space time transmit code and receive filter design for colocated mimo radar this paper deals with the design of multiple input multiple output radar space time transmit code sttc and space time receive filter strf to enhance moving targets detection in the presence of signal dependent interferences an iterative procedure whose convergence is analytically proved is devised to maximize the signal to interference plus noise ratio sinr accounting for both a similarity constraint and a constant modulus requirement on the probing waveform each iteration of the algorithm involves the solution of hidden convex problems specifically both a convex problem whose solution is provided in closed form and a set of fractional programming problems that can be globally solved in polynomial time via the dinkelback s procedure are settled the computational complexity is linear with the number of iterations and polynomial with the sizes of the sttc and the strf in particular the proposed technique provides a monotonic sinr improvement without limitations on the size of the similarity constraint and ensures convergence to a stationary point filling these important gaps in the open literature besides the reported results highlight that the new devised procedure outperforms both in the optimized sinr value and the computational complexity than the available counterparts

fourier based transmit beampattern design using mimo radar in multiple input multiple output mimo radar settings it is often desirable to transmit power only to a given location or set of locations defined by a beampattern transmit waveform design is a topic that has received much attention recently involving synthesis of both the signal covariance matrix r as well as the actual waveforms current methods involve a two step process of designing r via iterative solutions and then using r to generate waveforms that fulfill practical constraints such as having a constant envelope or drawing from a finite alphabet in this paper a closed form method to design r for a uniform linear array is proposed that utilizes the discrete fourier transform dft coefficients and toeplitz matrices the resulting covariance matrix fulfills the practical constraints such as positive semidefiniteness and the uniform elemental power constraint and provides performance similar to that of iterative methods which require a much greater computation time next a transmit architecture is presented that exploits the orthogonality of frequencies at discrete dft values to transmit a sum of orthogonal signals from each antenna the resulting waveforms provide a lower mean square error than current methods at a much lower computational cost and a simulated detection scenario demonstrates the performance advantages achieved

on the design of constant modulus probing signals for mimo radar probing signal waveforms play a central role in the signal processing performance of a multiple input multiple output mimo radar in practice for a given desired beam pattern we need to design a probing signal waveform whose beam pattern closely matches the desired one and whose autocorrelation and cross correlation sidelobes are kept low the latter properties are important to mitigate undesirable interference caused by multiple targets or scatterers in this correspondence we present an efficient optimization method to design a constant modulus probing signal which can synthesize a desired beam pattern while maximally suppressing both the autocorrelation and cross correlation sidelobes at between given spacial angles we formulate this problem as an unconstrained minimization of a fourth order trigonometric polynomial and propose an efficient quasi newton iterative algorithm to solve it besides we provide an analysis of the local minima of the fourth order trigonometric polynomial and prove that any local minima is a 1 2 approximation of its global optimal solution numerical examples show that the proposed approach compares favorably with the existing approach

codes based on a trellis cut set transformation i rotationally invariant codes let a trellis section spl tscr generate a trellis code spl cscr we study two trellis sections based on spl tscr a cut set trellis section spl tscr sub cs and a differential encoder trellis section spl tscr sub de we show that spl tscr can be transformed to a cut set trellis section spl tscr sub cs which is equivalent to spl tscr in the sense that both spl tscr and spl tscr sub cs generate spl cscr and both spl tscr and spl tscr sub cs have the same decoding complexity a differential encoder trellis section is equivalent to the trellis section obtained by following spl tscr with a differential encoder it is shown that both spl tscr sub cs and spl tscr sub de have inverse transform trellis sections a differential encoder trellis section generates a rotationally invariant ri code in a particularly simple and straightforward way but an ri code need not have a differential encoder trellis section however for all of the ri codes examined here we show that the cut set trellis section can be arranged into a differential encoder trellis section this means that these codes can be decomposed into an encoder followed by a differential encoder further we show that when spl tscr is formed using a linear binary convolutional encoder and a mapping by set partitioning then spl tscr followed by a differential encoder gives an ri code which in some cases is as good as the best previously known codes after applying the inverse transform to spl tscr sub de

unconstrained synthesis of covariance matrix for mimo radar transmit beampattern multiple input multiple output mimo radars have many advantages over their phased array counterparts improved spatial resolution better parametric identifiably and greater flexibility to design the transmit beampattern the design of the transmit beampatterns generally requires the waveforms to have arbitrary auto and cross correlation properties the correlation covariance matrix r of the waveforms must be positive semidefinite therefore synthesis of a desired beampattern is usually a constrained optimization problem in this paper to simplify the constrained optimization problem two algorithms are proposed to synthesize the waveform covariance matrix for the desired beampattern in the first proposed algorithm the elements of a square root matrix of the covariance matrix r are parameterized using the coordinates of a hypersphere that implicitly fulfil the constraints this yields an iterative algorithm whose convergence speed can be increased significantly by providing good initial values in the second algorithm the constraints and redundant information in the covariance matrix r are exploited to find a closed form solution the drawback of the second algorithm is that it may yield a pseudocovariance matrix pseudo cm that is not guaranteed to be positive semidefinite however the pseudo cm can be easily converted into a covariance matrix using eigenvalue decomposition and or shrinkage methods moreover a pseudo cm can also be used to provide good initial values for the first algorithm to enable faster convergence

mimo radar transmit beampattern matching design in this paper a novel transmit beampattern matching design method is proposed for a multiple input multiple output mimo radar system whereas previous approaches first solve the correlation matrix r and then use r to design the transmit signal s the proposed one step method obtains the transmit signal s directly through solving the waveform matrix p since mimo radar transmit beampattern matching design is an inverse problem of the transmit beampattern in this paper we first discuss some important properties of the transmit beampattern and standardize the beampattern matching design problem we then theoretically explain the value of the signal waveform matrix p followed by an unconstrained optimization problem to solve p numerical examples demonstrate the advantages of the proposed one step method for the mimo radar transmit beampattern matching design problem including uniform and non uniform linear arrays

codes based on a trellis cut set transformation ii codes for noncoherent detection for pt i see ibid vol 46 p 610 20 1998 we continue the study of cut set trellis section t sub cb codes begun in part i and show that such codes are well suited for use with multiple symbol noncoherent detection and conventional two symbol differential detection the information bit assignment for a cut set trellis section can be modified to make the resulting code noncatastrophic for noncoherent detection by this procedure many existing two dimensional 2 d and multidimensional mpsk and qam trellis codes designed for coherent detection can be easily converted into codes for both coherent and noncoherent detection compared to m dpsk trellis codes it is shown that 2 d mpsk cut set trellis section codes have improved performance for multiple symbol noncoherent detection and comparable performance for conventional two symbol differential detection thus cut set trellis section codes are a class of universal channel codes since a single encoder performs well for coherent detection multiple symbol noncoherent detection and conventional two symbol differential detection

bonnesen style inequalities for minkowski relative geometry on obtient deux inegalites du style bonnesen pour le rayon interieur relatif d un corps convexe par rapport a un autre espace a n dimensions

design of phase codes for radar performance optimization with a similarity constraint this paper deals with the design of coded waveforms which optimize radar performances in the presence of colored gaussian disturbance we focus on the class of phase coded pulse trains and determine the radar code which approximately maximizes the detection performance under a similarity constraint with a prefixed radar code this is tantamount to forcing a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the prefixed sequence we consider the cases of both continuous and finite phase alphabet and formulate the code design in terms of a nonconvex np hard quadratic optimization problem in order to approximate the optimal solutions we propose techniques with polynomial computational complexity based on the method of semidefinite program sdp relaxation and randomization moreover we also derive approximation bounds yielding a ldquomeasure of goodnessrdquo of the devised algorithms at the analysis stage we assess the performance of the new encoding techniques both in terms of detection performance and ambiguity function under different choices for the similarity parameter we also show that the new algorithms achieve an accurate approximation of the optimal solution with a modest number of randomizations

mimo radar waveform design with constant modulus and similarity constraints we consider the problem of waveform design for multiple input multiple output mimo radar in the presence of signal dependent interference embedded in white gaussian disturbance we present two sequential optimization procedures to maximize the signal to interference plus noise ratio sinr accounting for a constant modulus constraint as well as a similarity constraint involving a known radar waveform with some desired properties e g in terms of pulse compression and ambiguity the presented sequential optimization algorithms based on a relaxation method yield solutions with good accuracy their computational complexity is linear in the number of iterations and trials in the randomized procedure and polynomial in the receive filter length finally we evaluate the proposed techniques by considering their sinr performance beam pattern as well as pulse compression property via numerical simulations

successive qcqp refinement for mimo radar waveform design under practical constraints the authors address the problem of designing a waveform for multiple input multiple output mimo radar under the important practical constraints of constant modulus and waveform similarity incorporating these constraints in an analytically tractable manner is a longstanding open challenge this is due to the fact that the optimization problem that results from signal to interference plus noise ratio sinr maximization subject to these constraints is a hard non convex problem the authors develop a new analytical approach that involves solving a sequence of convex quadratically constrained quadratic programing qcqp problems which they prove converges to a sub optimal solution because an improvement in sinr results via solving each problem in the sequence they call the method successive qcqp refinement sqr furthermore the proposed sqr method can be easily extended to incorporate emerging requirements of spectral coexistence as shown briefly in this paper the authors evaluate sqr against other candidate techniques with respect to sinr performance beam pattern and pulse compression properties in a variety of scenarios results show that sqr outperforms state of the art methods that also employ constant modulus and or similarity constraints while being computationally less burdensome

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

intersections of dilatates of convex bodies we initiate a systematic investigation into the nature of the function k l that gives the volume of the intersection of one convex body k in r n and a dilatate l of another convex body l in r n as well as the function k l that gives the n 1 dimensional hausdor measure of the intersection of k and the boundary l of l the focus is on the concavity properties of k l of particular interest is the case when k and l are symmetric with respect to the origin in this situation there is an interesting change in the concavity properties of k l between dimension 2 and dimensions 3 or higher when l is the unit ball an important special case with connections to e lutwak s dual brunn minkowski theory we prove that this change occurs between dimension 2 and dimensions 4 or higher and conjecture that it occurs between dimension 3 and dimension 4 we also establish an isoperimetric inequality with equality condition for subsets of equatorial zones in the sphere s 2 and apply this and the brunn minkowski inequality in the sphere to obtain results related to this conjecture as well as to the properties of a new type of symmetral of a convex body which we call the equatorial symmetral

a positive answer to the busemann petty problem in three dimensions we prove that in e3 the busemann petty problem concerning cen tral sections of centrally symmetric convex bodies has a positive answer together with other results this settles the problem in each dimension

tractable transmit mimo beampattern design under a constant modulus constraint multiple input multiple output mimo radar systems allow each antenna element to transmit a different waveform this waveform diversity can be exploited to enhance the beampattern design in particular effective management of radar radiation power in directions of interest we address the problem of designing a beampattern for mimo radar which in turn is determined by the transmit waveform while unconstrained design is straightforward a key open challenge is enforcing the constant modulus constraint on the radar waveform it is well known that the problem of minimizing deviation of the designed beampattern from an idealized one subject to the constant modulus constraint constitutes a hard nonconvex problem existing methods that address constant modulus invariably lead to a stiff tradeoff between analytical tractability achieved by relaxations and approximations and realistic design that exactly achieves constant modulus but is computationally burdensome a new approach is proposed in our paper which involves solving a sequence of convex equality constrained quadratic programs each of which has a closed form solution and such that constant modulus is achieved at convergence we further prove that the converged solution satisfies the karush kuhn tucker optimality conditions of the aforementioned hard nonconvex problem we evaluate the proposed successive closed forms scf algorithm against the state of the art mimo beampattern design techniques in both narrowband and wideband setups and show that the scf breaks the tradeoff between desirable performance and the associated computation cost

mimo radar waveform covariance matrix for high sinr and low side lobe levels multiple input multiple output mimo radar has better parametric identifiability but compared to phased array radar it shows loss in signal to noise ratio due to noncoherent processing to exploit the benefits of both mimo radar and phased array a waveform covariance matrix is proposed to generate the proposed covariance matrix the values of the cosine function between 0 and with a step size of nt are used to form a positive semi definite toeplitz matrix where nt is the number of transmit antennas the proposed covariance matrix yields gain in the signal to interference plus noise ratio sinr compared to mimo radar and have lower sidelobe levels slls compared to phased array mimo radar and the recently proposed phased mimo scheme moreover in contrast to the phased mimo scheme where each antenna transmits a different power our proposed scheme allows same power transmission from each antenna simulation results validate our analytical results

on mimo detection under non gaussian target scattering in this paper we consider a multiple input multiple output mimo detection problem with m widely spaced transmit antennas and l widely spaced receive antennas and we study the problem of designing the signal waveforms transmitted by each source node under non gaussian target scattering and temporally correlated gaussian clutter two figures of merit are investigated for space time code stc optimization under a semidefinite rank constraint 1 the lower chernoff bound lcb to the detection probability for fixed probability of false alarm and 2 the mutual information mi between the observations available at the receive nodes and the channel response generated by a point like target assumed present tout court both receive and transmit power constraints are discussed if the scattering distribution possesses some suitably defined properties of unitary invariance see both mi optimal and lcb optimal stcs have a simple canonical structure the same set of clutter dependant temporal codewords are employed at the transmit nodes the only difference among the many solutions being the amount of power radiated by each antenna such a spatial power allocation critically depends upon the adopted figure of merit the specified power constraint and the underlying scattering model sufficient conditions to determine the optimal power allocation for all design criteria are provided asymptotic power distributions are also derived in the limit of vanishingly small and increasingly large signal to clutter ratios proving that assuming gaussian scattering at the design stage is a robust choice a case study of relevant practical interest is examined in depth so as to compare the proposed design criteria and to assess the impact of signal non gaussianity on the system performances

fourier transform null variety and laplacian s eigenvalues we consider a quantity the distance to the origin from the null variety of the fourier transform of the characteristic function of we conjecture rstly that is maximized among all convex balanced domains r d of a xed volume by a ball and also that is bounded above by the square root of the second dirichlet eigenvalue of we prove some weaker versions of these conjectures in dimension two as well as their validity for domains asymptotically close to a disk and also discuss further links between and the eigenvalues of the laplacians

finite alphabet constant envelope waveform design for mimo radar the design of waveforms with specified auto and cross correlation properties has a number of applications in multiple input multiple output mimo radar beampattern design in this work two algorithms are proposed to generate finite alphabet constant envelope ce waveforms with required cross correlation properties the first algorithm proposes a closed form solution to find the finite alphabet ce waveforms to realize the given covariance matrix here gaussian random variables rv s are mapped onto binary phase shift keying bpsk and quadrature phase shift keying qpsk symbols using nonlinear functions and the cross correlation relationship between the gaussian rv s and bpsk qpsk rv s is established this cross correlation relationship is exploited to convert the problem of finding the bpsk qpsk waveforms to realize the covariance matrix corresponding to the given beampattern into finding the gaussian rv s to realize another covariance matrix that can be easily found in the second algorithm by exploiting the results of first algorithm a generalized algorithm to generate bpsk waveforms to approximate the given beampattern is proposed simulation results show that proposed finite alphabet ce waveforms outperform the existing algorithms to approximate the desired beampattern

rotationally invariant nonlinear trellis codes for two dimensional modulation a general parity check equation is presented that defines rotationally invariant trellis codes of rate k k 1 for two dimensional signal sets this parity check equation is used to find rate k k 1 codes for 4psk 8psk 16psk and qam signal sets by systematic code searches the mpsk codes exhibit smaller free euclidean distances than nonrotationally invariant linear codes with the same number of states however since the nonlinear codes have a smaller number of nearest neighbors their performance at moderate signal to noise ratios is close to that of the best linear codes the rotationally invariant qam codes with 8 32 64 and 256 states achieve the same free euclidean distance as the best linear codes transparency of user information under phase rotations is accomplished either by conventional differential encoding and decoding or by integrating this function directly into the code trellis

constant modulus waveform design for mimo radar transmit beampattern a multiple input multiple output radar has great flexibility to design the transmit beampattern via selecting the probing waveform the idea of current transmit beampattern design is to approximate the disired transimit beampattern and minimize the cross correlation sidelobes in this paper under the constant modulus constraint two algorithms are proposed to design the probing waveform directly in the first algorithm the optimization criterion is minimizing the squared error between the designed beampattern and the given beampattern since the objective function is a nonconvex fourth order polynomial and the constant modulus constraint can be regarded as many nonconvex quadratic equality constraints an efficient alternating direction method of multipliers admm algorithm whose convergence speed is very fast is proposed to solve it in the second algorithm the criterion is minimizing the absolute error between the designed beampattern and the given beampattern this nonconvex problem can be formulated as l 1 norm problem which can be solved through a double admm algorithm finally we assess the performance of the two proposed algorithms via numerical results

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

constant modulus mimo radar waveform design with minimum peak sidelobe transmit beampattern this paper addresses the problem of minimum peak sidelobe transmit beampattern design for colocated multiple input multiple output radar systems two new methods are proposed to achieve it via designing constant modulus cm waveforms directly one is to maximize the ratio of the minimum mainlobe level to the peak sidelobe level psl without specifying pattern masks and another is to minimize psl under the mainlobe ripple and unspecified waveform modulus constraints the resultant optimization problems are difficult to solve due to the coupled numerator and denominator of the quadratic fractional programming problem formulation in the former and double sided quadratic constraints and unknown waveform modulus in the latter for the former we decouple the numerator and denominator by equivalent transformation and introduction of boundary type auxiliary variables in order that feasible methods are derived to tackle them independently and efficiently for the latter we simplify the coupled double sided quadratic constraints on the same waveform vector into those on different single auxiliary variables to tackle the corresponding nonconvex optimization problem numerical examples show that the proposed algorithms can attain sufficient low peak sidelobe beampattern levels under the cm constraints

ambiguity function shaping for cognitive radar via complex quartic optimization in this paper we propose a cognitive approach to design phase only modulated waveforms sharing a desired range doppler response the idea is to minimize the average value of the ambiguity function of the transmitted signal over some range doppler bins which are identified exploiting a plurality of knowledge sources from a technical point of view this is tantamount to optimizing a real and homogeneous complex quartic order polynomial with a constant modulus constraint on each optimization variable after proving some interesting properties of the considered problem we devise a polynomial time waveform optimization procedure based on the maximum block improvement mbi method and the theory of conjugate partial symmetric conjugate super symmetric fourth order tensors at the analysis stage we assess the performance of the proposed technique showing its capability to properly shape the range doppler response of the transmitted waveform

trellis coded multidimensional phase modulation a 2l dimensional multiple phase shift keyed l mpsk signal set is obtained by forming the cartesian product of l two dimensional mpsk signal sets a systematic approach to partitioning l mpsk signal sets that is based on block coding is used an encoder system approach is developed it incorporates the design of a differential precoder a systematic convolutional encoder and a signal set mapper trellis coded l 4psk l 8psk and l 16psk modulation schemes are found for 1

generation of correlated finite alphabet waveforms using gaussian random variables various examples of methods and systems are provided for generation of correlated finite alphabet waveforms using gaussian random variables in e g radar and communication applications in one example a method includes mapping an input signal comprising gaussian random variables rvs onto finite alphabet non constant envelope fance symbols using a predetermined mapping function and transmitting fance waveforms through a uniform linear array of antenna elements to obtain a corresponding beampattern the fance waveforms can be based upon the mapping of the gaussian rvs onto the fance symbols in another example a system includes a memory unit that can store a plurality of digital bit streams corresponding to fance symbols and a front end unit that can transmit fance waveforms through a uniform linear array of antenna elements to obtain a corresponding beam pattern the system can include a processing unit that can encode the input signal and or determine the mapping function

linear transmit processing in mimo communications systems we examine and compare the different types of linear transmit processing for multiple input multiple output systems where we assume that the receive filter is independent of the transmit filter contrary to the joint optimization of transmit and receive filters we can identify three filter types similar to receive processing the transmit matched filter the transmit zero forcing filter and the transmit wiener filter we show that the transmit filters are based on similar optimizations as the respective receive filters with an additional constraint for the transmit power moreover the transmit wiener filter has similar convergence properties as the receive wiener filter i e it converges to the matched filter and the zero forcing filter for low and high signal to noise ratio respectively we give closed form solutions for all transmit filters and present the fundamental result that their mean square errors are equal to the errors of the respective receive filters if the information symbols and the additive noise are uncorrelated however our simulations reveal that the bit error ratio results of the transmit filters differ from the results for the respective receive filters