Radar Waveform Design Based on Optimization Theory
2: University of Naples Federico II, Naples, Italy
3: Department of Electronic and Electrical Engineering, University College London, London, UK
4: Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL, USA
This book provides an overview of radar waveform synthesis obtained as the result of computational optimization processes and covers the most challenging application fields. The book balances a practical point of view with a rigorous mathematical approach corroborated with a wealth of numerical study cases and some real experiments. Additionally, the book has a crossdisciplinary approach because it exploits crossfertilization with the recent research and discoveries in optimization theory. The material of the book is organized into ten chapters, each one completed with a comprehensive list of references. The following topics are covered: recent advances of binary sequence designs and their applications; quadratic optimization for unimodular sequence synthesis and applications; a computational design of phaseonly (possibly binary) sequences for radar systems; constrained radar code design for spectrally congested environments via quadratic optimization; robust transmit code and receive filter design for extended targets detection in clutter; optimizing radar transceiver for Doppler processing via nonconvex programming; radar waveform design via the majorizationminimization framework; Lagrange programming neural network for radar waveform design; cognitive local ambiguity function shaping with spectral coexistence and experiments; and relative entropy based waveform design for MIMO radar. Targeted at an audience of radar engineers and researchers, this book provides thorough and uptodate coverage of optimisation theory for radar waveform design.
Inspec keywords: quadratic programming; filtering theory; Doppler radar; waveform analysis; entropy codes; binary sequences; radar transmitters; MIMO radar; radar receivers; radar clutter; radar theory; cognitive radar; minimisation; concave programming; radar computing; neural nets; radar detection
Other keywords: nonconvex programming; unimodular sequence synthesis; quadratic optimization; radar transceiver optimization; cognitive local ambiguity function shaping; phaseonly sequences; extended targets detection; robust transmit code; Doppler processing; clutter; computational design; optimization theory; MIMO radar; Lagrange programming neural network; relative entropybased waveform design; spectral coexistence; radar waveform design; constrained radar code design; spectrally congested environments; radar systems; receive filter design; binary sequence designs; majorizationminimization framework
Subjects: Signal processing theory; Codes; Radar equipment, systems and applications; General and management topics; Signal detection; General electrical engineering topics; Optimisation techniques; Mathematical analysis; Neural computing techniques; Filtering methods in signal processing; Mathematical analysis; Neural nets (theory); Optimisation techniques; Electrical engineering computing; Radar theory
 Book DOI: 10.1049/SBRA533E
 Chapter DOI: 10.1049/SBRA533E
 ISBN: 9781785619434
 eISBN: 9781785619441
 Page count: 348
 Format: PDF

Front Matter
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1 On recent advances of binary sequence designs and their applications
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Sequence design can find diverse applications in areas, including active sensing, communications, and medical imaging. In recent years, computational methods have been devised to synthesize arbitrary waveforms under various practical constraints, including the constant modulus constrains and spectrum containment restrictions. Indeed, as the hardware cost of arbitrary waveform generators decreases, arbitrary waveforms become widely popular in diverse systems, such as cognitive active sensing systems. However, for massive commercial markets, the active sensing systems, such as the automotive radar systems, must be rather inexpensive. Therefore, highly diverse binary sequences with arbitrary lengths are primary candidates for these systems due to the lowcost hardware advantages of generating these sequences. In this chapter, we will review wellknown binary sequences and their properties. We will also discuss recent advances of binary sequence designs and their applications.

2 Quadratic optimization for unimodular sequence synthesis and applications
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In this chapter, the non polynomial (NP) hard quadratic optimization problem under the constant modulus and similarity constraints is considered. Low complexity algorithms for both the continuous and finite alphabet phase cases are developed. Specifically, an iterative algorithm for the continuous phase case referred to as iterative algorithm for continuous phase case (IA CPC) is introduced to successively optimize the objective function with closed form solutions and its convergence is proven analytically. The computational complexity of IA CPC is only linear in the number of iterations and polynomial with the size of the design variables. Moreover, for the discrete phase case, the design variables are divided into K blocks. In each iteration, the exhaustive search is leveraged to sequentially optimize each block assuming the remaining K  1 blocks are fixed. The computational burden is related to the size of the design variables, the number of design variables in each block and the number of discrete phases. Finally, the performance of the new techniques in comparison with the existing approaches available in the open literature are assessed by considering radar code design applications. Numerical results are provided to show the effectiveness of the proposed algorithms in terms of the achieved objective function value and computational complexity. More examples related to radar applications can be found.

3 A computational design of phaseonly (possibly binary) sequences for radar systems
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This chapter is framed in the mentioned context with the goal of providing a comprehensive study on the design of unimodular (in particular binary) sequences possessing good aperiodic autocorrelation properties for radar systems. To this end, a new technically sound procedure aimed at designing continuous/discrete phase sequences with good aperiodic autocorrelation function (in terms of PSL and ISL) is introduced in this chapter. Specifically, resorting to the Pareto framework, the weighted sum of PSL and ISL is considered as an objective function to optimize under the phase only constraint on the probing waveform. Hence, an iterative procedure based on the coordinate descent (CD) method is introduced to deal with the resulting non deterministic polynomial time hardness (NP hard) optimization problem. Each iteration of the devised method requires the solution of a nonconvex min max problem. It is handled either through a novel bisection or a fast Fourier transform (FFT)based method, respectively, for the continuous and the discrete phase constraint. Several numerical examples will illustrate the performance enhancement (especially in the highly important binary case) of the devised algorithm.

4 Constrained radar code design for spectrally congested environments via quadratic optimization
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In the present chapter, we have addressed radar waveform optimization in spectrally crowded scenarios via constrained quadratic programming techniques. First of all, we have defined a taxonomy of performance measures which are to be optimized and/or controlled in the waveform synthesis: SINR, signal energy, similarity constraint, interference constraint for spectral compatibility, and quality constraint for bandwidth selection. Then, we have formulated the radar waveform design as a nonconvex QCQP problem and have exploited a specific methodology leveraging rankone matrix decomposition tools to prove the hidden convexity of some instances of the waveform optimization or to build solution procedures ensuring good quality radar waveforms.

5 Robust transmit code and receive filter design for extended targets detection in clutter
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In this chapter, the problem of joint transmit code and receive filter design has been considered to optimize the achieved SINR of extended targets in the presence of signal dependent interference. Due to uncertainties about the TIR, a worst case optimization framework has been proposed over two different T1R uncertainty sets. The former contains T1R's samples corresponding to some TAAs, whereas the latter is equivalent to bounding the actual T1R within a spherical set. Additionally, a PAR constraint has been imposed on the transmitted waveform so as to ensure its practical implementation. The obtained non convex optimization problem has been divided into two sequential relaxed SDP problems. Their corresponding solutions have then been exploited to synthesize the code filter pair via randomization methods.

6 Optimizing radar transceiver for Doppler processing via nonconvex programming
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This chapter addresses the performance improvement of a radar system in typical scenarios involving moving targets in the presence of clutter. The transmitted waveform and the receiver architecture form the key elements of the radar system. Hence, the goal is the joint design of the transmitter receiver pair for Doppler processing application via structured non convex programming optimization. Specifically, a bank of filters is considered at the receiver end to deal with the unknown Doppler shift of the target and the focus is on the joint design of the transmit waveform and the filter bank. To this end, the worst case signal to interference plus noise ratio (SINR) at the output of the filter bank is selected as the figure of merit to optimize. Besides, some suitable constraints are forced on the sought waveform. To handle the resulting non convex max min optimization problem, an appropriate reformulation is provided that paves the way to the design of an innovative iterative procedure, exhibiting a monotonic improvement of the worst case SINR. With regards to algorithm features, each iteration involves both a convex and a generalized fractional programming (GFP) problem that can be globally solved via generalized Dinkelbach's procedure with a polynomial computational complexity.

7 Radar waveform design via the majorizationâ€“minimization framework
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A multiple input multiple output (MIMO) radar transmits several probing waveforms simultaneously from its transmit antennas. It can improve the interference rejection capability and parameter identifiability as well as provide fl exibility for transmit beam pattern design . In addition, a cognitive approach for radar systems, which capitalizes on the information obtained from the surrounding environment or the prior knowledge stored in the platform, was proposed . The significance of MIMO radar and the cognitive approach has recently motivated active research into the waveform design. A well designed waveform can allow for more accurate detection and estimation.

8 Lagrange programming neural network for radar waveform design
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9 Cognitive local ambiguity function shaping with spectral coexistence and experiments
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In this chapter, the authors cope with the joint optimization of fasttime LAF and waveform spectrum. A novel objective function with a weighting factor to tradeoff the WISL with respect to rangeDoppler sidelobes and the spectral stopband energy is proposed. The energy and PAR constraints are forced on the designed signal, which further generalize the problem we developed. To handle the resulting nonconvex quadratic problem, a new iterative sequential quadratic optimization (ISQO) algorithm is proposed [30]. In each iteration, it transforms the optimization problem into a tractable quadratic subproblem that is further converted to a linear optimization problem with a closedform solution by resorting to the firstorder Taylor expansion. The proposed algorithm has polynomial time complexity that is linear with the number of iterations and polynomial with the size of the designed waveform and the number of the considered rangeDoppler bins. The simulation results highlight the superiority of the proposed algorithm contrast to AISO algorithm in terms of the objective value, computational time, and the capability of prohibiting the sidelobes of a strong return and narrowband spectral interferences from masking weak targets. The experimental results show that the hardware limitation influences the level suppressed in both energy spectral density (ESD) and WISL and confirms the effectiveness of spectral coexistence in prohibiting the sidelobes suppressed in the LAF rising.

10 Relative entropybased waveform design for MIMO radar
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In this chapter, the author considered the waveform design problem for MIMO radars in the presence of signal dependent clutter. This paper aim at improving the detection performance of the radar system. To this end, the author have employed the relative entropy associated with the detection problem as the design metric. The author studied the optimal structure of the energyconstrained waveforms and shown that the optimal left singular vectors of the waveform matrix should be aligned with the least interfered interference subspace. Two algorithms, which are both devised based on the MM techniques, are proposed to tackle the nonconvex design problem. This paper proved the convergence of the objective values for both algorithms. For the twostage algorithm, the associated minorizer is tighter to the objective function than that of the onestage algorithm. Thus, it converges faster than the onestage algorithm but with a much higher periteration computational complexity.

Back Matter
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