separate joint optimization of error feedback and coordinate transformation for roundoff noise minimization in two dimensional state space digital filters this paper is concerned with the minimization of roundoff noise subject to l sub 2 norm dynamic range scaling constraints in two dimensional 2 d state space digital filters two methods are proposed with the first one using error feedback alone and the second one using joint error feedback and coordinate transformation optimization in the first method several techniques for the determination of optimal full scale block diagonal diagonal and scalar error feedback matrices for a given 2 d state space digital filter are proposed in the second method an iterative approach for minimizing the roundoff noise under l sub 2 norm dynamic range scaling constraints is developed by jointly optimizing a scalar error feedback matrix and a coordinate transformation matrix which may be regarded as an alternative approach to the conventional method for synthesizing the optimal 2 d filter structure with minimum roundoff noise an analytical method for the joint optimization of a general error feedback matrix and a coordinate transformation matrix under the scaling constraints is also proposed a numerical example is presented to illustrate the utility of the proposed techniques

jointly optimized error feedback and realization for roundoff noise minimization in state space digital filters roundoff noise rn is known to exist in digital filters and systems under finite precision operations and can become a critical factor for severe performance degradation in infinite impulse response iir filters and systems in the literature two classes of methods are available for rn reduction or minimization one uses state space coordinate transformation the other uses error feedback feed forward of state variables in this paper we propose a method for the joint optimization of error feedback feed forward and state space realization it is shown that the problem at hand can be solved in an unconstrained optimization setting with a closed form formula for gradient evaluation and an efficient quasi newton solver the unconstrained minimization problem can be solved efficiently with the infinite precision solution as a reference point we then move on to derive a semidefinite programming sdp relaxation method for an approximate solution of optimal error feedback matrix with sum of power of two entries under a given state space realization simulations are presented to illustrate the proposed algorithms and demonstrate the performance of optimized systems

noise reduction strategies for digital filters error spectrum shaping versus the optimal linear state space formulation the noise reduction performance of error spectrum shaping ess structures and of the optimal linear state space lss structure are compared for second order digital filter sections it is shown that optimal direct form 1 and direct form 2 ess realizations have a higher signal to noise ratio than the optimal lss structure in practice suboptimal ess structures with simple hardware implementations are of greater interest several of these implementations are considered and optimal values for the ess coefficients in these structures are derived for filters with zeros at z 1 it is shown that some of the simple ess structures can outperform the optimal lss structure for elliptic filters it is shown that several of the suboptimal ess structures perform poorly but that others still perform well

a new scheme for designing iir filters with finite wordlength coefficients a scheme for designing iir digital filters with specified coefficient wordlengths based on successive digitization and reoptimization is presented and compared to related approaches via a number of design examples the ordered digitization of the coefficients based on their sensitivity with each digitization followed by reoptimization of the remaining coefficients results in a much reduced computation effort in comparison to related design approaches

generation of cross covariance sequences we propose a linear method for the generation of cross covariance sequences in multichannel systems described by rational transfer functions the method relies on the determination of a linear discrete time filter whose impulse response is the desired cross covariance sequence we also prove existence and uniqueness of the solution

nonlinear vibrations of shallow shells with complex boundary r functions method and experiments geometrically nonlinear vibrations of shallow circular cylindrical panels with complex shape of the boundary are considered the r functions theory and variational methods are used to study the problem the r functions method rfm allows constructing in analytical form the sequence of basis functions satisfying the given boundary conditions in case of complex shape of the boundary the problem is reduced to a single second order differential equation with quadratic and cubic nonlinear terms the method developed has been initially applied to study free vibrations of shallow circular cylindrical panels with rectangular base for different boundary conditions i clamped edges ii in plane immovable simply supported edges iii classically simply supported edges and iv in plane free simply supported edges then the same approach is applied to a shell with complex shape of the boundary experiments have been conducted on an aluminum panel with complex shape of the boundary in order to identify the nonlinear response of the fundamental mode these experimental results have been compared to numerical results

a note on the evaluation of complex integrals using filtering interpretations the evaluation of complex integrals of the form r k frac 1 2 pij oint h z h z 1 z k frac dz z for all k is frequently required in control communications and digital filtering problems the major drawback of all existing methods is that the evaluation of the integral has to be preceded or followed by a check of the stability of the transfer function h z to ensure that r k min infin max infin is a true correlation sequence in this correspondence we modify the method proposed in 1 to yield a simple linear algorithm that does not require a separate cheek for the stability of h z the filtering interpretations proposed in 1 are extended to the evaluation of more general complex integrals with integrands of the form h z g z 1 finally an algorithm using the levinson recursions is presented as a fast method to generate correlation sequences in the arma case

a ritz vibration analysis of doubly curved rectangular shallow shells using a refined first order theory integral expressions for strain and kinetic energies in vibration analysis of shear deformable doubly curved shallow shells are presented although the formulation follows the first order shear deformation theory the consideration of lame parameters for the transverse shear strain through shell thickness which have been hitherto neglected by other researchers shows linear distribution functions instead of constant values if the lame parameters were dropped finite higher order two dimensional orthogonal polynomials pb 2 functions previously used in thin shell analyses 3 6 have been generalized from three to five degrees to account for the additional rotation fields of a moderately thick shell these global shape functions satisfy the kinematic boundary conditions at the outset the excellent performance and versatility of the computational methodology of the pb 2 ritz method are illustrated in representative numerical simulations the resultant matrix size is far smaller than other discretization methods and yet accurate solutions can be obtained the validity of the results is verified through direct comparisons the discrepancy of various formulations with and without the lame parameters is presented and discussed

from max plus algebra to nonexpansive mappings a nonlinear theory for discrete event systems discrete event systems provide a useful abstraction for modelling a wide variety of systems digital circuits communication networks manufacturing plants etc their dynamics stability equilibrium states cyclical behaviour asymptotic average delays are of vital importance to system designers however in marked contrast to continuous dynamical systems there has been little systematic mathematical theory that designers can draw upon in this paper we survey the development of such a theory based on the dynamics of maps which are nonexpansive in the l norm this has its origins in linear algebra over the max plus semiring but extends to a nonlinear theory that encompasses a variety of problems arising in other mathematical disciplines we concentrate on the mathematical aspects and set out several open problems

effects of initial twist and thickness variation on the vibration behaviour of shallow conical shells abstract a thorough computational investigation is presented into the effects of initial twist and thickness variation on the vibratory characteristics of cantilevered pretwisted thin shallow conical shells with generally varying thickness an energy approach with the ritz minimization procedure is employed to arrive at a governing eigenvalue equation the admissiblepb 2shape functions which comprise sets of mathematically complete two dimensional orthogonal polynomials and a basic function are introduced to account for the boundary constraints and to approximate the three dimensional displacements of the conical shell the effects of angle of twist thickness variation and geometric parameters on the vibration response are presented in a comprehensive parametric study which provides useful new design data selected vibration mode shapes are also included to enhance the existing literature it is found that an increase in the angle of twist does not necessarily increase the torsional stiffness on the contrary the fundamental frequencies is reduced in the present analysis it is also observed that non zero angle of twist and non uniform chordwise thickness variation cause the absent of vibration mode symmetry

jointly optimal high order error feedback and realization for roundoff noise minimization in 1 d and 2 d state space digital filters joint optimization of high order error feedback and state space realization for minimizing roundoff noise at filter output subject to l2 scaling constraints is investigated for one and two dimensional state space digital filters linear algebraic techniques that convert the problems at hand into an unconstrained optimization problem are explored and an efficient quasi newton algorithm is then applied to solve the unconstrained optimization problem iteratively in this connection closed form formulas are derived for fast and accurate gradient evaluation finally case studies are presented to demonstrate that the high order error feedback does offer much improved performance and that the proposed joint optimization is superior relative to a sequentially optimized system where the state space coordinate transformation and high order error feedback matrices are optimized separately

noise reduction in recursive digital filters using high order error feedback the problem of solving the optimal minimum noise error feedback coefficients for recursive digital filters is addressed in the general high order case it is shown that when minimum noise variance at the filter output is required the optimization problem leads to set of familiar wiener hopf or yule walker equations demonstrating that the optimal error feedback can be interpreted as a special case of wiener filtering as an alternative to the optimal solution the formulas for suboptimal error feedback with symmetric or antisymmetric coefficients are derived in addition the design of error feedback using power of two coefficients is discussed the efficiency of high order error feedback is examined by test implementations of the set of standard filters it is concluded that error feedback is a very powerful and versatile method for cutting down the quantization noise in any classical infinite impulse response iir filter implemented as a cascade of second order direct form sections the high order schemes are attractive for use with high order direct form sections

an interpretation of error spectrum shaping in digital filters error spectrum shaping is used in digital filters to reduce roundoff noise there are several other filter structures which can also reduce roundoff noise are these methods of noise reduction in any way related this paper shows that optimal error spectrum shaping is a method of increasing the word length of accumulators and internal variables i e it is in its optimum form extended precision arithmetic furthermore there is no hardware advantage to be gained with optimum ess as compared to traditional methods of implementing extended precision arithmetic

efficient solution of a toeplitz plus hankel coefficient matrix system of equations there are well known fast algorithms such as the levinson recursion for solving linear equations with a toeplitz or hankel coefficient matrix this paper extends the saving obtained by the levinson recursion for inversion of a toeplitz matrix to problems involving the inversion of matrices which are the sum of toeplitz and hankel matrices

oversampled sigma delta modulation oversampled sigma delta modulation has been proposed as a practical implementation for high rate analog to digital conversion because of its simplicity and its robustness against circuit imperfections to date mathematical developments of the basic properties of such systems have been based either on simplified continuous time approximate models or on linearized discrete time models where the quantizer is replaced by an additive white uniform noise source in this paper we rigorously derive several basic properties of a simple discrete time single integrator loop sigma delta modulator with an accumulate and dump demodulator the derivation does not require any assumptions on the correlation or distribution of the quantizer error and hence involves no linearization of the nonlinear system but it does show that when the input is constant the state sequence of the integrator in the encoder loop can be modeled exactly as a linear system in an appropriate space two basic properties are developed 1 the behavior of the sigma delta quantizer when driven by a constant input and its relation to uniform quantization and 2 the rate distortion tradeoffs between the oversampling ratio and the average mean squared quantization error

the perron frobenius theorem for homogeneous monotone functions if a is a nonnegative matrix whose associated directed graph is strongly connected the perron frobenius theorem asserts that a has an eigenvector in the positive cone r n we associate a directed graph to any homogeneous monotone function f r n r n and show that if the graph is strongly connected then f has a nonlinear eigenvector in r n several results in the literature emerge as corollaries our methods show that the perron frobenius theorem is really about the boundedness of invariant subsets in the hilbert projective metric they lead to further existence results and open problems

solution of vibration problems for shallow shells of arbitrary form by the r function method an effective method for the free vibration of arbitrary plan form shallow shells is proposed the algorithm is based on the r function theory and variational ritz method the effectiveness of the method offered is illustrated by examples of shallow shells of a complex plan form at different boundary conditions three types of curvatures are considered which are the spherical circularly cylindrical and of hyperbolic paraboloid shape a number of the test problems were solved to check the veracity of the proposed method the comparison between the obtained results and those available in the literature was carried out

the contraction mapping approach to the perron frobenius theory why hilbert s metric the perron frobenius theorem says that if a is a nonnegative square matrix some power of which is positive then there exists an x0 such that anx anx converges to xn for all x 0 there are many classical proofs of this theorem all depending on a connection between positively of a matrix and properties of its eigenvalues a more modern proof due to garrett birkhoff is based on the observation that every linear transformation with a positive matrix may be viewed as a contraction mapping on the nonnegative orthant this observation turns the perron frobenius theorem into a special ease of the banach contraction mapping theorem furthermore it applies equally to linear transformations which are positive in a much more general sense the metric which birkhoff used to show that positive linear transformations correspond to contraction mappings is known as hilbert s projective metric the definition of this metric is rather complicated it is therefore natural to try to define another less complicated m

a pb 2 ritz formulation for flexural vibration of shallow cylindrical shells of rectangular planform abstract this paper presents a free flexural vibration analysis of shallow cylindrical shells of rectangular planform with arbitrary combinations of free simply supported and clamped boundaries the pb 2 ritz method previously used in the plate analysis by one of the authors is now extended to shallow shell studies the ritz approach along with the in plane and transverse displacements assumed in the form of orthogonal polynomials is used to formulate the problems numerical results for the vibration frequencies for wide ranges of aspect ratio shallowness ratio and thickness ratio are presented several selected contour plots of the mode shapes for six different combinations of boundary conditions are presented as usual the accuracy of the present method is illustrated by convergence tests and comparison with the well established results from the literature

error feedback in a class of orthogonal polynomial digital filter structures a method of applying error feedback in three digital ladder and lattice filter structures derived from the theory of orthogonal polynomials is proposed it is shown that error feedback leads to significant improvement in both the roundoff noise characteristics and limit cycle behavior in particular the error compensated one multiplier structure is superior to the uncompensated normalized ladder this results in significant savings in both hardware and speed of implementation

spectral theorem for convex monotone homogeneous maps and ergodic control we consider convex maps f rn rn that are monotone i e that preserve the product ordering of rn and nonexpansive for the sup norm this includes convex monotone maps that are additively homogeneous i e that commute with the addition of constants we show that the fixed point set of f when it is nonempty is isomorphic to a convex inf subsemilattice of rn whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f this yields in particular an uniqueness result for the bias vector of ergodic control problems this generalizes results obtained previously by lanery romanovsky and schweitzer and federgruen for ergodic control problems with finite state and action spaces which correspond to the special case of piecewise affine maps f we also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters

roundoff noise minimization and pole zero sensitivity in fixed point digital filters using residue feedback the problem of designing a finite wordlength fixed point realization of an nth order digital filter which uses residue feedback to minimize the output roundoff noise subject to l 2 scaling is considered the new structures require n extra additions but no more multiplications than the earlier low noise structures of mullis and roberts and have lower roundoff noise for sufficiently narrow bandwidth filters a new set of filter invariants called the residue modes are defined which characterize the new low noise structures and determine the output noise variance if the sum of the residue modes is less than the sum of the second order modes of mullis and roberts then lower roundoff noise is achieved every filter structure is shown to define a unique symmetrizing matrix q pole zero sensitivities and a new noise measure are defined in terms of q numerical results are included which compare the noise and pole zero sensitivity characteristics of different filter structures

on low roundoff noise and low sensitivity digital filter structures this note presents a simple alternative form for the error feedback digital filter structure which with some minor modification of assumptions can also represent the low sensitivity and low noise structures of agarwal and burrus this form can thus explain why the agarwal burrus structures have low roundoff noise and can be shown to have low sensitivity finally this form is generalized to define a new class of low noise and low sensitivity realization structures for digital filters

mean squared error analysis of quantizers with error feedback quantization is a fundamental process in digital signal processing delta sigma modulators are often utilized for quantization which can be easily implemented with static uniform quantizers and error feedback filters in this paper we analyze the mean squared quantization error of the quantizer with error feedback including the delta sigma modulators first we study the quantizer with an ideal optimal error feedback filter that minimizes the mean squared error mse of quantization we show that the amplitude response of the optimal error feedback filter can be parameterized by one parameter this parameterization enables us to find the optimal error feedback filter numerically second the relationship between the number of bits used for the quantizer and the achievable mse is clarified by using the optimal error feedback filter this makes it possible to investigate the efficiency of the quantizer with the optimal error feedback filter in terms of mse then ideal optimal error feedback filters are approximated by practically implementable filters using the yule walker method and the linear matrix inequality based method numerical examples are provided for demonstrating our analysis and synthesis

on the design of fir filters with powers of two coefficients fir filters with powers of two coefficients are attractive for cost effective implementations the performance of one such structure the difference routing digital filter 8 is discussed using an snr criterion an optimal algorithm to select the value of the filter coefficients is presented results indicate that a 2 3 db improvement in the snr can be obtained with this new approach in addition the study gives some insight into the relationship between the filter sampling rate the number of coefficients and the size of the coefficient alphabet in particular it is seen that a tradeoff between the filter sampling rate and the size of the coefficient alphabet exists

roundoff noise minimization for 2 d state space digital filters using joint optimization of error feedback and realization the joint optimization problem of error feedback and realization for two dimensional 2 d state space digital filters to minimize the effects of roundoff noise at the filter output subject to l2 norm dynamic range scaling constraints is investigated it is shown that the problem can be converted into an unconstrained optimization problem by using linear algebraic techniques the unconstrained optimization problem at hand is then solved iteratively by applying an efficient quasi newton algorithm with closed form formulas for key gradient evaluation analytical details are given as to how the proposed technique can be applied to the cases where the error feedback matrix is a general block diagonal diagonal or block scalar matrix a case study is presented to illustrate the utility of the proposed technique