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

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

whitehead 3 is a slice link we show that the whitehead double t wh l of a two component link l is topologically flat slice if and only if the linking number of l is zero when they exist the slices may be chosen so that the complement b slices is homotopy equivalent to a wedge of two circles s 1 v s 1 with certain meridinal loops of wh l freely generating b4 slices compare this with if1 where it is shown that the whitehead double of boundary links are slice this is consistent with the surgery conjecture but seems not to illuminate the key 3component case see i f2 in particular wh3 wh who is slice we will describe the argument quite explicitly for this case and indicate in closing remarks what modifications are needed for the general result first we deal with non existence let s be the obvious two component seifert surface for wh l a component of s is obtained by composing either of the two punctured tori tz cslx d 2 pictured below with an untwisted identification of s i 2 with a neighborhood of a component of l add a band to s s su band so that s becomes the seifert surface for some knot k the seifert forms for s and s are identical and in a natural basis will be

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

the trivial connection contribution to witten s invariant and finite type invariants of rational homology spheres we derive an analog of the melvin morton bound on the power series expansion of the colored jones polynomial of algebraically split links and boundary links this allows us to produce a simple formula for the trivial connection contribution to witten s invariant of rational homology spheres we show that then th term in the 1 k expansion of the logarithm of this contribution is a finite type invariant of ohtsuki order 3n and of at most garoufalidis ordern

milnor numbers spanning trees and the alexander conway polynomial abstract we study relations between the alexander conway polynomial l and milnor higher linking numbers of links from the point of view of finite type vassiliev invariants we give a formula for the first non vanishing coefficient of l of an m component link l all of whose milnor numbers i 1 i p vanish for p n we express this coefficient as a polynomial in milnor numbers of l depending on whether the parity of n is odd or even the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3 graphs into pairs of spanning trees our results complement determinantal formulas of traldi and levine obtained by geometric methods

on finite wordlength fir filter design a recent paper has investigated the use of integer linear programming for the design of chebyshev optimal finite wordlength fir digital filters we point out that for some time domain error norms coefficient rounding provides the best approximation to the infinite precision filter

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

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

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

covering sato levine invariants two covering versions of the sato levine invariant are constructed which provide obstructions to certain two component oriented links in the 3sphere being link concordant to boundary links these covering invariants are rational functions one of which detects both nonamphicheirality and noninvertibility of oriented links

on normal approximations of distributions in terms of dependency graphs l auteur interprete les bornes de l erreur introduites par stein dans l approximation normale de sommes de variables aleatoires dependantes en termes de graphes de dependance ceci mene a des ameliorations d un theoreme de limite centrale de petrovskaya et leontovich et de recentes applications de baldi et rinatt en particulier on obtient des bornes pour des taux de convergence on etudie ensuite l approximation normale du nombre de maximums locaux d une fonction aleatoire sur un graphe

fir filter design based on successive approximation of vectors we present a novel method for the design of finite im pulse response fir filters with discrete coefficients that belong in thesumofpowers of two pot space theimportanceofthisclass of filters cannot be overstated given the ever increasing number of applications for which a specific hardware implementation is needed filters that have coefficients that belong to such a class are also referred to as multiplierless filters given that the oper ations performed by the filter can all be implemented by using appropriately designed shifts of the input data making them a perfect choice whenever implementation simplicity and processing speed are the ultimate goal to produce suc had esign we employ a vector successive approximation technique successfully used in data compression that has a very low computational complexity the matching pursuits generalized bitplanes algorithm mpgbp we derive optimality conditions for the approximation dictionary we compare filters obtained with the proposed method with those derived in previous works based on this comparative analysis we show that this new and powerful way of producing the filters co efficients is also among the simplest available in the literature indexterms signalprocessing filterdesign successiveapprox imation methods vq

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

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

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

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

the arf and sato link concordance invariants the ke rvaire arf invariant is a z 2 valued concordance invariant of knots and proper links the b invariant or sato s invariant is a z valued concordance invariant of two component links of linking number zero discovered by j levine and studied by sato cochran and daniel ruberman cochran has found a sequence of invariants bi associated with a two component link of linking number zero where each bi is a z valued concordance invariant and bo 3 in this paper we demonstrate a formula for the arf invariant of a two component link l x u y of linking number zero in terms of the b invariant of the link arf x u y arf x arf y b x u y mod 2 this leads to the result that the arf invariant of a link of linking number zero is independent of the orientation of the link s components we then establish a formula for i bi in terms of the link s alexander polynomial a x y x1 y1 f x v if l l f 1 finally we find a relationship between the b invariants and linking numbers of lifts of x and y in a z 2 cover of the compliment of x u y

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

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

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

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

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

central limit theorem for exponentially quasi local statistics of spin models on cayley graphs central limit theorems for linear statistics of lattice random fields including spin models are usually proven under suitable mixing conditions or quasi associativity many interesting examples of spin models do not satisfy mixing conditions and on the other hand it does not seem easy to show central limit theorem for local statistics via quasi associativity in this work we prove general central limit theorems for local statistics and exponentially quasi local statistics of spin models on discrete cayley graphs with polynomial growth further we supplement these results by proving similar central limit theorems for random fields on discrete cayley graphs taking values in a countable space but under the stronger assumptions of alpha mixing for local statistics and exponential alpha mixing for exponentially quasi local statistics all our central limit theorems assume a suitable variance lower bound like many others in the literature we illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology statistical physics and random networks examples of clustering spin models include quasi associated spin models with fast decaying covariances like the off critical ising model level sets of gaussian random fields with fast decaying covariances like the massive gaussian free field and determinantal point processes with fast decaying kernels examples of local statistics include intrinsic volumes face counts component counts of random cubical complexes while exponentially quasi local statistics include nearest neighbour distances in spin models and betti numbers of sub critical random cubical complexes

a lattice basis reduction approach for the design of finite wordlength fir filters many applications of finite impulse response fir digital filters impose strict format constraints on the filter coefficients such requirements increase the complexity of determining optimal designs for the problem at hand we introduce a fast and efficient method based on the computation of good nodes for polynomial interpolation and euclidean lattice basis reduction experiments show that it returns quasi optimal finite wordlength fir filters compared to previous approaches it also scales remarkably well length 125 filters are treated in 9 s it also proves useful for accelerating the determination of optimal finite wordlength fir filters

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

homotopy on spatial graphs and the sato levine invariant edge homotopy and vertex homotopy are equivalence relations on spatial graphs which are generalizations of milnor s link homotopy we introduce some edge resp vertex homotopy invariants of spatial graphs by applying the sato levine invariant for the 2 component constituent algebraically split links and show examples of non splittable spatial graphs up to edge resp vertex homotopy all of whose constituent links are link homotopically trivial

milnor s invariants and the completions of link modules let l be a tame link of k 2 components in s3 h the abelianization of its group 7tj s3 l and ih the augmentation ideal of the integral group ring zh the ih adic completions of the alexander module and alexander invariant of l are shown to possess presentation matrices whose entries are given in terms of certain integers t4 i1 iq introduced by j milnor various applications to the theory of the elementary ideals of these modules are given including a condition on the alexander polynomial necessary for the linking numbers of the components of l with each other to all be zero in the special case 2 it is shown that the various milnor invariants f r 1 s 1 are determined up to sign by the alexander polynomial of l and that this alexander polynomial is 0 iff i r 1 s 1 0 for all r s 0 with r s even also the chen groups of l are determined up to isomorphism by those nonzero i r 1 s 1 with r s minimal in contrast it is shown by example that for k 3 the alexander polynomials of a link and its sublinks do not determine its chen groups

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

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

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

efficient special purpose linear programming for fir filter design a high efficiency special purpose linear programming algorithm for fir filter design is introduced this algorithm incorporates fft and fir zero phase filters to assist in the computation compared to the general purpose linear programming this special purpose linear programming produces large savings in the computing cost when used to design fir filters the savings increase with the length of the filter being designed

filter design with low complexity coefficients we introduce a heuristic for designing filters that have low complexity coefficients as measured by the total number of nonzeros digits in the binary or canonic signed digit csd representations of the filter coefficients while still meeting a set of design specifications such as limits on frequency response magnitude phase and group delay numerical examples show that the method is able to attain very low complexity designs with only modest relaxation of the specifications

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

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

correlation functions for random complex zeroes strong clustering and local universality we prove strong clustering of k point correlation functions of zeroes of gaussian entire functions in the course of the proof we also obtain universal local bounds for k point functions of zeroes of arbitrary nondegenerate gaussian analytic functions in the second part of the paper we show that strong clustering yields the asymptotic normality of fluctuations of some linear statistics of zeroes of gaussian entire functions in particular of the number of zeroes in measurable domains of large area this complements our recent results from the paper fluctuations in random complex zeroes

a new technique for the design of two dimensional fir and iir filters the design problem of two dimensional digital filters is the subject of this paper the problem is formulated for both fir and iir filters in such a way that the performance index is a convex function of the optimization vector the design technique is based on newton s method each iteration consists of n easily computed steps where n is the number of parameters to be optimized the coefficients can be obtained in the quantization levels according to the desired number representation illustrative examples and comparisons with existing techniques are also included