symmetries and time evolution in discrete phase spaces a soluble model calculation abstract using the flexibility and constructive definition of the schwinger bases we developed different mapping procedures to enhance different aspects of the dynamics and of the symmetries of an extended version of the two level lipkin model the classical limits of the dynamics are discussed in connection with the different mappings discrete wigner functions are also calculated

cramer rao bound for doa estimators under the partial relaxation framework derivation and comparison a class of computationally efficient doa estimators under the partial relaxation pr framework has recently been proposed conceptually different from conventional doa estimation methods in the literature the estimators under the pr framework rely on the non complete relaxation of the array manifold while performing a spectral search in the field of view this particular type of relaxation essentially implies a modified signal model with partial information loss due to the relaxation the information loss and its impact on the doa estimation performance have not yet been analytically quantified in the literature in this paper the information loss induced by the relaxation of the array manifold is investigated through the cramer rao bound crb the closed form expression of the crb for doa estimation under the pr model on the one hand provides insight on the information loss in the asymptotic region where the number of snapshots tends to infinity on the other hand the proposed crb characterizes the lower bound for the doa estimation performance of all pr estimators we prove that under the assumptions of gaussian source signal and noise the crb of the pr signal model is lower bounded by the conventional stochastic crb we also prove that the previously proposed weighted subspace fitting estimator under the pr framework asymptotically achieves the crb of the pr signal model furthermore it is shown that the asymptotic mean squared errors of all weighted subspace fitting estimators under the pr framework for any positive definite weighting matrix are identical

gaussian cramer rao bound for direction estimation of noncircular signals in unknown noise fields this paper focuses on the stochastic cramer rao bound crb on direction of arrival doa estimation accuracy for noncircular gaussian sources in the general case of an arbitrary unknown gaussian noise field parameterized by a vector of unknowns explicit closed form expressions of the stochastic crb for doa parameters alone are obtained directly from the slepian bangs formula for general noncircular complex gaussian distributions as a special case the crb under the nonuniform white noise assumption is derived our expressions can be viewed as extensions of the well known results by stoica and nehorai ottersten et al weiss and friedlander pesavento and gershman and gershman et al some properties of these crbs are proved and finally these bounds are numerically compared with the conventional crbs under the circular complex gaussian distribution for different unknown noise field models

erasure list decodable codes from random and algebraic geometry codes erasure list decoding was introduced to correct a larger number of erasures by outputting a list of possible candidates in this paper we consider both random linear codes and algebraic geometry codes for list decoding from erasures the contributions of this paper are twofold first for arbitrary 0 and epsilon 0 r and epsilon are independent we show that with high probability a q ary random linear code of rate r is an erasure list decodable code with constant list size q o 1 epsilon that can correct a fraction 1 r epsilon of erasures i e a random linear code achieves the information theoretic optimal tradeoff between information rate and fraction of erasures second we show that algebraic geometry codes are good erasure list decodable codes precisely speaking a q ary algebraic geometry code of rate r from the garcia stichtenoth tower can correct 1 r 1 sqrt q 1 1 q epsilon fraction of erasures with list size o 1 epsilon this improves the johnson bound for erasures applied to algebraic geometry codes furthermore list decoding of these algebraic geometry codes can be implemented in polynomial time note that the code alphabet size q in this paper is constant and independent of epsilon

jacobi ϑ functions and discrete fourier transforms properties of the jacobi 3 function and its derivatives under discrete fourier transforms are investigated and several interesting results are obtained the role of modulo n equivalence classes in the theory of functions is stressed an important conjecture is studied

analytical performance assessment of multi dimensional matrix and tensor based esprit type algorithms in this paper we present a generic framework for the asymptotic performance analysis of subspace based parameter estimation schemes it is based on earlier results on an explicit first order expansion of the estimation error in the signal subspace obtained via an svd of the noisy observation matrix we extend these results in a number of aspects firstly we demonstrate that an explicit first order expansion of the higher order svd hosvd based subspace estimate can be derived secondly we show how to obtain explicit first order expansions of the estimation error of arbitrary esprit type algorithms and provide the expressions for r d standard esprit r d unitary esprit r d standard tensor esprit as well as r d unitary tensor esprit thirdly we derive closed form expressions for the mean square error mse and show that they only depend on the second order moments of the noise hence to apply this framework we only need the noise to be zero mean and possess finite second order moments additional assumptions such as gaussianity or circular symmetry are not needed

list decoding from erasures bounds and code constructions we consider the problem of list decoding from erasures we establish lower and upper bounds on the rate of a binary linear code that can be list decoded with list size l when up to a fraction p of its symbols are adversarially erased such bounds already exist in the literature albeit under the label of generalized hamming weights and we make their connection to list decoding from erasures explicit our bounds show that in the limit of large l the rate of such a code approaches the capacity 1 p of the erasure channel such nicely list decodable codes are then used as inner codes in a suitable concatenation scheme to give a uniformly constructive family of asymptotically good binary linear codes of rate spl omega spl epsiv sup 2 log 1 spl epsiv that can be efficiently list decoded using lists of size o 1 spl epsiv when an adversarially chosen 1 spl epsiv fraction of symbols are erased for arbitrary spl epsiv 0 this improves previous results in this vein which achieved a rate of spl omega spl epsiv sup 3 log 1 spl epsiv

extended cahill glauber formalism for finite dimensional spaces i fundamentals the cahill glauber approach for quantum mechanics on phase space is extended to the finite dimensional case through the use of discrete coherent states all properties and features of the continuous formalism are appropriately generalized the continuum results are promptly recovered as a limiting case the jacobi theta functions are shown to have a prominent role in the context

discrete quantum phase spaces and the mod n invariance instituto de fisica teorica universidade estadual paulista rua pamplona 145 cep 01405 sao paulo sp

deterministic cramer rao bound for strictly non circular sources and analytical analysis of the achievable gains recently several high resolution parameter estimation algorithms have been developed to exploit the structure of strictly second order so non circular nc signals they achieve a higher estimation accuracy and can resolve up to twice as many signal sources compared to the traditional methods for arbitrary signals as a benchmark for these nc methods we derive the closed form deterministic r d nc cramer rao bound nc crb for the multi dimensional parameter estimation of strictly non circular rectilinear signal sources in this paper assuming a separable centro symmetric r d array we show that in some special cases the deterministic r d nc crb reduces to the existing deterministic r d crb for arbitrary signals this suggests that no gain from strictly non circular sources nc gain can be achieved under the deterministic data assumption in these cases for more general scenarios finding an analytical expression of the nc gain for an arbitrary number of sources is very challenging thus in this paper we simplify the derived nc crb and the existing crb for the special case of two closely spaced strictly non circular sources captured by a uniform linear array ula subsequently we use these simplified crb expressions to analytically compute the maximum achievable asymptotic nc gain for the considered two source case the resulting expression only depends on the various physical parameters and we find the conditions that provide the largest nc gain our analysis is supported by extensive simulation results

graph states in phase space the phase space for a system of n qubits is a discrete grid of 2n 2n points whose axes are labeled in terms of the elements of the finite field to endow it with proper geometrical properties we analyze the representation of graph states in that phase space showing that these states can be identified with a class of nonsingular curves we provide an algebraic representation of the most relevant quantum operations acting on these states and discuss the advantages of this approach

results on parity check matrices with optimal stopping and or dead end set enumerators the performance of iterative decoding techniques for linear block codes correcting erasures depends very much on the sizes of the stopping sets associated with the underlying tanner graph or equivalently the parity check matrix representing the code in this correspondence we introduce the notion of dead end sets to explicitly demonstrate this dependency the choice of the parity check matrix entails a tradeoff between performance and complexity we give bounds on the complexity of iterative decoders achieving optimal performance in terms of the sizes of the underlying parity check matrices further we fully characterize codes for which the optimal stopping set enumerator equals the weight enumerator

the decoding error probability of linear codes over the erasure channel in this paper we study the decoding error probability of linear codes over the erasure channel under the list decoding the notion of the q ell incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding by calculating the q ell incorrigible set distributions the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions for the ensemble of all n k linear codes the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined furthermore the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all n nr linear codes

unitary esprit how to obtain increased estimation accuracy with a reduced computational burden esprit is a high resolution signal parameter estimation technique based on the translational invariance structure of a sensor array previous esprit algorithms do not use the fact that the operator representing the phase delays between the two subarrays is unitary the authors present a simple and efficient method to constrain the estimated phase factors to the unit circle if centro symmetric array configurations are used unitary esprit the resulting closed form algorithm has an esprit like structure except for the fact that it is formulated in terms of real valued computations throughout since the dimension of the matrices is not increased this completely real valued algorithm achieves a substantial reduction of the computational complexity furthermore unitary esprit incorporates forward backward averaging leading to an improved performance compared to the standard esprit algorithm especially for correlated source signals like standard esprit unitary esprit offers an inexpensive possibility to reconstruct the impinging wavefronts signal copy these signal estimates are more accurate since unitary esprit improves the underlying signal subspace estimates simulations confirm that even for uncorrelated signals the standard esprit algorithm needs twice the number of snapshots to achieve a precision comparable to that of unitary esprit thus unitary esprit provides increased estimation accuracy with a reduced computational burden

dynamics in discrete phase spaces and time interval operators abstract the von neumann liouville time evolution equation is represented in a discrete quantum phase space the mapped liouville operator and the corresponding wigner function are explicitly written for the problem of a magnetic moment interacting with a magnetic field and the precessing solution is found the propagator is also discussed and a time interval operator associated to a unitary operator which shifts the energy levels in the zeeman spectrum is introduced this operator is associated to the particular dynamical process and is not the continuous parameter describing the time evolution the pair of unitary operators which shifts the time and energy is shown to obey the weyl schwinger algebra

stopping set distributions of some reed muller codes stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel bec let c be a binary n k linear code with parity check matrix h where the rows of h may be dependent a stopping set s of c with parity check matrix h is a subset of column indices of h such that the restriction of h to s does not contain a row of weight one the stopping set distribution ti h i 0n enumerates the number of stopping sets with size i of c with parity check matrix h note that stopping sets and stopping set distribution are related to the parity check matrix h of c let h be the parity check matrix of c which is formed by all the nonzero codewords of its dual code c a parity check matrix h is called bec optimal if ti h ti h i 0 1 n and h has the smallest number of rows in this paper we study stopping sets stopping set distributions and bec optimal parity check matrices of binary linear codes using finite geometry in combinatorics we obtain bec optimal parity check matrices and then determine the stopping set distributions for the simplex codes the hamming codes the first order reed muller codes and the extended hamming codes which are some reed muller codes or their shortening or puncturing versions

music like estimation of direction of arrival for noncircular sources this paper examines the asymptotic performance of music like algorithms for estimating directions of arrival doa of narrowband complex noncircular sources using closed form expressions of the covariance of the asymptotic distribution of different projection matrices it provides a unifying framework for investigating the asymptotic performance of arbitrary subspace based algorithms valid for gaussian or non gaussian and complex circular or noncircular sources we also derive different robustness properties from the asymptotic covariance of the estimated doa given by such algorithms these results are successively applied to four algorithms to two attractive music like algorithms previously introduced in the literature to an extension of these algorithms and to an optimally weighted music algorithm proposed in this paper numerical examples illustrate the performance of the studied algorithms compared to the asymptotically minimum variance amv algorithms introduced as benchmarks

generalized column distances the notion of generalized hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography in this paper we extend this concept to the context of convolutional codes in particular we focus on column distances and introduce the novel notion of generalized column distances gcd we first show that the hierarchy of gcd is strictly increasing we then provide characterizations of such distances in terms of the truncated parity check matrix of the code that will allow us to determine their values finally the case in which the parity check matrix is in systematic form is treated

stochastic crame spl acute r rao bound for noncircular signals with application to doa estimation after providing an extension of the slepian bangs formula for general noncircular complex gaussian distributions this paper focuses on the stochastic crame spl acute r rao bound crb on direction of arrival doa estimation accuracy for noncircular sources we derive an explicit expression of the crb for doa parameters alone in the case of noncircular complex gaussian sources by two different methods one of them consists of computing the asymptotic covariance matrix of the maximum likelihood ml estimator and the other is obtained directly from our extended slepian bangs formula some properties of this crb are proved and finally it is numerically compared with the crbs under circular complex gaussian and complex discrete distributions of sources

simultaneous schur decomposition of several nonsymmetric matrices to achieve automatic pairing in multidimensional harmonic retrieval problems this paper presents a new jacobi type method to calculate a simultaneous schur decomposition ssd of several real valued nonsymmetric matrices by minimizing an appropriate cost function thereby the ssd reveals the average eigenstructure of these nonsymmetric matrices this enables an r dimensional extension of unitary esprit to estimate several undamped r dimensional modes or frequencies along with their correct pairing in multidimensional harmonic retrieval problems unitary esprit is an esprit type high resolution frequency estimation technique that is formulated in terms of real valued computations throughout for each of the r dimensions the corresponding frequency estimates are obtained from the real eigenvalues of a real valued matrix the ssd jointly estimates the eigenvalues of all r matrices and thereby achieves automatic pairing of the estimated r dimensional modes via a closed form procedure that neither requires any search nor any other heuristic pairing strategy moreover we describe how r dimensional harmonic retrieval problems with r spl ges 3 occur in array signal processing and model based object recognition applications

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

the higher weight enumerators of the doubly even self dual 48 24 12 code an efficient algorithm is presented for calculating higher weight enumerators of linear codes given generator matrices by this algorithm the higher weight enumerators of the unique doubly even self dual code are calculated the algorithm is based on a previously shown relationship between tutte polynomials and higher weight enumerators

eigenvalues and eigenvectors of the finite fourier transform the eigenvalues and eigenvectors of the n n unitary matrix of finite fourier transform whose j k element is 1 n 1 2 exp 2 i n jk i 1 1 2 is determined in doing so a multitude of identities some of which may be new are encountered a conjecture is advanced

r dimensional esprit type algorithms for strictly second order non circular sources and their performance analysis high resolution parameter estimation algorithms designed to exploit the prior knowledge about incident signals from strictly second order so noncircular nc sources allow for a lower estimation error and can resolve twice as many sources in this paper we derive the r mathchar 702dd nc standard esprit and the r mathchar 702dd nc unitary esprit algorithms that provide a significantly better performance compared to their original versions for arbitrary source signals they are applicable to shift invariant r mathchar 702dd antenna arrays and do not require a centro symmetric array structure moreover we present a first order asymptotic performance analysis of the proposed algorithms which is based on the error in the signal subspace estimate arising from the noise perturbation the derived expressions for the resulting parameter estimation error are explicit in the noise realizations and asymptotic in the effective signal to noise ratio snr i e the results become exact for either high snrs or a large sample size we also provide mean squared error mse expressions where only the assumptions of a zero mean and finite so moments of the noise are required but no assumptions about its statistics are necessary as a main result we analytically prove that the asymptotic performance of both r mathchar 702dd nc esprit type algorithms is identical in the high effective snr regime finally a case study shows that no improvement from strictly non circular sources can be achieved in the special case of a single source

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

the angle angular momentum quantum phase space the angle angular momentum quantum phase space is considered and the corresponding heisenberg weyl group is studied when 2j 1 is a power of a prime z 2j 1 is a galois field and stronger results can be proved sl 2 z 2j 1 transformations are explicitly constructed and various properties of the displacement operators are studied central extensions of the abelian group by z 2j 1 are studied and they provide all the ways of constructing the heisenberg weyl group from g and z 2j 1

performance study of conditional and unconditional direction of arrival estimation a numerical and analytical study of conditional and unconditional direction of arrival doa estimation is presented explicit expressions for the unconditional cramer rao bounds on the doa estimation accuracy and the covariance matrix of the conditional maximum likelihood method are given it is shown that many doa estimation methods have the same asymptotic statistical properties under conditional and unconditional models the situation of two narrowband plane signals impinging on a uniformly spaced linear array is discussed

cramer rao bounds of doa estimates for bpsk and qpsk modulated signals this paper focuses on the stochastic cramer rao bound crb of direction of arrival doa estimates for binary phase shift keying bpsk and quaternary phase shift keying qpsk modulated signals corrupted by additive circular complex gaussian noise explicit expressions of the crb for the doa parameter alone in the case of a single signal waveform are given these crbs are compared on the one hand with those obtained with different a priori knowledge and on the other hand with crbs under the noncircular and circular complex gaussian distribution and with different deterministic crbs it is shown in particular that the crbs under the noncircular respectively circular complex gaussian distribution are tight upper bounds on the crbs under the bpsk respectively qpsk distribution at very low and very high signal to noise ratios snrs only finally these results and comparisons are extended to the case of two independent bpsk or qpsk distributed sources where an explicit expression of the crb for the doa parameters alone is given for large snr

bounds on the error probability of block codes over the q ary erasure channel in this paper tight bounds on the block error probability of linear block codes over order q finite fields for the q ary erasure channel under maximum likelihood ml decoding are developed upper bounds are obtained for uniform parity check ensembles sparse parity check ensembles general parity check ensembles e g gallager regular nonbinary low density parity check ensembles and for any given linear code with known distance spectrum the tightness of the upper bounds is confirmed both by the comparison with simple lower bounds and for gallager low density parity check ensembles by extensive monte carlo simulations exploiting the derived bounds it is shown how already for short blocks and small q 2 sparse ensembles attain block error probabilities close to those of idealized maximum distance separable mds codes down to low error probabilities whereas in the same regime binary codes show visible losses with respect to the singleton bound thanks to the accurate performance estimates the developed bounds can support the design of near optimum erasure correcting codes with short and moderate lengths

geometrical approach to the discrete wigner function in prime power dimensions we analyse the wigner function in prime power dimensions constructed on the basis of the discrete rotation and displacement operators labelled with elements of the underlying finite field we separately discuss the case of odd and even characteristics and analyse the algebraic origin of the non uniqueness of the representation of the wigner function explicit expressions for the wigner kernel are given in both cases

on the undetected error probability of binary matrix ensembles in this paper an ensemble analysis of the undetected error probability of a standard error detection scheme based on a sparse binary parity check matrix is presented the bernoulli ensemble the members of which are considered to be matrices generated from an i i d bernoulli source is primarily considered herein the main contributions of the present study are i the derivation of the error exponent of the average undetected error probability and ii closed form expressions for the variance of the undetected error probability the behavior of the exponent for an ensemble of sparse matrices is shown to be somewhat different from that for an ensemble of dense matrices furthermore as a byproduct of the proof of the variance formula a simple covariance formula of the weight distribution is derived

performance analysis of multi dimensional esprit type algorithms for arbitrary and strictly non circular sources with spatial smoothing spatial smoothing is a widely used preprocessing scheme to improve the performance of high resolution parameter estimation algorithms in case of coherent signals or if only a small number of snapshots is available in this paper we present a first order performance analysis of the spatially smoothed versions of r d standard esprit and r d unitary esprit for sources with arbitrary signal constellations as well as r d nc standard esprit and r d nc unitary esprit for strictly second order so non circular nc sources the derived expressions are asymptotic in the effective signal to noise ratio snr i e the approximations become exact for either high snrs or a large sample size moreover no assumptions on the noise statistics are required apart from a zero mean and finite so moments we show that both r d nc esprit type algorithms with spatial smoothing perform asymptotically identical in the high effective snr regime generally the performance of spatial smoothing based algorithms depends on the number of subarrays which is a design parameter that needs to be chosen beforehand in order to gain more insights into the optimal choice of the number of subarrays we simplify the derived analytical r d mean square error mse expressions for the special case of a single source the obtained mse expression explicitly depends on the number of subarrays in each dimension which allows us to analytically find the optimal number of subarrays for spatial smoothing based on this result we additionally derive the maximum asymptotic gain from spatial smoothing and compute the asymptotic efficiency for the single source case in closed form all the analytical results are verified by simulations

wigner weyl isomorphism for quantum mechanics on lie groups the wigner weyl isomorphism for quantum mechanics on a compact simple lie group g is developed in detail several features are shown to arise which have no counterparts in the familiar cartesian case notable among these is the notion of a semiquantized phase space a structure on which the weyl symbols of operators turn out to be naturally defined and figuratively speaking located midway between the classical phase space t g and the hilbert space of square integrable functions on g general expressions for the star product for weyl symbols are presented and explicitly worked out for the angle angular momentum case

a new upper bound on the block error probability after decoding over the erasure channel motivated by cryptographic applications we derive a new upper bound on the block error probability after decoding over the erasure channel the bound works for all linear codes and is in terms of the generalized hamming weights it turns out to be quite useful for reed muller codes for which all the generalized hamming weights are known whereas the full weight distribution is only partially known for these codes the error probability is related to the cryptographic notion of algebraic immunity we use our bound to show that the algebraic immunity of a random balanced m variable boolean function is of order m 2 1 o 1 with probability tending to 1 as m goes to infinity