vector quantization of speech and speech like waveforms an algorithm for the design of vector quantizers that are locally optimum in the sense of minimizing an average quantitative distortion measure is used to design 1 and 2 bit sample vector quantizers for both real sampled speech and a simulated speech like auto regressive random process both weighted and unweighted squared error distortion measures are considered several comparisons are made and discussed based on the average distortions of the vector quantization schemes the results for the simulated speech are compared to mathematical performance bounds from information theory to provide an indication of how nearly globally optimal vector quantization is for such highly correlated sources a comparison of the results for the real speech and the simulated speech provides a quantitative measure of the accuracy of such models and hence of the applicability of information theory bounds and code designs based on probabilistic models the signal to quantization noise ratios of vector quantizers designed to minimize squared error distortion are compared to those of several popular speech waveform coding systems of similar rates

a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

lesche stability of κ entropy abstract the lesche stability condition for the shannon entropy j stat phys 27 1982 419 represents a fundamental test for its experimental robustness for systems obeying the maxwell boltzmann statistical mechanics of course this stability condition must be satisfied by any entropic functional candidate to generate non conventional statistical mechanics in the present effort we show that the entropy recently introduced in literature phys rev e 66 2002 056125 satisfies the lesche stability condition

advances in residual vector quantization a review advances in residual vector quantization rvq are surveyed definitions of joint encoder optimality and joint decoder optimality are discussed design techniques for rvqs with large numbers of stages and generally different encoder and decoder codebooks are elaborated and extended fixed rate rvqs and variable rate rvqs that employ entropy coding are examined predictive and finite state rvqs designed and integrated into neural network based source coding structures are revisited successive approximation rvqs that achieve embedded and refinable coding are reviewed a new type of successive approximation rvq that varies the instantaneous block rate by using different numbers of stages on different blocks is introduced and applied to image waveforms and a scalar version of the new residual quantizer is applied to image subbands in an embedded wavelet transform coding system

properties of the non extensive gaussian entropy the present work investigates the lesche stability experimental robustness the thermodynamic stability the legendre structure of thermodynamics and derives the maximum entropy distribution of the one parametric non extensive gaussian entropy we show that this entropy definition fulfills both stability conditions for all values of its parameter q r the entropy maximizer contains the lambert w function which allows the preservation of the legendre transformations

generalized thermostatistics based on deformed exponential and logarithmic functions abstract the equipartition theorem states that inverse temperature equals the log derivative of the density of states this relation can be generalized by introducing a proportionality factor involving an increasing positive function x it is shown that this assumption leads to an equilibrium distribution of the boltzmann gibbs form with the exponential function replaced by a deformed exponential function in this way one obtains a formalism of generalized thermostatistics introduced previously by the author it is shown that tsallis thermostatistics with a slight modification is the most obvious example of this formalism and corresponds with the choice x x q

self similarity degree of deformed statistical ensembles abstract we consider self similar statistical ensembles with the phase space whose volume is invariant under the deformation that squeezes expands the coordinate and expands squeezes the momentum the related probability distribution function is shown to possess a discrete symmetry with respect to manifold action of the jackson derivative to be a homogeneous function with a self similarity degree q fixed by the condition of invariance under n 1 fold action of the related dilatation operator in slightly deformed phase space we find the homogeneous function is defined with the linear dependence at n 0 whereas the self similarity degree equals the gold mean at n 1 and q n in the limit n dilatation of the homogeneous function is shown to decrease the self similarity degree q at n 0

choice of quantization interval for finite energy fields we consider in this paper how to encode a finite energy random field via a finite number of bits we present a method giving rise to a finite bit representation for a given field we analyze the overall reconstruction error associated with this method and see that it is a summation which consists of the error coming from the sampling and quantization domains then we search for the best sampling and quantization parameters to get the optimal reconstruction error for a given number of bits this optimization leads to the number of bits versus reconstruction error curve consisting of best achievable points which is reminiscent of rate distortion curve in information theory

variable rate vector quantization for medical image compression three techniques for variable rate vector quantizer design are applied to medical images the first two are extensions of an algorithm for optimal pruning in tree structured classification and regression due to breiman et al the code design algorithms find subtrees of a given tree structured vector quantizer tsvq each one optimal in that it has the lowest average distortion of all subtrees of the tsvq with the same or lesser average rate since the resulting subtrees have variable depth natural variable rate coders result the third technique is a joint optimization of a vector quantizer and a noiseless variable rate code this technique is relatively complex but it has the potential to yield the highest performance of all three techniques

constrained clustering as an optimization method a deterministic annealing approach to clustering is derived on the basis of the principle of maximum entropy this approach is independent of the initial state and produces natural hierarchical clustering solutions by going through a sequence of phase transitions it is modified for a larger class of optimization problems by adding constraints to the free energy the concept of constrained clustering is explained and three examples are are given in which it is used to introduce deterministic annealing the previous clustering method is improved by adding cluster mass variables and a total mass constraint the traveling salesman problem is reformulated as constrained clustering yielding the elastic net en approach to the problem more insight is gained by identifying a second lagrange multiplier that is related to the tour length and can also be used to control the annealing process the open path constraint formulation is shown to relate to dimensionality reduction by self organization in unsupervised learning a similar annealing procedure is applicable in this case as well

simple transitive 2 representations of small quotients of soergel bimodules in all finite coxeter types but i2 12 i2 18 and i2 30 we classify simple transitive 2 representations for the quotient of the 2 category of soergel bimodules over the coinvariant algebra which

macroscopic thermodynamics based on composable nonextensive entropies abstract composability of entropy implies that entropy of the total system composed of independent subsystems is given by a function of entropies of the subsystems it is shown that the zeroth law of thermodynamics naturally leads to tsallis type pseudoadditivity of the composable entropies then macroscopic thermodynamics is discussed based on the simplest example of such entropies it is found that even if nonextensive entropy is initially employed the resulting thermodynamic entropy is necessarily extensive at the macroscopic level in accordance with caratheodory s theorem this result depends crucially on the assumptions that the subsystems are independent and the total energy is the sum of the energies of the subsystems

legendre structure of κ thermostatistics revisited in the framework of information geometry information geometry is a powerful framework in which to study families of probability distributions or statistical models by applying differential geometric tools it provides a useful framework for deriving many important structures in probability theory by identifying the space of probability distributions with a differentiable manifold endowed with a riemannian metric in this paper we revisit some aspects concerning the thermostatistics based on the entropy s in the framework of information geometry after introducing the dually flat structure associated with the distribution we show that the dual potentials derived in the formalism of information geometry correspond to the generalized massieu function and the generalized entropy s characterizing the legendre structure of the deformed statistical mechanics in addition we obtain several quantities such as escort distributions and canonical divergence relevant for the further development of the theory

application of generalized thermostatistics to fully developed turbulence abstract we apply the formalism of non extensive statistical mechanics to fully developed turbulent flows we obtain analytical formulas for probability density functions of velocity differences depending on distance r and reynolds number r which are in very good agreement with turbulence experiments the skewness is taken into account by analysing universal contributions due to higher order correlations of the chaotic dynamics we also calculate the moment scaling exponents m which also agree well with experimental data

deformed exponentials and logarithms in generalized thermostatistics abstract criteria are given that deformed logarithmic and exponential functions should satisfy with a pair of such functions one can associate another function called the deduced logarithmic function it is shown that generalized thermostatistics can be formulated in terms of deformed exponential functions together with the associated deduced logarithmic functions

general pseudoadditivity of composable entropy prescribed by the existence of equilibrium the concept of composability states that entropy of the total system composed of independent subsystems is a function of entropies of the subsystems here the most general pseudoadditivity rule for composable entropy is derived based only on the existence of equilibrium

the design of joint source and channel trellis waveform coders the generalized lloyd algorithm is applied to the design of joint source and channel trellis waveform coders to encode discrete time continuous amplitude stationary and ergodic sources operating over discrete memoryless noisy channels experimental results are provided for independent and autoregressive gaussian sources binary symmetric channels and absolute error and squared error distortion measures performance of the joint codes is compared with the tandem combination of a trellis source code and a trellis channel code on the independent gaussian source using the squared error distortion measure operating over an additive white gaussian noise channel it is observed that the jointly optimized codes achieve performance close to or better than that of separately optimized tandem codes of the same constraint length performance improvement via a predictive joint source and channel trellis code is demonstrated for the autoregressive gaussian source using the squared error distortion measure

a fuzzy vector quantization approach to image compression the use of fuzzy clustering analysis in the early stages of a vector quantization process is able to make this process less sensitive to initialization this is justified by the fact that fuzzy clustering provides a framework for the quantitative formulation of the uncertainty typically involved in a training vector space this paper proposes a fuzzy clustering based vector quantization algorithm which employs an effective vector assignment strategy for the transition from fuzzy mode where each training vector is assigned to more than one clusters to crisp mode where each training vector is assigned to only one cluster this transition is controlled by analytical conditions that are obtained by minimizing a modified objective function for the fuzzy c means algorithm the application to image compression shows that the proposed approach is able to achieve a very efficient performance while maintaining the computational capabilities of other methods reported in the literature

a new one parameter deformation of the exponential function abstract recently in kaniadakis physica a 296 2001 405 a new one parameter deformation for the exponential function exp x 1 2 x 2 x 1 exp 0 x exp x which presents a power law asymptotic behaviour has been proposed the statistical distribution f z 1 exp e has been obtained both as stable stationary state of a proper nonlinear kinetics and as the state which maximizes a new entropic form in the present contribution starting from the algebra and after introducing the analysis we obtain the exponential exp x as the eigenstate of the derivative and study its main mathematical properties

two parameter deformations of logarithm exponential and entropy a consistent framework for generalized statistical mechanics a consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace form entropy and by requiring that physically motivated mathematical properties are preserved the emerging differential functional equation yields a two parameter class of generalized logarithms from which entropies and power law distributions follow these distributions could be relevant in many anomalous systems within the specified range of parameters these entropies possess positivity continuity symmetry expansibility decisivity maximality concavity and are lesche stable the boltzmann shannon entropy and some one parameter generalized entropies already known belong to this class these entropies and their distribution functions are compared and the corresponding deformed algebras are discussed

an algorithm for the design of labeled transition finite state vector quantizers a finite state vector quantizer fsvq is a switched vector quantizer where the sequence of quantizers selected by the encoder can be tracked by the decoder it can be viewed as an adaptive vector quantizer with backward estimation a vector generalization of an aqb system recently a family of algorithms for the design of fsvq s for waveform coding application has been introduced these algorithms first design an initial set of vector quantizers together with a next state function giving the rule by which the next quantizer is selected the codebooks of this initial fsvq are then iteratively improved by a natural extension of the usual memoryless vector quantizer design algorithm the next state function however is not modified from its initial form in this paper we present two extensions of the fsvq design algorithms first the algorithm for fsvq design for waveform coders is extended to fsvq design of linear predictive coded lpc speech parameter vectors using an itakura saito distortion measure second we introduce a new technique for the iterative improvement of the next state function based on an algorithm from adaptive stochastic automata theory the design algorithms are simulated for an lpc fsvq and the results are compared with each other and to ordinary memoryless vector quantization several open problems suggested by the simulation results are presented

resolutions over symmetric algebras with radical cube zero abstract let be a finite dimensional indecomposable weakly symmetric algebra over an algebraically closed field k satisfying j 3 0 let s 1 s r be representatives of the isomorphism classes of simple modules and let e be the r r matrix whose i j entry is dim k ext 1 s i s j if there exists an eigenvalue of e satisfying 2 then the minimal resolution of each non projective finitely generated module has exponential growth with radius of convergence 1 2 2 4 on the other hand if all eigenvalues of e satisfy 2 then the dimensions of the modules in the minimal projective resolution of each finitely generated module are either bounded or grow linearly in this case we classify the possibilities for the matrix e the proof is an application of the perron frobenius theorem

schur weyl duality for infinitesimal q schur algebras sq 2 r 1 abstract using the result of s r doty d k nakano k m peters polynomial representations of frobenius kernels of gl 2 in contemp math vol 194 1996 pp 57 67 s konig c xi when is a cellular algebra quasi hereditary math ann 315 1999 281 293 we prove that a non semisimple infinitesimal schur algebra s 2 r 1 is not cellular furthermore we determine the structure of the endomorphism ring of tensor space as a module for the infinitesimal schur algebra s 2 r 1 up to morita equivalence both results generalize to the quantum case

stability of tsallis entropy and instabilities of renyi and normalized tsallis entropies a basis for q exponential distributions the q exponential distributions which are generalizations of the zipf mandelbrot power law distribution are frequently encountered in complex systems at their stationary states from the viewpoint of the principle of maximum entropy they can apparently be derived from three different generalized entropies the renyi entropy the tsallis entropy and the normalized tsallis entropy accordingly mere fittings of observed data by the q exponential distributions do not lead to identification of the correct physical entropy here stabilities of these entropies i e their behaviors under arbitrary small deformation of a distribution are examined it is shown that among the three the tsallis entropy is stable and can provide an entropic basis for the q exponential distributions whereas the others are unstable and cannot represent any experimentally observable quantities

thermodynamic stabilities of the generalized boltzmann entropies abstract we consider the thermodynamic stability conditions tsc on the boltzmann entropies generalized by tsallis q and kaniadakis deformed logarithmic functions it is shown that the corresponding tscs are not necessarily equivalent to the concavity of the generalized boltzmann entropies with respect to internal energy nevertheless both the tscs are equivalent to the positivity of standard heat capacity

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

vector predictive coding of speech at 16 kbits s vector quantization in its simplest form may be regarded as a generalization of pcm independent quantization of each sample of a waveform to what might be called vector pcm where a block of consecutive samples a vector is simultaneously quantized as one unit in theory a performance arbitrarily close to the ultimate rate distortion limit is achievable with waveform vector quantization if the dimension of the vector k is large enough the main obstacle in effectively using vector quantization is complexity a vector quantizer of dimension k operating at a rate of r bits sample requires a number of computations on the order of k2 kr and a memory of the same order however a low dimensional vector quantizer dimensions 4 8 achieves a remarkable improvement over scalar quantization pcm consequently using the vector quantizer as a building block and imbedding it with other waveform data compression techniques may lead to the development of a new and powerful class of waveform coding systems this paper proposes and analyzes a waveform coding system adaptive vector predictive coding avpc in which a low dimensionality vector quantizer is used in an adaptive predictive coding scheme in the encoding process a locally generated prediction of the current input vector is subtracted from the current vector and the resulting error vector is coded by a vector quantizer each frame consisting of many vectors is classified into one of m statistical types this classification determines which one of m fixed predictors and of m vector quantizers will be used for encoding the current frame

stabilities of generalized entropies the generalized entropic measure which is maximized by a given arbitrary distribution under the constraints on normalization of the distribution and the finite ordinary expectation value of a physical random quantity is considered to examine if it can be of physical relevance its experimental robustness is discussed in particular lesche s criterion is analysed which states that an entropic measure is stable if its change under an arbitrary weak deformation of the distribution representing fluctuations of experimental data remains small it is essential to note the difference between this criterion and thermodynamic stability a general condition under which the generalized entropy becomes stable is derived examples known in the literature including the entropy for the stretched exponential distribution the quantum group entropy and the entropy are discussed

thermodynamic equilibrium and its stability for microcanonical systems described by the sharma taneja mittal entropy it is generally assumed that the thermodynamic stability of equilibrium states is reflected by the concavity of entropy we inquire in the microcanonical picture about the validity of this statement for systems described by the two parametric entropy s kappa r of sharma taneja and mittal we analyze the composability rule for two statistically independent systems a and b described by the entropy s kappa r with the same set of the deformation parameters it is shown that in spite of the concavity of the entropy the composability rule modifies the thermodynamic stability conditions of the equilibrium state depending on the values assumed by the deformation parameters when the relation s kappa r a union b s kappa r a s kappa r b holds superadditive systems the concavity condition does imply thermodynamics stability otherwise when the relation s kappa r a union b s kappa r a s kappa r b holds subadditive systems the concavity condition does not imply thermodynamical stability of the equilibrium state

a possible deformed algebra and calculus inspired in nonextensive thermostatistics we present a deformed algebra related to the q exponential and the q logarithm functions that emerge from nonextensive statistical mechanics we also develop a q derivative and consistently a q integral for which the q exponential is an eigenfunction the q derivative and the q integral have a dual nature that is also presented

convergence of an adaptive linear estimation algorithm in this work we prove the almost sure convergence of an adaptive linear estimator governed by a stochastic gradient algorithm with decreasing step size in the presence of correlated observations two complementary contributions are added to the famous 1977 ljung theorem first we drop the condition of nondivergence of the algorithm assumed by ljung while that condition can be ensured by adding a barrier the convergence of the suitably bounded algorithm itself is not established even on the basis of ljung theorem here the barrier problem is overcome by proving that it is not necessary for the convergence our second contribution is to generalize the model describing the correlated observations no state space model is used and no linear relationship between the observations and the signal to be estimated needs to be assumed instead we use a decreasing covariance model that agrees with a very wide class of practical applications

a κ entropic approach to the analysis of the fracture problem abstract in order to study the relation between the ohmic resistance measured in a thin conducting ribbon and the length of a transversal cut we employ a one parameter deformed exponential and logarithm that were recently introduced phys rev e 66 2002 056125 in the framework of a generalized statistical mechanics the analytical results have been compared with the data that was experimentally obtained and numerically computed with the boundary element method remarkably the new deformed functions that interpolate between the standard functions and the power law functions allow the best fit of the experimental data to be obtained for a wide range of the cut length

gradient algorithms for designing predictive vector quantizers a predictive vector quantizer pvq is a vector extension of a predictive quantizer it consists of two parts a conventional memoryless vector quantizer vq and a vector predictor two gradient algorithms for designing a pvq are developed in this paper the steepest descent sd algorithm and the stochastic gradient sg algorithm both have the property of improving the quantizer and the predictor in the sense of minimizing the distortion as measured by the average mean squared error the differences between the two design approaches are the period and the step size used in each iteration to update the codebook and predictor the sg algorithm updates once for each input training vector and uses a small step size while the sd updates only once for a long period possibly one pass over the entire training sequence and uses a relatively large step size code designs and tests are simulated for both gauss markov sources and for sampled speech waveforms and the results are compared to codes designed using techniques that attempt to optimize only the quantizer for the predictor and not vice versa

remarks about the tsallis formalism in the present paper the conditions for the validity of the tsallis statistics are analyzed the same has been done following the analogy with the traditional case starting from the microcanonical description of the systems and taking into account their self similarity scaling properties in the thermodynamic limit it is analyzed the necessary conditions for the equivalence of microcanonical ensemble with the tsallis generalization of the canonical ensemble it is shown that the tsallis statistics is appropriate for the macroscopic description of systems with potential scaling laws of the asymptotic accessible states density of the microcanonical ensemble our analysis shows many details of the tsallis s formalism the q expectation values the generalized legendre transformations between the thermodynamic potentials as well as the conditions for its validity having a priori the possibility to estimate the value of the entropic index without the necessity of appealing to the computational simulations or the experiment on the other hand the definition of physical temperature received a modification that differs from the toral result for the case of finite systems we have generalized the microcanonical thermostatistics of gross with the generalization of the curvature tensor for this kind of description

predictive vector quantizer design using deterministic annealing a new approach is proposed for predictive vector quantizer pvq design which is inherently probabilistic and is based on ideas from information theory and analogies to statistical physics the approach effectively resolves three longstanding fundamental shortcomings of standard pvq design the first complication is due to the pvq prediction loop which has a detrimental impact on the convergence and the stability of the design procedure the second shortcoming is due to the piecewise constant nature of the quantizer function which makes it difficult to optimize the predictor with respect to the overall reconstruction error finally a shortcoming inherited from standard vq design is the tendency of the design algorithm to terminate at a locally rather than the globally optimal solution we propose a new pvq design approach that embeds our previous asymptotic closed loop acl approach within a deterministic annealing da framework the overall da acl method profits from its two main components in a complementary way acl is used to overcome the first difficulty and offers the means for stable quantizer design as it provides an open loop design platform yet allows the pvq design algorithm to asymptotically converge to optimization of the closed loop performance objective da simultaneously mitigates or eliminates the remaining design shortcomings its probabilistic framework replaces hard quantization with a differentiable expected cost function that can be jointly optimized for the predictor and quantizer parameters and its annealing schedule allows the avoidance of many poor local optima substantial performance gains over traditional methods have been achieved in the simulations

chebyshev polynomials on symmetric matrices in this paper we evaluate chebyshev polynomials of the second kind on a class of symmetric integer matrices namely on adjacency matrices of simply laced dynkin and extended dynkin diagrams as an application of these results we explicitly calculate minimal projective resolutions of simple modules of symmetric algebras with radical cube zero that are of finite and tame representation type

deformed logarithms and entropies by solving a differential functional equation inposed by the maxent principle we obtain a class of two parameter deformed logarithms and construct the corresponding two parameter generalized trace form entropies generalized distributions follow from these generalized entropies in the same fashion as the gaussian distribution follows from the shannon entropy which is a special limiting case of the family we determine the region of parameters where the deformed logarithm conserves the most important properties of the logarithm and show that important existing generalizations of the entropy are included as special cases in this two parameter class

non linear kinetics underlying generalized statistics abstract the purpose of the present effort is threefold firstly it is shown that there exists a principle that we call kinetical interaction principle kip underlying the non linear kinetics in particle systems independently on the picture kramers boltzmann used to describe their time evolution secondly the kip imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy thirdly the kip allows on one hand to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and on the other hand suggests how we can introduce naturally new distributions finally as a working example of the approach to the non linear kinetics here presented a new non extensive statistics is constructed and studied starting from a one parameter deformation of the exponential function holding the relation f x f x 1

a mechanism to derive multi power law functions an application in the econophysics framework abstract it is generally recognized that economical systems and more in general complex systems are characterized by power law distributions sometime these distributions show a changing of the slope in the tail so that more appropriately they show a multi power law behavior we present a method to derive analytically a two power law distribution starting from a single power law function recently obtained in the frameworks of the generalized statistical mechanics based on the sharma taneja mittal information measure in order to test the method we fit the cumulative distribution of personal income and gross domestic production of several countries obtaining a good agreement for a wide range of data

next state functions for finite state vector quantization in this paper the finite state vector quantization scheme called dynamic finite state vector quantization dfsvq is investigated with regard to its subcodebook construction in dfsvq each input block is encoded by a small codebook called subcodehook which is created from a much larger codebook called supercodebook each subcodebook is constructed by selecting using a reordering procedure a set of appropriate codevectors from the supercodebook the performance of the dfsvq depends on this reordering procedure therefore several reordering procedures are introduced and their performances are evaluated in this paper the reordering procedures that are investigated are based on the conditional histogram of the codevectors index prediction vector prediction nearest neighbor design and frequency usage of the code vectors the performances of the reordering procedures are evaluated by comparing their hit ratios the number of blocks encoded by the subcodebook and their computational complexity experimental results are presented and it is found that the reordering procedure based on the vector prediction performs the best when compared with the other reordering procedures

legendre structure of the thermostatistics theory based on the sharma taneja mittal entropy abstract the statistical proprieties of complex systems can differ deeply for those of classical systems governed by boltzmann gibbs entropy in particular the probability distribution function observed in several complex systems shows a power law behavior in the tail which disagrees with the standard exponential behavior showed by gibbs distribution recently a two parameter deformed family of entropies previously introduced by sharma taneja and mittal stm has been reconsidered in the statistical mechanics framework any entropy belonging to this family admits a probability distribution function with an asymptotic power law behavior in the present work we investigate the legendre structure of the thermostatistics theory based on this family of entropies we introduce some generalized thermodynamical potentials study their relationships with the entropy and discuss their main proprieties specialization of the results to some one parameter entropies belonging to the stm family are presented

vector quantizers and predictive quantizers for gauss markov sources low rate vector quantizers are designed and simulated for highly correlated gauss markov sources and the resulting performance is compared with arnstein s optimized predictive quantizer and with huang and schultheiss optimized transform coder two implementations of vector quantizers are considered full search vector quantizers which are optimal but require large codebook searches and tree searched vector quantizers which are suboptimal but require far less searching the various systems are compared on the basis of performance complexity and generality of design techniques