",isbn:"978-1-83968-760-0",printIsbn:"978-1-83968-759-4",pdfIsbn:"978-1-83968-761-7",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"cc49d6034d85f8f2e2890c6acc3cc629",bookSignature:"Dr. Abhijit Biswas",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10285.jpg",keywords:"Mott Insulators, Semi Metals, Polycrystals, Single Crystals, Electronic Properties, Magnetic Properties, PLD, MBE, Topological Insulators, Topological Hall Effect, Devices Applications, Catalysis",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"September 9th 2020",dateEndSecondStepPublish:"October 7th 2020",dateEndThirdStepPublish:"December 6th 2020",dateEndFourthStepPublish:"February 24th 2021",dateEndFifthStepPublish:"April 25th 2021",remainingDaysToSecondStep:"5 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"A pioneering researcher in the field of tailoring metal oxide crystal surfaces and growth as well as engineering of thin films for various emergent phenomena and energy applications. Dr. Biswas received his Ph.D. from POSTECH, South Korea.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"194151",title:"Dr.",name:"Abhijit",middleName:null,surname:"Biswas",slug:"abhijit-biswas",fullName:"Abhijit Biswas",profilePictureURL:"https://mts.intechopen.com/storage/users/194151/images/system/194151.png",biography:"Dr. Abhijit Biswas is a research associate at the Indian Institute of Science Education and Research (IISER) Pune, in India. His research goal is to design and synthesize highest quality epitaxial heterostructures and superlattices, to play with their internal degrees of freedom to exploit the structure–property relationships, in order to find the next-generation multi-functional materials, in view of applications and of fundamental interest. His current research interest ranges from growth of novel perovskite oxides to non-oxides epitaxial films, down to its ultra-thin limit, to observe unforeseeable phenomena. He is also engaged in the growth of high quality epitaxial layered carbides and two-dimensional non-oxide thin films, to exploit the strain, dimension, and quantum confinement effect. His recent work also includes the metal-insulator transitions and magneto-transport phenomena in strong spin-orbit coupled epitaxial perovskite oxide thin films by reducing dimensionality as well as strain engineering. He is also extremely interested in the various energy related environment friendly future technological applications of thin films. In his early research career, he had also extensively worked on the tailoring of metal oxide crystal surfaces to obtain the atomic flatness with single terminating layer. Currently, he is also serving as a reviewer of several reputed peer-review journals.\nDr. Biswas received his B.Sc. in Physics from Kalyani University, followed by M.Sc in Physics (specialization in experimental condensed matter physics) from Indian Institute of Technology (IIT), Bombay. His Ph.D., also in experimental condensed matter physics, was awarded by POSTECH, South Korea for his work on the transport phenomena in perovskite oxide thin films. Before moving back to India as a national post-doctoral fellow, he was a post-doc at POSTECH working in the field of growth and characterizations of strong spin-orbit coupled metal oxide thin films.",institutionString:"Indian Institute of Science Education and Research Pune",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"Indian Institute of Science Education and Research Pune",institutionURL:null,country:{name:"India"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"20",title:"Physics",slug:"physics"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"205697",firstName:"Kristina",lastName:"Kardum Cvitan",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/205697/images/5186_n.jpg",email:"kristina.k@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"8356",title:"Metastable, Spintronics Materials and Mechanics of Deformable Bodies",subtitle:"Recent Progress",isOpenForSubmission:!1,hash:"1550f1986ce9bcc0db87d407a8b47078",slug:"solid-state-physics-metastable-spintronics-materials-and-mechanics-of-deformable-bodies-recent-progress",bookSignature:"Subbarayan Sivasankaran, Pramoda Kumar Nayak and Ezgi Günay",coverURL:"https://cdn.intechopen.com/books/images_new/8356.jpg",editedByType:"Edited by",editors:[{id:"190989",title:"Dr.",name:"Subbarayan",surname:"Sivasankaran",slug:"subbarayan-sivasankaran",fullName:"Subbarayan Sivasankaran"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1591",title:"Infrared Spectroscopy",subtitle:"Materials Science, Engineering and Technology",isOpenForSubmission:!1,hash:"99b4b7b71a8caeb693ed762b40b017f4",slug:"infrared-spectroscopy-materials-science-engineering-and-technology",bookSignature:"Theophile Theophanides",coverURL:"https://cdn.intechopen.com/books/images_new/1591.jpg",editedByType:"Edited by",editors:[{id:"37194",title:"Dr.",name:"Theophanides",surname:"Theophile",slug:"theophanides-theophile",fullName:"Theophanides Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3092",title:"Anopheles mosquitoes",subtitle:"New insights into malaria vectors",isOpenForSubmission:!1,hash:"c9e622485316d5e296288bf24d2b0d64",slug:"anopheles-mosquitoes-new-insights-into-malaria-vectors",bookSignature:"Sylvie Manguin",coverURL:"https://cdn.intechopen.com/books/images_new/3092.jpg",editedByType:"Edited by",editors:[{id:"50017",title:"Prof.",name:"Sylvie",surname:"Manguin",slug:"sylvie-manguin",fullName:"Sylvie Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"371",title:"Abiotic Stress in Plants",subtitle:"Mechanisms and Adaptations",isOpenForSubmission:!1,hash:"588466f487e307619849d72389178a74",slug:"abiotic-stress-in-plants-mechanisms-and-adaptations",bookSignature:"Arun Shanker and B. 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\n
1. Introduction
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Multiple-input multiple-output (MIMO) radar emits multiple probing signals via its transmit antennas, which provides the greater flexibility for the design of the whole radar system, and boosts the development of more sophisticated signal processing algorithms [1]. On the basis of the configurations of transmitter/receiver antennas, MIMO radar systems can be classified into two categories: widely distributed [2, 3] and colocated [4, 5]. The former has different angles of view on the target owing to widely separated antennas, and this feature can be used to improve the performance of target detection and angle estimation, as well as the capabilities of target identification and classification [6]. The latter shares the same aspect angle of the target by using tightly spaced antennas. However, colocated MIMO radar exploits the waveform diversity to form a long virtual array, thus providing better results concerning spatial resolution, target localization, and the interference rejection, as well as obtaining the degrees of freedom for the design of transmit beam pattern [1, 7, 8].
\n
Recently, colocated MIMO radar waveform design is a hot and challenging topic and has received significant attention. In general, these works can be divided into two categories. The first category focuses on the fast-time waveforms design exploiting some a priori information. In particular, in [6], by using the a priori knowledge of target power spectral density, the minimax robust waveforms are designed based on the rules of the mutual information (MI) and minimum mean-square error (MMSE). In [9], MIMO waveforms for the case of an extended target are devised based on the maximization of signal-to-interference plus-noise ratio (SINR) through a gradient-based algorithm assuming the knowledge of both the target and signal-dependent clutter statistics. In [10], by considering MMSE as figure of merit, MIMO radar waveforms are synthesized under signal-dependent clutter. The join design of the transmit waveform and the receive filter is addressed for improving the extended target delectability in the presence of signal-dependent clutter, by employing a cycle iteration algorithm with ensuring convergence [11]. In [12], by designing the transmit waveform and the receive filter, two sequential optimization algorithms are proposed to maximize SINR subject to the constant modulus and similarity constraints. Based on the rule of the worst-case output SINR in the presence of unknown target angle, the robust joint design of transmit waveform and the receive filter is considered [13]. Some more works can be found in [7, 8, 14, 15].
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The second category addresses the MIMO radar space-(slow) time code design for moving target scenarios. In particular, in [16], MIMO radar slow-time code shares the ability of improving the resolution in angle-Doppler images and obtaining enhanced moving target detection performance. In [17], the signal-dependent interference is alleviated by the space-time coding framework based on a beamspace space-time adaptive processing (STAP). In [18], based on the max-min SINR optimization criteria, the time-division beamforming signal is designed for a multiple target scenario. For a moving point-like target detection, based on the worst case SINR over the actual and signal-dependent clutter statistics, the robust joint design of the space-time transmit code (STTC) satisfying the energy and similarity constraints and the space-time receive filter (STRF) is addressed in [19].
\n
This chapter handles the joint design of the STTC and STRF with the aim of enhancing the moving target detectability under signal-dependent interferences and white Gaussian noise. Unlike [19, 20], some knowledge of target and clutter statistics is assumed to be available. In particular, the SINR is considered as figure of merit to maximize subject to a constant modulus constraint on the transmit signal in addition to a similarity constraint. To deal with the resulting nonconvex design problem, an iterative algorithm ensuring convergence is proposed. Each iteration of the proposed algorithm involves the solution of hidden convex problems. Specifically, both a convex problem with closed-form solution and a set of fractional programming problems, which can be globally solved through the Dinkelback’s algorithm, are solved. The resulting computational complexity is linear with the number of iterations and polynomial with the sizes of the STTC and the STRF.
\n
The remainder of the chapter is organized as follows. In Section 2, the system model is formalized. In Section 3, the constrained optimization problem under constant modulus and similarity constraints is formulated. In Section 4, the new optimization algorithm is presented. In Section 5, the performance of the new procedure is evaluated. Finally, in Section 6, concluding remarks and possible future research tracks are provided.
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\n
\n
2. System model
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We focus on a colocated narrow band MIMO radar system consisting of \n\n\nN\nT\n\n\n transmitters antennas and \n\n\nN\nR\n\n\n receivers. Each transmitter emits a slow-time phase-coded coherent train with \n\nK\n\n pulses. Let \n\ns\n\nk\n\n=\n\n\n\n\ns\n1\n\n\nk\n\n\n\n\ns\n2\n\n\nk\n\n\n⋯\n\n\ns\n\nN\nT\n\n\n\nk\n\n\n\nT\n\n∈\n\nC\n\nN\nT\n\n\n\n, \n\nk\n=\n1\n,\n2\n,\n⋯\n,\nK\n\n denote the transmitted space code vector at the \n\nk\n\n th transmission interval, where \n\n\n\ns\n\nn\nt\n\n\n\nk\n\n\n denotes the \n\nk\n\n th transmitted phase-code pulse of the \n\n\nn\nt\n\n\n th transmitting antenna, for \n\n\nn\nt\n\n=\n1\n,\n2\n,\n⋯\n,\n\nN\nT\n\n\n, \n\n\n\n⋅\n\nT\n\n\n stands for the transpose, and \n\n\nC\nN\n\n\n is the set of \n\nN\n\n-dimensional vectors of complex numbers. At each receiver, the received waveform is downconverted to baseband, undergoes a pulse matched filtering operation, and then is sampled. Hence, the observations of the \n\nk\n\n th slow-time sample for a far-field moving target at the azimuth angle \n\n\nθ\n0\n\n\n can be expressed as [21]
\n\n\n\nα\n0\n\n\n is a complex parameter taking into account the target radar cross section (RCS), channel propagation effects, and other terms involved into the radar range equation.
\n\n\n\nv\n\nd\n0\n\n\n\n denotes the normalized target Doppler frequency, which is related to the radial velocity \n\n\nv\nr\n\n\n via the equation \n\n\nv\n\nd\n0\n\n\n=\n2\n\nv\nr\n\nT\n/\nλ\n\n with \n\nλ\n\n being the carrier wavelength and \n\nT\n\n being the pulse repetition time (PRT).
\n\n\nA\n\nθ\n\n=\n\na\nr\n∗\n\n\nθ\n\n\n\na\nt\n\n†\n\n\nθ\n\n\n, in which \n\n\n\na\nt\n\n\nθ\n\n\n and \n\n\n\na\nr\n\n\nθ\n\n\n denote the transmit spatial steering vector and the receive spatial steering vector at the azimuth angle \n\nθ\n\n, respectively, and \n\n\n\n⋅\n\n∗\n\n\n and \n\n\n\n⋅\n\n†\n\n\n are the conjugate and the conjugate transpose operators, respectively. In particular, for the uniform linear arrays (ULAs), they are given by
with \n\n\nd\nT\n\n\n and \n\n\nd\nR\n\n\n being the array interelement spacing of the transmitter and the receiver, respectively.
\n
\n\n\n\nd\n\n\nk\n\n∈\n\nC\n\nN\nR\n\n\n,\nk\n=\n1\n,\n2\n,\n⋯\n,\nK\n\n, considering \n\nM\n\n signal-dependent uncorrelated point-like interfering scatterers. Specifically, as shown in Figure 1, the angle space is discretized as \n\nΘ\n=\n\n0\n1\n⋯\nL\n\n×\n\n\n2\nπ\n\n\n\nL\n+\n1\n\n\n\n\n. For the \n\nm\n\n th interfering source located at the range-azimuth bin \n\n\n\nr\nm\n\n\nl\nm\n\n\n\n, \n\n\nr\nm\n\n∈\n\n0\n1\n⋯\n\nK\n−\n1\n\n\n\n, \n\n\nl\nm\n\n∈\n\n0\n1\n⋯\nL\n\n\n, the received interfering vector \n\n\nd\n\n\nk\n\n\n can be expressed as the superposition of the returns from \n\nM\n\n interference sources, i.e.,
with \n\n\nρ\nm\n\n\n, \n\n\nv\n\nd\nm\n\n\n\n, and \n\n\nθ\nm\n\n\n, respectively, the complex amplitude, the normalized Doppler frequency, and the look angle, given by \n\n\nθ\nm\n\n=\n\n\n2\nπ\n\n\n\nL\n+\n1\n\n\n\n\nl\nm\n\n\n, of the \n\nm\n\n th interferences. Furthermore, \n\nM\n\n is nominally equal to \n\nK\n×\n\n\nL\n+\n1\n\n\n\n.
\n
\n\n\n\nv\n\n\nk\n\n∈\n\nC\n\nN\nR\n\n\n,\nk\n=\n1\n,\n2\n,\n⋯\n,\nK\n\n denotes additive noise, modeled as independent and identically distributed (i.i.d.) complex circular zero-mean Gaussian random vector, i.e., \n\nv\n\nk\n\n∼\nCN\n\n0\n\n\nσ\n2\n\n\nI\n\nN\nR\n\n\n\n\n\n, where \n\n\n\nI\n\nN\nR\n\n\n\n denotes \n\n\nN\nR\n\n×\n\nN\nR\n\n\n-dimensional identity matrix.
\n
Figure 1.
Range-azimuth bins (the target of interest is represented by the red (solid) circle).
\n
Let \n\nx\n=\n\n\n\n\nx\nT\n\n\n1\n\n\n⋯\n\n\nx\nT\n\n\nK\n\n\n\nT\n\n\n, \n\ns\n=\n\n\n\n\ns\nT\n\n\n1\n\n\n⋯\n\n\ns\nT\n\n\nK\n\n\n\nT\n\n\n, \n\nd\n=\n\n\n\n\nd\nT\n\n\n1\n\n\n⋯\n\n\nd\nT\n\n\nK\n\n\n\nT\n\n\n, and \n\nv\n=\n\n\n\n\nv\nT\n\n\n1\n\n\n⋯\n\n\nv\nT\n\n\nK\n\n\n\nT\n\n\n. Then, Eq. (1) can be expressed in a compact form as
with \n\np\n\n\nv\nd\n\n\n=\n\n\n1\n\ne\n\nj\n2\nπ\n\nv\nd\n\n\n\n⋯\n\ne\n\nj\n2\nπ\n\n\nK\n−\n1\n\n\n\nv\nd\n\n\n\n\nT\n\n\n being the temporal steering vector, \n\n⊗\n\n denotes the Kronecker product, and \n\nDiag\n\n⋅\n\n\n denotes the diagonal matrix formed by the entries of the vector argument. Additionally, we assume that the noise vector \n\n\nv\n\n\n is a zero-mean circular complex Gaussian random vector with covariance matrix \n\n\nΣ\nv\n\n=\nE\n\n\n\nv\n\n\nv\n†\n\n\n\n=\n\nσ\nv\n2\n\n\n\nI\n\n\nN\nR\n\nK\n\n\n\n. Finally, interference vector \n\n\nd\n\n\n can be expressed as
\n\n\nr\n∈\n\n0\n1\n⋯\n\nK\n−\n1\n\n\n\n and \n\n\n\nk\n1\n\n\nk\n2\n\n\n∈\n\n\n1\n2\n⋯\nK\n\n2\n\n\n. In particular, we assume that \n\n\nρ\nm\n\n\n, \n\nm\n=\n1\n,\n2\n,\n⋯\n,\nM\n\n, and \n\n\nα\n0\n\n\n are a zero-mean uncorrelated random variables with, respectively, \n\n\nσ\nm\n2\n\n=\nE\n\n\n\nρ\nm\n2\n\n\n\n\n and \n\n\nσ\n0\n2\n\n=\nE\n\n\n\n\nα\n0\n\n\n2\n\n\n\n. As to the normalized Doppler frequency of the interfering signals, we model \n\n\nv\n\nd\nm\n\n\n\n as a random variable uniformly distributed around a mean Doppler frequency \n\n\n\nv\n¯\n\n\nd\nm\n\n\n\n, i.e.,
where \n\n\nε\nm\n\n\n accounts for the uncertainty on \n\n\nv\n\nd\nm\n\n\n\n. Basing on the previous assumptions, the interference vector \n\n\nd\n\n\n has zero mean and covariance matrix
and \n\n\nϒ\nt\n\n=\n\n1\nt\n\n\n\n1\nt\n\nT\n\n\n with \n\n\n1\nt\n\n=\n\n\n1\n1\n⋯\n1\n\nT\n\n\n being the \n\n\nN\nT\n\n×\n1\n\n vector, \n\n⊙\n\n and \n\nE\n\n⋅\n\n\n denote the Hadamard product and the statistical expectation, respectively. This expression, for the covariance matrix \n\n\nΣ\nd\n\n\ns\n\n\n, follows from the results obtained in ([19], Appendix 1).
\n
Inspection of (11) and (12) reveals that the interference covariance matrix \n\n\nΣ\nd\n\n\ns\n\n\n requires the knowledge of \n\n\nθ\nm\n\n\n and \n\n\nσ\nm\n2\n\n\n as well as \n\n\n\nv\n¯\n\n\nd\nm\n\n\n\n and \n\n\nε\nm\n\n\n, for \n\nm\n=\n1\n,\n2\n,\n⋯\n,\nM\n\n. These information can be obtained according to a cognitive paradigm [22, 23, 24] through exploiting a site-specific (possible dynamic) environment database, which involves a geographical information system (GIS), digital terrain maps, previous scans, tracking files, clutter models (in terms of electromagnetic reflectivity and spectral density), and meteorological information.
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\n
\n
3. Problem formulation
\n
This section formulates the joint design problem of the STTC and STRF based on the maximization of the output SINR considering practical constraints.
\n
\n
3.1. Output SINR
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Letting the observations \n\n\nx\n\n\n be processed via the STRF \n\n\nw\n∈\n\nC\n\n\nN\nR\n\nK\n\n\n\n, the SINR \n\n\nρ\n̂\n\n\n\n\ns\n\n\nw\n\n\n\n\n at the output of the receiver can be expressed as
and assume \n\n\nw\n≠\n0\n\n and the independence between the disturbance and the noise random processes.
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In particular, the numerator in (14) denotes the useful energy at the output of the STRF, \n\n\nw\n†\n\n\nΣ\nd\n\n\ns\n\nw\n\n and \n\n\nσ\nv\n2\n\n\n\nw\n†\n\n\nw\n\n\n represent the clutter energy and noise energy, respectively, at the output of \n\n\nw\n\n\n. Observe that the clutter energy \n\n\nw\n†\n\n\nΣ\nd\n\n\ns\n\nw\n\n functionally relies on the STTC \n\n\nw\n\n\n and the STRF \n\n\ns\n\n\n through \n\n\nΣ\nd\n\n\ns\n\n\n as well as the useful energy. Furthermore, we note that the objective function \n\n\nρ\n̂\n\n\n\n\ns\n\n\nw\n\n\n\n\n requires that the exact angle \n\n\nθ\n0\n\n\n and normalized Doppler frequency \n\n\nv\n\nd\n0\n\n\n\n are known. However, from a practical point of view, the explicit knowledge of \n\n\nθ\n0\n\n\n and \n\n\nv\n\nd\n0\n\n\n\n cannot be available. To circumvent this drawback, the averaged SINR defined as \n\nρ\n\n\n\ns\n\n\nw\n\n\n\n=\nE\n\n\n\nρ\n̂\n\n\n\n\ns\n\n\nw\n\n\n\n\n\n\n as figure of merit is exploited. More specifically, we suppose that \n\n\nv\n\nd\n0\n\n\n\n and \n\n\nθ\n0\n\n\n are independent random variables uniformly distributed around a mean Doppler frequency \n\n\n\nv\n¯\n\n\nd\n0\n\n\n\n and a mean azimuth \n\n\n\nθ\n¯\n\n0\n\n\n, respectively, i.e., \n\n\nv\n\nd\n0\n\n\n∼\nU\n\n\n\n\nv\n¯\n\n\nd\n0\n\n\n−\n\n\nε\n0\n\n2\n\n\n\n\n\nv\n¯\n\n\nd\n0\n\n\n+\n\n\nε\n0\n\n2\n\n\n\n\n, \n\n\nθ\n0\n\n∼\nU\n\n\n\n\nθ\n¯\n\n0\n\n−\n\n\nϑ\n0\n\n2\n\n\n\n\n\nθ\n¯\n\n0\n\n+\n\n\nϑ\n0\n\n2\n\n\n\n\n, where \n\n∼\n\n means “\n\ndistribute\n\n” and \n\nU\n\n represents uniform distribution and \n\n\nε\n0\n\n\n and \n\n\nϑ\n0\n\n\n accounts for the uncertainty on \n\n\nv\n\nd\n0\n\n\n\n and \n\n\nθ\n0\n\n\n, respectively. Interestingly, after some algebraic manipulations, the objective function \n\nρ\n\n\n\ns\n\n\nw\n\n\n\n\n shares the following two equivalent expressions,
While \n\nS\n=\n\nss\n†\n\n∈\n\nH\n\nKN\nT\n\n\n\n and \n\nW\n=\n\nww\n†\n\n∈\n\nH\n\nKN\nR\n\n\n\n, \n\n\nΞ\nm\n\n\n is given by (12), E denotes the energy of \n\n\ns\n\n\n, \n\n\n\nΞ\n¯\n\nm\n\n=\n\nσ\nm\n2\n\n\nΨ\n\nε\nm\n\n\n\nv\n¯\n\n\nd\nm\n\n\n\n⊗\n\nϒ\nr\n\n\n, \n\n\nΨ\n\nε\nm\n\n\n\nv\n¯\n\n\nd\nm\n\n\n\n\n\nk\n1\n\n\nk\n2\n\n\n=\n\n\n\n\nΦ\n\nε\nm\n\n\n\nv\n¯\n\n\nd\nm\n\n\n\n\n\nk\n1\n\n\nk\n2\n\n\n\n\n∗\n\n\n, \n\n∀\n\n\nk\n1\n\n\nk\n2\n\n\n∈\n\n\n1\n2\n⋯\nK\n\n2\n\n\n and \n\n\nϒ\nr\n\n=\n\n1\nr\n\n\n\n1\nr\n\nT\n\n\n with \n\n\n1\nr\n\n=\n\n\n1\n1\n⋯\n1\n\nT\n\n∈\n\nC\n\nN\nR\n\n\n\n, and \n\ntr\n\n⋅\n\n\n denotes the trace of square matrix. These expressions follow from the results obtained in ([19], Appendix 3).
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Note that \n\nΓ\n\n\n\nS\n\n\n\n\n and \n\nΘ\n\n\n\nW\n\n\n\n\n can be rewritten in block matrix form, i.e.,
where \n\n\nΓ\n\n\nm\n1\n\n\nm\n2\n\n\n\n∈\n\nC\n\n\nN\nR\n\n×\n\nN\nR\n\n\n\n\n and \n\n\nΘ\n\n\ni\n1\n\n\ni\n2\n\n\n\n∈\n\nC\n\n\nN\nT\n\n×\n\nN\nT\n\n\n\n\n can be computed by (38) and (46) respectively, \n\n∀\n\n\nm\n1\n\n\nm\n2\n\n\ni\n1\n\n\ni\n2\n\n\n∈\n\n\n1\n2\n⋯\nK\n\n4\n\n\n, as shown in Appendix \n\nA\n\n.
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3.2. Constant modulus and similarity constraints
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In practical applications, the designed STTC is enforced to be unimodular (i.e., constant modulus) since the nonlinear property of radar amplifiers [24, 25]. To this end, we limit the modulus of each element of the code \n\n\ns\n\n\n as a constant. Precisely, the \n\ni\n\n th element \n\n\ns\ni\n\n\n of \n\n\ns\n\n\n can be written as
with \n\n\nφ\ni\n\n\n denoting the phase of \n\n\ns\ni\n\n\n. Furthermore, \n\nK\n\n different similarity constraints are enforced on the \n\n\nN\nT\n\n\n transmitting waveforms, namely
where \n\n\n\ns\n0\n\n\nk\n\n∈\n\nC\n\nN\nT\n\n\n\n is the reference code vector at the \n\nk\n\n th transmission interval, \n\n\nξ\nk\n\n\n is a real parameter ruling the extent of the similarity, and \n\n∥\nx\n\n\n∥\n∞\n\n\n denotes the infinite norm.
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Without loss of generality, we assume the same similarity parameter \n\n\nξ\n0\n\n\n (i.e., \n\n\nξ\n0\n\n=\n\nξ\n1\n\n=\n⋯\n=\n\nξ\nK\n\n\n) [12, 26, 28, 29, 30] on the sought STTC. Thus, Eq. (24) can be written as \n\n∥\ns\n−\n\ns\n0\n\n\n∥\n∞\n\n≤\n\nξ\n0\n\n\n, where \n\n\ns\n0\n\n=\n\n\n\n\ns\n0\nT\n\n\n1\n\n\n⋯\n\n\ns\n0\nT\n\n\nK\n\n\n\nT\n\n\n is the reference code vector. Several reasons are presented to show the motivation to exploit the similarity constraints on radar codes. Actually, an arbitrary optimization of SINR via designing an STTC does not offer any kind of control on the shape of the resulting designed waveforms. Specifically, an pure optimization of the SINR can cause signals sharing high peak sidelobe levels and, in general, with an undesired ambiguity function feature. To this end, by exploiting the similarity constraint, when \n\n\n\ns\n0\n\n\n possesses suitable properties, such as low peak sidelobe levels, and reasonable Doppler resolutions, the designed STTC can enjoy some of the good ambiguity function feature of \n\n\n\ns\n0\n\n\n. In other words, the similarity constraint compromises the performance between SINR improvement and suitable waveform features [31].
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3.3. Design problem
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Summarizing, the joint design of the STTC and the STRF can be formulated as the following constrained optimization problem:
where \n\n∣\n⋅\n∣\n\n and \n\n∥\n⋅\n∥\n\n, respectively, represent the modulus and the Euclidean norm. Without loss of generality, we add the constraint \n\n∥\nw\n\n\n∥\n2\n\n=\n1\n\n. \n\n\nP\n1\n\n\n is a NP-hard problem [12, 28] whose optimal solution cannot be found in polynomial time. Next, we develop a new iterative algorithm to offer high-quality solution to the NP-hard problem (25).
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4. STTC and STRF design procedure
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This section focuses on the design of an iterative algorithm ensuring convergence properties, which is capable of offering high-quality solutions to the NP-hard problem \n\n\nP\n1\n\n\n by sequentially improving the SINR. In particular, we exploit the pattern search framework to cyclically optimize the design variables \n\n\nw\n\ns\n1\n\n\ns\n2\n\n⋯\n\ns\n\n\nN\nT\n\nK\n\n\n\n\n.
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4.1. STRF optimization
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In this subsection, we deal with the STRF optimization for a fixed STTC \n\n\ns\n\n\n. Specifically, we handle the optimization problem
i.e., to a generalized eigenvector of the matrices \n\nΓ\n\n\n\ns\n\n\ns\n†\n\n\n\n\n and \n\n\nΣ\ndv\n\n\n\n\ns\n\n\ns\n†\n\n\n\n\n corresponding to the maximum generalized eigenvalue. Thus, a closed-form solution to \n\n\nP\nw\n\n\n can be obtained by normalizing \n\n\n\nw\no\n\n\n.
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4.2. STTC optimization
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This subsection is devoted to the optimization of the STTC under a fixed STRF. Precisely, each code element in \n\n\ns\n\n\n is sequentially optimized under the fixed remaining \n\n\nN\nT\n\nK\n−\n1\n\n elements. Performing some algebraic manipulations to similarity constraints [26], the optimization problem \n\n\nP\n\n\ns\n¯\n\ni\n\n\n\n with respect to the \n\ni\n\n th STTC variable, \n\ni\n=\n1\n,\n…\n,\n\nN\nT\n\nK\n\n, is written by,
where \n\ns\n=\n\n\n\ns\n1\n\n\ns\n2\n\n⋯\n\ns\n\ni\n−\n1\n\n\n\n\ns\n¯\n\ni\n\n\ns\n\ni\n+\n1\n\n\n⋯\n\ns\n\nKN\nT\n\n\n\nT\n\n\n, \n\n\nγ\ni\n\n=\narg\n\ns\n\n0\ni\n\n\n−\narccos\n\n\n1\n−\n\nξ\n2\n\n/\n2\n\n\n\n, \n\nδ\n=\n2\narccos\n\n\n1\n−\n\nξ\n2\n\n/\n2\n\n\n\n, \n\nξ\n=\n\n\n\nN\nT\n\nK\n\n\n\nξ\n0\n\n\n with \n\n0\n≤\nξ\n≤\n2\n\n, and \n\n\ns\n\n0\ni\n\n\n\n is the \n\ni\n\n th element of \n\n\n\ns\n0\n\n\n. Notice that for \n\nξ\n=\n0\n\n, the code \n\n\ns\n\n\n is equal to the reference code \n\n\n\ns\n0\n\n\n, whereas the similarity constraint would become the constant modulus constraint with \n\nξ\n=\n2\n\n.
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\n\n\nRemark\n\n: This procedure by resorting to pattern search framework offers a new strategy to address the code design problem under a fixed filter. In addition, this STTC optimization problem can be efficiently but approximatively settled by semidefinite relaxation (SDR) and randomization procedure with the computational complexity of \n\nO\n\n\n\n\n\nN\nT\n\nK\n\n\n3.5\n\n\n+\nO\n\n\nL\n\n\n\n\nN\nT\n\nK\n\n\n2\n\n\n\n\n, where \n\nL\n\n is the number of randomization trials. However, the SDR technique usually shares a huge computational complexity, especially in large dimension \n\n\nN\nT\n\nK\n\n, thus limiting its applications in real-time systems; moreover, the existing approach also needs the reasonable selection of \n\nL\n\n. On the other hand, it is shown that a higher quality solution can be further obtained via a sequential iteration optimization algorithm, which is capable of monotonically increasing the SINR value and achieving a stationary point of the formulated NP-hard problem [27].
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Next, we focus on the proposed iteration algorithm to solve problem (27) in a polynomial time. In particular, performing some algebraic manipulations to the objective function in (27), \n\n\nP\n\n\ns\n¯\n\ni\n\n\n\n can be equivalently rewritten as a fractional programming optimization problem by the following proposition.
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Proposition 4.1 The problem\n\n\nP\n\n\ns\n¯\n\ni\n\n\n\nis equivalent to
and\n\n\na\n\nk\n,\ni\n\n\n,\n\nb\n\nk\n,\ni\n\n\n\nare constants for\n\nk\n=\n0\n,\n1\n,\n2\n\n, \n\nℜ\n\nx\n\n\ndenotes the real part of\n\nx\n\n.
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Proof. See Appendix \n\nB\n\n.
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Problem (28) is solvable [32] since the objective function is continuous with \n\nℜ\n\n\n\nb\n\n1\n,\ni\n\n\n\n\ns\n¯\n\ni\n\n\n\n+\n\nb\n\n3\n,\ni\n\n\n>\n0\n\n and the constraint is a compact set (closed and bounded set of \n\nC\n\n). Thus, we consider the following parametric problem [32],
where \n\n\nc\ni\n\n=\n\na\n\n1\n,\ni\n\n\n−\nμ\n\nb\n\n1\n,\ni\n\n\n\n and the constant \n\n\na\n\n3\n,\ni\n\n\n−\nμ\n\nb\n\n3\n,\ni\n\n\n\n do not affect the optimal value.
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Interestingly, problem (31) shares a closed-form solution whose phase \n\n\nφ\n∗\n\n\n is given by,
We observe that problems (28) and (30) are relevant in each other via Lemma 2.1 of [32]. Specifically, we can find a solution to problem (28) by obtaining a solution of the equation \n\nϱ\n\nμ\n\n=\n0\n\n concerning \n\n\n\ns\n¯\n\ni\n\n\n. To this end, the Dinkelbach-type procedure [32, 33] summarized in Algorithm 1 is introduced to solve problem (27).
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Algorithm 1. Dinkelbach-type algorithm for solving \n\n\nP\n\n\ns\n¯\n\ni\n\n\n\n
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Input:\n\n\na\n\n1\n,\ni\n\n\n\n, \n\n\na\n\n3\n,\ni\n\n\n\n, \n\n\nb\n\n1\n,\ni\n\n\n\n, \n\n\nb\n\n3\n,\ni\n\n\n\n, \n\n\nγ\ni\n\n\n and \n\nδ\n\n;
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Output: An optimal solution \n\n\n\ns\n̂\n\ni\n\n\n to \n\n\nP\n\n\ns\n¯\n\ni\n\n\n\n;\n
Randomly generate \n\n\n\ns\n¯\n\n\ni\n,\n0\n\n\n\n within the feasible sets;
Compute \n\n\nμ\n1\n\n=\n\n\nℜ\n\n\n\na\n\n1\n,\ni\n\n\n\n\ns\n¯\n\n\ni\n,\n0\n\n\n\n\n+\n\na\n\n3\n,\ni\n\n\n\n\nℜ\n\n\n\nb\n\n1\n,\ni\n\n\n\n\ns\n¯\n\n\ni\n,\n0\n\n\n\n\n+\n\nb\n\n3\n,\ni\n\n\n\n\n\n and let \n\nk\n≔\n1\n\n;
Find the optimal solution \n\n\n\ns\n¯\n\n\ni\n,\nk\n\n\n\n by solving problem (30),
If \n\nϱ\n\n\nμ\nk\n\n\n=\n0\n\n, then \n\n\n\ns\n¯\n\n\ni\n,\nk\n\n\n\n is an optimal solution of \n\n\nP\n\n\ns\n¯\n\ni\n\n\n\n with optimal value \n\n\nμ\nk\n\n\n and stop. Otherwise, go to step 5;
Let \n\n\nμ\nk\n\n=\n\n\nℜ\n\n\n\na\n\n1\n,\ni\n\n\n\n\ns\n¯\n\n\ni\n,\nk\n\n\n\n\n+\n\na\n\n3\n,\ni\n\n\n\n\nℜ\n\n\n\nb\n\n1\n,\ni\n\n\n\n\ns\n¯\n\n\ni\n,\nk\n\n\n\n\n+\n\nb\n\n3\n,\ni\n\n\n\n\n\n and \n\nk\n≔\nk\n+\n1\n\n; Then go to step 2.
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Algorithm 1 sharing a linear convergence rate [34] is needed to handle the problem (30) in each iteration. The objective value of the generated sequence of points has a monotonic convergence property, and the optimal value of (28) can be achieved eventually. We set the exit condition \n\nϱ\n\nμ\n\n=\n0\n\n, actually, which can be replaced by \n\nϱ\n\nμ\n\n≤\nς\n\n, with \n\nς\n\n being a prescribed accuracy.
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4.3. Transmit-receive system design
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This subsection reports the iteration optimization procedure for the STTC and STRF in Algorithm 2. In particular, Algorithm 2 guarantees that the SINR monotonically increases2. Furthermore, we need to point out that the maximum block improvement (MBI) [24] framework could be used to ensure the convergence to a stationary point of problem \n\n\nP\n1\n\n\n.
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The global computation consume of the Algorithm 2 is linear to the number of iterations and polynomial with the sizes of the STTC and the STRF. More specifically, each iteration of the proposed algorithm involves the computational cost associated with the solution to problems (26) and \n\n\nP\n\n\ns\n¯\n\ni\n\n\n\n, for \n\ni\n=\n1\n,\n2\n,\n⋯\n,\n\nN\nT\n\nK\n\n. The former requires to solve the generalized eigenvalue decomposition with the order of \n\nO\n\n\n\n\n\nN\nR\n\nK\n\n\n3\n\n\n\n (see [35], p. 500). Similarly, the latter is linear to polynomial with the size of the STTC, while each iteration needs the solution of a generalized fractional programming problem with the computational complexity of \n\nO\n\n\n\n\n\nN\nT\n\nK\n\n\n2\n\n\n\n. We need to point out that SOA2, based on the SDR and randomization method, can also be used to the solution of problem (25). However, it cannot guarantee the convergence to a stationary point due to the use of randomized approximations. Moreover, from computational complexity, each iteration of SOA2 has the order of \n\nO\n\n\n\n\n\nN\nR\n\nK\n\n\n3\n\n\n\n+ \n\nO\n\n\n\n\n\nN\nT\n\nK\n\n\n3.5\n\n\n\n+ \n\nO\n\n\nL\n\n\n\n\nN\nT\n\nK\n\n\n2\n\n\n\n\n, whereas Algorithm 2 is \n\nO\n\n\n\n\n\nN\nR\n\nK\n\n\n3\n\n\n\n+ \n\nO\n\n\n\n\n\nN\nT\n\nK\n\n\n3\n\n\n\n.
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Algorithm 2. Algorithm for the joint STTC \n\n\ns\n\n\n and STRF \n\n\nw\n\n\n design
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Input:\n\n\n\nθ\n¯\n\n0\n\n\n, \n\n\nϑ\n0\n\n\n, \n\n\n\ns\n0\n\n\n, \n\nξ\n\n, \n\n\nσ\nm\n\n,\n\nr\nm\n\n,\n\n\nv\n¯\n\n\nd\nm\n\n\n,\n\nε\nm\n\n\n, for \n\nm\n=\n0\n,\n1\n,\n⋯\n,\nM\n\n, and \n\n\nθ\np\n\n\n, for \n\np\n=\n1\n,\n2\n,\n⋯\n,\nM\n\n;
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Output: An optimal solution \n\n\n\n\n\ns\n∗\n\n\n\nw\n∗\n\n\n\n to \n\n\nP\n1\n\n\n;\n
For \n\nn\n=\n0\n\n and initialize \n\n\n\ns\n\nn\n\n\n=\n\ns\n0\n\n\n;
Compute \n\n\n\nw\n\n0\n\n\n=\n\n\n\n\nw\no\n\n0\n\n\n\n\n∥\n\nw\no\n\n0\n\n\n∥\n\n\n\n and \n\n\nρ\n0\n\n=\nρ\n\n\n\n\ns\n\n0\n\n\n\n\nw\n\n0\n\n\n\n\n;
\n\n\nn\n≔\nn\n+\n1\n\n and \n\ni\n=\n0\n\n;
Compute \n\n\n\nΣ\n¯\n\ndv\n\n\n\n\n\nw\n\nn\n\n\n\n\nw\n\n\nn\n\n†\n\n\n\n\n\n and \n\nΘ\n\n\n\n\nw\n\nn\n\n\n\n\nw\n\n\nn\n\n†\n\n\n\n\n\n by (20) and (22), respectively;
\n\n\ni\n≔\ni\n+\n1\n\n;
Compute \n\n\na\n\nk\n,\ni\n\n\n\n and \n\n\nb\n\nk\n,\ni\n\n\n\n by (50) and (51), \n\nk\n=\n0\n,\n1\n,\n2\n\n, respectively;
Find \n\n\na\n\n3\n,\ni\n\n\n\n and \n\n\nb\n\n3\n,\ni\n\n\n\n by (29);
Exploit Algorithm 1 to update \n\n\ns\ni\n\n\n by maximizing the problem (27);
If \n\ni\n=\n\nN\nT\n\nK\n\n, output \n\n\n\ns\n\nn\n\n\n=\n\n\n\ns\n1\n\n\ns\n2\n\n⋯\n\ns\n\nKN\nT\n\n\n\nT\n\n\n. Otherwise, return to step 7;
Compute \n\n\nΣ\ndv\n\n\n\n\n\ns\n\nn\n\n\n\n\ns\n\n\nn\n\n†\n\n\n\n\n\n and \n\nΓ\n\n\n\n\ns\n\nn\n\n\n\n\ns\n\n\nn\n\n†\n\n\n\n\n\n by (18) and (21), respectively;
Find the generalized eigenvector \n\n\n\nw\no\n\nn\n\n\n\n of matrices \n\nΓ\n\n\n\n\ns\n\nn\n\n\n\n\ns\n\n\nn\n\n†\n\n\n\n\n\n and \n\n\nΣ\ndv\n\n\n\n\n\ns\n\nn\n\n\n\n\ns\n\n\nn\n\n†\n\n\n\n\n\n corresponding to the maximum generalized eigenvalue;
Compute \n\n\n\nw\n\nn\n\n\n=\n\n\n\n\nw\no\n\nn\n\n\n\n\n∥\n\nw\no\n\nn\n\n\n∥\n\n\n\n and \n\n\nρ\nn\n\n=\nρ\n\n\n\n\ns\n\nn\n\n\n\n\nw\n\nn\n\n\n\n\n;
If \n\n∣\n\nρ\nn\n\n−\n\nρ\n\nn\n−\n1\n\n\n∣\n≤\nκ\n\n, where \n\nκ\n\n is a user selected parameter to control convergence, output \n\n\n\ns\n∗\n\n=\n\ns\n\nn\n\n\n\n and \n\n\n\nw\n∗\n\n=\n\nw\n\nn\n\n\n\n; Otherwise, repeat step 5 until convergence.
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5. Numerical results
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This section focuses on assessing the capability of the proposed algorithm for designing optimized STTC and STRF in signal-dependent interference for both a nonuniform and an uniform point-like clutter environment. In particular, for both scenarios, we consider an L-band radar with operating frequency \n\n\nf\nc\n\n=\n1.4\n\n GHz, which is equipped with an ULA of \n\n\nN\nT\n\n=\n4\n\n transmit elements and \n\n\nN\nR\n\n=\n8\n\n receive elements under an interelement spacing \n\n\nd\nt\n\n=\n\nd\nr\n\n=\nλ\n/\n2\n\n. We set the code length \n\nK\n=\n13\n\n for each transmitter and the orthogonal linear frequency modulation (LFM3) is used as the reference waveform \n\n\n\ns\n0\n\n\n [12] with the \n\n\n\nn\nt\n\nk\n\n\n th entry of the reference \n\n\n\nS\n\n0\n\n\n\n given by,
where \n\n\nn\nt\n\n=\n1\n,\n2\n,\n⋯\n,\n\nN\nT\n\n\n and \n\nk\n=\n1\n,\n2\n,\n⋯\n,\nK\n\n. Hence, the reference code is derived as \n\n\ns\n0\n\n=\nvec\n\n\nS\n\n0\n\n\n\n\n. Moreover, we assume the target located at range-azimuth bin of interest (0,0) with power \n\n\nσ\n0\n2\n\n=\n10\n\n dB. In addition, we set a mean azimuth \n\n\n\nθ\n¯\n\n0\n\n=\n\n0\n∘\n\n\n with azimuth uncertainty \n\nϑ\n/\n2\n=\n\n1\n∘\n\n\n, and a normalized mean Doppler frequency \n\n\n\nv\n¯\n\n\nd\n0\n\n\n=\n0.4\n\n with Doppler uncertainty \n\n\nε\n0\n\n/\n2\n=\n0.04\n\n for the presence of target. We set the noise variance to \n\n\nσ\nv\n2\n\n=\n0\n\n dB. Finally, the exit condition4\n\nς\n=\n\n10\n\n−\n3\n\n\n\n for Algorithms 1 and 2 is \n\nκ\n=\n\n10\n\n−\n3\n\n\n\n, i.e.,
All simulations are performed using Matlab 2010a version, running on a standard PC (with a 3.3 GHz Core i5 CPU and 8 GB RAM).
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5.1. Nonuniform point-like clutter environment
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This subsection focuses on a scenario where three disturbances, respectively, are located at the spatial angles \n\n\nθ\n1\n\n=\n−\n\n55\n°\n\n,\n\nθ\n2\n\n=\n−\n\n20\n°\n\n,\n\nθ\n3\n\n=\n\n40\n°\n\n\n, with corresponding range bins \n\n\nr\ni\n\n=\n0\n,\ni\n=\n1\n,\n2\n,\n3\n\n and powers \n\n\nσ\n1\n2\n\n=\n30dB\n\n, \n\n\nσ\n2\n2\n\n=\n28dB\n\n, \n\n\nσ\n3\n2\n\n=\n25\n\n dB. Moreover, we suppose \n\n\n\nv\n¯\n\n\nd\n1\n\n\n=\n−\n0.35\n,\n\n\nv\n¯\n\n\nd\n2\n\n\n=\n−\n0.15\n,\n\n\nv\n¯\n\n\nd\n3\n\n\n=\n0.25\n,\n\nε\nm\n\n/\n2\n=\n0.04\n,\nm\n=\n1\n,\n2\n,\n3\n\n for the presence of the disturbances.
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For comparison purpose, we also perform simulations for the SOA2 with constant modulus and similarity constraints as well as the algorithm in [19] with energy constraint (i.e., \n\n∥\ns\n\n\n∥\n2\n\n=\n1\n\n), respectively. In particular, Figure 2 shows the SINR versus the iteration number for different \n\nξ\n\n by also comparing the results obtained via Algorithm 2 and SOA2 considering L = 100 and exploiting the CVX toolbox [36] to handle the semidefinite programming (SDP) involved in SOA2. The results exhibit that the SINR values achieved using Algorithm 2 and SOA2 increase as the iteration number increases. In addition, the SINR increases as \n\nξ\n\n increases owing to the higher degrees of freedom available at the design stage. Precisely, Algorithm 2 is superior to SOA2 for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n\n. It is interesting to note that Algorithm 2 and SOA2 share almost the same SINR for \n\nξ\n=\n2\n\n, whereas both obtain lower SINR than the case considering energy constraint. Finally, it is worth pointing out that a loss of SINR caused by constant constraint can be observed since the gap of SINR between \n\nξ\n=\n2\n\n and energy constraint is about 1 dB.
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Figure 2.
The SINR behavior versus iteration number assuming a target with \n\n−\nπ\n/\n180\n≤\n\nθ\n0\n\n≤\nπ\n/\n180\n\n, \n\n0.36\n≤\n\nv\n\nd\n0\n\n\n≤\n0.44\n\n for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n,\n2\n\n, \n\n\n\ns\n0\n\n\n as the initial point.
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Table 1 reports the achieved SINR values, iterations number, and global computation time of Algorithm 2 and SOA2 supposing a target with \n\n−\nπ\n/\n180\n≤\n\nθ\n0\n\n≤\nπ\n/\n180\n\n, \n\n0.36\n≤\n\nv\n\nd\n0\n\n\n≤\n0.44\n\n for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n,\n2\n\n and setting the same exit condition for SOA2. We observe that Algorithm 2 and SOA2 both converge very fast. Additionally, Algorithm 2 is superior to SOA2 concerning the achieved SINR value for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n\n and concerning the required computational cost for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n,\n2\n\n.
SINR values (in dB), iterations number, and global computation time (in seconds) of Algorithm 2 and SOA2 assuming a target with \n\n−\nπ\n/\n180\n≤\n\nθ\n0\n\n≤\nπ\n/\n180\n\n, \n\n0.36\n≤\n\nv\n\nd\n0\n\n\n≤\n0.44\n\n for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n,\n2\n\n, \n\n\n\ns\n0\n\n\n as the initial point.
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In the following, the joint frequency and azimuth behavior of STTC and STRF are considered corresponding to \n\nξ\n=\n2\n\n supposing \n\n−\nπ\n/\n180\n≤\n\nθ\n0\n\n≤\nπ\n/\n180\n\n, \n\n0.36\n≤\n\nv\n\nd\n0\n\n\n≤\n0.44\n\n for different iteration numbers, by using the contour map of the slow-time cross ambiguity function (CAF) [19],
where \n\n\nA\n̂\n\n\nv\nθ\n\n\n and \n\n\nP\nr\n\n\n are obtained by exploiting Eqs. (6) and (8), respectively. Figure 3 plots the contour map of the Doppler-azimuth plane of CAF at \n\nr\n=\n0\n\n versus the iteration number \n\nn\n=\n\n0\n1\n4\n15\n\n\n for Algorithm 2. As expected, the lower and lower values in the regions of (highlighted by black ellipses) \n\n\nθ\n1\n\n=\n−\n\n55\n°\n\n\n and \n\n−\n0.39\n≤\nv\n≤\n−\n0.31\n\n, \n\n\nθ\n2\n\n=\n−\n\n20\n°\n\n\n and \n\n−\n0.19\n≤\nv\n≤\n−\n0.11\n\n, and \n\n\nθ\n3\n\n=\n\n40\n°\n\n\n and \n\n0.21\n≤\nv\n≤\n0.29\n\n are achieved, with the increase of \n\nn\n\n. Thus, it is worth pointing out that the proposed algorithm can suitably shape the CAF to resist interferences.
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Figure 3.
Doppler-azimuth plane of CAF at \n\nr\n=\n0\n\n for \n\nξ\n=\n2\n\n of Algorithm 2 for \n\nn\n=\n\n0\n1\n4\n15\n\n\n assuming a target with \n\n−\nπ\n/\n180\n≤\n\nθ\n0\n\n≤\nπ\n/\n180\n\n, \n\n0.36\n≤\n\nv\n\nd\n0\n\n\n≤\n0.44\n\n (black ellipses represent the locations of three interference sources), \n\n\n\ns\n0\n\n\n as the initial point.
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For the uniform distribution, we define both standard deviations \n\n\nσ\n\nv\n\nd\n0\n\n\n\n\n and \n\n\nσ\n\nθ\n0\n\n\n\n of target Doppler and azimuth as, respectively,
Figure 4 shows the SINR behaviors versus the standard deviations \n\n\nσ\n\nv\n\nd\n0\n\n\n\n\n (Figure 4a) and \n\n\nσ\n\nθ\n0\n\n\n\n (Figure 4b) supposing \n\n\n\nθ\n¯\n\n0\n\n=\n\n0\n°\n\n\n, \n\n\n\nv\n¯\n\n\nd\n0\n\n\n=\n0.4\n\n, respectively. Our curves highlight that the proposed algorithm can further improve SINR gain in comparison with SOA2 for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n\n. We also observe that the higher \n\n\nσ\n\nv\n\nd\n0\n\n\n\n\n and \n\n\nσ\n\nθ\n0\n\n\n\n and the lower SINR can be obtained due to the larger inaccuracies on the knowledge of Doppler and azimuth of the actual target. Finally, we need to point out that the proposed design procedure still has the better robustness against a large uncertain set in comparison with SOA2.
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Figure 4.
The SINR behaviors versus the standard deviations \n\n\nσ\n\nv\n\nd\n0\n\n\n\n\n (Figure 4a) and \n\n\nσ\n\nθ\n0\n\n\n\n (Figure 4b) of Doppler and azimuth of target with \n\n\n\nθ\n¯\n\n0\n\n=\n\n0\n°\n\n\n, \n\n\n\nv\n¯\n\n\nd\n0\n\n\n=\n0.4\n\n considering \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n,\n2\n\n, respectively, \n\n\n\ns\n0\n\n\n as the initial point.
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5.2. Uniform clutter environment
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This subsection focuses on a scenario where we consider a homogeneous range-azimuth ground clutter interfering with the range-azimuth bin of interest (0,0). Specifically, for each range-azimuth ground clutter bin, a clutter to noise ratio (CNR) of 25 dB and a normalized Doppler frequency \n\n\nv\n¯\n\n=\n0\n\n with Doppler uncertainty \n\nε\n/\n2\n=\n0.04\n\n are considered. We suppose \n\nM\n=\n50\n\n range-azimuth ground clutter bins located within the azimuth angular sector \n\n\n\n−\nπ\n/\n2\n\n\nπ\n/\n2\n\n\n\n. Moreover, we set the range ring \n\n\nr\ni\n\n=\n0\n\n for all range-azimuth ground clutter bins.
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In Figure 5, we show the SINR of Algorithm 2 and SOA2 for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n,\n2\n\n supposing a target \n\n−\nπ\n/\n180\n≤\n\nθ\n0\n\n≤\nπ\n/\n180\n\n, \n\n0.36\n≤\n\nv\n\nd\n0\n\n\n≤\n0.44\n\n. The SINR values increases both for Algorithm 2 and SOA2 with the increasing iteration number \n\nn\n\n. Furthermore, we observe the higher \n\nξ\n\n, the better SINR values reflecting the larger and larger feasible set. Interestingly, Algorithm 2 significantly outperforms SOA2 for all the considered \n\nξ\n\n, except for \n\nξ\n=\n2\n\n where they both achieve the same SINR value. In particular, we see that the gap between \n\nξ\n=\n2\n\n and energy constraint is about 1.1 dB because of the introduction of constant modulus constraint. We also observe that in this scenario, Algorithm 2 needs a higher number of iterations to achieve convergence compared with that in Figure 2. For instance, for \n\nξ\n=\n0.1\n\n, Algorithm 2 converges with about 12 iterations in Figure 5, whereas in Figure 2 after about 2 iterations.
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Figure 5.
The SINR behavior versus iteration number assuming a target with \n\n−\nπ\n/\n180\n≤\n\nθ\n0\n\n≤\nπ\n/\n180\n\n, \n\n0.36\n≤\n\nv\n\nd\n0\n\n\n≤\n0.44\n\n in uniform clutter environment for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n,\n2\n\n, \n\n\n\ns\n0\n\n\n as the initial point.
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In Table 2, we summarize the SINR values, iterations number, and the global computation time of Algorithm 2 and SOA2. In particular, Algorithm 2 shows a lower computational time for \n\nξ\n=\n0.1\n,\n2\n\n. Furthermore, it is observed that the gains of 2.3 and 3 dB are achieved using Algorithm 2 with a slightly higher computational cost for \n\nξ\n=\n0.5\n,\n1.3\n\n, respectively.
SINR values (in dB), iterations number, and global computation time (in seconds) of Algorithm 2 and SOA2 assuming a target with \n\n−\nπ\n/\n180\n≤\n\nθ\n0\n\n≤\nπ\n/\n180\n\n, \n\n0.36\n≤\n\nv\n\nd\n0\n\n\n≤\n0.44\n\n in uniform clutter environment for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n,\n2\n\n, \n\n\n\ns\n0\n\n\n as the initial point.
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Figure 6 shows the joint frequency and azimuth behavior of STTC and STRF concerning CAF. Specifically, the contour map of the Doppler-azimuth plane of CAF at \n\nr\n=\n0\n\n against the iteration number (\n\nn\n=\n\n0\n10\n30\n82\n\n\n) considering \n\nξ\n=\n2\n\n for Algorithm 2 is plotted. We observe that \n\n\ng\n\nn\n\n\n\n\n\n\ns\n\nn\n\n\n\n\nw\n\nn\n\n\nr\nv\nθ\n\n\n obtains lower and lower values in the region of \n\n−\nπ\n/\n2\n≤\nθ\n≤\nπ\n/\n2\n\n, \n\n−\n0.04\n≤\nv\n≤\n0.04\n\n (highlighted by black rectangles) with the increase of iteration number \n\nn\n\n. This performance behavior highlights that the proposed algorithm of joint design STTC and STRF possesses the ability of sequentially refining the shape of the CAF to achieve better and better clutter suppression levels.
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Figure 6.
Doppler-azimuth plane of CAF at \n\nr\n=\n0\n\n for \n\nξ\n=\n2\n\n of Algorithm 2 for \n\nn\n=\n\n0\n10\n30\n82\n\n\n assuming a target with \n\n−\nπ\n/\n180\n≤\n\nθ\n0\n\n≤\nπ\n/\n180\n\n, \n\n0.36\n≤\n\nv\n\nd\n0\n\n\n≤\n0.44\n\n in uniform clutter environment (black rectangles represent the locations of uniform clutter), \n\n\n\ns\n0\n\n\n as the initial point of Algorithm 2 and SOA2.
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Figure 7 plots the SINR versus the standard deviations \n\n\nσ\n\nv\n\nd\n0\n\n\n\n\n (Figure 7a) and \n\n\nσ\n\nθ\n0\n\n\n\n (Figure 7b) of Doppler and azimuth of target with \n\n\n\nθ\n¯\n\n0\n\n=\n\n0\n°\n\n\n, \n\n\n\nv\n¯\n\n\nd\n0\n\n\n=\n0.4\n\n, respectively. Again, we see that Algorithm 2 obtains a higher SINR gain than SOA2 for \n\nξ\n=\n0.1\n,\n0.5\n,\n1.3\n\n, whereas they both fulfill the near same gain at \n\nξ\n=\n2\n\n. Interestingly, we also observe that a decreasing trend in gain with the increase in standard deviation. This is reasonable due to that the larger standard deviation results in the larger uncertainty on the knowledge of target.
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Figure 7.
The SINR behaviors versus the standard deviations \n\n\nσ\n\nv\n\nd\n0\n\n\n\n\n (a) and \n\n\nσ\n\nθ\n0\n\n\n\n () of Doppler and azimuth of target with \n\n\n\nθ\n¯\n\n0\n\n=\n\n0\n°\n\n\n, \n\n\n\nv\n¯\n\n\nd\n0\n\n\n=\n0.4\n\n, respectively, \n\n\n\ns\n0\n\n\n as the initial point of Algorithm 2 and SOA2.
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6. Conclusions
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This chapter has considered the joint STTC and STRF design for MIMO radar under signal-dependent interference. We focus on a narrow band colocated MIMO radar with a moving point-like target considering imprecise a prior knowledge including Doppler and azimuth. Summarizing,
We have devised an iterative algorithm to maximize the SINR accounting for both a similarity constraint and constant modulus requirements on the probing waveform. Each iteration of the algorithm requires the solution of hidden convex problems. The consequent computational complexity is linear with the number of iterations and polynomial with the sizes of the STTC and the STRF.
We have assessed the performance of the proposed iteration algorithm through numerical simulations. The results have manifested that the larger the similarity parameter (i.e., the weaker the similarity constraint), the larger the output SINR due to the expanded feasible set. Moreover, we observed that the devised iteration procedure can provide a monotonic improvement of SINR and ensuring convergence to a stationary point, which possesses excellent superiority in computation complexity and performance gain compared with the related SOA2. The numerical examples also have revealed the capability of the developed procedure to sequentially refine the shape of the CAF both in nonuniform point-like clutter environment and uniform clutter environment.
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Possible future work tracks might extend the proposed framework to consider spectral constraint [37] and MIMO radar beampattern design by optimizing integrated sidelobe level (ISL) with practical constraints.
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Acknowledgments
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This work was supported by the National Natural Science Foundation of China under Grants 61771109 and 61501083. The authors like to thank Dr. Augusto Aubry for his constructive comments.
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\n\n\n\n
Let us denote \n\n\nS\n\n\n in block matrix form, i.e.,
Hence, exploiting the fact that \n\n\nv\n\nd\n0\n\n\n\n and \n\n\nθ\n0\n\n\n are statistically independent random variables, the block matrix \n\n\nΓ\n\n\nm\n1\n\n\nm\n2\n\n\n\n\n of \n\nΓ\n\n\n\nS\n\n\n\n\n in (21) can be expressed as
Since \n\n\nv\n\nd\n0\n\n\n\n is a uniformly distributed random variable, e.g., \n\n\nv\n\nd\n0\n\n\n∼\nU\n\n\n\n\nv\n¯\n\n\nd\n0\n\n\n−\n\n\nε\n0\n\n2\n\n\n\n\n\nv\n¯\n\n\nd\n0\n\n\n+\n\n\nε\n0\n\n2\n\n\n\n\n, the first expectation of (38) can be computed as
Based on \n\n\nθ\n0\n\n\n as a uniformly distributed random variable, e.g., \n\n\nθ\n0\n\n∼\nU\n\n\n\n\nθ\n¯\n\n0\n\n−\n\n\nϑ\n0\n\n2\n\n\n\n\n\nθ\n¯\n\n0\n\n+\n\n\nϑ\n0\n\n2\n\n\n\n\n, the \n\n\n\nq\n1\n\n\nq\n2\n\n\n\n entry of expectation \n\n\n\nΦ\n¯\n\n\n\nq\n1\n\n\nq\n2\n\n\n\n\n can be computed as
As a consequence, based on the statistical independence of \n\n\nv\n\nd\n0\n\n\n\n and \n\n\nθ\n0\n\n\n, the block matrix \n\n\nΘ\n\n\ni\n1\n\n\ni\n2\n\n\n\n\n of \n\nΘ\n\n\n\nW\n\n\n\n\n in (22) is
where \n\n\na\nn\n\n=\n\n\n\nα\n\nn\n,\n1\n\n\n\nα\n\nn\n,\n2\n\n\n⋯\n\nα\n\nn\n,\n\nKN\nT\n\n\n\n\nT\n\n∈\n\nC\n\nKN\nT\n\n\n\n, for \n\nn\n=\n1\n,\n2\n,\n⋯\n,\n\nKN\nT\n\n\n. Hence, the \n\n\ns\n†\n\nΘ\n\n\nww\n†\n\n\ns\n\n can be expressed as
Using the property \n\n\nα\n\nn\n,\ni\n\n\n=\n\nα\n\ni\n,\nn\n\n∗\n\n\n since \n\nΘ\n\n\nww\n†\n\n\n\n is a positive semidefinite matrix, (48) can be computed as
where \n\n\nβ\n\nm\n,\nn\n\n\n\n denotes the \n\n\nm\nn\n\n\n th entry of \n\n\n\nΣ\n¯\n\ndv\n\n\n\nww\n†\n\n\n\n.
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\n',keywords:"multiple input multiple output (MIMO), space-time transmit code (STTC), space-time receive filter (STRF), signal-dependent interferences, signal to interference plus noise ratio (SINR)",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/57895.pdf",chapterXML:"https://mts.intechopen.com/source/xml/57895.xml",downloadPdfUrl:"/chapter/pdf-download/57895",previewPdfUrl:"/chapter/pdf-preview/57895",totalDownloads:692,totalViews:178,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"June 12th 2017",dateReviewed:"October 25th 2017",datePrePublished:"December 20th 2017",datePublished:"May 16th 2018",dateFinished:null,readingETA:"0",abstract:"This chapter deals with the design of multiple input multiple-output (MIMO) radar space-time transmit code (STTC) and space-time receive filter (STRF) to enhance moving targets detection in the presence of signal-dependent interferences, where we assume that some knowledge of target and clutter statistics are available for MIMO radar system according to a cognitive paradigm by using a site-specific (possible dynamic) environment database. Thus, an iterative sequential optimization algorithm with ensuring the convergence is proposed to maximize the signal to interference plus noise ratio (SINR) under the similarity and constant modulus constraints on the probing waveform. In particular, each iteration of the proposed algorithm requires to solve the hidden convex problems. The computational complexity is linear with the number of iterations and polynomial with the sizes of the STTW and the STRF. Finally, the gain and the computation time of the proposed algorithm also compared with the available methods are evaluated.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/57895",risUrl:"/chapter/ris/57895",book:{slug:"topics-in-radar-signal-processing"},signatures:"Guolong Cui, Xianxiang Yu and Lingjiang Kong",authors:[{id:"213949",title:"Prof.",name:"Guolong",middleName:null,surname:"Cui",fullName:"Guolong Cui",slug:"guolong-cui",email:"cuiguolong@uestc.edu.cn",position:null,institution:{name:"University of Electronic Science and Technology of China",institutionURL:null,country:{name:"China"}}},{id:"213973",title:"Mr.",name:"Xianxiang",middleName:null,surname:"Yu",fullName:"Xianxiang Yu",slug:"xianxiang-yu",email:"xianxiangy@gmail.com",position:null,institution:null},{id:"213974",title:"Prof.",name:"Lingjiang",middleName:null,surname:"Kong",fullName:"Lingjiang Kong",slug:"lingjiang-kong",email:"ljkong@uestc.edu.cn",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. System model",level:"1"},{id:"sec_3",title:"3. Problem formulation",level:"1"},{id:"sec_3_2",title:"3.1. Output SINR",level:"2"},{id:"sec_4_2",title:"3.2. Constant modulus and similarity constraints",level:"2"},{id:"sec_5_2",title:"3.3. Design problem",level:"2"},{id:"sec_7",title:"4. STTC and STRF design procedure",level:"1"},{id:"sec_7_2",title:"4.1. STRF optimization",level:"2"},{id:"sec_8_2",title:"4.2. STTC optimization",level:"2"},{id:"sec_9_2",title:"4.3. Transmit-receive system design",level:"2"},{id:"sec_11",title:"5. Numerical results",level:"1"},{id:"sec_11_2",title:"5.1. Nonuniform point-like clutter environment",level:"2"},{id:"sec_12_2",title:"5.2. Uniform clutter environment",level:"2"},{id:"sec_14",title:"6. Conclusions",level:"1"},{id:"sec_15",title:"Acknowledgments",level:"1"},{id:"sec_16",title:"",level:"1"}],chapterReferences:[{id:"B1",body:'Li J, Stoica P. MIMO Radar Signal Processing. 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IEEE Journal of Selected Topics in Signal Processing. 2007;1:147-155. DOI: 10.1109/JSTSP.2007.897056\n'},{id:"B7",body:'Aubry A, De Maio A, Huang Y. MIMO radar Beampattern design via PSL/ISL optimization. IEEE Transactions on Signal Processing. 2016;64:3955-3967. DOI: 10.1109/TSP.2016.2543207\n'},{id:"B8",body:'Ahmed S, Alouini MS. MIMO-radar waveform covariance matrix for high SINR and low side-lobe levels. IEEE Transactions on Signal Processing. 2014;62:2056-2065. DOI: 10.1109/TSP.2014.2307282\n'},{id:"B9",body:'Friedlander B. Waveform design for MIMO radars. IEEE Transactions on Aerospace and Electronic Systems. 2007;43:1227-1238. DOI: 10.1109/TAES.2007.4383615\n'},{id:"B10",body:'Naghibi T, Behnia F. MIMO radar waveform design in the presence of clutter. IEEE Transactions on Aerospace and Electronic Systems. 2011;47:770-781. DOI: 10.1109/TAES.2011.5751224\n'},{id:"B11",body:'Chen CY, Vaidyanathan PP. MIMO radar waveform optimization with prior information of the extended target and clutter. IEEE Transactions on Signal Processing. 2009;57:3533-3544. DOI: 10.1109/TSP.2009.2021632\n'},{id:"B12",body:'Cui G, Li H, Rangaswamy M. MIMO radar waveform design with constant modulus and similarity constraints. IEEE Transactions on Signal Processing. 2014;62:343-353. DOI: 10.1109/TSP.2013.2288086\n'},{id:"B13",body:'Zhu W, Tang J. Robust design of transmit waveform and receive filter for colocated MIMO radar. IEEE Signal Processing Letters. 2015;22:2112-2116. DOI: 10.1109/LSP.2015.2461460\n'},{id:"B14",body:'Jiu B, Liu H, Wang X, Zhang L, Wang Y, Chen B. Knowledge-based spatial-temporal hierarchical MIMO radar waveform design method for target detection in heterogeneous clutter zone. IEEE Transactions on Signal Processing. 2015;63:543-554. DOI: 10.1109/TSP.2014.2366714\n'},{id:"B15",body:'Imani S, Ghorashi SA. Transmit signal and receive filter design in co-located MIMO radar using a transmit weighting matrix. IEEE Signal Processing Letters. 2015;22:1521-1524. DOI: 10.1109/LSP.2015.2411676\n'},{id:"B16",body:'Xue M, Zhu X, Li J, Vu D, Stoica PMIMO. Radar waveform design. In: De Maio A, Gini FL P, editors. Waveform Design and Diversity for Advanced Radar Systems. 2012. pp. 89-120. DOI: 10.1049/PBRA022.ch4\n'},{id:"B17",body:'Mecca VF, Krolik JL, Robey FC. Beamspace slow-time MIMO radar for multipath clutter mitigation. IEEE International Conference on Acoustics, Speech and Signal Processing 31 March-4 April 2008. Las Vegas, Nevada, USA: IEEE; 2008. p. 2313-2316\n'},{id:"B18",body:'Duly AJ, Krogmeier JV. Time-division beamforming for MIMO radar waveform design. IEEE Transactions on Aerospace and Electronic Systems. 2013;49:1210-1223. DOI: 10.1109/TAES.2013.6494408\n'},{id:"B19",body:'Karbasi SM, Aubry A, Carotenuto V, Naghsh MM, Bastan MH. Knowledge-based design of space-time transmit code and receive filter for a multiple-input-multiple-output radar in signal-dependent interference. IET Radar, Sonar, Navigation. 2015;9:1124-1135. DOI: 10.1049/iet-rsn.2014.0527\n'},{id:"B20",body:'Yu X, Cui G, Kong L, Carotenuto V. Space-time transmit code and receive filter Design for Colocated MIMO radar. IEEE radar conference; 2-6 May 2016; Philadelphia, Pennsylvania, USA. IEEE. 2016:1-6\n'},{id:"B21",body:'Cui G, Yu X, Carotenuto V, Kong L. Space-time transmit code and receive filter design for colocated MIMO radar. IEEE Transactions on Signal Processing. 2017;65:1116-1129. DOI: 10.1109/TSP.2016.2633242\n'},{id:"B22",body:'Guerci JR. Cognitive radar: The knowledge aided fully adaptive approach. IEEE radar conference; 10-14 May 2010; Washington, DC, USA. IEEE. 2010:1365-1370\n'},{id:"B23",body:'Aubry A, De Maio A, Farina A, Wicks M. Knowledge-aided (potentially cognitive) transmit signal and receive filter design in signal-dependent clutter. IEEE Transactions on Aerospace and Electronic Systems. 2013;49:93-117. DOI: 10.1109/TAES.2013.6404093\n'},{id:"B24",body:'Aubry A, De Maio A, Jiang B, Zhang S. Ambiguity function shaping for cognitive radar via complex quartic optimization. IEEE Transactions on Signal Processing. 2013;61:5603-5619. DOI: 10.1109/TSP.2013.2273885\n'},{id:"B25",body:'Cui G, Fu Y, Yu X, Li J. Local ambiguity function shaping via unimodular sequence design. IEEE Signal Processing Letters. 2017;24:977-981. DOI: 10.1109/LSP.2017.2700396\n'},{id:"B26",body:'De Maio A, De NS, Huang Y, Luo Z, Zhang S. Design of phase codes for radar performance optimization with a similarity constraint. IEEE Transactions on Signal Processing. 2009;57:610-621. DOI: 10.1109/TSP.2008.2008247\n'},{id:"B27",body:'Aubry A, Carotenuto V, De Maio A. Forcing multiple spectral compatibility constraints in radar waveforms. IEEE Signal Processing Letters. 2016;23:483-487. DOI: 10.1109/LSP.2016.2532739\n'},{id:"B28",body:'Aubry A, De Maio A, Piezzo M, Farina A, Wicks M. Cognitive design of the receive filter and transmitted phase code in reverberating environment. IET Radar, Sonar, Navigation. 2012;6:822-833. DOI: 10.1049/iet-rsn.2012.0029\n'},{id:"B29",body:'Cui G, Yu X, Foglia G, Huang Y, Li J. Quadratic optimization with similarity constraint for unimodular sequence synthesis. IEEE Transactions on Signal Processing. 2017;65:4756-4769. DOI: 10.1109/TSP.2017.2715010\n'},{id:"B30",body:'Yu X, Cui G, Ge P, Kong L. Constrained radar waveform design algorithm for spectral coexistence. IET Electronic Letters. 2017;53:558-560. DOI: 10.1049/el.2016.4524\n'},{id:"B31",body:'Li J, Guerci JR, Signal XL. Waveforms optimal-under-restriction design for active sensing. IEEE Signal Processing Letters. 2016;13:565-568. DOI: 10.1109/SAM.2006.1706159\n'},{id:"B32",body:'Barros AI, Frenk JBG, Schaible SA. New algorithm for generalized fractional programs. Mathematical Programming. 1996;72:147-175. DOI: 10.1109/CC.2017.7942327\n'},{id:"B33",body:'Crouzeix JP, Ferland JA, Schaible S. An algorithm for generalized fractional programs. Journal of Optimization Theory and Applications. 1985;47:35-49. DOI: 10.1109/IPDPS.2017.22\n'},{id:"B34",body:'Aubry A, De Maio A, Naghsh MM. Optimizing radar waveform and Doppler filter bank via generalized fractional programming. IEEE Journal of Selected Topics in Signal Processing. 2015;9:1387-1399. DOI: 10.1109/JSTSP.2015.2469259\n'},{id:"B35",body:'Golub G H and Loan C F V. Matrix Computations. 4th ed. Baltimore, MD: The Johns Hopkins University Press; 2013\n'},{id:"B36",body:'Grant M and Boyd S. CVX Package [Internet]. March 2017. Available from: http://www.cvxr.com/cvx.r\n\n'},{id:"B37",body:'Aubry A, De Maio A, Huang Y, Piezzo M, Farina A. A new radar waveform design algorithm with improved feasibility for spectral coexistence. IEEE Transactions on Aerospace and Electronic Systems. 2015;52:1029-1038. DOI: 10.1109/TAES.2014.140093\n'}],footnotes:[{id:"fn1",explanation:"Notice that based on its definition, the shift matrix satisfies the condition \n\n\nJ\nr\n\n=\n\nJ\n\n−\nrT\n\n\n\n."},{id:"fn2",explanation:"Notice that the similar convergence analysis can be obtained in [23]."},{id:"fn3",explanation:"Notice that LFM waveforms have good properties in the pulse compression and ambiguity feature."},{id:"fn4",explanation:"Notice that we consider the exit condition \n\nA\n/\n\n10\n4\n\n\n both for Algorithms 1 and 2, where \n\nA\n\n denotes the upper bound of the objective function neglecting the signal-dependent interference (for example, \n\nA\n=\n10\n\n is considered in this simulation)."}],contributors:[{corresp:"yes",contributorFullName:"Guolong Cui",address:"cuiguolong@uestc.edu.cn",affiliation:'
School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, China
School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, China
'}],corrections:null},book:{id:"6347",title:"Topics in Radar Signal Processing",subtitle:null,fullTitle:"Topics in Radar Signal Processing",slug:"topics-in-radar-signal-processing",publishedDate:"May 16th 2018",bookSignature:"Graham Weinberg",coverURL:"https://cdn.intechopen.com/books/images_new/6347.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"104720",title:"Dr.",name:"Graham",middleName:"V",surname:"Weinberg",slug:"graham-weinberg",fullName:"Graham Weinberg"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},chapters:[{id:"57773",title:"Waveform Design and Related Processing for Multiple Target Detection and Resolution",slug:"waveform-design-and-related-processing-for-multiple-target-detection-and-resolution",totalDownloads:1037,totalCrossrefCites:0,signatures:"Gaspare Galati and Gabriele Pavan",authors:[{id:"213346",title:"Prof.",name:"Gaspare",middleName:null,surname:"Galati",fullName:"Gaspare 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1. Introduction
In recent years, technological advances in the field of echocardiography have allowed for a faster acquisition of images with an improved spatial and temporal resolution. As part of these advances, the advent of speckle tracking imaging has resulted in an explosion of investigations into myocardial deformation, as evidenced by more than 5000 articles on PubMed, increasing exponentially since 2005 (https://pubmed.ncbi.nlm.nih.gov/?term=speckle+tracking, accessed 7th of May 2020). The past two decades has also seen in a shift in “stress echocardiography” from being dominated by acute drug-based interventions to primarily exercise challenges. Therefore, this chapter focuses on the current knowledge related to myocardial deformation during acute exercise stress. Instead of just summarizing the current literature, a careful selection of articles is presented that is then used to provide the reader with a narrative that highlights important general principles of cardiac physiology, including the responses to exercise. To achieve this aim, first a brief overview of the principles and mechanisms governing myocardial deformation will be provided summarised and the key terminology will be defined. Then, the general role of exercise stress testing will be discussed, before the benefits of obtaining myocardial deformation during exercise in health and disease will be reviewed.
2. Principles of myocardial deformation
During contraction of the heart, deformation of the whole muscle occurs in four quantifiable dimensions. In general, these have been identified as: longitudinal shortening (=longitudinal strain, %), circumferential shortening (circumferential strain, %), radial lengthening (=radial strain, %) and rotation (apical − basal rotation = net twist angle, degrees), as well as the diastolic reversal of all of these indices. In addition, the rate of systolic shortening and diastolic lengthening can be measured, which is referred to as strain rate, twisting rate, and untwisting rate. An important distinction must be made between myocardial deformation and pure “velocities”, which do not consider the relative shortening (contraction) or lengthening (relaxation) of heart muscle itself but only consider the linear displacement of single myocardial points. Although myocardial velocities can also be measured, they are not representative of the contraction and relaxation of heart muscle. For these reasons, parameters such as E’ (“E prime”), which typically represent myocardial velocities in a single location on the mitral annulus, are not discussed in this chapter.
The conventional categorizations of deformation into strain and twist are logical from a biophysics and bioengineering perspective, since deformation of the heart can indeed be detected in these distinct 2-dimensional echocardiographic imaging planes. However, as will be reviewed in the following section on the anatomy and electrical conductance, the structure of the heart is far from symmetrical and—to achieve the final coordination of all components with each heartbeat—important functional differences in the various regions within the heart are present. These intricate deformational patterns can be conceptually simplified by considering the region-specific deformation in a 2-dimensional plane, allowing for easier evaluation of cardiac mechanics in both the laboratory and the clinic. However, one must consider the 3D deformation of the heart muscle, where the deformation of the four imaging planes occur simultaneously and with many of these aspects anatomically and functionally interwoven. This anatomical complexity is the focus of the next section.
2.1 Anatomy
Historical reviews have often credited Leonardo da Vinci’s observations in the 15th century as some of the first to describe the gross anatomy of the heart and his speculations about the resulting function. In his drawings1, da Vinci refers to the importance of vortices, which necessitate the presence of helical structures and/or motions that were apparent as “clockwise and counterclockwise spirals within the aorta as the outlet of the left ventricle” [1]. More than a century after da Vinci’s death, William Harvey published his seminal book Exercitatio Anatomica De Motu Cordis Et Sanguinis In Animalibus (An Anatomical Study on the Motion of the Heart and Blood in Living Beings, 1628 [2]), in which he established the circulation—including the anatomy and motion of the heart—as we mostly know it today, thereby also popularizing the previous work by Ibn al-Nafis [3]. In 1669, Richard Lower provided remarkable detail on the anatomy of the heart in his publication of Tractatus de Corde… (Treatise on the Heart. … [4]). Despite these early discoveries, it wasn’t until the contributions by McCallum and then Mall in the early twentieth century that there were new advancements in this field [5, 6]. During the second World War, Robb & Robb provided an exceptionally detailed overview of the accumulated knowledge that covered five centuries of discoveries [7]. Then, 27 years later, in 1969, Streeter et al. published the much-cited myocardial fiber distribution of the left ventricle (LV) in dogs, and Greenbaum et al. confirmed the observations in human cadavers [8, 9].
Today, after centuries of observations, there is still debate on the exact origins and arrangements of the heart [10]. However, general consensus exists that the mammalian LV consists of oblique fibers in the endocardium that gradually change into circumferential fibers in the midwall and continue to oblique fibers in the subepicardium, orientated in the opposite direction to those in the endocardium, thus creating what is often referred to as a helical arrangement [11, 12, 13, 14]. Noteworthy insight has also been provided by the description of sheets and laminae, which may not only impact the effect of individual myofibres but also the electrical propagation across the myocardium [15, 16]. With regard to the latter, the coordinated sequence of electrical propagation and activation of the LV occurs in a specific apex-to-base and endocardial-to-epicardial order during systole [17]. Due to these different electrical activation times, each part of the heart muscle is activated for different durations, therefore shortening and lengthening velocities (or systolic and diastolic “strain rates”) vary significantly in the different regions of the LV and are not associated with the overall heart rate [18]. A significant addition to the longstanding knowledge on oblique and circumferential fibers was provided by Lunkenheimer et al., who provided evidence for the existence of transmural myofibres that may be of fundamental relevance to the regulation of forces associated with normal myocardial contraction and relaxation [19]. Finally, there is important structural diversity on the myocyte level that contributes to the overall elasticity of the cardiomyocyte, as revealed by different isoforms of the giant protein titin, which may influence myocardial deformation in systole and diastole, not least during exercise [20, 21]. Collectively, the current knowledge indicates a non-uniform, complex mesh of diverse cardiac myofibre arrangements which may be grouped in sheets and laminae, influencing the electrical activation sequence of the heterogeneously distributed autonomic nerves in the heart (Figure 1, [22]). In comparison to the LV, the macro-structure of the right ventricle (RV) is not cone-shaped but resembles that of a crescent, almost wrapping around the LV. Yet, the underlying micro-structure is similar to the LV, albeit with some key differences. Like the LV, the epicardial and endocardial fibers are arranged helically, but with a smaller range of oblique angles [23]. The main difference to the LV seems to be in the myofiber arrangement of the midwall. Here, “the circumferentially arranged middle fibres are confined to the LV and septum” [8] and “without such beneficial architectural remodeling […] seem unsuited structurally to sustain a permanent increase in afterload” [23]. It is probably because of the overall crescent shape (that makes echocardiographic image acquisition in any plane other than the longitudinal challenging), and the lack of an obvious torsional motion, that the assessment of right ventricular deformation has largely focused on longitudinal strain.
Figure 1.
LV anatomy, strain and twist. (A) Although the detailed anatomy of the heart is still a matter of debate, the most comprehensive, evidence-based model includes a mesh of oblique, circumferential and transmural fibers (1–5). (B) LV strain is typically assessed in three planes, the longitudinal plane (from the apex to the base, L), the circumferential plane, C, and the radial plane (from the endocardium to epicardium, R). Owing to the specific anatomy, contraction of the LV results in a twisting motion around the long-axis, with an opposing rotational movement at the base compared with the apex that is rapidly released in diastole. Resultant twist and twist velocity curves produce a clear signal for peak LV twist and early diastolic untwisting rate (red arrows). Please see further details and the original figures in Refs. [14, 24].
2.2 Definitions and selection of myocardial deformation parameters
Because of the increasing number of studies focused on myocardial deformation mentioned in the introduction to this chapter, it has been inevitable that some inconsistencies exist regarding the nomenclature in the literature. Here, a summary of the most common definitions is provided and the reader is also referred to previous review articles for further details on the terminology [24, 25, 26].
With regard to the LV, three strain components have been established: longitudinal, circumferential and radial strain [25]. Systolic strain rate was once thought to reflect contractility; however, these hopes have not been sustained. Furthermore, the anatomy of the heart does not support the measurement of radial strain since there are no radial fibers in the LV or RV. Although the transmural fibers may somewhat relate to this type of strain, they maximally constitute ~20% to overall deformation and do not seem to run strictly in the radial direction. Second, the classification of twist or torsion as a “shear strain” or fourth dimension of deformation does not fit the underlying anatomy of the heart either. There is currently no empirical evidence for the existence of a meaningful number of longitudinal fibers that could determine longitudinal deformation of the ventricles. Instead, the oblique fibers that make up most of the fibers within the left ventricular walls are likely responsible for deformation in the longitudinal direction. Consequently, it does not seem appropriate to calculate twist or torsion from the longitudinal and circumferential shear angle, also because this approach does not capture the potential regional differences that exist between the base and apex in both the LV and RV. Despite these drawbacks to the radial and longitudinal parameters, it must be acknowledged that longitudinal strain has become the most established measure as a clinical marker with diagnostic potential [27]. For these reasons, in the context of this chapter, it seems appropriate to ignore LV radial strain but include LV longitudinal and circumferential strain as well as twist and untwisting rate. Since no clear circumferential fibers or twisting motion have been detected in the RV, the focus for that chamber will be exclusively on longitudinal strainFigure 2.
Figure 2.
RV strain. The measurement of RV strain at rest (left) and during exercise (right) in a patient with hypertrophic cardiomyopathy. Because of the anatomical arrangement of the RV, longitudinal strain is the most commonly investigated parameter, although further clarity is required whether to always include or exclude the septum [28]. From a functional perspective, there is strong evidence that the septal deformation is more similar to that of the LV than the RV free wall, as supported by evidence of a shared morphology [29, 30]. Please see further details and the original figure in: Wu et al. [31].
Parameter (unit)
Description
Circumferential strain (%)
Percentage shortening of the circumference
Global longitudinal strain (%)
Typically, the average strain of multiple walls obtained from different echocardiographic windows (4-chamber, 2-chamber, 3-chamber)
Longitudinal strain (%)
Shortening along the long-axis of the ventricles in a single 2-dimensional imaging plane (for example a 4-chamber view)
Shear strain
The strain resulting from two different normal strains, for example “longitudinal-circumferential shear strain”
Strain (rate) imaging
Generic term that can refer to strain data obtained with either tissue Doppler or speckle tracking echocardiography
Strain rate (/s)
The rate of shortening (strain) or lengthening (strain) of each strain
Tissue Doppler strain (%)
Strain obtained with tissue Doppler echocardiography, which is more angle-dependent than speckle tracking echocardiography
Tissue velocity imaging (%)
Echocardiographic imaging based upon Doppler modality, often synonymous with tissue Doppler strain
Twist (degrees)
Also called the net twist angle, obtained from the net difference in rotation between the left ventricular base and apex. Not to be confused with torsion or rotation, the latter referring to the local angular deformation at the base and apex
Untwisting rate (°/s)
The maximal early diastolic rate of reversal of twist
Table 1.
Deformation parameters.
3. Echocardiographic assessment of myocardial deformation during exercise
3.1 Why exercise?
Even if all humans were elite athletes, we would spend most of the time in a day in a biological state of rest—or certainly in a state of low physical activity that only constitutes a fraction of the total capacity of our cardiovascular system. Accordingly, the routine clinical practice of examining cardiac function at rest is a good representation of the condition we find ourselves in most of the time. However, when a person requires an echocardiographic examination, it is typically for clinical reasons initiated by the presence of negative symptoms, often presenting as “exertional dyspnea” or angina. If an echocardiographic examination then detects structural and functional abnormalities of the heart that are congruent with the individual’s symptoms, the diagnosis of heart disease is likely. However, resting assessment of cardiac function often fails to recapitulate conditions of exertional dyspnea, and thus can sometimes lead to misdiagnosis. Equally, waiting until the emergence of symptoms postpones clinical treatment. For this reason, “stress testing” has been suggested to offer the opportunity of a “window into the future”. By taking the person out of their typical state of rest or low physical activity and stressing the full range of their cardiovascular system until maximum effort, underlying abnormalities may be detected that remain otherwise unknown. Examples for the benefit of exercise testing have been presented in relation to “unmasking masked hypertension” [32, 33]. Similarly, in pregnancy it has been proposed that the cardiovascular responses to exercise tests prior to conception may be indicators of the presence or absence of complications during future pregnancies [34, 35, 36, 37]. Furthermore, the complex etiology of heart failure has justified detailed exercise testing to identify the most important contributors out of the numerous cardiac or peripheral factors that may be involved in the development and/or the state of heart failure [38, 39, 40].
It is now recognized among clinical practitioners that the investigation of myocardial deformation during exercise can provide additive value, since previous research studies have revealed new (and sometimes surprising) insight into the behavior of the heart during exercise. As will be discussed in detail in Section 3.3, these findings have informed our basic understanding of cardiac function and sometimes guided future clinical investigation. Since myocardial function, including parameters of myocardial deformation, are influenced by the general loading state of the heart, any exercise responses must be seen in the context of general cardiovascular responses, as discussed in the next section.
3.2 General cardiovascular responses to exercise
In the context of myocardial deformation, the most relevant cardiovascular and cardiopulmonary responses to a standardized exercise test pertain to stroke volume, cardiac output, end-diastolic volume, blood pressure, arterial resistance, lactate, and maximal oxygen consumption (VO2max). In healthy individuals, a clear change in these parameters can be expected at the onset of low intensity dynamic exercise that should continue to change linearly up to moderate intensities. Importantly, dynamic exercise tests cause a disproportionate peripheral vasodilation in relation to the increase in cardiac output, and hence total peripheral resistance drops sharply at the onset of exercise and then remains constant across moderate and high exercise intensities [41]. From a diastolic perspective, end-diastolic volume has been shown to increase in some studies while others have not observed any change with exercise. This is not trivial since an acute increase in end-diastolic volume has been associated with an increased stroke volume, an effect also known as the Frank-Starling mechanism [42]. However, the overall contribution of end-diastolic volume to stroke volume is still relatively low because most of the increase in stroke volume has been attributed to the enhanced contractility that reduces the end-systolic volume.
At workloads above moderate intensity, several important physiological changes occur in healthy individuals. Blood lactate concentrations increase exponentially and CO2 production rises above O2 consumption, both reflecting the greater contribution of anaerobic metabolic pathways to overall energy utilization and causing a strong stimulus for vasodilation not least in the cerebral circulation. During the highest effort, stroke volume and VO2 have been reported to plateau and even decrease, but the exact pattern and the underlying mechanisms to this response remain a matter of debate [43]. Fortunately, this does not seem to impact the interpretation of cardiovascular responses to exercise in patients, since the sub-maximal data are currently thought to be of sufficient clinical value to determine whether exercise performance is normal or impaired [44].
One important distinction between the LV and RV responses to exercise is the potential for a “disproportionate load” on the RV [45], which is perhaps explained by both a greater relative rise in pulmonary blood pressure compared with that in the aorta, and differences in RV intrinsic factors such as force development. The differences between the LV and RV responses to exercise highlight the specific impact exercise has on the cardiovascular system. Consequently, determining the true origin of exercise limitations is challenging because many components of the cardiovascular system may be affected. For example, studies have shown that an exaggerated rise in blood pressure during exercise may be associated with negative outcomes, but whether this is caused by the heart or the periphery may be more difficult to determine [46, 47, 48]. Even in heart failure, the reduced exercise tolerance has been suggested to be a result of both central and non-cardiac limitations [38, 39, 40, 49]. Consequently, assessing myocardial deformation in relation to conventional exercise responses is essential for the quantification of the contributions of the heart muscle itself.
3.3 Myocardial deformation during exercise
Whatever myocardial parameter one chooses to examine during exercise, the interpretation of the responses can be tricky. For example, an increase in myocardial deformation with sub-maximal cycle exercise along with a typical drop in arterial resistance and concomitant reductions in end-systolic volume, in the presence of no adverse structural remodeling would be reflective of a “healthy” response. Equally, it is theoretically possible that the absence of a clear increase in myocardial deformation—which could be interpreted to represent myocardial dysfunction—may be a normal response if the increase in blood pressure and peripheral resistance were excessively high (or the exercise test did in fact create a condition of increased afterload). In this case, it is conceivable that the origin of the exercise limitation may not be cardiac despite the attenuated deformation, but perhaps peripheral in nature causing an exercise failure before the cardiac reserve is fully used [39]. Therefore, this section provides an overview of the general trend of myocardial deformation during exercise, but the reader is alerted that a qualitative interpretation must be performed after consideration of the wider physiology. Articles in this section were included if the studies had obtained data with echocardiography during exercise (tissue Doppler and tissue velocity imaging data were mostly excluded because both techniques are angle-dependent and typically represent only data from a single segment within the mitral annulus). Although a promising and exciting alternative to echocardiography, myocardial deformation during exercise obtained using MRI is not the focus of this chapter [50, 51]. Studies were also excluded if they obtained data immediately following exercise effort, as discussed in more detail in the section on methodological considerations. Finally, the avid reader is referred to some excellent review articles that cover more of the literature than this book chapter can accommodate [52, 53, 54, 55].
3.3.1 Physiological insight from healthy individuals
The physiology of myocardial deformation during exercise in healthy people is the fundamental basis upon which to interpret the responses in patient populations. Although many clinical research studies also include a healthy control group, sometimes these are matched to the patient groups in their demographics and, therefore, may not represent truly “healthy” individuals. Wherever possible, the data presented here will be from populations purposefully recruited as young healthy reference groups. To date, studies have revealed a variety of new perspectives that may be of great importance for the interpretation of clinical populations.
A decade ago, two studies revealed the strain and twist responses during incremental exercise. First, Doucende et al. showed that left ventricular twist and circumferential strain increased linearly up to moderate exercise intensities, while longitudinal strain increased initially but then plateaued at low exercise efforts [56]. This study also highlighted the interdependence of systolic and diastolic deformation, the role of untwisting rate in LV filling during exercise and the contribution of the LV apex to the overall myocardial response. Second, it was shown that LV twist and untwisting rate increased linearly up to near-maximal efforts, correlating with stroke volume and, thus, perhaps contributing to maximal exercise capacity in humans [57]. The importance of regional LV deformation, at the LV apex, was again highlighted. Several other studies have revealed similar patterns of LV twist during exercise in pre- and postmenopausal women, in athletes and of humans ascending to high altitude [58, 59, 60, 61, 62]. Consequently, it is now generally accepted that an increase in LV twist with exercise up to moderate intensities can be expected as a normal response (Figure 3). Surprisingly few studies dedicated to healthy individuals have measured LV strain during exercise, but they agree in general that longitudinal strain also increases with exercise [56, 61, 62, 63, 64]. Because of the risk of potential confounders, it is not possible to directly compare the response in LV twist and strain obtained in different studies. But in general, it is of great importance to note that the patters of the responses to exercise are not always the same for the two parameters, reminding us that they do not represent the same myocardial deformation. In agreement with the general physiological response to incremental exercise, LV twist increases linearly while longitudinal strain seems to plateau at low exercise efforts. This was more recently confirmed by Williams et al., who reported the same disparity between parameters in young healthy men [62]. Interestingly, in the same study, women seemed to have more of a linear response in longitudinal strain akin to LV twist. The disparity between LV twist and longitudinal strain has also been noted in studies on aging where LV twist consistently increases, but longitudinal strain does not change or decreases. Considering the well-established progression of aortic stiffness with aging [65], longitudinal strain appears to be at odds again with general physiology. Future studies should not only examine the parameters in relation to their sensitivity as a clinical marker but also consider the fit with general physiology.
Figure 3.
Myocardial deformation to incremental exercise. LV twist curves during incremental exercise, revealing a linear increase up to 70–80% of maximal individual exercise effort for both peak systolic LV twist (highest value in black lines top row) and peak diastolic untwisting rate (lowest value within black lines bottom row). Red lines represent myocardial deformation at the LV apex, blue lines at the LV base. Black lines are the composite of apical and basal data. Please see further details and the original figure in Ref. [57].
Studying the acute effects of exercise on myocardial deformation may be influenced by the chronic remodeling that humans have experienced. In this regard, Burns et al. showed that aging seems to be associated with a reduced LV twist reserve during exercise in a population of 60-year old individuals [66]. Similarly, “female aging”, as represented by the menopause, seems to impact the myocardial response to exercise, which may be further altered by exercise training [58]. One of the more surprising observations has been that of Cooke et al. who proposed that endurance trained athletes with enlarged “athlete’s heart” and a greater stroke volume had a similar systolic LV function, including LV twist, during submaximal exercise compared to untrained humans with smaller stroke volume [67]. Similar to the results presented by Doucende and Williams discussed above [56, 62], this particular exercise response strongly suggests that the mechanical2 systolic function of the heart may not be strictly associated with its output (stroke volume). Some mathematical calculations support the potentially poor linear association between systolic LV mechanical function and ejection fraction while others suggest a strong relationship [68]. In any case, the previous findings suggest that future investigations into the interaction between systolic deformation and ejection, and diastolic deformation and filling are needed to clarify the current uncertainty. One reason for the existing disagreement between mechanical function and associated hemodynamics may be the technical limitations causing restricted views from a 2D echocardiographic window. In the case of exercise responses, this may be particularly evident at the LV apex, since the apex has been proposed as an important contributor to exercise responses (in particular in diastolic function) [56, 57, 69]. However, in the echocardiographic images relevant for the measurement of global longitudinal strain, the representation of the apical segments is proportionately small and their contribution to longitudinal strain and strain rate may be underestimated compared with short-axis views [18]. Thus, some of the insight provided by myocardial deformation during exercise in healthy people relates to our more general understanding of cardiac function.
Compared with LV strain, RV longitudinal strain seems to be ~10 percentage points higher in healthy young humans at rest, likely reflecting the different anatomy combined with a lower pulmonary resistance compared with the aorta. Most studies reporting RV strain in healthy individuals during exercise have done so by including healthy controls as comparators to cardiac patients. From those studies, some patterns have emerged that suggest a consistently increased RV longitudinal strain during submaximal exercise in healthy individuals [28, 31, 70]. The mechanisms for this are probably similar to those of the LV, where an increased sympathetic state increases contractility while peripheral (pulmonary) vasodilation decreases downstream resistance [71]. However, during intense exercise, it seems that right ventricular myocardial deformation increases perhaps less than the LV, and it has even been shown to decrease. Given that both ventricles should produce approximately the same stroke volume under stable conditions, the lower RV strain during exercise is another indicator that the interaction between the mechanical function of the ventricles and the circulation may depend as much on the local arterial resistance as it may depend on the muscular performance (and therefore health) of the ventricles, and thus fitting the long-standing concept of a greater afterload-sensitivity of the RV. Recent studies in advanced heart failure patients who were surgically implanted with left ventricular assist devices (LVAD) may support this, since the mechanical pumps “unload” the LV and shift blood volume to the rest of the circulation, maybe creating “A Different Kind of Stress Test for the RV” [72, 73]. The accurate measurement of pulmonary and aortic resistance beyond the measurement or estimation of blood pressure is certainly going to elucidate the differential exposure and performance of the two ventricles [74]. At present, it seems that exercise does indeed cause a greater afterload challenge for the RV compared with the LV. In fact, it is worth noting that the exercise modalities used in the studies presented so far in this section have mostly employed “dynamic” exercise (see Section 3.4). In this context, it is essential to point out that this type of exercise increases sympathetic activation of the myocardium and reduces arterial resistance compared with the resting state, therefore creating an environment for the LV (and at low intensities for the RV) that is characterized by reduced afterload. During higher exercise intensities, pulmonary resistance can increase during dynamic exercise and create an augmented afterload challenge [45]. Strength exercise, also called resistance exercise, and isometric handgrip exercise are two other modalities that can provide an afterload challenge for the LV [75]. Interestingly, studies employing these exercise modalities in a number of different populations have consistently shown that the reduced systolic deformation is in part compensated for by an increase in heart rate, but can also be uncoupled from diastolic function [76, 77, 78, 79]. Given that resistance exercise produces a very different challenge to dynamic exercise, and that strength training is an important addition to rehabilitation, future research should consider incorporating responses during high resistive efforts [80, 81].
3.3.2 Exercise responses in patients with cardiovascular disease
In a seminal study, Notomi et al. provided mechanistic insight into the complex interdependence between systolic and diastolic function in hypertrophic cardiomyopathy [20]. Although the study used Tissue Doppler Imaging, it is a landmark study that has provided new insight and has popularized the use of exercise testing for both basic science and new insight into cardiac performance in patients. The study revealed that LV twist during exercise was significantly reduced in patients with hypertrophic cardiomyopathy. Similarly, two other studies concluded that systolic deformation reserve is reduced in patients with hypertrophic cardiomyopathy [82, 83]. However, one challenge in patient populations is that the change in heart rate is often different compared with control groups, and therefore it is possible that the groups experienced different physiological stimuli. This is a recurring problem in exercise studies that currently reduces the confidence in some conclusions. Equally, sometimes the matching of the change in heart rate between groups may lead to unequal workloads or changes in blood pressure, highlighting again the need to interpret myocardial deformation during exercise in the context of general physiological responses. Notwithstanding, the overall trend is that LV myocardial deformation in patients is reduced in response to an acute exercise challenge, including in cardiac amyloidosis, hypertension, cancer, coronary artery disease, as well as in patients with valve disease before and after surgical correction [84, 85, 86, 87, 88, 89]. Some subtle observations, however, are worthy of discussion. For example, in patients with microvascular angina, only the subendocardial strain was reduced, and diastolic function during exercise was more severely affected than systolic reserve [90]. Similarly, myocardial regions can respond differently during exercise in coronary artery disease patients, as shown by differential basal vs. apical rotational mechanics [89]. In an elegant study in patients with hypertrophic cardiomyopathy, Soullier et al. showed that there was significant heterogeneity in the response of the different deformation parameters to exercise, and that resting twist was even increased in patients while diastolic untwisting rate was less affected [83]. In patients with a prior heart transplant, the age of the recipients and donors seem to influence the longitudinal and circumferential strain response to exercise [91, 92]. All these observations highlight the very subtle changes that can occur between parameters, and between systolic and diastolic function. To determine the full significance of such differences should be the focus of future investigations. Furthermore, it will be essential to relate myocardial deformation more often to parameters like cardiac output, to enable the meaningful interpretation of deformation indices and their contribution to the overall capacity of the heart. When this was done in previous studies, the myocardial deformation during exercise provided a clear advancement of our general understanding of the etiology and/or progression of cardiac disease [93].
Because of the prevalence and importance of pulmonary hypertension, and the exercise limitations of heart failure patients, myocardial deformation of the RV during exercise has received heightened attention [94]. Similar to the LV response, the expected increase in RV myocardial deformation during exercise is generally blunted, not just in pulmonary hypertension but also in tetralogy of Fallot, systemic sclerosis, and hypertrophic cardiomyopathy [31, 95, 96, 97]. Most often, there is clear evidence that pulmonary artery pressures increased disproportionately in the groups that had a blunted increase in RV longitudinal strain during exercise. Importantly, these patients often have normal pulmonary artery pressures at rest, which not only emphasizes the diagnostic value of exercise testing, it also highlights the possibility that patients with suspected LV pathology should be tested for the RV myocardial response to exercise.
3.4 Important practical considerations
Any echocardiographic examination consists of two main parts: (1) the acquisition of standardized echocardiographic images, and (2) the analysis of images for the quantification of relevant parameters [98, 99]. When conducting echocardiography during exercise, both parts require modified approaches to ensure that the conclusions drawn remain valid. Here, based upon our extensive practical experience, we present some “take-home-messages” that we consider essential for the echocardiographic assessment of myocardial deformation during exercises.
Typically, exercise tests are performed in a stepwise (constant intensity for some minutes, then increasing) or incremental (gradually increasing intensity with every second) manner. Because different protocols provoke different physiological responses, the correct protocol must be selected carefully.
Exercise responses depend on the relative workload of an individual. Therefore, exercise intensities should be adjusted to an individual’s anticipated capacity and patients’ myocardial deformation interpreted in relation to the relative workload [100].
The individual adjustment of workload increments during the test should also acknowledge fitness, age, sex, medical history, and acute or chronic injuries.
For the assessment of myocardial deformation during exercise, running or cycling modalities are the most common. For the reason of improved image quality and because it is relatively safe/feasible, the preferred choice for exercise echocardiography may be supine cycling.
While it is generally accepted that gentle end-expiratory breath holds can be performed to obtain images, it is preferable to obtain echocardiographic cine loops during free breathing and average some cardiac cycles during inspiration and expiration.
It is important to distinguish between the physiological demands of different exercise modalities, categorized as: dynamic, static, and impact [101]. Consequently, certain types of exercise can be considered more as an “afterload challenge” than others, and the responses of myocardial deformation may vary greatly between these types of exercise. In this context, the reader is reminded that exercise training interventions for health will need to consider the same complexities, as evidenced by the potential for differential effects of moderate continuous exercise training versus high-intensity interval training in some cardiac patients [102].
One concern with regard to exercise testing is the risk of triggering adverse events. Although this will depend on the specific individual being tested and must be decided by qualified personnel on a case-by-case basis, as evidenced by a comprehensive study performed by Rognmo et al. [103], the overall risk for serious adverse events seems to be relatively low. Particular health and safety precautions should be taken in patients with overt or suspected arrhythmia and the decision “not allowed to perform an exercise test” may have to be taken.
Standardization of echocardiographic data acquisition during exercise is absolutely necessary. Sonographers should minimize the sector width and depth, maximize imaging frame rates, only use one focal point and position this in the optimal location, and optimize the overall image to maximize the visibility of the endocardial border for speckle tracking analysis. Although 3D echocardiography may solve some of the limitations of 2D echocardiography, at present the frame rates are too low to obtain the necessary temporal resolution for quantification of myocardial deformation during exercise, although this is expected to change in the near future.
During exercise, when respiration and heart are increased, the quick location of the optimal echocardiographic window is necessary. Marking up the location on the chest after the resting assessment serves as a “quick help” during exercise. The sonographer must, however, still optimize the image and perhaps move the transducer slightly during exercise.
Since heart rate increases during exercise but imaging frame rates are already maximized, the effective frame rate (data points per cardiac cycle) decreases. Although this cannot be fully corrected, it seems advisable to perform cubic spline interpolation to attenuate some of these limitations [104]. Note that cubic spline interpolation will not only add points in time (for example for the more confident assessment of dyssynchrony), it also slightly adjusts the peak values.
Data acquisition immediately following exercise is not the same as “during” exercise. With the cessation of exercise, especially after a strenuous effort with strong muscular contractions, instant changes in whole-body hemodynamics set in [105]. Hence, these data do not reflect an exercise challenge but a “exercise recovery” state.
For the acquisition of LV twist, apical data must be obtained by moving the transducer close to the point of obtaining a 4-chamber view, otherwise severely misrepresentative data will be collected [24].
4. Summary and conclusions
The assessment of LV and RV myocardial deformation during exercise is feasible and has contributed unique insight into cardiac physiology in health and disease. Inherent methodological challenges require appropriate training and a careful approach to image acquisition, analysis and interpretation. However, ongoing technological advancements and an increasing knowledge suggest that the echocardiographic assessment of myocardial deformation during exercise will play an ever-increasing role in future research and the clinical examination of the cardiac patient.
Acknowledgments
The authors express their sincere gratitude to the publisher, IntechOpen, for their very kind and generous financial support of this chapter.
\n',keywords:"exercise, heart, stress testing, diagnostics, imaging, echocardiography, VO2max, CPET, strain, twist, torsion, untwisting rate, blood pressure, LVAD, heart failure, speckle tracking, hypertension",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/72623.pdf",chapterXML:"https://mts.intechopen.com/source/xml/72623.xml",downloadPdfUrl:"/chapter/pdf-download/72623",previewPdfUrl:"/chapter/pdf-preview/72623",totalDownloads:124,totalViews:0,totalCrossrefCites:0,dateSubmitted:"March 4th 2020",dateReviewed:"May 25th 2020",datePrePublished:"June 26th 2020",datePublished:null,dateFinished:null,readingETA:"0",abstract:"The human heart is an asymmetrical structure that consists of oblique, circumferential, and transmural fibers, as well as laminae and sheets. Sequential electrical activation of all the muscle fibers ultimately results in a coordinated contraction of the heart muscle also referred to as “deformation.” This is immediately followed by myocardial relaxation, when the preceding deformation is reversed, and the ventricles fill with blood. Given the complexity of these repetitive motions, it is not surprising that there is great diversity in the myocardial deformation between different individuals and between distinct populations. Exercise presents a natural challenge to determine the full capacity of an individual’s heart, and modern imaging technologies allow for the non-invasive assessment of myocardial deformation during exercise. In this chapter, the most relevant anatomical basis for myocardial deformation is summarized and definitions of the most relevant parameters are provided. Then, the general cardiac responses to exercise are highlighted before the current knowledge on myocardial deformation during exercise is discussed. The literature clearly indicates that the echocardiographic evaluation of myocardial deformation during exercise holds great promise for the identification of sub-clinical disease. Future studies should aim to determine the mechanisms of differential expression of myocardial deformation during exercise in health and disease.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/72623",risUrl:"/chapter/ris/72623",signatures:"Eric J. Stöhr and T. Jake Samuel",book:{id:"9581",title:"Endocarditis",subtitle:null,fullTitle:"Endocarditis",slug:null,publishedDate:null,bookSignature:"",coverURL:"https://cdn.intechopen.com/books/images_new/9581.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Principles of myocardial deformation",level:"1"},{id:"sec_2_2",title:"2.1 Anatomy",level:"2"},{id:"sec_3_2",title:"2.2 Definitions and selection of myocardial deformation parameters",level:"2"},{id:"sec_5",title:"3. Echocardiographic assessment of myocardial deformation during exercise",level:"1"},{id:"sec_5_2",title:"3.1 Why exercise?",level:"2"},{id:"sec_6_2",title:"3.2 General cardiovascular responses to exercise",level:"2"},{id:"sec_7_2",title:"3.3 Myocardial deformation during exercise",level:"2"},{id:"sec_7_3",title:"3.3.1 Physiological insight from healthy individuals",level:"3"},{id:"sec_8_3",title:"3.3.2 Exercise responses in patients with cardiovascular disease",level:"3"},{id:"sec_10_2",title:"3.4 Important practical considerations",level:"2"},{id:"sec_12",title:"4. Summary and conclusions",level:"1"},{id:"sec_13",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'Buckberg GD. 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