adaptive recovery of a chirped signal using the rls algorithm this paper studies the performance of the recursive least squares rls algorithm in the presence of a general chirped signal and additive white noise the chirped signal which is a moving average ma signal deterministically shifted in frequency at rate spl psi can be used to model a frequency shift in a received signal general expressions for the optimum wiener hopf coefficients one step recovery and estimation errors noise and lag misadjustments and the optimum adaptation constant spl beta sub opt are found in terms of the parameters of the stationary ma signal the output misadjustment is shown to be composed of a noise spl xi sub 0 m spl beta 2 and lag term spl kappa spl beta sup 2 spl psi sup 2 and the optimum adaptation constant is proportional to the chirp rate as spl psi sup 2 3 the special case of a chirped first order autoregressive ar1 process with correlation spl alpha is used to illustrate the effect the bandwidth 1 spl alpha of the chirped signal on the adaptation parameters it is shown that unlike for the chirped tone where the spl beta sub opt increases with the filter length m the adaptation constant reaches a maximum for m near the inverse of the signal correlation 1 spl alpha furthermore there is an optimum filter length for tracking the chirped signal and this length is less than 1 spl alpha

some new results on products of apostol bernoulli and apostol euler polynomials abstract we perform a further investigation for the apostol bernoulli and apostol euler polynomials and numbers by making use of an elementary idea used by euler in the discovery of his famous pentagonal number theorem we establish some new formulae for products of an arbitrary number of apostol bernoulli and apostol euler polynomials and numbers these results are the corresponding generalizations of some known formulae including the higher order convolution ones discovered by agoh and dilcher 2014 5 on the classical bernoulli and euler polynomials

a review of morphology elasticity relationships in human trabecular bone theories and experiments abstract in the perspective of predicting mechanical from morphological properties of human trabecular bone the theoretical and experimental relationships between volume fraction fabric and elastic properties were reviewed five data sets of human trabecular bone and two data sets of idealized cells were obtained from various investigators and analyzed statistically with one isotropic and four anisotropic models for each model multiple linear regressions were performed to fit the components of both the compliance and the stiffness tensors using volume fraction and in some cases fabric the adjusted coefficients of determination of the regressions and the average relative errors of the reported versus the predicted tensor norms were calculated the three anisotropic models that implied a log transformation of the data showed the best results excluding the idealized cell data the adjusted coefficients of determination of these models ranged from 0 80 to 0 95 for the compliance and from 0 80 to 0 94 for the stiffness tensors while the average relative errors varied between 16 and 55 for the compliance and between 25 and 62 for the stiffness data the use of volume fraction alone in the isotropic model decreased the adjusted coefficients of determination by 0 03 0 25 and increased the average relative errors by 5 27 this review confirms the potential of morphology elasticity relationships for estimation of elastic properties of human trabecular bone using peripheral quantitative computed tomography or magnetic resonance imaging but emphasizes the need for standardized measurements of mechanical properties at both continuum and tissue level

pade approximation and apostol bernoulli and apostol euler polynomials using the pade approximation of the exponential function we obtain recurrence relations between apostol bernoulli and between apostol euler polynomials as applications we derive some new lacunary recurrence relations for bernoulli and euler polynomials with gap of length 4 and lacunary relations for bernoulli and euler numbers with gap of length 6

comparative tracking performance of the lms and rls algorithms for chirped narrowband signal recovery this paper studies the comparative tracking performance of the recursive least squares rls and least mean square lms algorithms for time varying inputs specifically for linearly chirped narrowband input signals in additive white gaussian noise it is shown that the structural differences in the implementation of the lms and rls weight updates produce regions where the lms performance exceeds that of the rls and other regions where the converse occurs these regions are shown to be a function of the signal bandwidth and signal to noise ratio snr lms is shown to place a notch in the signal band of the mean lag filter thus reducing the lag error and improving the tracking performance for the chirped signal it is shown that this produces smaller tracking error for small snr for high snr there is a region of signal bandwidth for which rls will provide lower error than lms but even for these high snr inputs lms always provides superior performance for very narrowband signals

some relationships between the apostol bernoulli and apostol euler polynomials recently srivastava and pinter 1 investigated several interesting properties and relationships involving the classical as well as the generalized or higher order bernoulli and euler polynomials they also showed among other things that the main relationship proven earlier by cheon 2 can easily be put in a much more general setting the main object of the present sequel to these earlier works is to derive several general properties and relationships involving the apostol bernoulli and apostol euler polynomials some of these general results can indeed be suitably specialized in order to deduce the corresponding properties and relationships involving the generalized bernoulli and generalized euler polynomials other relationships associated with the stirling numbers of the second kind are also considered

tracking properties and steady state performance of rls adaptive filter algorithms adaptive signal processing algorithms derived from ls least squares cost functions are known to converge extremely fast and have excellent capabilities to track an unknown parameter vector this paper treats analytically and experimentally the steady state operation of rls recursive least squares adaptive filters with exponential windows for stationary and nonstationary inputs a new formula for the estimation noise has been derived involving second and fourth order statistics of the filter input as well as the exponential windowing factor and filter length furthermore it is shown that the adaptation process associated with lag effects depends solely on the exponential weighting parameter in addition the calculation of the excess mean square error due to the lag for an assumed markov channel provides the necessary information about tradeoffs between speed of adaptation and steady state error it is also the basis for comparison to the simple lms algorithm in a simple case of channel identification it is shown that the lms and rls adaptive filters have the same tracking behavior finally in the last part we present new rls restart procedures applied to transversal structures for mitigating the disastrous results of the third source of noise namely finite precision arithmetic

limitations of the continuum assumption in cancellous bone most existing stress analyses of the skeleton which consider cancellous bone assume that it can be modelled as a continuum in this paper we develop a criterion for the validity of this assumption the limitations of the continuum assumption appear in two areas near biologic interfaces and in areas of large stress gradients these limitations are explored using a probabilistic line scanning model for density measurement resulting in an estimate of density accuracy as a function of line length which is experimentally verified within three to five trabeculae of an interface a continuum model is suspect when results as predicted using continuum analyses vary by more than 20 30 over a distance spanning three to five trabeculae the results are suspect

behavior of the partial correlation coefficients of a least squares lattice filter in the presence of a nonstationary chirp input this paper studies the performance of the a posteriori recursive least squares lattice filter in the presence of a nonstationary chirp signal the forward and backward partial correlation parcor coefficients for a wiener hopf optimal filter are shown to be complex conjugates for the general case of a nonstationary input with constant power such an optimal filter is compared to a minimum mean square error based least squares lattice adaptive filter expressions are found for the behavior of the first stage of the adaptive filter based on the least squares algorithm for the general nth stage the parcor coefficients of the previous stages are assumed to have attained wiener hopf optimal steady state the parcor coefficients of such a least squares adaptive filter are compared with the optimal coefficients for such a nonstationary input the optimal lattice fitter is seen to track a chirp input without any error and the tracking lag in such an adaptive filter is due to the least squares update procedure the expression for the least squares based parcor coefficients are found to contain two terms a decaying convergence term due to the weighted estimation procedure and a tracking component that asymptotically approaches the optimal coefficient value the rate of convergence is seen to depend inversely on the forgetting factor the tracking lag of the filter is derived as a function of the rate of nonstationarity and the forgetting factor it is shown that for a given chirp rate there is a threshold adaptation constant below which the total tracking error is negligible for forgetting factors above this threshold the error increases nonlinearly further this threshold forgetting factor decreases with increasing chirp rate simulations are presented to validate the analysis

optimization of adaptive identification for time varying filters adaptive identification in a time varying context is studied when controlled by the lms algorithm with constant gain under the assumption of correlated successive input vectors it is well known by experience that the tracking mean square error mse epsilon mu results from the tradeoff between the gradient part which is increasing and the lag contribution which is decreasing in this note we clarify the relative roles of the gradient and lag errors by proving their decoupled character this property relies upon independence between the additive noise at the output of the plant to be identified and the information vector at the plant input convergence of the mse is established rather than assumed quantitative evaluations of upper and lower bounds allow an approximate optimization of the gain in two important cases the optimum is exact one of these cases is slow variations it is defined in a quantitative manner thanks to the ratio of the variation noise to the output additive noise

the adaptation of bone apparent density to applied load a phenomenological theory of bone remodeling was developed with improved spatial stability compared to some of the more standard formulations the improved stability was created by changing the nature of the remodeling differential equation to have an exponential character as a result the theoretical predictions are consistent with the experimental observation that changes in bone density during disuse after hip surgery during growth and during aging are all consistent with an exponential dependence of density on time the new theory and the standard theory were both used to model the time course of bone changes in two animal models of bone loss during disuse the new theory was better able to model the results of the experiments than the standard theory the basic continuum theory underlying the remodeling theory was presented in some detail this presentation was used to motivate the development of the new theory as the standard theories can predict non smooth distributions of bone density rather than the expected smooth distributions it was shown that these non smooth distributions are a violation of the continuum assumption one of the bases for the theory of finite element stress analysis the new model s stability was investigated using example problems and shown to be improved compared to the standard model

closed form analytical results for the rejection of narrow band interference in pn spread spectrum systems part ii linear interpolation filters the performance of pn spread spectrum communication systems in the presence of narrow band interference is studied when linear interpolation filters are employed for the estimation and subsequent suppression of the interference closed form analytical expressions for the system s performance are established for a broad class of interference processes the advantages of linear interpolation filters over predictive filters with identical number of taps are examined analytically and some unexpected results are obtained the analytical results are illustrated by examples

tracking performance of the rls algorithm applied to an antenna array in a realistic fading environment in this paper frequency domain techniques are used to derive the tracking properties of the recursive least squares rls algorithm applied to an adaptive antenna array in a mobile fading environment expanding the use of such frequency domain approaches for nonstationary rls tracking to the interference canceling problem that characterizes the use of antenna arrays in mobile wireless communications the analysis focuses on the effect of the exponential weighting of the correlation estimation filter and its effect on the estimations of the time variant autocorrelation matrix and cross correlation vector specifically the case of a flat rayleigh fading desired signal applied to an array in the presence of static interferers is considered with an ar2 fading process approximating the jakes fading model the result is a mean square error mse performance metric parameterized by the fading bandwidth and the rls exponential weighting factor allowing optimal parameter selection the analytic results are verified and demonstrated with a simulation example

some generalizations of the apostol bernoulli and apostol euler polynomials abstract the main object of this paper is to give analogous definitions of apostol type see t m apostol on the lerch zeta function pacific j math 1 1951 161 167 and h m srivastava some formulas for the bernoulli and euler polynomials at rational arguments math proc cambridge philos soc 129 2000 77 84 for the so called apostol bernoulli numbers and polynomials of higher order we establish their elementary properties derive several explicit representations for them in terms of the gaussian hypergeometric function and the hurwitz or generalized zeta function and deduce their special cases and applications which are shown here to lead to the corresponding results for the classical bernoulli numbers and polynomials of higher order

size effect method application for modeling of human cancellous bone using geometrically exact cosserat elasticity abstract in the present study the method of size effect was used to describe the hierarchical multi scale behavior of trabecular human bone using micro ct image the micro ct images extracted from a 98 year old male patient were used to provide the 3d model of cancellous bone which involves one of the most complex natural pore network the non linear or so called geometrically exact cosserat elasticity including augmented lagrangian eulerian ale method has been used to handle the multi scale behavior of trabecular human bone involving size effect in explicit manner we have chosen the conformal energy density within micro randomness issue in which the heterogeneous deformations at small scales are inherently sustained

tracking model of an adaptive lattice filter for a linear chirp fm signal in noise the paper studies the behavior of the partial correlation parcor coefficients and the output misadjustment of the stochastic gradient adaptive lattice filter in response to a complex linear chirp fm signal in white gaussian noise analytic expressions for the optimal parcor coefficients of the filter are derived analytic as well as iterative models for a three stage filter are also derived the analytic expressions show that the tracking and convergence properties of the filter are separate phenomena simulation results also show that the spectral contents of the parcor coefficients for the stochastic gradient update algorithm consist of a stationary and a linearly swept component a single stage model is developed to explain this behavior finally output misadjustment plots for the filter show that an optimum value for the forgetting factor can be obtained to minimize the misadjustment but the value required to achieve local minimum misadjustment varies with each stage of the filter it is shown that in applications where the input has a high signal to noise ratio snr the misadjustment decreases rapidly at each successive stage thus implying that relatively short filter lengths are sufficient to provide effective tracking

adaptive tracking of linear time variant systems by extended rls algorithms we exploit the one to one correspondences between the recursive least squares rls and kalman variables to formulate extended forms of the rls algorithm two particular forms of the extended rls algorithm are considered one pertaining to a system identification problem and the other pertaining to the tracking of a chirped sinusoid in additive noise for both of these applications experiments are presented that demonstrate the tracking superiority of the extended rls algorithms compared with the standard rls and least mean squares lms algorithms

adaptive recovery of a chirped sinusoid in noise ii performance of the lms algorithm the authors study the ability of the exponentially weighted recursive least square rls algorithm to track a complex chirped exponential signal buried in additive white gaussian noise power p sub n the signal is a sinusoid whose frequency is drifting at a constant rate psi lt is recovered using an m tap adaptive predictor five principal aspects of the study are presented the methodology of the analysis proof of the quasi deterministic nature of the data covariance estimate r k a new analysis of rls for an inverse system modeling problem a new analysis of rls for a deterministic time varying model for the optimum filter and an evaluation of the residual output mean square error mse resulting from the nonoptimality of the adaptive predictor the misadjustment in terms of the forgetting rate beta of the rls algorithm it is shown that the misadjustment is dominated by a lag term of order beta sup 2 and a noise term of order beta thus a value beta sub opt exists which yields a minimum misadjustment it is proved that beta sub opt m 1 rho psi sup 2 sup 1 3 and the minimum misadjustment is equal to 3 4 p sub n m 1 beta sub opt where rho is the input signal to noise ratio snr

asymptotic estimates for apostol bernoulli and apostol euler polynomials we analyze the asymptotic behavior of the apostol bernoulli polynomials bn x in detail the starting point is their fourier series on 0 1 which it is shown remains valid as an asymptotic expansion over compact subsets of the complex plane this is used to determine explicit estimates on the constants in the approximation and also to analyze oscillatory phenomena which arise in certain cases these results are transferred to the apostol euler polynomials en x via a simple relation linking them to the apostol bernoulli polynomials

fourier expansions and integral representations for the apostol bernoulli and apostol euler polynomials we investigate fourier expansions for the apostol bernoulli and apostol euler polynomials using the lipschitz summation formula and obtain their integral representations we give some explicit formulas at rational arguments for these polynomials in terms of the hurwitz zeta function we also derive the integral representations for the classical bernoulli and euler polynomials and related known results

a homogenization sampling procedure for calculating trabecular bone effective stiffness and tissue level stress a homogenization sampling procedure is introduced which allows computation of effective trabecular bone stiffness and individual trabecula level stress based on precise models of trabecular bone architecture three dimensional digitized images of 53 trabecular bone specimens with a resolution of 50 f per voxel were directly converted into three dimensional finite element meshes by making each voxel an 8 node isoparametric brick element owing to the large mesh of 8ooo elements an element by element pre conditioned conjugate gradient ebepcg program was written to solve the local homogenization finite element equations predicted effective stiffness measures correlated well with experimental results r2 z 0 73 the predicted effective stiffnesses tended to under estimate the experimental values average absolute errors in effective stiffness estimates ranged between 31 and 38 for the sampling procedure compared to a range 49 150 for a regression fit to volume fraction squared trabecula level stress ranged between 200 and 300 times that predicted by analyzing trabecular bone as a continuum both tensile and compressive tissue stresses were engendered by a continuum compressive stress trabecula level strain energy density sed ranged between 0 and 100 times the continuum sed value for two trabecular specimens in conclusion the homogenization sampling procedure consistently predicted the influence of trabecular bone architecture on effective stiffness it can also provide trabecular tissue stress and strain estimates for arbitrary global loading of whole bones tissue stresses and strains showed large variations compared to corresponding continuum level quantities