reduces the number of adjacent elements that contribute to the rotated lobe and leads eventually to blind spots between them.

Even without blind spots, we expect narrow lobes to lead to aliasing phenomena, and in this section we will describe and quantify this aliasing. The grid in Fig. It is as if the bearing calculation were attracted to the antenna directions. Sequences of sketches in Figs. The input lobe Fig. Finally, it makes a jump toward the next element, and we see in the fourth slide that the algorithm distorts the interpolated input lobe quite desperately as it tries to cross the gap between the elements.

There are always at least two physical points along each side of the lobe possibly counting the top point once for each side , and even though the top of the input lobe follows the nearest element, the sides of the lobes are firmly lined up, and the aliasing is negligible. When that is the case, there are always at least two physical points along the sides, and the direction of the rotated lobe is firmly defined.

We quantify the aliasing as the maximum extent to which the curve in Fig. We perform these calculations for a continuous range of lobe widths, for different counts of 40 2 Radiolocation with Multiple Directional Antennas Fig. Some typical results are shown in Fig. The aliasing amplitude shows a sharply defined threshold, a barrier in lobe width, below which the bearing calculation quickly becomes useless.

Cubic and fifth-order interpolations change the shape of the barrier slightly, in the range of widths where the aliasing error is already growing large. It is clear from the discussion of Figs. This is of course consistent with the fact that the barrier is simply caused by the inadequate overlap of the antenna elements, but more importantly, it confirms that our intuitive criterion — having the lobe half-width as large as the inter-element angle — is a valid, even somewhat conservative, criterion for avoiding the aliasing.

We conclude this section with the observation that, since the aliasing pattern in Figs. Such compensation works for aliasing amplitudes within a degree or so. However, flatness of the curve around antenna directions Fig. This figure shows that keeping the lobe half-width above the inter-element angle upper plot is more than sufficient to avoid aliasing.

Synonyms and antonyms of radiolocation in the English dictionary of synonyms

Both coordinates are measured in degrees directions, on the other hand, amplify the measurement errors of the input, resulting in a bias toward one element or the other. It is best to minimize the aliasing through the physical design of the antenna, as much as the engineering considerations allow. In this section we estimate the robustness of our radiolocation algorithm under such variations.

Planar radiolocation can not determine that elevation, but it is important to estimate how accurately it can reproduce the azimuth for increasing elevation angles. Similar to the aliasing analysis in Sect. We carry out this investigation in simulation, and we limit it to rotationally axially symmetrical antenna lobes. For an elevated source, the planar rotated lobe L.

As the elevation increases, the intersection of the cone and the main lobe becomes smaller and narrower, and it disappears when " reaches the first null. In fact, the radiolocated azimuth exhibits an aliasing pattern, similar to that in Fig. We illustrate the above observations in Fig. The main lobe grows smaller and narrower for higher elevations, but the side lobes do not diminish in size 44 2 Radiolocation with Multiple Directional Antennas Figure 2.

We see that there is a threshold elevation, beyond which the radiolocation breaks down rapidly, and this threshold is lower for narrower lobes, as one would expect. Significantly, Fig. The model lobe shape is that of the linear array, described in Sect. Units of both coordinates are degrees 2. Test source was a 6 cm dipole, at the distance of 12 ft, and the operating frequency was 2. That scaling is compensated for by the optimization algorithm Sect.

For a fundamentally planar optimization algorithm, this range is quite satisfactory. We next want to decide on the width of the element lobes, and we know from calculations illustrated in Figs. Next, we need to decide on the form of the error function. Using all of this, we perform simulated bearing calculations to determine the average processing time tavg , average number of iterations Iavg , and the aliasing amplitude.

We summarize the results in Table 2. Table 2. The reader will appreciate our concern for the time budget of the calculations, since the shortest processing time of 0. Timing calculations in this example were carried out on a 2. The antenna is a element ring, consisting of helices described in Sect. The helical elements were designed according to standard engineering formulas for helical antennas see e. In the course of design work, we prototyped and tested a helix with a somewhat wider lobe; this alternative antenna had a troubling propensity to switch from its proper end-fire mode into a wide-radiating, dipole-like mode.

It is known that helical antennas have this second, broadside mode see Balanis , which is more stable for shorter, wide-lobed helices. This placed a practical limit on the width of the lobe that we could achieve with the helical design. We see a slight aliasing pattern, which is within the range that can be compensated numerically; however, elements with somewhat broader lobes should be used in commercial devices of this type.

Single-Packet Radiolocation of SIAM J. We have seen that the directional sensitivity of individual sensors is essential, and that their fields must be somewhat overlapped. We have also seen in Sect. The crux of radiolocating the sources lies in accurately determining the direction of the rotated lobe, on the basis of somewhat sparse data points, which are the antenna signals. We entrusted the optimization to a straightforward iterative minimization algorithm, and while this approach was successfully implemented and proved reliable in a networking device, we explore in this chapter an alternative method of image synthesis.

The iterative optimization can be implemented only in sequential code, although the calculation of the variational error integral 2. The algorithm is nevertheless of variable duration, and is susceptible to the usual optimization derailments, which we discussed briefly in Sect. Furthermore, it would be difficult to extend this iterative optimization to detect multiple sources simultaneously. For example, independently optimizing the rotated lobes for just two sources doubles the number of variational parameters, and much more than doubles the duration and likely computational difficulties.

One alternative is to subtract the signal of a found source from the total input, then repeat the optimization in order to find the next one; this algorithm is more tractable, but the computing time nevertheless grows proportionally to the number of sources, and the approach has an ad hoc air about it. Simple thresholding and maximum-finding will then locate all of the sources in that signal at once.

Here we will investigate an approach that can detect multiple sources, and which can be implemented as a computing process of fixed duration. This is because two coherent plane waves, traveling in different directions, fill the space with an advancing interference pattern. Signals received on an antenna add as phasors, and the power delivered is, in general, not equal to the sum of powers of the individual waves.

If the phase between the two sources changes sufficiently rapidly during the power measurement, the detected signal strengths are additive on the average, and the sources can be distinguished. We will see in Chap. Two physically distinct sources, separated by a distance distinguishable by radiolocation, are very unlikely to maintain coherent phase over so many periods.

The only realistic interference that can arise is from multi-path reception of several waves from the same source. We may add that while compound eyes spatially distributed arrays of directional sensors are vulnerable to this type of interference, aperture instruments are not. The same linear additivity of waves, which gives rise to the interference, also allows the camera to collimate the two wavefronts onto two separate points detectors in the image plane, and their relative phase becomes immaterial.

This is why our eyes can see an object and its image in the mirror at the same time, and see them as two separate things! We described this function for a single source as the rotated lobe in Chap. We know its actual values only in the directions of the physical antenna elements, but if we had infinitely many elements, densely covering all directions, we would know it completely.

If any arbitrary function can be expressed as a linear combination of the functions from a basis set, possibly using infinitely many basis functions, the basis set is complete. There are infinitely many complete basis sets of functions on any domain of continuous variables, but the expansion is likely to be useful only if the choice of basis reflects some inner structure of the problem that we are applying it to. For the interested reader, almost any introductory text on quantum mechanics e. Let us now choose as our basis set a collection of identical lobes L.

We can also discern that the basis set must cover the directions densely infinitely densely, ideally because lobes in directions far from any mi need not be linear combinations of those pointing in directions mi. We truncated the expansion to a finite number of terms, as must be the case in practice, but conceptually this is an infinite series, and can even be an integral over continuous parameters, as we shall see. The trick is to find the expansion coefficients. The matrix S is therefore the so-called Gram matrix matrix of scalar products of the basis Li , and for a set of linearly independent vectors the Gram matrix is positive definite; see, for example, Gantmacher for the details of the proof.

Therefore, if there is a unique solution of the linear system, 3. This is true even if the expansion in 3. For a complete finite basis, the expansion coefficients satisfy 3. On the face of it, we have outlined a promising approach to forming multisource images of incoming waves. Once the basis set is chosen, the overlap matrix in 3.

We solve the system 3. We can speak of the vector a as the image formed by the compound antenna. The difficulty lies in the numerical properties of the system 3. In order to be complete, our basis set must contain many functions of identical shape, which differ only slightly from each other in their direction. Although the equations in the system 3. But any subset of independent vectors must also be linearly independent; therefore all principal minors of the above Gram matrix also have positive determinants.

It follows from an algebraic theorem Sylvester that the Gram matrix is positive definite, i. No physical quantity in this arrangement calls for numbers of such magnitude; rather, the system of equations is almost singular because of the close similarity of the basis functions. Generic equation-solving algorithms fail to solve 3.

We must approach the image formation with a bit more subtlety, and in the sections that follow, we will see that the very symmetry of the vision field opens an elegant way to not only calculating the image, but to understanding its structure as well. The overlap between any two lobes the matrix element Sij depends only on the magnitude of the angle between the lobes, not on their absolute positions. First row column of S contains the overlaps of the arbitrarily chosen zero-indexed element with itself and all the others the self-overlap S00 will be the largest of them.

Because the overlap depends only on the angular difference, all the successive rows columns must be successive circular shifts of the first one, and S is what is known as a circulant matrix. This is really a considerable simplification — a circulant matrix is fully determined by a single column — and it is brought about in this case by a particular symmetry of the arrangement, the invariance of the overlap under a discrete set of rotations.

As a remarkable consequence of this simplification, all circulant matrices of order n are diagonalized by the same unitary transformation. It is easily shown that U is unitary, i. Algebraic proof of this result is not difficult, and can be found in Gray ; Davis , but for our purposes the following reasoning by analogy will prove more useful: Let us suppose that the overlap S is not a discrete matrix, but an operator acting upon functions on the unit circle.

Crucially, the operator S depends on the difference of angles only. A broad class2 of operators can be diagonalized by convolution with an appropriate unitary transformation U; over both of its variables. We will tentatively parameterize this basis with a parameter k, and write the double convolution as: SF.

An operator is normal if it commutes with its adjoint see e. Since S is real and symmetric, it is certainly normal. Its diagonal elements comprise the unnormalized Fourier transform of its values for one variable held constant ' D 0 , i. Periodic condition on the circle requires that k D 0; 1; 2; : : : We can replace the integral in 3.

Introducing this discretization into 3. Returning now to the solution of 3. The vector Sa in 3. For a practical and thorough engineering-level account of the discrete Fourier transform, see Smith Tables of better-known transforms can be found in formula handbooks e. We mentioned in Sect. We may ask naively whether 3.

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The answer is: yes, but only for the single source, and even then the overlap vectors will not match perfectly, and we will be thrown back at a discretized aliased version of pattern matching. Should we follow that route, the full-blown iterative method described in Chap. We saw in 3. In this chapter, we look at some of the prevalent wireless standards, and draw conclusions about feasible radiolocation strategies. Skip to main content. Advertisement Hide. Chapter First Online: 11 November This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, log in to check access. Antolovic, D. Bray, J. Gast, M. Litwin, L.