Principles of Modern Radar. Volume 2.pdf

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Preface

What this Book Addresses

Why this Book was Written

We are very proud to say that the chapters in this volume have been written by leading experts in the field of radar, all of whom are active researchers in their areas of expertise and most of whom are also short course instructors for practicing engineers. . We are grateful to each of the contributing authors who share our vision for a long-needed advanced radar textbook that covers a diverse range of topics in a clear, coherent, and consistent framework.

How the Content was Developed

The History of the POMR Series

Acknowledgements

Most importantly, we are grateful to our families for their patience, love and support as we prepare, review, revise, coordinate and iterate material.

To our Readers

Publisher Acknowledgments

Technical Reviewers

Editors and Contributors

Volume Editors

Chapter Contributors

He is a member of the Society for Industrial and Applied Mathematics (SIAM) and the Institute of Electrical and Electronics Engineers (IEEE). He is a member of the Scientific Committee of SESAR (Single European Sky ATM Research) European Commission and Eurocontrol.

Overview: Advanced 1

Techniques in Modern Radar

INTRODUCTION
RADAR MODES
RADAR AND SYSTEM TOPOLOGIES
TOPICS IN ADVANCED TECHNIQUES

Waveforms and Spectrum
Synthetic Aperture Radar
Array Processing and Interference Mitigation Techniques Section 1.2 suggests that measurement diversity and the ability to adapt to the chang-
Post-Processing Considerations
Emerging Techniques

COMMENTS
REFERENCES

Due to the maturity of subsystem technology – especially antenna and computing capabilities – the class of targets of interest to air-to-ground radar has evolved rapidly. System performance is determined by a number of factors, including the two-dimensional cross-correlation function (that is, the ambiguity function) for the passive waveform.

AAAA

PART I

Waveforms and Spectrum

Advanced Pulse Compression Waveform Modulations and Techniques

Optimal and Adaptive MIMO Waveform Design

MIMO Radar

Radar Applications of Sparse Reconstruction and Compressed Sensing

Advanced Pulse Compression 2

Waveform Modulations and Techniques

INTRODUCTION

Organization
Key Points
Notation

Common Variables
Stretch Processing
Stepped Chirp Waveforms
Nonlinear Frequency Modulated Waveforms Variables associated with NFLM waveforms are as follows
Mismatched Filters

Acronyms

SNRBF signal-to-noise ratio at the output of a filter with bandwidth BF SNRD FT Signal-to-noise ratio at the output of the DFT. Yn(w) spectrum of the nth pulse with a matched filter applied to the receive bin size 8fD FT DFT.

STRETCH PROCESSING

Introduction
Processing Bandwidth
Technique Overview
Implementation

Transmit Waveform
Receiver
SNR Loss
Discrete Fourier Transform

Example System Parameters
Processing Gain

Now, consider a distributor placed at a time delay equal to the center of the range window (i.e., td =tr cvortd =0). The minimum number of samples to process is equal to the product of the sampling rate and the pulse width and is 100,000.

STEPPED CHIRP WAVEFORMS

Introduction
Transmit Waveform
Processing Options

Time Domain
Frequency Domain

Summary

The signal in equation (2.59) contains two phase terms, 2πf0td and 2πpf0td, which are constant over a pulse. In the signal processor, the first step is to upsample (i.e., interpolate) the baseband signals in equation (2.59) by a factor Nsc. Note that the phase term 2πpf td in equation (2.68) has been removed by the shift operation.

NONLINEAR FREQUENCY MODULATED WAVEFORMS

Inverse Functions
Parametric Equations
Fourier Approach
Design Example
Doppler Analysis
Summary

Discrete Fourier Transform
Range Sidelobe Suppression

Straddle Loss and Range Ambiguities

Straddle Loss
Waveform Parameter Trade Summary

Impacts of Doppler

The relationship in equation (2.81) suggests that the derivative of the phase function (that is, the instantaneous frequency) at a given value of time t determines the spectral response at a given frequency, and that and t are one-to-one, provided that φ (t) equal is monotone. For a point target located at range R0, the compressed response is a sampled instantiation of the digital sinc in (2.114) or . M The DFT is a linear operator; thus, the range profile associated with multiple scatterers is simply the superimposition of the individual responses. The product of the PRI and the number of pulses, NT, defines the dwell time associated with the waveform.

QUADRIPHASE SIGNALS

Introduction
BTQ Transformation
Quadriphase Waveform
Waveform Properties .1 Spectrum
Summary

Over a chip, the phase of the signal varies linearly, whereas the phase of a traditional phase-encoded waveform is constant. In a radar system, the amplitude and phase of the complex signal in equation (2.139) are used to modulate a sinusoid. With a phase-coded waveform, the shape of the chip determines the envelope of the waveform's spectrum [1].

MISMATCHED FILTERS

Introduction
Performance Metrics
Mismatched Filter Approaches

Minimum ISR Filter
Minimum ISR Example
Reduced Peak Sidelobe Filters

Summary

1 is a C matrix of dimension (K +M−1)× M whose rows contain delayed copies of the phase code, and z is a column vector containing the filter coefficients. 250-element MISR filter as a function of the number of Doppler shift cycles across the pulse. In the zero-region filter, the 40 sidelobes on either side of the main lobe are assigned a weight of 10, and the remaining sidelobes are assigned a weight of 1.

PROBLEMS

46] Nunn, C., “Constrained optimization applied to pulse compression codes and filters,” in Proceedings of the 2005 IEEE International Radar Conference, p. and the increase in peak lateral cowl caused by Doppler shift. A filter is needed to suppress the two sidelobes on either side of the main lobe.

Optimal and Adaptive MIMO 3

Waveform Design

INTRODUCTION

Organization
Key Points

The concept of constrained MIMO waveform design is addressed in Section 3.5 to account for important real-world constraints such as constant modulus. Finally, in Section 3.6, the idea of adaptive MIMO waveform design is introduced when the channel needs to be estimated on the fly. Note that for the case of additive Gaussian colored noise (AGCN) this receiver is also statistically optimal [7].

Optimum MIMO Waveform Design for the Additive Colored Noise Case 91 is given by [7]

Next, we illustrate the application of the previously given optimal design equations to the additive colored noise problem arising from a broadband multipath interference source. This example illustrates the optimal transmit/receive configuration for maximizing output SINR in the presence of colored noise interference arising from a multipath broadband noise source. Note that in both cases the optimal pulse tries to anti-match the colored noise spectrum under the frequency resolution constraint set by the total pulse width.

Optimum MIMO Design for Maximizing Signal-to-Clutter Ratio 95 That is, the optimum transmit waveform is the maximum eigenfunction associated with the

OPTIMUM MIMO DESIGN FOR MAXIMIZING SIGNAL-TO-CLUTTER RATIO

Referring to Figure 3-5, the corresponding SCR at the receiver input is given by . Equation (3.24) is a generalized Rayleigh quotient [8] that is maximized when s is a solution to the generalized eigenvalue problem. Due to space constraints, we will instead consider its application to the sidelobe target suppression problem, which is very closely related to the ground clutter interference problem.

Optimum MIMO Design for Maximizing Signal-to-Clutter Ratio 97

Note the presence of transmit antenna pattern nulls in the directions of competing targets as desired. In this simple example, we rigorously verify an intuitively obvious result regarding the pulse shape and detection of a point target in uniform clutter: the best waveform for detecting a point target in independent and distributed clutter identically (i.d.) is itself an impulse (i.e., a maximum-resolution waveform), a well-known result rigorously proven by Manasse [21] using a different method.

Optimum MIMO Design for Target Identification 99 where

OPTIMUM MIMO DESIGN FOR TARGET IDENTIFICATION

Optimum MIMO Design for Target Identification 101 where

Note that Sopt(ω) places more energy in spectral regions where H(ω) is large (i.e., spectral regions where the difference between targets is large, which is again an intuitively attractive result). Although pulse modulation was used to illustrate the optimal transmit design equations, theoretically we could have used any transmit DOF (e.g. polarization). Note that the optimal pulse emphasizes the parts of the spectrum where the two targets differ the most.

Optimum MIMO Design for Target Identification 103 Multitarget Case Given L targets in general, we wish to ensure that the L-target re-

This compares to 0.47 for LFM in Example 3.4, an improvement of 6.5 dB, which is slightly less than the previous two-target example. As expected, the optimal waveform significantly outperforms the unoptimized pulse waveform, such as LFM.

CONSTRAINED OPTIMUM MIMO RADAR

The stationary phase method has been applied to the problem of creating a non-linear frequency modulated (NLFM) pulse (and thus constant modulus in the sense that the modulus of the baseband complex envelope is constant, i.e. |s(t)| = constant ) with a prescribed size spectrum [5, 12]. Here we use the stationary-phase method to design a constant-module NLFM pulse that matches the magnitude spectrum of the optimal pulse derived for the multipath interference problem addressed in Example 3.1. It is natural to ask whether an NLFM waveform with the same spectral magnitude as the optimal pulse (but not necessarily the same phase) will enjoy some (if not all) of the matching gains.

ADAPTIVE MIMO RADAR

Dynamic MIMO Calibration

As a result, an estimate of the covariance matrix is used in (3.5) for the whitening filter rather than the actual covariance, as was the case in Section 3.2. Plotted in Figure 3-19 is the overall output SINR loss relative to the optimum for the short-pulse case of example 3.1 as a function of the number of independent samples used in (3.65). By repeating this process for different orientations, a detailed lookup table for the steering vector vectors can be constructed.

PROBLEMS

Maximizing the norm of the separation metric d= y1−y2in equation (3.42) is statistically optimal for the additive white noise case assuming a unimodal PDF and monotonic distribution function. The energy in the white target echo for the case of infinite duration is given by 1. Derive an expression for the waveform sp that lies in the matched subspace spanned by the k best eigenvectors.

MIMO Radar

INTRODUCTION

Organization
Notation

Assume that only a single target is illuminated and detected by the elements of a MIMO radar. In a MIMO radar with widely separated antennas, each transmit/receive pair is assumed to observe an independent realization of the target reflectivity. For these reasons, MIMO radar with widely separated antennas will not be covered in the following discussion.

THE MIMO VIRTUAL ARRAY

MIMO Signal Correlation Matrix
MIMO Spatial Beamforming
MIMO Gain
Phased Array versus Orthogonal Waveforms

The signal correlation matrix of the spoiled phased array, Rφ/Spoilt, is rank-1, just as in the case of the pristine phased array. We have shown that the transmission gain of an array antenna can be controlled by designing the MIMO signal correlation matrix. Let GRX(θ;θ0) be the gain of the receiving array in the direction θ when digitally sent to the angle θ0.

WAVEFORMS FOR MIMO RADAR

Classes of Waveforms for MIMO Radar
MIMO Range Response
Example: Up- and Down-Chirp

Note that our analysis of the spatial characteristics of a MIMO radar focused on the case where τ = 0, the range bin of interest. As shown in Figure 4-9, each waveform has a desirable autocorrelation, and the cross-correlation peak is well below that of the autocorrelation. This is evident from the range response, which for a broadside target is the sum of the auto and cross correlations, as reported in (4.34).

APPLICATIONS OF MIMO RADAR

MIMO SAR
MIMO GMTI
Distributed Apertures
Beampattern Synthesis

Note that the effective length of the dense MIMO array is the same as that of the corresponding Vernier array. However, a higher trackside sampling rate can be achieved by the dense MIMO array if it uses the same PRF as the Vernier array. Note, however, that the angular resolution of the phased array is not improved by decay.

SUMMARY

The SINR loss for the orthogonal waveforms is significantly lower than that for the conventional phased array for many frequencies, so a distinctly improved minimum speed is expected. It also allows the traditional phased array architecture to be analyzed as a subset of MIMO radar. Since all radars are MIMO radars, the question is not whether to use a phased array or a MIMO radar, but rather which signal correlation matrix is optimal.

Reconstruction and Compressed Sensing

INTRODUCTION

Organization
Notation
Acronyms

CS performance guarantees are based on conditions of the forward operator, such as the restricted isometric property or mutuality. N =number of elements in the unknown vector M =number of measurements collected s=xtrue0, the sparsity of the unknown vector. K =number of slow time pulses L=number of range gates in fast time T = slow time sampling period (seconds).

CS THEORY

The Linear Model

Radio Frequency Tomography Example
The Ambiguity Function
Multichannel Example
Comments

Regularization of the Linear Model through Sparsity
Performance Guarantees

The Restricted Isometry Property
Matrices that Satisfy RIP
Mutual Coherence

In section 5.3.1 we will discuss different formulations of the problem described in (5.12) and algorithms for its solution. From a CS perspective, these randomization techniques can be seen as attempts to reduce the coherence of the forward operator A [17]. A manageable but conservative limit for the RIC can be obtained from the interdependence of the columns of A defined as.

SR ALGORITHMS

Equivalent Optimization Problems and the Pareto Frontier
Solvers
Thresholding Algorithms

Soft Thresholding
Hard Thresholding

Iterative Reweighting Schemes

The convex relaxation of the reconstruction problem 0 given in (5.12) can be considered as a penalized least squares problem. The fact that the solution BP is the limit of the solution QPλ is important. Briefly, determining λ from the solution of one of the constrained problems is quite difficult.