sigma delta modulation with i i d gaussian inputs the response of a single loop sigma delta modulator to an independent identically distributed i i d gaussian input signal is analyzed a continuous time stochastic model is developed and the connection of the model to the system is described a condition is given so that the difference between the behavior of the model and that of the true system can be made arbitrarily small theories from renewal and wiener processes are applied to show the convergence and mixing properties of the output sequence also derived is the power spectrum of the quantization noise compared to the spectrum when the input is dc the i i d gaussian random process smears the discrete spectrum into band structures

multiplier free iir filter realization using periodically time varying state space structure i structure and design a multiplier free realization for iir filters is proposed a given target iir filter is implemented in cascade or parallel form of biquad units each biquad unit is realized with a periodically time varying ptv state space system the system uses a higher order ptv state space ptv ss structure operating at an elevated signal rate to realize each biquad the coefficients of the ptv ss structure used in the realization are restricted to a few power of two values therefore such a structure can be implemented without any hardware multiplier a brief analysis of precision and round off error for the realization is presented examples show that the frequency response of the proposed realization matches very well with that of the target filter

peak error constrained sparse fir filter design using iterative socp in this paper a novel algorithm is proposed to design sparse fir filters it is known that this design problem is highly nonconvex due to the existence of norm of a filter coefficient vector in its objective function to tackle this difficulty an iterative procedure is developed to search a potential sparsity pattern which is then used to compute the final solution by solving a convex optimization problem in each iterative step the original sparse filter design problem is successively transformed to a simpler subproblem it can be proved that under a weak condition globally optimal solutions of these subproblems can be attained by solving their dual problems in this case the overall iterative procedure converges to a locally optimal solution of the original design problem the design procedure described above can be repeated for several times to further improve the sparsity of design results the output of the previous stage can be used as the initial point of the subsequent design the performance of the proposed algorithm is evaluated by two sets of design examples and compared to other sparse fir filter design algorithms

λ aluthge transforms and schatten ideals abstract let t l h and let t u t t u be the polar decomposition of t then for every 0 1 the aluthge transform is defined by t t u t 1 we show that several properties which are known for the usual aluthge transform i e the case 1 2 also hold for aluthge transforms with 0 1 moreover we get several results which are new even for the usual aluthge transform

fully sigma delta modulation encoded fir filters the author suggests an fir filter structure where both the input signal and the impulse response are encoded using sigma delta modulation as a result the convolution can be obtained by summing a collection of binary or ternary numbers depending on the quantizer in the sigma delta modulator this filter structure can accept analog input signals directly and can be built using modular hardware the overall error spectrum due to the encoding decoding processes is derived and the implementation issues of this filter structure are discussed an example of its performance with different input signals is provided

new initialization methods for discrete coefficient fir filter design with coefficient scaling and the use of scale factor in the design process the initialization of filter coefficients in discrete coefficient finite impulse response fir filter design with coefficient scaling using coefficient value assignment based optimization techniques is considered a common weakness of existing initialization measures a total square error tse measure and a maximum error me measure is described new tse and me measures that overcome the weakness are introduced as opposed to the current knowledge it is revealed that tse and me measures do not necessarily provide different initial coefficient set information and that the initial coefficient set information by the me measure is not unique in general on a parallel track the treatment of coefficient scaling in the optimization process is considered the closed form expression of the objective function normalized peak ripple magnitude nprm which was defined implicitly in the literature is given in contrast to existing methods the computation of nprm accordingly does not require a scale factor parameter independent of filter coefficients although coefficient scaling persists then tse and me measures become practically equivalent although in general they indicate different initial scale factors since an explicit scale factor parameter is no longer needed

convexity and matrix means abstract in this article we present some mean inequalities for convex functions that lead to some generalized inequalities treating the arithmetic geometric and harmonic means for numbers and matrices our first main inequality will be 1 f 0 f 1 f 1 f 0 f 1 f 1 1 for the convex function f when 1 and 0 1 moreover when 1 the inequality will be valid for operator convex functions then by selecting an appropriate convex function we obtain certain matrix inequalities in particular we obtain several mixed mean inequalities for operators using real and operator convexity our discussion will lead to new multiplicative refinements and reverses of the heinz and holder inequalities for matrices new and refined trace and determinant inequalities the significance of this work is its general treatment where convexity is the only needed property

a class of high precision multiplier free fir filter realizations with periodically time varying coefficients proposes a class of high precision multiplier free realizations for fir filters these realizations use upsampling and downsampling in conjunction with a periodically time varying system to achieve time invariant multiplier free fir filter operation nonbinary encoding schemes are used for obtaining the filter coefficients which are periodically time varying ptv i e they vary in a periodic fashion each target filter coefficient is directly mapped into a set of ptv coefficients so that the realizations are easy to obtain the values of the ptv coefficients are restricted to either the ternary set

quantization noise in single loop sigma delta modulation with sinusoidal inputs an exact nonlinear difference equation is derived and solved for a simple sigma delta modulator consisting of a discrete time integrator and a binary quantizer inside a single feedback loop and an arbitrary input signal it is shown that the system can be represented as an affine operation discrete time integration of a biased input followed by a memoryless nonlinearity an extension of the transform method for the analysis of nonlinear systems is applied to obtain formulas for first and second order time average moments of the binary quantization noise including the sample mean energy and autocorrelation the results are applied to the special case of a sinusoidal input signal to evaluate these time averages and the power spectrum in the limit of large oversampling ratios the marginal moments behave as if the quantization noise had a uniform distribution the spectrum is neither white nor continuous however even in the limit of large oversampling ratios

a class of equilibrium problems with lower and upper bounds abstract in this paper we obtained some existence results of a class of abstract equilibrium problem our results give answers to the open problem raised by isac et al furthermore we studied the equilibrium problems with lower and upper bounds and restricted conditions and obtained more results finally we generalized these open problems to the case of two spaces

matrix young inequalities for the hilbert schmidt norm abstract let a b and x be n n complex matrices such that a and b are positive semidefinite if p q 1 with 1 p 1 q 1 it is shown that 1 p a p x 1 q xb q 2 2 1 r 2 a p x xb q 2 2 axb 2 2 where r max p q and 2 is the hilbert schmidt norm generalizations and applications of this inequality are also considered

some windows with very good sidelobe behavior some of the windows presented by harris 1 are not correct in terms of their reported peak sidelobes and optimal behavior we present corrected plots of harris windows and also derive additional windows with very good sidelobes and optimal behavior under several different constraints the temporal weightings are characterized as a sum of weighted cosines over a finite duration the plots enable the reader to select a window to suit his requirements in terms of bias due to nearby sidelobes and bias due to distant sidelobes

an explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings abstract let h be a hilbert space and c be a nonempty closed convex subset of h t i i n be a family of nonexpansive mappings from c into h g i c c r be a finite family of equilibrium functions i 1 2 k a be a strongly positive bounded linear operator with a coefficient and b lipschitzian relaxed cocoercive map of c into h moreover let r k n k 1 k n satisfy appropriate conditions and f k 1 k e p g k v i c b n n f i x t n 0 we introduce an explicit scheme which defines a suitable sequence as follows x n 1 n f x n 1 n a w n p c i s n b s r 1 n 1 s r 2 n 2 s r k n k x n n n and x n strongly converges to x f which satisfies the variational inequality a f x x x 0 for all x f the results presented in this paper mainly extend and improve a recent result of colao v colao an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings nonlinear analysis 2009 doi 10 1016 j na 2009 01 115 and qin x qin m shang y su a general iterative method for equilibrium problems and fixed point problems in hilbert spaces nonlinear analysis 69 2008 3897 3909

realization of finite impulse response filters using coefficients 1 0 and 1 new classes of nonrecursive linear digital filters are proposed as alternatives to conventional methods for digital signal processing devices in fact the traditional approach where signal samples are represented by 8 15 bit words can be very expensive to implement due to the complexity of multibit multipliers and adders filter structures consisting of a transversal filter with tap coefficients restricted to 1 0 or 1 and cascaded with simple recursive sections are proposed using a mean square error criterion dynamic programming methods to select optimum sets of coefficient values for the taps are presented and evaluated it is shown that these structures lend themselves to very simple hardware implementations and offer a valid alternative to the conventional filtering approach designs for several example filters are presented and their implementation complexity is examined

existence of solutions for abstract economic equilibrium problems and algorithm in this paper we study a new kind of vector equilibrium problem and we proved the existence of solutions for this kind of equilibrium problem by using the section theorem and fan kkm theorem then we generalize this kind of vector equilibrium problem to a more general case equilibrium problems with lower and upper bounds was an open problem proposed by isac sehgal and singh in 1999 j li o chadli y chiang and j c yao zhang cong jun derived some results of the equilibrium problems with lower and upper bounds under certain conditions in this paper we derive more results of this open problem under certain conditions constructs an iteration algorithm and discuses the convergence of the algorithm

design and hybrid realization of fir nyquist filters with quantized coefficients and low sensitivity to timing jitter a design of finite impulse response fir nyquist filters with zero intersymbol interference isi and low sensitivity to timing jitter is presented using an affine scaling linear programming algorithm a near optimum quantized coefficient set can be obtained in a feasible computational time by varying a parameter the design provides a tradeoff between the tail energy of the impulse response in the time domain and the stopband of the magnitude response in the frequency domain we also present a pipelined multiplier free fir filter realization with periodically time varying ptv coefficients based on a hybrid form the realizations exploit the coefficient symmetry to reduce the hardware by about one half by placing most of the shifts followed by addition toward the back end of the structure hardware is reduced due to the shorter wordlength of the adders the proposed structure has a provision to increase the speed by adjusting a design parameter but at the expense of more hardware

a generalized lattice filter for finite wordlength implementation with reduced number of multipliers the excellent finite wordlength fwl property of lattice digital filters is well known the four multiplier normalized lattice with signal power at all delay elements normalized to unity has particular advantage in its overflow property however when used to implement an nth order digital filter the normalized lattice implementation requires 5n 1 multipliers there exists another lattice structure with excellent fwl property called the injected numerator lattice structure in this paper we combine the injected numerator lattice and tapped numerator lattice to form a new hybrid lattice structure which is not only canonic in the number of multipliers resulting in a significant reduction in overall implementation cost but also exhibits much better fwl properties than the normalized lattice structure an improved peakedness measure is also introduced for application where the input signal has a strong time varying sinusoidal component the new structure requires a few additional adders it can be used to implement any causal and stable z transform transfer function two numerical examples are presented to demonstrate the performance of the proposed structure

arithmetic geometric mean and related inequalities for operators abstract in recent years certain arithmetic geometric mean and related inequalities for operators and unitarily invariant norms have been obtained by many authors based on majorization technique and so on we first point out that they are direct consequences of integral expressions of relevant operators furthermore we obtain related new inequalities theorems 4 5 and 6 based on our current approach

optimum binary windows for discrete fourier transforms a class of windows called binary windows for the frequency domain implementation of the discrete fourier transform is proposed while the conventional time domain windowing requires n 2 stored values of the data window and n multiplications the binary windows do not require multiplications or stored constants rather they are replaced by shift and add operations typical hardware structures are presented for efficient implementation of the binary windows

notes on matrix arithmetic geometric mean inequalities abstract for positive semi definite n n matrices the inequality 4 ab a b 2 isshown to hold for every unitarily invariant norm the connection of this with some other matrix arithmetic geometric mean inequalities and trace inequalities is discussed

improved young and heinz inequalities for matrices we give refinements of the classical young inequality for positive real numbers and we use these refinements to establish improved young and heinz inequalities for matrices

fir filters with sigma delta modulation encoding two methods are presented for building fir finite impulse response digital filters where either the impulse response or the input signal is encoded using sigma delta modulation for each method both single loop and two stage sigma delta modulation are used for encoding these systems are operated at a sampling rate higher than the nyquist rate of the signal and their performance depends on the oversampling ratio except in the decoding circuit these filters require no multiplication and hence can be built using simple hardware an analysis of the systems is provided and examples for filters realized using these approaches are given

lll algorithm and the optimal finite wordlength fir design in practical finite impulse response fir digital filter applications it is often necessary to represent the filter coefficients with a finite number of bits the finite wordlength restriction increases the filter deviation this increase can be reduced substantially if the optimal finite wordlength coefficients are used the time needed to compute these coefficients is greatly reduced with the help of a lower bound on the deviation increase derivation of an improved lower bound that uses the well known lll algorithm is presented in this correspondence

reduction of symmetric complex filters due to their linear phase property symmetric filters are an interesting class of finite impulse response fir filters moreover symmetric fir filters allow an efficient implementation in this paper we extend the classical definition of hermitian symmetry to a more general symmetry that is also applicable to complex filters this symmetry is called generalized hermitian symmetry we show the usefulness of this definition as it allows for a unified treatment of even and odd length filters central in this paper is a theorem on the reduction of generalized hermitian symmetric filters to hermitian symmetric filters both with finite precision coefficients a constructive proof of this theorem is presented and an associated procedure for reducing generalized hermitian symmetric filters is derived two of the examples show the application of the reduction procedure and the achieved savings on arithmetic costs finally all three examples show that a special instance of the generalized hermitian symmetric filters with finite precision coefficients may have lower arithmetic costs than the hermitian symmetric filter from which it is derived

equilibrium problems in hadamard manifolds abstract an equilibrium theory is developed in hadamard manifolds the existence of equilibrium points for a bifunction is proved under suitable conditions and applications to variational inequality fixed point and nash equilibrium problems are provided the convergence of picard iteration for firmly nonexpansive mappings along with the definition of resolvents for bifunctions in this setting is used to devise an algorithm to approximate equilibrium points

two stage sigma delta modulation an exact analysis of the nonlinear difference equation describing a discrete time two stage sigma delta quantizer with dc and sinusoidal inputs is presented it is shown that for dc inputs the binary quantizer error of the second quantizer which appears at the output of the modulator as a second order difference is asymptotically white uniformly distributed and uncorrelated with the input for sinusoidal inputs this result holds with one exception the autocorrelation which is a sum of weighted bessel functions does not correspond to that of a white noise process at large oversampling ratios the autocorrelation is approximately white and hence the white additive noise model is a good approximation for evaluating the first and second order moments it is found that for both a sinc sup 3 decoder filter and an ideal low pass filter decoder the average quantization noise power is inversely proportional to the fifth power of the oversampling ratio r for both types of inputs this represents an improvement over the single loop sigma delta quantizer by a factor of r sup 2 simulation results that agree with the theoretical analysis are presented

improved filter sharpening we propose a natural extension to kaiser hamming 1977 filter sharpening methods to allow for a piecewise linear desired amplitude change function acf the primary advantages of the proposed acf over piecewise constant acfs is that we obtain better control of selective improvement or degradation in either the passband or stopband or both and we are not restricted to applying our methods to filters with piecewise constant pass and stopbands since linear segments of slope 1 can be used to retain existing filter performance in either passband or stopband the proposed acf approximating polynomial ap is easy to compute may be constrained to have simple or integer coefficients and may be expressed as the ap of kaiser and hamming plus a correction polynomial we also provide applications for motivation

multistage sigma delta modulation a theoretical basis is provided for multistage sigma delta modulation msm which is a cascade realization of several single loop sigma delta modulators with a linear combinatorial network equations are derived describing the output and the quantization noise of msm for an arbitrary input signal and the noise shaping characteristic of msm is investigated the spectral characteristics of an m stage sigma delta modulator with both dc and sinusoidal inputs are developed for both types of inputs the binary quantizer noise of the mth m or 3 quantizer which appears at the output as an mth order difference is asymptotically white uniformly distributed and uncorrelated with the input level it is also found that for an m stage sigma delta quantizer with either an ideal low pass filter or a sinc sup m 1 filter decoder the average quantization noise of the system is inversely proportional to the 2m 1 th power of the oversampling ratio this implies that the high order systems are favourable in terms of the trade off between the quantization noise and oversampling ratio simulation results are presented to support the theoretical analysis

numerical radius inequalities via convexity abstract the main goal of this article is to present numerical radius inequalities for matrices based on convexity of certain numerical radius functions this simple approach extends some unitarily invariant norms inequalities such as heinz and young inequalities to the context of numerical radius however due to weak unitary invariance of the numerical radius these extended inequalities will be weaker than the corresponding unitarily invariant norms versions

young s inequality in trace class operators if a and b are trace class operators and if u is a partial isometry then where 1 denotes the norm in the trace class the present paper characterises the cases of equality in this young inequality and the characterisation is examined in the context of both the operator and the hilbert schmidt forms of young s inequality

new existence results for equilibrium problems we consider equilibrium problems in the framework of the formulation proposed by blum and oettli which includes variational inequalities nash equilibria in noncooperative games and vector optimization problems for instance as particular cases we establish new sufficient and or necessary conditions for existence of solutions of such problems our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems together with extensions to equilibrium problems of gap functions for variational inequalities then we apply our results to some particular instances of equilibrium problems obtaining results which include among others a new lemma of the alternative for convex optimization problems

design of discrete coefficient fir filters on loosely connected parallel machines this paper presents a new branch and bound mixed integer linear programming based algorithm for designing discrete coefficient finite impulse response fir filters using a cluster of workstations as the computation platform the discrete coefficient space considered is the sum of signed power of two space but the technique is also applicable to other discrete coefficient spaces the key issue determining the success of the algorithm is the ability to partition the original problem into several independent parts that can be distributed to a cluster of machines for solution the master slave model is adopted for the control of the machines test run results showed that super linear speedup i e the speedup factor is more than the number of machines running in parallel may be achieved

the multiplier penalty method for generalized nash equilibrium problems in banach spaces this paper deals with generalized nash equilibrium problems gneps in banach spaces we give an existence result for normalized equilibria of jointly convex gneps and then propose an augmented lag

design of cascade form fir filters with discrete valued coefficients the authors show that by cascading two direct form fir filters each with coefficients that are the sum or difference of two power of two terms it is possible to achieve very small peak ripple an iterative equalization strategy is used in the design of the cascaded filter the success of the method depends on the initial prototype filter being used two equally effective methods are presented for selecting the prototype filter each yields a final design with good roundoff noise property

dynamic programming methods for designing fir filters using coefficients 1 0 and 1 in this paper we describe an efficient design procedure for designing fir filters whose general structure consists of a transversal filter with tap coefficients restricted to 1 0 or 1 and cascaded with simple recursive sections due to the multiplication free nature of this transversal filter the configuration is particularly attractive from the standpoint of implementation with very simple hardware dynamic programming methods in conjunction with a mean square error functional are presented to select the set of coefficient values for the taps the approach is based on the assumption that the approximation sequence can be described as the output of a sequential machine whose state corresponds to the state space representation of the sections in cascade with the transveral filter

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

design and multiplier less implementation of a class of two channel pr fir filterbanks and wavelets with low system delay a new method for designing two channel pr fir filterbanks with low system delay is proposed it is based on the generalization of the structure previously proposed by phoong et al 1995 such structurally pr filterbanks are parameterized by two functions spl beta z and spl alpha z that can be chosen as linear phase fir or allpass functions to construct fir iir filterbanks with good frequency characteristics the case of using identical spl beta z and spl alpha z was considered by phoong et al with the delay parameter m chosen as 2n 1 in this paper the more general ease of using different nonlinear phase fir functions for spl beta z and spl alpha z is studied as the linear phase constraint is relaxed the lengths of spl beta z and spl alpha z are no longer restricted by the delay parameters of the filterbanks hence higher stopband attenuation can still be achieved at low system delay the design of the proposed low delay filterbanks is formulated as a complex polynomial approximation problem which can be solved by the remez exchange algorithm or analytic formula with very low complexity in addition the orders and delay parameters can be estimated from the given filter specifications using a simple empirical formula therefore low delay two channel pr filterbanks with flexible stopband attenuation and cutoff frequencies can be designed using existing filter design algorithms the generalization of the present approach to the design of a class of wavelet bases associated with these low delay filterbanks and its multiplier less implementation using the sum of powers of two coefficients are also studied

optimal design of finite precision fir filters using linear programming with reduced constraints an algorithm for the design of optimal one dimensional 1 d and two dimensional 2 d fir filters over a discrete coefficient space is proposed the algorithm is based on the observation that the equiripple frequencies of a subproblem sp in the branch and bound bab algorithm are closely related to those of neighboring sps by using the relationship among the sps the proposed algorithm reduces the number of constraints required for solving each sp thus the overall computational load for the design of fir filters with discrete coefficients is significantly alleviated compared with the conventional bab algorithm

singular value estimates of oblique projections abstract let w and m be two finite dimensional subspaces of a hilbert space h such that h w m and let p w m denote the oblique projection with range w and nullspace m in this article we get the following formula for the singular values of p w m 2 s k p w m 1 min f h x w m 2 where the minimum is taken over the set of all operator pairs f h on h such that r f w r h m and fh p w m and k 1 dim w we also characterize all the pairs where the minimum is attained

a discrete filled function method for the design of fir filters with signed powers of two coefficients in this paper we consider the optimal design of finite impulse response fir filters with coefficients expressed as sums of signed powers of two spt terms where the normalized peak ripple npr is taken as the performance measure this problem is formulated as a mixed integer programming problem based on a transformation between two different integer spaces and the computation of the optimal scaling factor for a given set of coefficients this mixed integer programming problem is transformed into an equivalent integer programming problem then an efficient algorithm based on a discrete filled function is developed for solving this equivalent problem for illustration some numerical examples are solved

moving horizon design of discrete coefficient fir filters we show how the discrete coefficient filter design problem can be solved with a moving horizon optimization approach the computation time of this procedure is determined by the optimization horizon and does not grow exponentially with the filter length spl sigma spl delta design methods are a special case of the proposed procedure

complexity reduction of digital filters using shift inclusive differential coefficients we present a graph theoretical methodology that reduces the implementation complexity of the multiplication of a constant vector and a scalar the complexity of implementation is defined as the required amount of computations like additions the proposed approach is called minimally redundant parallel mrp optimization and is mainly presented in a finite impulse response fir filtering framework to obtain a low complexity multiplierless implementation the key idea is to expand the design space using shift inclusive differential coefficients sidcs together with computation reordering using a graph theoretic approach to obtain maximal computation sharing the problem is formulated using a graph in which vertices and edges represent coefficients and computational cost number of resources the multiplierless solution is obtained by solving a set cover problem on the vertices in the graph a simple polynomial run time algorithm based on a greedy approach is presented the proposed approach is compared with common subexpression elimination to show slightly better results and is combined with it for further reduction of complexity simulation results show that 50 60 complexity reduction is achieved by only applying the mrp algorithm and 70 complexity reduction is obtainable by combining it with common subexpression elimination under a delay constraint of two or three

spectral analysis of quantization noise in a single loop sigma delta modulator with dc input an exact discrete time analysis of the moments and spectra of the quantization noise of a discrete time single loop sigma delta modulator with a dc input is presented an exact difference equation for the discrete time nonlinear system is used to evaluate the first and second order moments and power spectrum of the binary quantizer noise and the binary quantizer output for a single loop sigma delta encoder with a dc input it is shown that the sample mean and power of the binary quantization noise are consistent with the common uniform distribution assumption but that the autocorrelation and power spectrum are not consistent with the white noise assumption the results are used to evaluate the overall sample average mean squared quantization error as a function of the decimation filter used

an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings abstract let h be a hilbert space consider on h a sequence of nonexpansive mappings t n with common fixed points a finite family of equilibrium functions g i i 1 k a contraction f with coefficient 0 1 and a strongly positive linear bounded operator a with coefficient 0 let 0 assuming there are common equilibrium points of the family g i i 1 k which are also fixed points for t n we define a suitable sequence which strongly converges to the unique such point which also satisfies the variational inequality a f x x x 0 for all the x in the intersection of the equilibrium points and the common fixed points of the sequence t n

operator inequalities involving improved young and its reverse inequalities abstract this note aims to present some operator inequalities and matrices inequalities for the hilbert schmidt norm first the refined young and its reverse inequalities for scalars are given then based on these inequalities operator inequalities and matrices inequalities on the hilbert schmidt norm are established

a new approach to the design of discrete coefficient fir digital filters in this correspondence the iterative lagrange multiplier approach is proposed for designing discrete coefficient fir digital filters the method associates the conventional lagrange multiplier approach and a tree search algorithm for each branch of the tree the lagrange multiplier approach is used to optimize the remaining unquantized coefficients of the designed fir filter in the least squares sense when one or more of the coefficients takes on discrete values design examples including general low pass filters and nyquist filters are presented to demonstrate the effectiveness of the method also the method can be extended to design discrete coefficient 2 d fir filters

commutator inequalities for the hilbert schmidt norm abstract two types of commutator inequalities for the hilbert schmidt norm are established the first type of these inequalities is related to a classical inequality of clarkson and the second type is related to the unitary approximation of positive and invertible operators

a douglas rachford splitting method for solving equilibrium problems abstract we propose a splitting method for solving equilibrium problems involving the sum of two bifunctions satisfying standard conditions we prove that this problem is equivalent to find a zero of the sum of two appropriate maximally monotone operators under a suitable qualification condition our algorithm is a consequence of the douglas rachford splitting applied to this auxiliary monotone inclusion connections between monotone inclusions and equilibrium problems are studied

on scalar equilibrium problem in generalized convex spaces in this paper we prove the existence of a solution to the scalar equilibrium problem in generalized convex space as applications we derive results on the best approximations and simultaneous approximations the results of this paper generalize some known results in the literature

interpolated inequalities for unitarily invariant norms abstract in this article we interpolate the well known young and heinz inequalities for unitarily invariant norms and some of their known refinements then we prove new interpolated refinements in the end we use this interpolation idea to prove a hidden monotonicity behavior these inequalities obey

a bundle method for solving equilibrium problems we present a bundle method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle first we consider a general algorithm that we prove to be convergent then we explain how to make this algorithm implementable the strategy is to approximate the nonsmooth convex functions by piecewise linear convex functions in such a way that the subproblems are easy to solve and the convergence is preserved in particular we introduce a stopping criterion which is satisfied after finitely many iterations and which gives rise to stationary points finally we apply our implementable algorithm for solving the particular case of singlevalued and multivalued variational inequalities and we find again the results obtained recently by salmon et al 18

finite word length fir filter design using integer programming over a discrete coefficient space it is demonstrated that the improvement achieved by using integer programming over simple coefficient rounding in the design of finite impulse response fir filters with discrete coefficients is most significant when the discrete coefficient space is the powers of two space or when a specification is to be met with a given coefficient word length by increasing the filter length both minimax and least square error criteria are considered

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

a dual view of equilibrium problems the aim of this paper is to introduce and study a dual problem associated to a generalized equilibrium problem gep we show that the solutions of gep and its dual are strictly related to the saddle points of an associated lagrangian function and under some suitable conditions to the solutions of a family of parametric optimization problems and their dual problems our results allow us to show that well known concepts and results from duality theory of some important particular cases of gep like variational inequalities and optimization problems can be recovered

pseudomonotone variational inequality problems existence of solutions necessary and sufficient conditions for the set of solutions of a pseudomonotone variational inequality problem to be nonempty and compact are given

on a generalized proximal point method for solving equilibrium problems in banach spaces abstract we introduce a regularized equilibrium problem in banach spaces involving generalized bregman functions for this regularized problem we establish the existence and uniqueness of solutions these regularizations yield a proximal like method for solving equilibrium problems in banach spaces we prove that the proximal sequence is an asymptotically solving sequence when the dual space is uniformly convex moreover we prove that all weak accumulation points are solutions if the equilibrium function is lower semicontinuous in its first variable we prove under additional assumptions that the proximal sequence converges weakly to a solution

performance limit of finite wordlength fir digital filters in many practical situations it is necessary to represent the coefficients of a finite impulse response fir digital filter by a finite number of bits this not only degrades the filter frequency response but also introduces a theoretical limit on the performance of the filter derivation of a lower bound on filter degradation is the purpose of this paper we consider a general case of a length n filter with a discrete set of allowable coefficients a theorem that gives the lower bound on the increase in minimax approximation error that is caused by the finite wordlength restriction is presented its extension and application to filter design cases is demonstrated the importance of this bound is not only theoretical its practical effectiveness is shown in the algorithm for optimal finite wordlength fir filter design where it significantly reduces the amount of computation

general existence theorems for unilateral problems in continuum mechanics the problem of minimizing a possibly non convex and non coercive functional is studied either necessary or sufficient conditions for the existence of solutions are given involving a generalized recession functional whose properties are discussed thoroughly the abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region

on certain conditions for the existence of solutions of equilibrium problems the main purpose of this paper is the study of sufficient and or necessary conditions for existence of solutions of equilibrium problems we discuss some of the assumptions of the problem under which the introduced conditions are sufficient and or necessary and also analyze the effect of these assumptions on the connection between the solution sets of the equilibrium problem and of a related convex feasibility problem

an interpolation technique for computing the dft of a sparse sequence an interpolation technique for the approximate computation of the dft of a sparse sequence is presented this new method is more efficient than fft pruning particularly for long sequences where the effectiveness of fft pruning diminishes this interpolation method for computing the dft of a sparse sequence is very useful in filter design and spectrum analysis

matrix form of the inverse young inequalities abstract we use operator monotone and operator convex functions to prove an inverse to the young inequality for eigenvalues of positive definite matrices and then apply it to obtain a matrix inverse young inequality which can be considered as a complement of a result of t ando also we give a necessary and sufficient condition for the equality

well posed equilibrium problems abstract in this paper we introduce some notions of well posedness for scalar equilibrium problems in complete metric spaces or in banach spaces as equilibrium problem is a common extension of optimization saddle point and variational inequality problems our definitions originates from the well posedness concepts already introduced for these problems we give sufficient conditions for two different kinds of well posedness and show by means of counterexamples that these have no relationship in the general case however together with some additional assumptions we show via ekeland s principle for bifunctions a link between them finally we discuss a parametric form of the equilibrium problem and introduce a well posedness concept for it which unifies the two different notions of well posedness introduced in the first part

double loop sigma delta modulation with dc input a discrete time model having a two bit 2 b quantizer is analyzed exactly and an analytical expression for the quantizer noise sequence is found rigorous answers are then provided to two fundamental questions for a double loop sigma delta modulation system with dc input 1 what is the long term statistical behavior of the internal quantizer noise 2 how does the asymptotic mean square sigma delta quantization error vary as a function of the oversampling ratio

on some operator inequalities abstract for hilbert space operators s t with s invertible and self adjoint corach porta and recht recently proved that sts 1 s 1 ts 2 t a generalization of this inequality to larger classes of operators and norms is obtained as an immediate consequence of the operator form of the arithmetic geometric mean inequality some related inequalities are also discussed

exact and approximate algorithms for the filter design optimization problem the filter design optimization fdo problem is defined as finding a set of filter coefficients that yields a filter design with minimum complexity satisfying the filter constraints it has received a tremendous interest due to the widespread application of filters assuming that the coefficient multiplications in the filter design are realized under a shift adds architecture the complexity is generally defined in terms of the total number of adders and subtractors in this paper we present an exact fdo algorithm that can guarantee the minimum design complexity under the minimum quantization value but can only be applied to filters with a small number of coefficients we also introduce an approximate algorithm that can handle filters with a large number of coefficients using less computational resources than the exact fdo algorithm and find better solutions than existing fdo heuristics we describe how these algorithms can be modified to handle a delay constraint in the shift adds designs of the multiplier blocks and to target different filter constraints and filter forms experimental results show the effectiveness of the proposed algorithms with respect to prominent fdo algorithms and explore the impact of design parameters such as the filter length quantization value and filter form on the complexity and performance of filter designs

decimation for sigma delta modulation decimation is an important component of oversampled analog to digital conversion it transforms the digitally modulated signal from short words occurring at high sampling rate to longer words at the nyquist rate here we are concerned with the initial stage of decimation where the word rate decreases to about four times the nyquist rate we show that digital filters comprising cascades of integrate and dump functions can match the structure of the noise from sigma delta modulation to provide decimation with negligible loss of signal to noise ratio explicit formulas evaluate particular tradeoffs between modulation rate signal to noise ratio length of digital words and complexity of the modulating and decimating functions

a complete refinement of young s inequality abstract in this article we present multiple term refinements of young s inequality for real numbers and operators in particular given a natural number n we find n positive terms refining young s inequality detailed properties of this refinement are studied leading to a new interesting mean

design of nonuniformly spaced linear phase fir filters using mixed integer linear programming an optimization problem for designing a nonuniformly spaced linear phase fir filter with minimal complexity is formulated and solved by mixed integer linear programming milp examples illustrate that the proposed method is useful for designing a wide range of filter types and can outperform subset selection based design methods

an explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings let h be a hilbert space t i i n a family of nonexpansive mappings from h into itself g i cxc r a finite family of equilibrium functions email protected 1 2 k a a strongly positive bounded linear operator with coefficient email protected and f an a contraction on h let w n be the mapping generated by t i and l n as in 1 5 let s r k n k be the resolvent generated by g k and r k n as in lemma 2 4 moreover let r k n k 1 k e n and l n satisfy appropriate conditions and email protected k 1 ksep g k n nfix t n email protected we introduce an explicit scheme which defines a suitable sequence as follows z n email protected email protected z n email protected na w ns r 1 n 1s r 2 n 2 s r k n kz email protected email protected n and z n strongly converges to x f which satisfies the variational inequality 0 for all email protected f the results presented in this paper mainly extend and improve some recent results in vittorio colao et al an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings nonlinear anal 71 2009 2708 2715 s plubtieng r punpaeng a general iterative method for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 336 2007 455 469 s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2007 506 515

fir filter design over a discrete powers of two coefficient space fir digital filters with discrete coefficient values selected from the powers of two coefficient space are designed using the methods of integer programming the frequency responses obtained are shown to be superior to those obtained by simply rounding the coefficients both the weighted minimax and the weighted least square error criteria are considered using a weighted least square error criterion it is shown that it is possible to predict the improvement that can be expected when integer quadratic programming is used instead of simple coefficient rounding

a use of double integration in sigma delta modulation sigma delta modulation is viewed as a technique that employs integration and feedback to move quantization noise out of baseband this technique may be iterated by placing feedback loop around feedback loop but when three or more loops are used the circuit can latch into undesirable overloading modes in the desired mode a simple linear theory gives a good description of the modulation even when the quantization has only two levels a modulator that employs double integration and two level quantization is easy to implement and is tolerant of parameter variation at sampling rates of 1 mhz it provides resolution equivalent to 16 bit pcm for voiceband signals digital filters that are suitable for converting the modulation to pcm are also described

noncoercive optimization problems this paper modifies a recent sufficient condition for the existence of optimal solutions proposed by buttazzo and tomarelli to obtain a condition that is necessary and sufficient applications to mathematical programming are given and stability results established

dithering and its effects on sigma delta and multistage sigma delta modulation the spectrum of the quantization error in a dithered sigma delta modulator is derived under the constraint that the dithering signal does not cause overload the results apply to dc sinusoidal and more general quasi stationary signals it is shown in the case of a simple sigma delta modulation that no overload dithering can smooth the error spectrum and can make the quantization error asymptotically uncorrelated with the input it does not however make the quantization error white in the case of multistage sigma delta modulation with the appropriate dithering the quantization error becomes white even for a system with only two stages the signal to quantization noise ratio sqnr is derived for sigma delta and multistage sigma delta oversampled analog to digital conversion with additive dithering simulation results are presented to support the theoretical analysis

a tikhonov type regularization for equilibrium problems in hilbert spaces abstract in this paper we consider a general equilibrium problem in a hilbert space defined on a closed and convex set we show a tikhonov type regularization method that can be extended for the equilibrium problem under mild assumptions we establish the equivalence between the existence of solution of the original problem and the boundedness of the sequence generated by regularized problems

a note on the arithmetic geometric mean inequality for matrices abstract we give a simple proof of the inequality aa x xbb 2 a ab where a b and x are arbitrary n n matrices and is any unitarily invariant norm for the schatten p norms p with 1 p aa x xbb p 2 a xb p if and only if aa x xbb

projection algorithms for solving nonmonotone equilibrium problems in hilbert space we propose two projection algorithms for solving an equilibrium problem where the bifunction is not required to be satisfied any monotone property under assumptions on the continuity convexity of the bifunction and the nonemptyness of the solution set of the minty equilibrium problem we show that the sequences generated by the proposed algorithms converge weakly and strongly to a solution of the primal equilibrium problem respectively