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

a very hard log space counting class abstract we consider the logarithmic space counting and optimization classes l span l and opt l which are defined analogously to their polynomial time counterparts we obtain complete functions for these three classes in terms of graphs and finite automata we show that l and opt l are both included in nc 2 but that surprisingly span l seems to be a much harder class than l and opt l we demonstrate that span l functions can be computed in polynomial time if and only if p p and all the classes of the polynomial time hierarchy are included in p this result follows from the fact that span l and p are very similar span l p and any function in p can be represented as the difference of a function in fl and a function in span l nevertheless the inclusion p span l would imply nl p np we furthermore investigate restrictions of the classes opt l and span l

on simple oversampled a d conversion in shift invariant spaces it has been found that the quantization error e for a conventional oversampled analog to digital a d conversion behaves like spl par e spl par sup 2 o spl tau sup 2 with respect to the sampling rate spl tau recently conventional a d conversion has been extended to a d conversion based on shift invariant spaces as consequences of such extension it offers rich choices to build a nonideal a d conversion system of high accuracy and low computational complexity as well as reduces the noise sensitivity and computational complexity in digital to analog d a conversion therefore it is necessary to establish the estimate of quantization error for the extended a d conversion based on shift invariant spaces in this paper we introduce a constructive method to establish an estimate of the quantization error as e sup 2 o spl tau sup 2 for oversampled a d conversion in shift invariant spaces meanwhile we demonstrate that the bit rate required to encode the converted digital signal in such a d conversion scheme only increases as the logarithm of the sampling ratio therefore the quantization error is an exponentially decaying function of the bit rate in order to establish such an estimate we need the nonuniform sampling theorem for shift invariant spaces which as the necessary preparation is studied prior to introducing the constructive method

double interpolation for digital to analog conversion interpolative digital to analog converters generate an output that has only a few analog levels they provide fine resolution by oscillating rapidly between these levels in such a manner that the average output represents the value of the applied code here we describe an improved method of interpolating that results in reduced noise in the signal band a theory of the interpolation confirmed by experiments demonstrates that switching between only two levels at 1 3 mhz could provide 16 bit resolution for telephone signals

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 properties of sigma delta modulators with dc inputs new properties of single and double loop sigma delta modulators with constant inputs are derived by exploiting the inherent structure os the output sequences or codewords that the modulators are capable of producing upper bounds are derived on the number of n bit codewords for the single and double loop modulators analytical lower bounds on the mean squared error mse obtainable by any decoder linear or nonlinear in approximating the constant input are also derived optimal nonlinear decoders for constant inputs based on a table lookup approach which operates directly on the nonuniform quantization intervals are considered using simulations is is found that the optimal nonlinear decoders perform better than linear decoders by about 3 and 20 db for the single and double loop modulators respectively a cascade structure specifically for constant inputs is introduced and its corresponding decoding algorithm is derived it is shown that for a fixed latency the mse performance of the cascade structure is 12 db superior and its throughput is twice that of the conventional two stage mash modulator

sigma delta modulation with leaky integration and constant input exact descriptions of the behavior of quantization noise for single loop multistage cascade and multiloop sigma delta modulators for a variety of input signals have been found during recent years under the assumption of ideal integration the techniques used to solve the ideal integrator case do not easily extend to the more realistic model of a leaky integrator sigma delta in this paper a dynamical system representation for the leaky integrator sigma delta with a constant input is developed several properties of the resulting piecewise monotone and piecewise linear transformation t on the interval

shape preserving surfaces with constraints on tension parameters the c1 rational bi cubic local interpolation schemes are presented for the shape preservation of convex monotone and positive surface data the shape of the surface is controlled locally with the help of eight tension parameters over each rectangular patch data dependent constraints are developed on half of the tension parameters to preserve the intrinsic shapes of the surface data the remaining ones are unconstrained thus free to be used to obtain the smoothest surface

chain codes and their linear reconstruction filters a differential chain code implementation of the basic eight directional and n ring chain code for the encoding of line drawings is considered several distortion and rate criteria are considered and they are evaluated for the special case of encoding infinite straight lines these results indicate that the differential code outperforms ring codes and the ring delta code using a linear decoding filter for reconstructing a smoothed approximation to the original line is also considered optimal reconstruction filters for infinite straight lines at various angles are derived an angle invariant reconstruction filter that minimizes the expected square distortion taken over all angles is derived the overall performance using this filter is close to that of the optimum filter furthermore the impulse response of the angle invariant filter is approximately rectangular which can lead to simple hardware implementations

inverse halftoning and kernel estimation for error diffusion two different approaches in the inverse halftoning of error diffused images are considered the first approach uses linear filtering and statistical smoothing that reconstructs a gray scale image from a given error diffused image the second approach can be viewed as a projection operation where one assumes the error diffusion kernel is known and finds a gray scale image that will be halftoned into the same binary image two projection algorithms viz minimum mean square error mmse projection and maximum a posteriori probability map projection that differ on the way an inverse quantization step is performed are developed among the filtering and the two projection algorithms map projection provides the best performance for inverse halftoning using techniques from adaptive signal processing we suggest a method for estimating the error diffusion kernel from the given halftone this means that the projection algorithms can be applied in the inverse halftoning of any error diffused image without requiring any a priori information on the error diffusion kernel it is shown that the kernel estimation algorithm combined with map projection provide the same performance in inverse halftoning compared to the case where the error diffusion kernel is known

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

refined error analysis in second order spl sigma spl delta modulation with constant inputs although the technique of sigma delta spl sigma spl delta modulation is well established in practice for performing high resolution analog to digital a d conversion theoretical analysis of the error between the input signal and the reconstructed signal has remained partial for modulators of order higher than 1 the only rigorous error analysis currently available that matches practical and numerical simulation results is only applicable to a very special configuration namely the standard and ideal k bit k loop spl sigma spl delta modulator moreover the error measure involves averaging over time as well as possibly over the input value at the second order it is known in practice that the mean squared error decays with the oversampling ratio spl lambda at the rate o spl lambda sup 5 in this paper we introduce two new fundamental results in the case of constant input signals we first establish a framework of analysis that is applicable to all second order modulators provided that the built in quantizer has uniformly spaced output levels and that the noise transfer function has its two zeros at the zero frequency in particular this includes the one bit case a rigorous and deterministic analysis of which is still not available this generalization has been possible thanks to the discovery of the mathematical tiling property of the state variables of such modulators the second aspect of our contribution is to perform an instantaneous error analysis that avoids infinite time averaging until now only an o spl lambda sup 4 type error bound was known to hold in this setting under our generalized framework we provide two types of squared error estimates one that is statistically averaged over the input and another that is valid for almost every input in the sense of lebesgue measure in both cases we improve the error bound to o spl lambda sup 4 5 up to a logarithmic factor for a general class of modulators including some specific ones that are covered in this paper in detail in the particular case of the standard and ideal two bit double loop configuration our methods provide a previously unavailable instantaneous error bound of o spl lambda sup 5 again up to a logarithmic factor

approximating a bandlimited function using very coarsely quantized data improved error estimates in sigma delta modulation sigma delta quantization is a method of representing bandlimited signals by 0 1 sequences that are computed from regularly spaced samples of these signals as the sampling density convolving these one bit sequences with appropriately chosen kernels produces increasingly close approximations of the original signals this method is widely used for analogto digital and digital to analog conversion because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine resolution quantization we present examples of how tools from number theory and harmonic analysis are employed in sharpening the error estimates in sigma delta quantization

fir differentiators for quantized signals the impact of quantization noise on a signal whose rate is to be estimated using a fir differentiator is analyzed concentrating on the important constant rate case in order that the filter be optimized for systems with low frequency rates of change formulae for the mean squared error of the filter the corresponding spectral characteristics and general formulae governing the filter coefficients are derived the characteristics of four specific differentiators including a representative wideband differentiator are examined and compared it is shown that a differentiator that is optimum in terms of its attenuation of white noise can also be considered optimum with respect to quantization noise attenuation in certain circumstances an elegant relationship is derived between worst case rms error and the fractional value of the rate at which this error occurs minimization of this worst case mean squared error is shown to be achieved with a simple differentiator however the corresponding average error is poor and a simple nonlinear filter that minimizes the worst ease error while retaining a similar average mean squared error to that of the optimum differentiator is proposed the equivalence between fir differentiators and the decoders used in single loop sigma delta modulators is also highlighted

the complexity of bayesian networks specified by propositional and relational languages abstract we examine the complexity of inference in bayesian networks specified by logical languages we consider representations that range from fragments of propositional logic to function free first order logic with equality in doing so we cover a variety of plate models and of probabilistic relational models we study the complexity of inferences when network query and domain are the input the inferential and the combined complexity when the network is fixed and query and domain are the input the query data complexity and when the network and query are fixed and the domain is the input the domain complexity we draw connections with probabilistic databases and liftability results and obtain complexity classes that range from polynomial to exponential levels we identify new languages with tractable inference and we relate our results to languages based on plates and probabilistic relational models

the pros and cons of democracy we introduce the concept of democracy in which the individual bits in a coarsely quantized representation of a signal are all given equal weight in the approximation to the original signal we prove that such democratic representations cannot achieve the same accuracy as optimal nondemocratic schemes

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

a saturation analysis of error spectrum shaping quantizers error spectrum shaping ess quantizers are analyzed in the context of realistic fir reconstruction filters and nonzero probability of saturation conditions for stable operation are derived and a suboptimal but effective and computationally efficient design method is proposed excellent agreement is obtained between the analysis and simulations even under these more stringent constraints ess quantizers are shown to achieve significant signal to quantization noise ratio sqnr gains when quantizing oversampled gaussian sources

error estimation of a kind of rational spline this paper deals with the approximation properties of a kind of rational spline with linear denominator when the function being interpolated is c 3 in an interpolating interval error estimate expressions of interpolating functions are derived convergence is established the optimal error coefficient c i is proved to be symmetric about the parameters of the rational interpolation and it is bounded finally the precise jump measurements of the second derivatives of the interpolating function at the knots are given

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

positivity preserving interpolation of positive data by rational cubics this work is a contribution towards the graphical display of data when it is positive the data are required to be represented in such a way that its visual display looks smooth and pleasant its positive shape is preserved everywhere and the computation cost is economical a c 1 piecewise rational cubic function in its most general form has been utilized for this objective the method is implemented for the 1d data initially and then it is extended to an interpolating rational bicubic form for the data arranged over a rectangular grid simple sufficient conditions are developed on the free parameters in the description of the rational function to visualize the positive data in the form of positive curves and surfaces

multiloop sigma delta quantization it is shown that for a multiloop sigma delta modulator driven by a dc input the statistical behavior described by long term time averages of the error process from an inner multibit quantizer is consistent with that of a signal independent white uniform noise the same result also holds for sinusoidal inputs provided that the modulator has three or more loops this behavior is exploited to obtain trade offs between asymptotic system performances and oversampling ratios for several sigma delta modulator decoding filter pairs for an l loop modulator with a simple sinc sup l 1 decoding filter it is shown that doubling the oversampling ratio can result in an improvement of 3 01 2l 1 db in the signal to quantization noise ratio

on the robustness of single loop sigma delta modulation sigma delta modulation a widely used method of analog to digital a d signal conversion is known to be robust to hardware imperfections i e bit streams generated by slightly imprecise hardware components can be decoded comparably well we formulate a model for robustness and give a rigorous analysis for single loop sigma delta modulation applied to constant signals dc inputs for n time cycles with an arbitrary small enough initial condition u sub o and a quantizer that may contain an offset error the mean square error mse of any decoding scheme for this quantizer with u sub o and the offset error known is bounded below by 1 96n sup 3 we also determine the asymptotically best possible mse as n spl rarr spl infin for perfect decoding when u sub o 0 and u sub o 1 2 the robustness result is the upper bound that a triangular linear filter decoder with both u sub o and the offset error unknown achieves an mse of 40 3n sup 3 these results establish the known result that the o 1 n sup 3 decay of the mse with n is optimal in the single loop case under weaker assumptions than previous analyses and show that a suitable linear decoder is robust against offset error these results are obtained using methods from number theory and fourier analysis

one bit sigma delta quantization with exponential accuracy one bit quantization is a method of representing bandlimited signals by 1 sequences that are computed from regularly spaced samples of these signals as the sampling density convolving these one bit sequences with appropriately chosen filters produces increasingly close approximations of the original signals this method is widely used for analog to digital and digital to analog conversion because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine resolution quantization however unlike fine resolution quantization the accuracy of one bit quantization is not well understood a natural error lower bound that decreases like 2 can easily be given using information theoretic arguments yet no one bit quantization algorithm was known with an error decay estimate even close to exponential decay in this paper we construct an infinite family of one bit sigma delta quantization schemes that achieves this goal in particular using this family we prove that the error signal for bandlimited signals is at most o 2 07 2003 wiley periodicals inc

c 2 positivity preserving rational interpolation splines in one and two dimensions abstract a class of rational quartic cubic interpolation spline with two local control parameters is presented which can be c 2 continuous without solving a linear system of consistency equations for the derivative values at the knots the effects of the local control parameters on generating interpolation curves are illustrated for c 2 interpolation the given interpolant can locally reproduce quadratic polynomials and has o h 2 or o h 3 convergence simple schemes for the c 2 interpolant to preserve the shape of 2d positive data are developed moreover based on the boolean sum of quintic interpolating operators a class of bi quintic partially blended rational quartic cubic interpolation surfaces is also constructed the given interpolation surface provides four local control parameters and can be c 2 continuous without using the second or higher mixed partial derivatives on a rectangular grid simple sufficient data dependent constraints are also derived on the local control parameters to preserve the shape of a 3d positive data set arranged over a rectangular grid

granular quantization noise in the first order delta sigma modulator a unified approach to analyzing the granular quantization error of the first order delta sigma modulator is presented the approach handles many of the previously analyzed input sequences in addition to a large class of new input sequences by averaging over the arbitrarily small amount of circuit noise assumed to be present at the analog input to the delta sigma modulator a simple expression for the autocorrelation of the quantization error is derived each term in the expression is formally equal to the quantization error of a nonoverloaded uniform quantizer operating upon a finite partial sum of consecutive input sequence samples hence existing results concerning uniform quantizers are directly applicable in evaluating the autocorrelation expression for specific input sequences the theory is also applicable to deterministic input sequences and has been applied to obtain a new closed form result for sinusoidal input sequences ergodic results which assert that under mild conditions the autocorrelation equals the time average autocorrelation in probability are presented

the structure of the limit cycles in sigma delta modulation it is shown that when the input to a sigma delta modulator is a dc level that can be expressed as a rational number b a when normalized with respect to the quantizer step the output bit string is periodic with a period that is multiple of the denominator a on the basis of number theory the structure of these cycles for single loop modulators is determined and the noise contribution is computed around such levels the noise has two peaks for which the maximum value and the width are proportional to the relative signal bandwidth and to the inverse of the period of the cycle respectively the effect of the limit cycles on the performance of the a d and d a converters using sigma delta modulation is discussed a comparison between single loop and double loop modulators from the point of view of this phenomena is made

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

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

functions computable in polynomial space consider nondeterministic polynomial time turing machine that on input x outputs a 3 3 matrix with entries from 1 0 1 on each of its paths define the function f where f x is the upper left entry in the product of all these matrices in an order of the paths to be made precise below we show that the class of functions f computable as just described is exactly the class fpspace of integer valued functions computable by polynomial space turing machines along the way we obtain characterizations of fpspace in terms of arithmetic circuits and straight line programs

convexity preserving piecewise rational quartic interpolation a family of piecewise rational quartic interpolants is given identified uniquely by the value of a tension parameter lambda i each interpolant of the family can be c 2 spline without solving a linear or nonlinear system of consistency equations for the derivative values at the knots the interpolant can preserve the local convexity concavity properties of the given data a proper choice of lambda i to guarantee shape preservation is given a convergence analysis establishes an error bound in terms of lambda i and shows that o h 3 accuracy is obtained for c 2 continuity several examples are supplied to support the practical value of the method

polynomial space counting problems the classes of functions textit pspace and natural textit pspace that are analogous to the class p are defined functions in textit pspace count the number of accepting computations of a nondeterministic polynomial space bounded turing machine and functions in natural textit pspace count the number of accepting computations of nondeterministic polynomial space bounded turing machines that on each computation path make at most a polynomial number of nondeterministic choices in contrast to what is known about p exact characterizations of both textit pspace and natural textit pspace are found in particular pspace fspace the class of functions computable in polynomial space and natural pspace fspace poly the class of functions computable in polynomial space with output length bounded by a polynomial both textit pspace and natural textit pspace can be characterized by counting problems related to alternating turing machines both te

performance in waveform quantization the quantization noise and dynamic range of commonly known analog digital converters or modulators are computed from the equations derived by analyzing the generalized quantizer which deploys integer coefficient transversal filters the results show that signal to noise ratio s n curves for high sampling frequency do not have a constant slope of one at a low signal level the s n curve has a slope of 1 5 from the results equations to approximate s n curves in the linear region are derived they can be used to plot a s n curve for any sampling frequency and any number of quantization steps

a fast block based nonlinear decoding algorithm for spl sigma spl delta modulators previous work has established that the digital output of a spl sigma spl delta modulator as an a d converter contains more information about the analog input than is extracted with conventional linear filtering under reasonable mathematical assumptions optimal nonlinear decoding of the digital output can achieve significantly larger signal to noise ratios than linear filtering however the hitherto proposed decoding algorithms only demonstrate conceptual feasibility and are impractical from a computational point of view we present a new block based decoding algorithm that like previous work employs projections onto convex sets the algorithm owes its speed to a change of projection norm an accelerated convergence scheme and a decimation like subsampling it is on the order of 10 sup 4 10 sup 5 times faster than one previously published algorithm for typical parameter values and about 2 10 times slower than linear decoding the new algorithm is applicable to all currently popular spl sigma spl delta architectures

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

deterministic analysis of oversampled a d conversion and decoding improvement based on consistent estimates discusses the deterministic analysis of oversampled a d conversion adc the properties derivable from such an analysis and the consequences on reconstruction using nonlinear decoding given a band limited input x producing a quantized version c the authors consider the set of all input signals that are band limited and produce c they call any element of this set a consistent estimate of x regardless of the type of encoder simple predictive or noise shaping they show that this set is convex and as a consequence any nonconsistent estimate can be improved they also show that the classical linear decoding estimates are not necessarily consistent numerical tests performed on simple adc single loop and multiloop spl sigma spl delta modulation show that consistent estimates yield a mean square error mse that decreases asymptotically with the oversampling ratio faster than the linear decoding mse by approximately 3 db octave this implies an asymptotic mse of the order of spl oscr r sup spl minus 2n 2 instead of spl oscr r sup spl minus 2n 1 in linear decoding where r is the oversampling ratio and n the order of the modulator methods of improvement of nonconsistent estimates based on the deterministic knowledge of the quantized signal are proposed for simple adc predictive adc single loop and multiloop spl sigma spl delta modulation

reaction systems and extremal combinatorics properties extremal combinatorics is the study of the size that a collection of objects must have in order to certainly satisfy a given property reaction systems are a recent formalism for computation inspired by chemical reactions this work is a first contribution to the study of the behavior of large reaction systems by means of extremal combinatorics we define several different properties that capture some basic and dynamical behaviors of a reaction system and we prove that they must necessarily be satisfied if the system is large enough explicit bounds and formulae are also provided

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

error diffused image compression using a binary to gray scale decoder and predictive pruned tree structured vector quantization the authors consider data compression of binary error diffused images the original contribution is using nonlinear filters to decode error diffused images to compress them in the gray scale domain this gives better image quality than directly compressing the binary images their method is of low computational complexity and can work with any halftoning algorithm

positivity preserving rational bi cubic spline interpolation for 3d positive data this paper deals with the shape preserving interpolation problem for visualization of 3d positive data a required display of 3d data looks smooth and pleasant a rational bi cubic function involving six shape parameters is presented for this objective which is an extension of piecewise rational function in the form of cubic quadratic involving three shape parameters simple data dependent constraints for shape parameters are derived to conserve the inherited shape feature positivity of 3d data remaining shape parameters are left free for designer to modify the shape of positive surface as per industrial needs the interpolant is not only local c 1 but also it is a computationally economical in comparison with existing schemes several numerical examples are supplied to support the worth of proposed interpolant

complexity of finite horizon markov decision process problems controlled stochastic systems occur in science engineering manufacturing social sciences and many other cntexts if the systems is modeled as a markov decision process mdp and will run ad infinitum the optimal control policy can be computed in polynomial time using linear programming the problems considered here assume that the time that the process will run is finite and based on the size of the input there are mny factors that compound the complexity of computing the optimal policy for instance there are many factors that compound the complexity of this computation for instance if the controller does not have complete information about the state of the system or if the system is represented in some very succint manner the optimal policy is provably not computable in time polynomial in the size of the input we analyze the computational complexity of evaluating policies and of determining whether a sufficiently good policy exists for a mdp based on a number of confounding factors including the observability of the system state the succinctness of the representation the type of policy even the number of actions relative to the number of states in almost every case we show that the decision problem is complete for some known complexity class some of these results are familiar from work by papadimitriou and tsitsiklis and others but some such as our pl completeness proofs are surprising we include proofs of completeness for natural problems in the as yet little studied classes np pp

on the stability of sigma delta modulators a framework for stability analysis of sigma delta modulators is presented it is argued that limit cycles for constant inputs are natural objects to investigate in this context a number of analytical and approximate techniques to aid the stability analysis of the double loop and interpolative modulators are discussed and lead to ways of improving the design that explicitly take stability into account

positive data modeling using spline function a rational cubic function with two parameters has been constructed to visualize the positive data the main focus of the work is the representation of the data in such a way that its view looks smooth and attractive in the first step simple data dependent constraints are derived on the parameters in the description of the rational cubic function to visualize the shape of positive data then it is extended to a rational bi cubic partially blended functions coons patches and derived constraints on parameters to visualize the shape of positive surface data the developed scheme is locally positive and economical the approximation order of rational cubic spline function is oh i 3

breaking the feedback loop of a class of spl sigma spl delta a d converters spl sigma spl delta modulation is the currently successful technique used to perform high resolution analog to digital conversion in spite of its practical success its theoretical signal analysis has remained limited because a spl sigma spl delta modulator contains of a feedback loop that includes a nonlinear operation i e the amplitude discretization or quantization the feedback allows us to use oversampling to compensate for the limitations of the quantizer in resolution and in precision which are typical of analog circuits however because of the lack of signal analysis it is still not clear how much resolution of conversion can be gained as a function of the oversampling we show that for a large class of spl sigma spl delta modulators the feedback loop theoretically yields an equivalent feedforward signal flow graph at least for constant inputs this is possible thanks to remarkable modulo properties of these modulators this equivalence can be asymptotically extrapolated to time varying inputs with increasing oversampling although the exact components of the equivalent graph are not currently known in general the theoretical structure of the feedforward graph is sufficient to point out misconceptions in the current knowledge on the final resolution of an nth order spl sigma spl delta modulator specifically except when the modulator is ideal the global resolution of conversion increases by n bits per octave of oversampling instead of the currently believed rate of n 1 2 bits octave

analysis of dc input response for a class of one bit feedback encoders a class of one bit feedback encoders that includes sigma delta modulation encoders data modulation encoders and some dpcm encoders is defined for any encoder from the class it is shown that the encoder output sequence in response to a dc input sequence is a typical sequence for a certain strict sense stationary ergodic process whose power spectral density is determined the result is applied to give the performance of an analog to digital conversion system operated at high sampling rates

shape preserving piecewise rational interpolant with quartic numerator and quadratic denominator an explicit representation of a piecewise rational interpolant with quartic numerator and quadratic denominator is presented for positivity data monotone data and convex data the shape preserving properties of the interpolant are given the interpolant is c 2 continuous spline with a shape parameter w i on each subinterval the values of w i to guarantee shape preservation are estimated a convergence analysis establishes an error bound in terms of w i and shows that the interpolant is o h 2 or o h 3 accurate several examples are supplied to support the practical value of the given interpolation method

ranks of strictly minimal reaction systems induced by permutations abstract reaction system is a computing model inspired by the biochemical interaction taking place within the living cells various extended or modified frameworks motivated by biological physical or purely mathematically considerations have been proposed and received significant amount of attention notably in the recent years this study however takes after particular early works that concentrated on the mathematical nature of minimal reaction systems in the context free basic framework and motivated by a recent result on the sufficiency of strictly minimal reaction systems to simulate every reaction system this paper focuses on the largest reaction system rank attainable by strictly minimal reaction systems where the rank pertains to the minimum size of a functionally equivalent reaction system precisely we provide a very detailed study for specific strictly minimal reaction system induced by permutations up to the quaternary alphabet along the way we obtain a general result about reaction system rank for cartesian product of functions specified by reaction systems

ancestors descendants and gardens of eden in reaction systems this paper analyses several problems related to finding and counting ancestor and descendant states as well as gardens of eden i e states without predecessors in reaction systems the focus is on the complexity of finding and counting preimages and ancestors that are minimal with respect to cardinality it turns out that the problems concerning gardens of eden seem to require the presence of an np oracle to be solved all the problems studied are intractable with a complexity that ranges form fp np log n to fpspace poly

positive quartic monotone quintic c 2 spline interpolation in one and two dimensions abstract this paper is concerned with shape preserving interpolation of discrete data by polynomial splines we show that positivity can be always preserved by quartic c2 splines and monotonicity by quintic c2 splines this is proved for one dimensional interpolation as well as for two dimensional interpolation on rectangular grids

a zero crossing algorithm for the estimation of the frequency of a single sinusoid in white noise presents a new algorithm for the estimation of the frequency of a single sinusoid in white noise based on the computation of the interval between zero crossings it is shown that for a high signal to noise ratio the output error spectrum is concentrated in the high frequency region theoretically the desired precision can be achieved using a proper low pass filter the spectrum of the error due to the interpolation of the zero crossing is computed simulation results show that a precision of 10 sup 7 10 sup 8 can be obtained with modest computational effort

high resolution distributed sampling of bandlimited fields with low precision sensors the problem of sampling a discrete time sequence of spatially bandlimited fields with a bounded dynamic range in a distributed communication constrained processing environment is studied a central unit having access to the data gathered by a dense network of low precision sensors is required to reconstruct the field snapshots to maximum accuracy both deterministic and stochastic field models are considered for stochastic fields results are established in the almost sure sense the feasibility of having a flexible tradeoff between the oversampling rate sensor density and the analog to digital converter adc precision while achieving an exponential accuracy in the number of bits per nyquist interval per snapshot is demonstrated this exposes an underlying conservation of bits principle the bit budget per nyquist interval per snapshot the rate can be distributed along the amplitude axis sensor precision and space sensor density in an almost arbitrary discrete valued manner while retaining the same exponential distortion rate characteristics

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

the complexity of computing the size of an interval given a p order a over a universe of strings i e a transitive reflexive antisymmetric relation such that if x y in a then x is polynomially bounded by y an interval size function of a returns for each string x in the universe the number of strings in the interval between strings b x and t x with respect to a where b x and t x are functions that are polynomial time computable in the length of x by choosing sets of interval size functions based on feasibility requirements for their underlying p orders we obtain new characterizations of complexity classes we prove that the set of all interval size functions whose underlying p orders are polynomial time decidable is exactly mp we show that the interval size functions for orders with polynomial time adjacency checks are closely related to the class fpspace poly indeed fpspace poly is exactly the class of all nonnegative functions that are an interval size function minus a polynomial time computable function we study two important functions in relation to interval size functions the function mdiv maps each natural number n to the number of nontrivial divisors of n we show that mdiv is an interval size function of a polynomial time decidable partial p order with polynomial time adjacency checks the function mmonsat maps each monotone boolean formula f to the number of satisfying assignments of f we show that mmonsat is an interval size function of a polynomial time decidable total p order with polynomial time adjacency checks finally we explore the related notion of cluster computation

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

the complexity of optimal small policies we investigate the complexity of problems concerned with partially observable markov decision processes which are run for a finite number of steps under small policies the calculation of the expected sum of rewards of a process under a small policy is shown to be complete for the complexity class pp a class which lies intermediate between np and pspace optimal small policy computation is shown to be complete for nppp the latter contrasts results of papadimitriou and tsitsiklis papadimitriou and tsitsiklis 1987 who showed that this problem is pspace complete if no assumptions about the representability of the policy are made and that it is p complete for fully observable processes

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 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

modulo sigma delta modulation a modulo sigma delta modulator is introduced and the behavior of the quantization error is derived the system consists of a modulo limiter followed by a sigma delta modulator the limiter confines the input to the no overload region of the sigma delta modulator and the modulo arithmetic performed by the limiter is amenable to recently developed techniques for the exact analysis of quantizer error behavior in sigma modulators with bounded inputs the quantization error behavior is derived for a modulator driven by a quasi stationary random process the limit distribution and the power of the quantization error are found except for some singular cases the normalized quantization error is uniformly distributed in 1 2 1 2 the power spectrum and the autocorrelation function of the quantization error with a causal arma p q process input are also derived it is shown that the quantization noise is white when the input is a random process with stationary independent increments simulation results support the theoretical analysis

reconstruction of oversampled band limited signals from spl sigma spl delta encoded binary sequences considers the application of spl sigma spl delta modulators to analog to digital conversion the authors have previously shown that for constant input signals optimal nonlinear decoding can achieve large gains in signal to noise ratio snr over linear decoding the present paper shows a similar result for band limited input signals the new nonlinear decoding algorithm is based on projections onto convex sets pocs and alternates between a time domain operation and a band limitation to find a signal invariant under both the time domain operation results in a quadratic programming problem the band limitation can be based on singular value decomposition of a certain matrix the authors show simulation results for the snr performance of a pocs based decoder and a linear decoder for the single loop double loop and two stage spl sigma spl delta modulators and for a specific fourth order interpolative modulator depending on the modulator and the oversampling ratio improvements in snr of up to 10 20 db can be achieved

rate distortion performance in coding bandlimited sources by sampling and dithered quantization the rate distortion characteristics of a scheme for encoding continuous time band limited stationary sources with a prescribed band is considered in this coding procedure the input is sampled at nyquist s rate or faster the samples undergo dithered uniform or lattice quantization using subtractive dither and the quantizer output is entropy coded the rate distortion performance and the tradeoff between the sampling rate and the quantization accuracy is investigated utilizing the observation that the coding scheme is equivalent to an additive noise channel it is shown that the mean square error of the scheme is fixed as long as the product of the sampling period and the quantizer second moment is kept constant while for a fixed distortion the coding rate generally increases when the sampling rate exceeds the nyquist rate finally as the lattice quantizer dimension becomes large the equivalent additive noise channel of the scheme tends to be white gaussian and both the rate and the distortion performance become invariant to the sampling rate

noise reduction in oversampled filter banks using predictive quantization we introduce two methods for quantization noise reduction in oversampled filter banks these methods are based on predictive quantization noise shaping or linear prediction it is demonstrated that oversampled noise shaping or linear predictive subband coders are well suited for subband coding applications where for technological or other reasons low resolution quantizers have to be used in this case oversampling combined with noise shaping or linear prediction improves the effective resolution of the subband coder at the expense of increased rate simulation results are provided to assess the achievable quantization noise reduction and resolution enhancement and to investigate the rate distortion properties of the proposed methods

quantization noise spectra several results describing the behavior of quantization noise in a unified and simplified manner are discussed exact formulas for quantizer noise spectra are developed they are applied to a variety of systems and inputs including scalar quantization pcm dithered pcm sigma delta modulation dithered sigma delta modulation two stage sigma delta modulation and second order sigma delta modulation