constructive techniques for labeling constant weight gray codes with applications to minimal generating sets of semigroups abstract the partition type conjecture a generalization of the middle levels conjecture of combinatorics states that for all positive integers n and r with n r 1 and every non exceptional partition type of weight r of x n there exists a constant weight gray code which admits an orthogonal labeling by partitions of type we prove results in combinatorics and finite semigroup theory providing the completion of the proof that the partition type conjecture is true for all types having more than one class of size greater than one and leaving open only those cases which are equivalent to the middle levels conjecture the rank of a finite semigroup s is the cardinality of a minimum generating set for s if s is idempotent generated the idempotent rank of s is the cardinality of a minimum idempotent generating set for s a semigroup of transformations of x n 1 n is said to be s n normal if s is closed under conjugation by the permutations of x n the results here concerning the partition type conjecture are used to determine a simple formula for the rank and idempotent rank of every s n normal semigroup

designing structured tight frames via an alternating projection method tight frames also known as general welch bound equality sequences generalize orthonormal systems numerous applications including communications coding and sparse approximation require finite dimensional tight frames that possess additional structural properties this paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems ieps which includes the frame design problem to apply this method one needs only to solve a matrix nearness problem that arises naturally from the design specifications therefore it is the fast and easy to develop versions of the algorithm that target new design problems alternating projection will often succeed even if algebraic constructions are unavailable to demonstrate that alternating projection is an effective tool for frame design the paper studies some important structural properties in detail first it addresses the most basic design problem constructing tight frames with prescribed vector norms then it discusses equiangular tight frames which are natural dictionaries for sparse approximation finally it examines tight frames whose individual vectors have low peak to average power ratio par which is a valuable property for code division multiple access cdma applications numerical experiments show that the proposed algorithm succeeds in each of these three cases the appendices investigate the convergence properties of the algorithm

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

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

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

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

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

mimo radar transmit beampattern design without synthesising the covariance matrix compared to phased array multiple input multiple output mimo radars provide more degrees of freedom dof that can be exploited for improved spatial resolution better parametric identifiability lower sidelobe levels at the transmitter receiver and design variety of transmit beampatterns the design of the transmit beampattern generally requires the waveforms to have arbitrary auto and cross correlation properties the generation of such waveforms is a two step complicated process in the first step a waveform covariance matrix is synthesized which is a constrained optimization problem in the second step to realize this covariance matrix actual waveforms are designed which is also a constrained optimization problem our proposed scheme converts this two step constrained optimization problem into a one step unconstrained optimization problem in the proposed scheme in contrast to synthesizing the covariance matrix for the desired beampattern nt independent finite alphabet constant envelope waveforms are generated and preprocessed with weight matrix w before transmitting from the antennas in this work two weight matrices are proposed that can be easily optimized for the desired symmetric and nonsymmetric beampatterns and guarantee equal average power transmission from each antenna simulation results validate our claims

optimal sequences and sum capacity of synchronous cdma systems the sum capacity of a multiuser synchronous cdma system is completely characterized in the general case of asymmetric user power constraints this solves the open problem posed by rupf and massey see ibid vol 40 p 1261 6 1994 which had solved the equal power constraint case we identify the signature sequences with real components that achieve sum capacity and indicate a simple recursive algorithm to construct them

idempotent rank in finite full transformation semigroups the subsemigroup sing n of singular elements of the full transformation semigroup on a finite set is generated by n n l 2 idempotents of defect one in this paper we extend this result to the subsemigroup k n r consisting of all elements of rank r or less we prove that the idempotent rank defined as the cardinality of a minimal generating set of idempotents of k n r is s n r the stirling number of the second kind

a quotient universal digital topology we study a special topology on z 2 and show that both the marcus topology and the khalimsky topology and also one more digital topology on z 2 may be obtained as three of its quotient topologies a quotient closure operator of the topology studied is discussed too and an analogue of the jordan curve theorem for this closure operator is proved

graphs with a path partition for structuring digital spaces we introduce the concept of graphs with a path partition and define a special type of connectedness in these graphs the connectedness is shown to have certain properties suitable for using graphs with a path partition as convenient background structures on digital spaces for the study of digital images we introduce a family of such graphs on z 2 and present a jordan curve theorem for them

border and surfacetracing theoretical foundations in this paper we define and study digital manifolds of arbitrary dimension and provide in particular a general theoretical basis for curve or surface tracing in picture analysis the studies involve properties such as the one dimensionality of digital curves and n 1 dimensionality of digital hypersurfaces that makes them discrete analogs of corresponding notions in continuous topology the presented approach is fully based on the concept of adjacency relation and complements the concept of dimension as common in combinatorial topology this work appears to be the first one on digital manifolds based on a graph theoretical definition of dimension in particular in the n dimensional digital space a digital curve is a one dimensional object and a digital hypersurface is an n 1 dimensional object as it is in the case of curves and hypersurfaces in the euclidean space relying on the obtained properties of digital hypersurfaces we propose a uniform approach for studying good pairs defined by separations and obtain a classification of good pairs in arbitrary dimension we also discuss possible applications of the presented definitions and results

dual coordinate ascent methods for non strictly convex minimization we consider a dual method for solving non strictly convex programs possessing a certain separable structure this method may be viewed as a dual version of a block coordinate ascent method studied by auslender 1 section 6 we show that the decomposition methods of han 6 7 and the method of multipliers may be viewed as special cases of this method we also prove a convergence result for this method which can be applied to sharpen the available convergence results for han s methods

mimo radar beampattern design via psl isl optimization we deal with the robust design of multiple input multiple output mimo waveform covariance matrices that optimize the worst case over steering mismatches transmit beampattern assuming either the peak sidelobe level psl or the integrated sidelobe level isl as figure of merit to this end we model the uncertainty set associated with each steering vector through two double sided potentially non convex quadratic constraints in addition we force two suitable constraints on the optimization variable the former accounts for the width of the mainbeam whereas the latter is either a uniform or a relaxed elemental power requirement allowing to control the amount of transmitted power we prove that both the mentioned waveform covariance designs lead to non convex optimization problems which remarkably share some hidden convexity properties hence we devise polynomial time procedures aimed at synthesizing the desired optimal mimo waveform covariance matrices finally at the analysis stage we assess the performance of the proposed techniques showing their capability to ensuring improved worst case performance than some counterparts available in open literature

numerical methods for inverse singular value problems3 two numerical methods one continuous and the other discrete are proposed for solving inverse singular value problems the first method consists of solving an ordinary differential equation obtained from an explicit calculation of the projected gradient of a certain objective function the second method generalizes an iterative process proposed originally by friedland nocedal and overton siam j numer anal 24 1987 pp 634 667 for solving inverse eigenvalue problems with the geometry understood from the first method it is shown that the second method also the method proposed by friedland nocedal and overton for inverse eigenvalue problems is a variation of the newton method while the continuous method is expected to converge globally at a slower rate in finding a stationary point of the objective function the discrete method is proved to converge locally at a quadratic rate if there is a solution some numerical examples are presented

a successive projection method it is of both theoretical and practical interest to find the projection of a point to the intersection of a finite number of closed convex sets by a sequence of projections to the individual sets successively in this paper we study such a method and analyze its convergence properties a main feature of the method is its capability to decompose the projection problem into several small ones for some structured sparse problems these small subproblems can be solved independently and the presented method has a potential use in parallel computation

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

counting techniques to label constant weight gray codes with links to minimal generating sets of semigroups let be a partition of the positive integer n a partition of the set xn 1 2 n is said to be of type if the sizes of its classes form the partition of n given 1 r n an r element subset a of xn and a partition of xn are said to be orthogonal if every class of meets a in exactly one element let gn r be the graph whose vertices are the r element subsets of xn with two sets being adjacent if they intersect in r 1 elements the graph gn r is hamiltonian hamiltonian cycles of gn r are early examples of error correcting codes where they came to be known as constant weight gray codes a hamiltonian cycle a1 a2 anr in gn r is said to be orthogonally labeled if there exists a list of distinct partitions 1 2 nr of type such that i is orthogonal to both ai and ai 1 where i 1 2 nr taken modulo nr for all but a finite class of partition types we present counting arguments to prove that any hamiltonian cycle in gn r can be orthogonally labeled the remaining cases with the exception of cases which are equivalent to the celebrated middle levels conjecture are treated via constructive arguments in a sequel a semigroup of transformations of xn is sn normal if it is closed under conjugation by the permutations of xn the combinatorics results here and in the sequel lead to a determination of the rank and idempotent rank of all sn normal semigroups thereby broadly generalizing the known result that the rank and idempotent rank of k n r is s n r a stirling number of the second kind

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

index system and separability of constant weight gray codes a number system is developed for the conversion of natural numbers to the codewords of the gray code g n k of length n and weight k and vice versa the focus is on the subcode g n k of g n consisting of those words of g n with precisely k 1 bits 0

a survey of combinatorial gray codes the term combinatorial gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified small way this notion generalizes the classical binary reflected gray code scheme for listing n bit binary numbers so that successive numbers differ in exactly one bit position as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families including permutations and combinations the area of combinatorial gray codes was popularized by herbert wilf in his invited address at the siam conference on discrete mathematics in 1988 and his subsequent siam monograph combinatorial algorithms an update 1989 in which he posed some open problems and variations on the theme this resulted in much recent activity in the area and most of the problems posed by wilf are now solved in this paper we survey the area of combinatorial gray codes describe recent results variations and trends and highlight some open problems

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

structured low rank approximation this paper concerns the construction of a structured low rank matrix that is nearest to a given matrix the notion of structured low rank approximation arises in various applications ranging from signal enhancement to protein folding to computer algebra where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system the task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix this paper addresses some of the theoretical and numerical issues involved in the problem two procedures for constructing the nearest structured low rank matrix are proposed the procedures are flexible enough that they can be applied to any lower rank any linear structure and any matrix norm in the measurement of nearness the techniques can also be easily implemented by utilizing available optimization packages the special case of symmetric toeplitz structure using the frobenius matrix norm is used to exemplify the ideas throughout the discussion the concept rather than the implementation details is the main emphasis of the paper

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

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

accelerating the convergence of the method of alternating projections the powerful von neumann halperin method of alternating projections map is an algorithm for determining the best approximation to any given point in a hilbert space from the intersection of a finite number of subspaces it achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the it individual subspaces which make up the intersection the main practical drawback of this algorithm at least for some applications is that the method is slowly convergent in this paper we consider a general class of iterative methods which includes the map as a special case for such methods we study an accelerated version of this algorithm which was considered earlier by gubin polyak and raik 14 and by gearhart and koshy 13 we show that the accelerated algorithm converges faster than the map in the case of two subspaces but is in general it not faster than the map for more than two subspaces however for a symmetric version of the map the accelerated algorithm always converges faster for any number of subspaces our proof seems to require the use of the spectral theorem for self adjoint mappings

synthesis of linear and planar arrays with minimum element selection a new method for the synthesis of linear and planar arrays having prescribed beamwidth and sidelobe levels and a minimum number of elements is proposed in the method the number of elements in an array is minimized while constraining the amplitude response error in the mainlobe region the attenuation in the sidelobe region and the array dimensions an iterative constrained optimization method is used where the amplituderesponse error is linearly approximated at each iteration while concurrently minimizing a re weighted l1 norm of the array coefficients to ensure robustness of the array we constrain a sensitivity parameter namely the white noise gain to be above a prescribed level furthermore the method also provides the additional flexibility of controlling the array dimensions symmetry properties and element positions of the array two variants have been developed in the first variant both the array coefficients and the positions of the elements are optimized in the second variant only the array coefficients are optimized while the elements are fixed at predefined positions experimental comparisons with several state of the art competing methods show that the proposed method provides greater flexibility of controlling the robustness beampattern response error array dimensions and element positions while at the same time the number of elements is less than or equal to that of the competing methods

low complexity moving target parameter estimation for mimo radar using 2d fft in multiple input multiple output radar to localize a target and estimate its reflection coefficient a given cost function is usually optimized over a grid of points the performance of such algorithms is directly affected by the grid resolution increasing the number of grid points enhances the resolution of the estimator but also increases its computational complexity exponentially in this paper two reduced complexity algorithms are derived based on capon and amplitude and phase estimation apes to estimate the reflection coefficient angular location and doppler shift of multiple moving targets by exploiting the structure of the terms the cost function is brought into a form that allows us to apply the two dimensional fast fourier transform 2d fft and reduce the computational complexity of estimation using low resolution 2d fft the proposed algorithm identifies suboptimal estimates and feeds them as initial points to the derived newton gradient algorithm in contrast to the grid based search algorithms the proposed algorithm can optimally estimate on and off the grid targets in very low computational complexity a new apes cost function with better estimation performance is also discussed generalized expressions of the cramer rao lower bound are derived to assess the performance of the proposed algorithm