cramer rao lower bounds for subcarrier snr estimates over multicarrier channels considering orthogonal frequency division multiplexing transmissions analytical expressions for the cramer rao lower bounds for the subcarrier signal to noise ratio snr estimates are derived the transmitted signal is assumed to be corrupted by additive white gaussian noise on the different subcarriers the channel is assumed to be slowly time varying over the observation interval so that it can be assumed constant it will be shown that the achievable performance on subcarrier snr estimation can be significantly improved by exploiting the mutual information the same experienced noise power between the different tones

stationary random fields with linear regressions we analyze and identify stationary fields with linear regressions and quadratic conditional variances we give sufficient conditions to determine one dimensional distributions uniquely as normal and as certain compactly supported distributions our technique relies on orthogonal polynomials which under our assumptions turn out to be a version of the so called continuous q hermite polynomials

a note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations abstract following the lead of carrillo arch ration mech anal 147 1999 269 361 recently several authors have used kru kov s device of doubling the variables to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations in all these results the second order differential operator is not allowed to depend explicitly on the spatial variable which certainly restricts the range of applications of entropy solution theory the purpose of this paper is to extend a version of carrillo s uniqueness result to a class of degenerate parabolic equations with spatially dependent second order differential operator the class is large enough to encompass several interesting nonlinear partial differential equations coming from the theory of porous media flow and the phenomenological theory of sedimentation consolidation processes

a comparison of snr estimation techniques for the awgn channel the performances of several signal to noise ratio snr estimation techniques reported in the literature are compared to identify the best estimator the snr estimators are investigated by the computer simulation of baseband binary phase shift keying psk signals in real additive white gaussian noise awgn and baseband 8 psk signals in complex awgn the mean square error is used as a measure of performance in addition to comparing the relative performances the absolute levels of performance are also established the simulated performances are compared to a published cramer rao bound crb for real awgn and a crb for complex awgn that is derived here some known estimator structures are modified to perform better on the channel of interest estimator structures for both real and complex channels are examined

towards a q analogue of the kibble slepian formula in 3 dimensions abstract we study a generalization of the kibble slepian ks expansion formula in 3 dimensions the generalization is obtained by replacing the hermite polynomials by the q hermite ones if such a replacement would lead to non negativity for all allowed values of parameters and for all values of variables ranging over certain cartesian product of compact intervals then we would deal with a generalization of the 3 dimensional normal distribution we show that this is not the case nevertheless we indicate other applications of so generalized ks formula namely we use it to sum certain kernels built of the al salam chihara polynomials for the cases that were not considered by other authors one of such kernels sums up to the askey wilson density disclosing its new interesting properties in particular we are able to obtain a generalization of the 2 dimensional poisson mehler formula we also pose several open questions

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

cramer rao bounds for snr estimation of oversampled linearly modulated signals most signal to noise ratio snr estimators use the receiver matched filter output sampled at the symbol rate an approach which does not preserve all information in the analog waveform due to aliasing thus it is relevant to ask whether avoiding aliasing could improve snr estimation to this end we compute the corresponding data aided da and nondata aided nda cramer rao bounds crbs we adopt a novel dual filter framework which is shown to be information preserving under suitable conditions and considerably simplifies the analysis it is shown that the crb can be substantially reduced by exploiting any available excess bandwidth depending on the modulation scheme the snr range and the estimator type da or nda

cramer rao lower bounds for nda snr estimates of square qam modulated transmissions in this paper we derive for the first time analytical expressions for the exact cramer rao lower bounds on the variance of unbiased non data aided nda signal to noise ratio snr estimators of square qam modulated signals the channel is assumed to be constant over the observation interval and the received signal is supposed to be corrupted by additive white gaussian noise awgn the derived expressions corroborate previous attempts to numerically compute the considered crlbs it will be shown that the nda crlbs differ widely from one modulation order to another especially at moderate snr levels

well posedness of higher order camassa holm equations abstract we consider higher order camassa holm equations describing exponential curves of the manifold of smooth orientation preserving diffeomorphisms of the unit circle in the plane we establish the existence of global weak solutions we also present some invariant spaces under the action of the equation moreover we prove a weak equals strong uniqueness result

l 1 framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations we develop a general l 1 framework for deriving continuous de pendence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the chen perthame kinetic approach 9 we ap ply our l 1 framework to establish an explicit estimate for continuous depen dence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method without imposition of bounded variation of the approximate solutions finally as an example of a direct application of this framework to numerical methods we focus on a linear convection diusi on model equation and derive an l 1 error estimate for an upwind central finite dierence scheme

moment based parameter estimation and blind spectrum sensing for quadrature amplitude modulation knowing accurate noise variance and signal power is crucial to most spectrum sensing algorithms such as energy detection matched filter detection and cyclostationary detection in this paper we consider a practical scenario when these two parameters are unknown and are needed to be estimated before the spectrum sensing this task is non trivial without knowing the status of the primary user and we categorize the related spectrum sensing as a blind one we develop the estimation algorithms for unknown parameters by exploiting the signal constellation of the primary user three different parameter estimators that do not require any training are then proposed based on the moments of the received signals since the secondary user may not know the primary user s signal constellation we develop a robust approach that approximates a finite quadrature amplitude modulation qam constellation by a continuous uniform distribution we also derive the modified cramer rao bound crb for noise variance estimation then the optimal moment pair is found from minimizing the mean squared error mse of the signal to noise ratio snr the method of choosing the spectrum sensing threshold by taking into consideration the estimation error is also discussed

the bi poisson process a quadratic harness this paper is a continuation of our previous research on quadratic harnesses that is processes with linear regressions and quadratic conditional variances our main result is a construction of a markov process from given orthogonal and martingale polynomials the construction uses a two parameter extension of the al salam chihara polynomials and a relation between these polynomials for different values of parameters

snr estimation for multilevel constellations using higher order moments the performance of existing moments based non data aided nda estimators of signal to noise ratio snr in digital communication systems substantially degrades with multilevel constellations we propose a novel moments based approach that is amenable to practical implementation and significantly improves on previous estimators of this class this approach is based on a linear combination of ratios of certain even order moments which allow the derivation of nda snr estimators without requiring memory costly lookup tables the weights of the linear combination can be tuned according to the constellation and the snr operation range as particular case we develop an eighth order statistics eos based estimator showing in detail the statistical analysis that leads to the weight optimization procedure the eos based estimators yield improved performance for multilevel constellations especially for those with two and three amplitude levels monte carlo simulations validate the new approach in a wide snr range

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

an envelope based feature for nda snr estimation signal to noise ratio snr estimation is essential in many communication systems moment based snr estimators are a type of envelope based estimator the performance of moment based estimators are known to decrease with increasing snr for nonconstant modulus signals in this paper an envelopebased feature is proposed to overcome the challenges posed by nonconstant modulus signals the feature extraction and estimation processes are presented and the results are shown the newly proposed estimator performs better than classical moment based estimators for modulations such as quadrature amplitude modulations qam and amplitude and phase shift keying apsk modulations

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

data aided and non data aided maximum likelihood snr estimators for cpm data aided da and non data aided nda maximum likelihood ml estimators for the snr of cpm in the presence of phase and frequency offset are derived and analyzed cramer rao bounds for both are obtained and compared to the simulated performance of these estimators for a full response example cpsfk and a partial response example gmsk analysis and simulations show that the performance of the da ml estimator suffers from an unremovable bias caused by uncompensated frequency offset due to frequency offset estimation errors as a consequence the estimator error variance of the da ml estimator is not able to achieve its lower bound in contrast the estimator error variance of the nda ml estimator is capable of achieving its lower bound although the lower bound for nda ml estimator is higher than the lower bound for the da ml estimator this is because the nda ml estimator is not burdened with the requirement of relying on estimates of nuisance parameters the nda estimator only achieves its lower bound when the observation length or true snr are sufficiently large

on three dimensional multivariate version of q normal distribution and probabilistic interpretations of askey wilson al salam chihara and q ultraspherical polynomials abstract we study properties of compactly supported 4 parameter 12 23 13 q 1 1 4 family of continuous type 3 dimensional distributions that have the property that for q 1 members of this family tend to members of some 3 dimensional normal family of distribution for q 0 we deal with 3 dimensional generalization of kesten mckay distributions in a very special case when 12 13 23 q all one dimensional marginals are identical semicircle distributions we find both all marginals as well as all conditional distributions moreover we find also families of polynomials that are orthogonalized by these one dimensional margins and one dimensional conditional distributions consequently we find moments of both conditional and unconditional distributions of dimensions one and two in particular we show that all one dimensional and two dimensional conditional moments of say order n and are polynomials of the same order n in the conditioning random variables finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials among them are the famous askey wilson al salam chihara polynomials considered in the complex but conjugate parameters as well as q hermite and rogers polynomials it seems that this paper is one of the first papers that give a probabilistic interpretation of rogers continuous q ultraspherical polynomials

askey wilson polynomials quadratic harnesses and martingales we use orthogonality measures of askey wilson polynomials to construct markov processes with linear regressions and quadratic conditional variances askey wilson polynomials are orthogonal martingale polynomials for these processes

snr estimation for nonconstant modulus constellations we propose a new technique to estimate the signal to noise ratio snr over the flat fading channel this is a nondata aided envelope based estimator that can be applied to nonconstant modulus constellations which is a feature not found in existing approaches we also analyze the performance of our estimators for both phase shift keying psk and non psk constellations by deriving their asymptotic variances and comparing with the crame spl acute r rao bounds crbs moreover we discuss how the snr estimates can be used to approximate the bit error rate ber and how the accuracy of the snr estimate is related to that of the ber estimate which justifies the necessity for the accurate estimation of snr the analytical performance is shown for both psk and non psk constellations such as 8 quadrature amplitude modulation qam and 16 qam monte carlo simulation results corroborate our analysis

stability of solutions of quasilinear parabolic equations abstract we bound the difference between solutions u and v of u t a u div x f h and v t b v div x g k with initial data and respectively by u t v t l p e a e t l r n 2 p b t a b x f x g f u g u h k p e p here all functions a f and h are smooth and bounded and may depend on u x r n and t the functions a and h may in addition depend on u identical assumptions hold for the functions that determine the solutions v furthermore e r n is assumed to be a bounded set and p and p are fractions that depend on n and p the diffusion coefficients a and b are assumed to be strictly positive and the initial data are smooth

continuous dependence on the nonlinearities of solutions of degenerate parabolic equations explicit estimates for the continuous dependence inl 0 t l1 rd of solutions of the equation lv v v in 0 rdwith initial conditionv 0 h with respect to the nonlinear continuously differential functions and are established

stability of parabolic problems with nonlinear wentzell boundary conditions abstract of concern is the nonlinear uniformly parabolic problem u t div a u u 0 x f x u t a u x u q lb u 0 for x r n and t 0 the last equation holds on the boundary here a a i j x i j is a real hermitian uniformly positive definite n n matrix c with 0 r r q 0 lb is the laplace beltrami operator on the boundary and a u is the conormal derivative of u with respect to a and everything is sufficiently regular the solution of this wellposed problem depends continuously on the ingredients of the problem namely a q f this is shown using semigroup methods in g m coclite a favini g r goldstein j a goldstein s romanelli continuous dependence on the boundary parameters for the wentzell laplacian semigroup forum 77 1 2008 101 108 here we prove explicit stability estimates of the solution u with respect to the coefficients a q and the initial condition f moreover we cover the singular case of a problem with q 0 which is approximated by problems with positive q

a nondata aided snr estimation technique for multilevel modulations exploiting signal cyclostationarity signal to noise ratio snr estimators of linear modulation schemes usually operate at one sample per symbol at the matched filter output in this paper we propose a new method for estimating the snr in the complex additive white gaussian noise awgn channel that operates directly on the oversampled cyclostationary signal at the matched filter input exploiting cyclostationarity proves to be advantageous due to the fact that a signal free euclidean noise subspace can be identified such that only second order moments of the received waveform need to be computed the proposed method is nondata aided nda as well as constellation and phase independent and only requires prior timing synchronization to fully exploit the cyclostationarity property the estimator can also be applied to nonconstant modulus constellations without requiring any tuning which is a feature not found in existing approaches implementation aspects and simpler suboptimal solutions are also provided

maximum likelihood snr estimation of linearly modulated signals over time varying flat fading simo channels in this paper we tackle for the first time the problem of maximum likelihood ml estimation of the signal to noise ratio snr parameter over time varying single input multiple output simo channels both the data aided da and the non data aided nda schemes are investigated unlike classical techniques where the channel is assumed to be slowly time varying and therefore considered as constant over the entire observation period we address the more challenging problem of instantaneous i e short term or local snr estimation over fast time varying channels the channel variations are tracked locally using a polynomial in time expansion first we derive in closed form expressions the da ml estimator and its bias the latter is subsequently subtracted in order to obtain a unbiased da estimator whose variance and the corresponding cramer rao lower bound crlb are also derived in closed form due to the extreme nonlinearity of the log likelihood function llf in the nda case we resort to the expectation maximization em technique to iteratively obtain the exact nda ml snr estimates within very few iterations most remarkably the new em based nda estimator is applicable to any linearly modulated signal and provides sufficiently accurate soft estimates i e soft detection for the unknown transmitted symbols therefore hard detection can be easily embedded in the iteration loop in order to improve its performance at low snr levels we show by extensive computer simulations that the new estimators are able to accurately estimate the instantaneous per antenna snrs as they coincide with the da crlb over a wide range of practical snrs

a few remarks on orthogonal polynomials knowing a sequence of moments of a given infinitely supported distribution we obtain quickly coefficients of the power series expansion of monic polynomials p n n 0 that are orthogonal with respect to this distribution coefficients of expansion of x n in the series of p j j n two sequences of coefficients of the 3 term recurrence of the family of p n n 0 the so called linearization coefficients i e coefficients of expansion of p n p m in the series of p j j m n conversely assuming knowledge of the two sequences of coefficients of the 3 term recurrence of a given family of orthogonal polynomials p n n 0 we express with their help coefficients of the power series expansion of p n coefficients of expansion of x n in the series of p j j n moments of the distribution that makes polynomials p n n 0 orthogonal further having two different families of orthogonal polynomials p n n 0 and q n n 0 and knowing for each of them sequences of the 3 term recurrences we give sequence of the so called connection coefficients between these two families of polynomials that is coefficients of the expansions of p n in the series of q j j n we are able to do all this due to special approach in which we treat vector of orthogonal polynomials p j x j 0 n as a linear transformation of the vector x j j 0 n by some lower triangular n 1 i n 1 matrix n

expansions of one density via polynomials orthogonal with respect to the other abstract we expand the chebyshev polynomials and some of its linear combination in linear combinations of the q hermite the rogers q utraspherical and the al salam chihara polynomials and vice versa we use these expansions to obtain expansions of some densities including q normal and some related to it in infinite series constructed of the products of the other density times polynomials orthogonal to it allowing deeper analysis and discovering new properties on the way we find an easy proof of expansion of the poisson mehler kernel as well as its reciprocal we also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

a parabolic system model for the formation of porous silicon existence uniqueness and stability in this paper a nonlinear weakly coupled parabolic system is constructed to model porous silicon formation the model considers physical and chemical factors such as charge carriers transport ions transport electrochemical reactions and material balance in order to prove the existence of a unique solution for the proposed parabolic model with zero neumann boundary we use a monotone method based on the upper and lower solutions the asymptotic stability of the steady state solution is also demonstrated and the behavior of this model is shown for specific cases where the analytical stationary solution is known finally a discussion of the results is carried out and we conclude that the proposed model reproduces important qualitative features which are observed experimentally

nevanlinna theory of the askey wilson divided difference operator abstract this paper establishes a version of nevanlinna theory based on askey wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane c a second main theorem that we have derived allows us to define an askey wilson type nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero pole scarce that is their zeros poles are indeed deficient in the sense of difference nevanlinna theory a natural consequence is a version of askey wilson type picard theorem we also give an alternative and self contained characterisation of the kernel functions of the askey wilson operator in addition we have established a version of unicity theorem in the sense of askey wilson this paper concludes with an application to difference equations generalising the askey wilson second order divided difference equation

blind signal to noise ratio estimation of real sinusoid in additive noise we consider the problem of estimation of signal to noise ratio snr with a real deterministic sinusoid with unknown frequency phase and amplitude in additive gaussian noise of unknown variance the method of moments a general method to derive estimators based on high order moments is used to derive a blind snr estimator that does not require the knowledge of the instantaneous frequency of the sinusoid through separate estimation of signal and noise power cramer rao lower bounds crlbs are also derived for estimators of signal and noise power and then for snr estimators we show through numerical simulations the statistical performances of the estimators that we compare to the corresponding crlbs and discuss their use in practical applications

more on the q oscillator algebra and q orthogonal polynomials properties of certain q orthogonal polynomials are connected to the q oscillator algebra the wall and q laguerre polynomials are shown to arise as matrix elements of q exponentials of the generators in a representation of this algebra a realization is presented where the continuous q hermite polynomials form a basis of the representation space various identities are interpreted within this model in particular the connection formula between the continuous big q hermite and the continuous q hermite polynomials is thus obtained and two generating functions for these last polynomials are derived algebraically

conditional moments of q meixner processes we show that stochastic processes with linear conditional expectations and quadratic conditional variances are markov and their transition probabilities are related to a three parameter family of orthogonal polynomials which generalize the meixner polynomials special cases of these processes are known to arise from the non commutative generalizations of the levy processes

optimal control of a nonlinear parabolic elliptic system abstract in this paper we shall study the problem of optimal control of the parabolic elliptic system u t f t x u x g t x u p x a t x u x x f 0 b and p x x p h t x u u x k t x u with initial data u t 0 u 0 the existence and uniqueness of weak solution to the parabolic elliptic system are given in a short interval according to the variational method optimal control theories and distributed parameter system control theories we can deduce that the norm of the solution is related to the control item and initial value in the special hilbert space the optimal control of the parabolic elliptic system with the initial data is given and the existence of an optimal solution to the parabolic elliptic system is proved

the attractor of a parabolic elliptic system abstract following coclite holden and karlsen g m coclite h holden and k h karlsen well posedness for a parabolic elliptic system discrete contin dyn syst 13 3 2005 659 682 and tian and fan lixin tian jinling fan the attractor on viscosity degasperis procesi equation nonlinear analysis real world applications 2007 we study the dynamical behaviors of the parabolic elliptic system u t f t x u x g t x u p x e u x x 0 and p x x p h t x u u x k t x u with initial data u t 0 u 0 the existence of global solution to the parabolic elliptic system in l 2 under the periodic boundary condition is discussed we also establish the existence of the global attractor of semi group to solutions on the parabolic elliptic system in h 2

gain control free near efficient phase acquisition for qam constellations this paper introduces a novel not data aided phase offset estimator for quadrature amplitude modulated qam signals contrarily to near efficient existing phase acquisition techniques this estimator does not require a preliminary gain adjustment stage while its accuracy preserves the slope of cramer rao bound for medium high signal to noise ratio snr ranges where it typically outperforms existing blind estimators with significant improvement for dense and cross qam constellations moreover it needs only a very rough estimate of the snr like other gain control free blind phase offset estimators it measures the amount of the cyclic shift by which the four folded phase probability density function pdf is rotated under an unknown phase offset estimation of the phase offset induced cyclic shift is conducted first by measuring the received data phase pdf by a canonical phase histogram procedure then by estimating the phase offset induced cyclic shift through a cyclic cross correlation based procedure between the measured phase histogram and a reference phase pdf evaluated within the zero phase offset hypothesis actually the estimation procedure is presented in a generalized version that considers a tomographic projection of the bidimensional magnitude phase pdf of suitable nonlinear transformations of the received data the tomographic projection performs a magnitude weighing on the pdf and this in turn results in an improved overall estimation accuracy as shown by theoretical analysis and numerical simulations here performed to assess the estimator performance

cramer rao lower bound and em algorithm for envelope based snr estimation of nonconstant modulus constellations signal to noise ratio snr estimation for linearly modulated signals is addressed in this letter focusing on envelope based estimators which are robust to carrier offsets and phase jitter and on the challenging case of nonconstant modulus constellations for comparison purposes the true cramer rao lower bound is numerically evaluated obtaining an analytical expression in closed form for the asymptotic case of high snr values which quantifies the performance loss with respect to coherent estimation as the maximum likelihood algorithm is too complex for practical implementation an expectation maximization em approach is proposed achieving a good tradeoff between complexity and performance for medium to high snrs finally a hybrid scheme based on em and moments based estimates is suggested which performs close to the theoretical limit over a wide snr range

on the structure and probabilistic interpretation of askey wilson densities and polynomials with complex parameters abstract we give equivalent forms of the askey wilson polynomials expressing them with the help of the al salam chihara polynomials after restricting parameters of the askey wilson polynomials to complex conjugate pairs we expand the askey wilson weight function in the series similar to the poisson mehler expansion formula and give its probabilistic interpretation in particular this result can be used to calculate explicit forms of q hermite moments of the askey wilson density hence enabling calculation of all moments of the askey wilson density on the way by setting certain parameter q to 0 we get some formulae useful in the rapidly developing so called free probability

stability estimates for parabolic problems with wentzell boundary conditions abstract of concern is the uniformly parabolic problem u t div a u u 0 x f x u t a u u q lb u 0 for x r n and t 0 here a a i j x i j is a real hermitian uniformly positive definite n n matrix c with 0 q 0 and a u is the conormal derivative of u with respect to a and everything is sufficiently regular the solution of this well posed problem depends continuously on the ingredients of the problem namely a q f this is shown using semigroup methods in g m coclite a favini g r goldstein j a goldstein s romanelli continuous dependence on the boundary parameters for the wentzell laplacian semigroup forum in press more precisely if we have a sequence of such problems with solutions u n and if a n a n etc in a suitable sense then u n u the solution of the limiting problem the abstract analysis associated with operator semigroup theory gives this conclusion but no rate of convergence determining how fast the convergence of the solutions is requires detailed estimates such estimates are provided in this paper