AngleofArrival Estimation Using Radar Interferometry: Methods and Applications
Radar interferometers provide a costeffective radar architecture to achieve enhanced angle accuracy for enhanced target tracking. Presenting a comprehensive understanding of various radar interferometer architectures, Angle of Arrival Estimation Using Radar Interferometry aims to quantify interferometer angle estimation accuracy by developing a general understanding of various radar interferometer architectures and presenting a comprehensive understanding of the effects of radarbased measurement errors on angleofarrival estimation. The interferometer architectures described include a basic digital interferometer, a monopulse interferometer, an orthogonal interferometer and signal processing algorithms. By exploring interferometry and beyond, this book offers a unique perspective and an in depth look at the derivation of angle error equations for a radar interferometer as affected not only by additive noise but by other error effects such as multipath, glint, and spectral distortion. As such this book is primarily directed toward tracking radars but will also discuss imaging applications as well.
Inspec keywords: signal processing; antenna arrays; radar interferometry; directionofarrival estimation; radar antennas
Other keywords: radar fundamentals; RF interferometry; radar angleofarrival estimation; radar waveforms; tropospheric effects; interferometer signal processing; interferometer angleofarrival error effects; radar interferometry; probability theory; sparsely populated antenna arrays
Subjects: Antenna arrays; Radar equipment, systems and applications; General electrical engineering topics; Signal processing and detection
 Book DOI: 10.1049/SBRA508E
 Chapter DOI: 10.1049/SBRA508E
 ISBN: 9781613531846
 eISBN: 9781613531853
 Page count: 360
 Format: PDF

Front Matter
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1 Applications of RF Interferometry
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The first known use of an interferometer was in the work of Michelson and Morley in 1890 and again in 1920, where they used interference patterns from light emitted by stars to measure the diameter of large stars. The stellar image creates an interference pattern related to the diameter of the star and the size of the optical aperture. Shortly thereafter, in 1946, Ryle and Vonberg transferred the principles of optical interferometry to radio waves for solar observations. These early interferometers combined coherent analog signal to create amplitude interference patterns to achieve enhanced angular resolution for stellar measurements and imaging. With advances in radio frequency transmitter and receiver technology and in analogtodigital converters, interferometers have entered the digital age. The digital interferometer spatially samples signals at rates greater than or equal to the Nyquist rate for the bandwidth limited signals and uses phase, as opposed to amplitude, information to measure angleofarrival with high precision. As a result, these digital radio frequency (RF) interferometers have found application in several areas that include military, commercial, and scientific endeavors. In this chapter, five applications are presented that illustrate the diversity and versatility of interferometry: military, sports, synthetic aperture radar (SAR), radio astronomy, and geostationary satellite tracking. Both SAR and radio astronomy are imaging interferometer techniques that take advantage of large aperture separations, whereas military and sports applications are tracking radars that use interferometry to achieve high angle accuracy as opposed to high angle resolution. In addition, radio astronomy is an example of a passive interferometer whose signal is generated from an external source (stellar objects). The other three interferometer applications are active radars that generate specific waveforms or signals that facilitate interferometer angle estimation. This book focuses on active tracking interferometers where angle accuracy is the driving requirement. The distinction between tracking and imaging interferometers is made in this chapter using the five examples.
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2 Probability Theory
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The theory of probability is a special case of general measure theory in mathematics and as such has its fundamental definitions and basic results derived from real analysis. In this chapter, we define the fundamentals of probability theory only to the extent necessary for its application to radar interferometry error analysis, while maintaining some fidelity to measure theory. The CramerRao lower bound is introduced and will be used in later chapters to provide an estimate for unbiased angleofarrival estimates. The WeissWeinstein, ZivZakai, and Bhattacharyya lower bounds for biased random processes are defined and applied in Chapter 7 to estimate the performance of angleofarrival in the presence of angle ambiguities.
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3 Radar Fundamentals
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In the midnineteenth century, James Clark Maxwell, a British mathematician, developed a unified theory of electricity and magnetism. The equations predicting the behavior of electromagnetic signals, known as Maxwell's equations, led to discoveries that include radar waves. Maxwell's equations can be used to derive representations of radar signals propagating in various media, and for this purpose we can use the equations to derive a free space propagation of radar signals. These signal representations will be useful in understanding the impact of error sources on angleofarrival estimation using interferometric radar.
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4 Radar AngleofArrival Estimation
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The paper focus on interferometer angle error and on the signal processing required for interferometry, but also introduce monopulse angleofarrival for comparison. This chapter, derive the basic equations for monopulse angle estimation and discuss the fundamentals of eigenbased superresolution techniques.
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5 Radar Waveforms
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Waveform selection is a critical component of an interferometer architecture design. Achieving angle accuracy in complex target and interference environments can be accomplished only with the proper choice of waveforms. Mediumto highresolution waveforms can mitigate the effects of clutter and multipath that can degrade phase integrity and angle accuracy. When the clutter and/or multipath are spatially distributed, the range sidelobe performance resulting from the matched filter is critical in reducing the effects of distributed clutter and multipath. Because an interferometer may use small distributed arrays with relatively large beamwidths, the selection of waveforms to mitigate the scattering effect or to resolve target features becomes an important part of an interferometer architecture. In this chapter, several types of frequencyand phasemodulated waveforms are defined, and their impact on postcompression angle accuracy is discussed.
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6 The Radar Interferometer
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In their classic book, Barton and Ward characterized angleofarrival errors for various antenna types, including interferometers. In this chapter, we focus on the effect of additive random noise on angleofarrival accuracy for several types of interferometers: monopulse interferometer, digital interferometer, orthogonal interferometer, amplitude interferometer, and bistatic interferometer. In each case, angleofarrival is computed as a difference in measurements and the error due to additive random noise affects each interferometer type differently. The basic angle accuracy equations are derived for each interferometer type for the case that noise is independent and identically distributed (IID) and for the case when it is not IID. For the monopulse and digital interferometer, the general case is considered where the noise error is neither independent nor identically distributed. These basic accuracy equations relate radar design parameters to angle accuracy and can be used to design interferometer architectures. For this chapter, we derive the accuracy equations assuming an interferometer that uses two antennas to measure angle in one dimension. The angle precision relationships presented in this chapter are derived from principles developed in Chapters 2, 3, and 4. In Chapter 7, we will consider interferometers with more than two antennas that are designed to estimate the twodimensional angleofarrival.
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7 Interferometer Signal Processing
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In this chapter, the fundamental signal processing required to implement an interferometer is introduced. The processing required for an interferometer is similar to the processing required for radars in general with the exception that angle estimation is implemented using phase comparison. Because phase is defined only over an interval [0 2p], the phase can wrap over many 2p intervals, creating an ambiguity in the phase measurement. For the interferometer, that relative phase ambiguity between two antennas must be resolved in order to estimate angleofarrival. In this chapter, we discuss methods to resolve interferometer phase ambiguities. In particular, all receive interferometer antennas or arrays must be calibrated to ensure that relative phase is consistent across the radar field of regard. The orthogonal interferometer has unique processing requirements due to its implementation of pseudoorthogonal waveforms. Calibration of an orthogonal interferometer requires that all antennas be calibrated for transmit as well as receive. The derivation of angleofarrival from relative phase measurements is presented for various farfield conditions. The firstorder derivation assumes that the constant phase contours that emanate from a target are essentially linear across the interferometer distributed antennas. This linear farfield condition is satisfied when the target is sufficiently far in range from the interferometer but depends on the interferometer baseline and radar operating frequency. Secondorder derivations are also presented for quadratic phase behavior across the interferometer antenna.
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8 Sparsely Populated Antenna Arrays
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The chapter begins with a brief introduction to sparse linear arrays. We use interval partitions to define minimum redundancy partitions and almost minimum redundancy partitions that achieve low sidelobe excitation patterns. These arrays assure that the spatial Nyquist condition is satisfied and also attempt to maximize the number of partition differences for optimized sidelobe control. Interval partitions are applied to coprime integers to generate coprime arrays. The spatial Nyquist condition and partition difference redundancy are used to develop a numerical sieve to generate arrays with low sidelobe antenna patterns. The linear Nyquist condition is generalized to two dimensions to generate 2D sparse arrays with low sidelobe performance. Angleofarrival estimation methods are developed first for linear sparse arrays that satisfy the spatial Nyquist condition and then generalized to arrays that do not satisfy the spatial Nyquist condition. Finally, the methods of Ishimaru [1962] and Mitra et al. [2004, 2005] are developed to illustrate two techniques that use formulations of the uniform linear array antenna pattern to create unequally spaced arrays elements.
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9 Interferometer AngleofArrival Error Effects
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The angle error equations derived to this point have reflected only the error performance due to thermal noise effects. However, angleofarrival performance can be affected by various sources of error other than thermal noise. In this chapter, we focus on errors that affect angle accuracy as opposed to angle precision. Because an interferometer determines angleofarrival from the output of phase or voltage from two antennas, the sensitivity of angle measurement to errors can be significant.
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10 Tropospheric Effects on AngleofArrival
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In this chapter, we characterize the effects of angle bias due to refraction and angle precision due to turbulence. As a result, angle bias due to refraction can be calibrated and removed, if possible, and any residual error can be quantified in the interferometer angle error budget. Also, the effects of angle precision due to turbulence can be determined and accounted for in the same angle error budget. For turbulence it will be important to understand the spatial correlation of errors that will determine the magnitude of the impact of turbulence effects on angle precision for an interferometer.
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Appendix A: Discrete Fourier Transform
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The discrete Fourier transform (DFT) is the algorithm of choice to implement a sampled version of the Fourier transform. The DFT can be interpreted as a sampled version of the discrete time Fourier transform (DTFT). The DFT is normally implemented in practice using the famous fast Fourier transform (FFT) algorithm. The FFT is an efficient way to implement the DFT and is not a transform in and of itself. The apparent loss of amplitude in the DFT when the input signal frequency does not match the DFT sample frequency is called straddle loss. Straddle loss can be mitigated by zero padding the FFT algorithm to increase the sample frequency and applying a weighting function to the data to degrade the resolution.
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Appendix B: The Matched Filter
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In this appendix, we derive the expression for the matched filter that optimizes the signaltonoise ratio for a given signal. We show that the matched filter is directly proportional to the signal shifted in time divided by the noise power and thus is matched to the signal and the noise.
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Appendix C: The Principle of Stationary Phase
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The principle of stationary phase states that the major contribution to the energy spectral density comes from when the phase is stationary. In other words, the spectral density energy of frequency is relatively large at a specific time when the rate of change of frequency at that time is relatively small.
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Appendix D: The Fundamental Theory of Binary Code
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This paper will begin with a review of the theory of binary codes. A binary shift register was also defined.
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Appendix E: Theoretical Development of Kasami Codes
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We present the mathematical theory that leads to Kasami codes using the notation developed in Appendix D.
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Appendix F: Relationship of the Continuous Power Spectrum and Discrete Variance
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We show the relationship between the continuous noise power spectrum and the sample variance for a discrete process.
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Appendix G: TimeofArrival CRLB (Alternative Approach)
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We take a less rigorous approach to establish the CLRB in order to provide an alternative derivation of the CRLB for timeofarrival angle estimation.
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Appendix H: TwoDimensional Trilateration Using CPT and RGS Ranging Methods  MATLAB® Code
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This program computes the location of a target in 2D using trilateration. The trilateration solution uses phase differencing with four receivers where the trilateration can be either CPT or RGS. The distinction is in which error model for range error is chosen to be used in the code.
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Appendix I: AngleofArrival Determination Using a Rotated Antenna Configuration
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Assume that interferometer antennas are configured in a triangle configuration and that the origin is chosen to be equidistant from each antenna. Also assume that the three interferometer antennas are located in the xy plane with y being up and that z is perpendicular to the array plane. Let d define the distance from any antenna to the origin and a1 and a2 be the rotation angle of the two left antennas with respect to the xaxis and a3 be the rotation of the right antenna with respect to the xaxis. Furthermore, we assume that the xy axes are rotated such that a1 = a2 = a.
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Appendix J: First and SecondOrder Interferometer Angle Measurements  MATLAB® Code
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This paper computes antennas lie in the xy plane, first order angle estimation, and second order angle estimation.
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Appendix K: Interferometer Angle Measurements for Distributed Transmit/Receive Antennas  MATLAB® Code
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This paper presents the interferometer angle measurement for distributed transmit/receive antenna.
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Back Matter
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