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

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

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

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

on constrained randomized quantization randomized dithered quantization is a method capable of achieving white reconstruction error independent of the source dithered quantizers have traditionally been considered within their natural setting of uniform quantization in this paper we extend conventional dithered quantization to nonuniform quantization via a subterfage dithering is performed in the companded domain closed form necessary conditions for optimality of the compressor and expander mappings are derived for both fixed and variable rate randomized quantization numerically mappings are optimized by iteratively imposing these necessary conditions the framework is extended to include an explicit constraint that deterministic or randomized quantizers yield reconstruction error that is uncorrelated with the source surprising theoretical results show direct and simple connection between the optimal constrained quantizers and their unconstrained counterparts numerical results for the gaussian source provide strong evidence that the proposed constrained randomized quantizer outperforms the conventional dithered quantizer as well as the constrained deterministic quantizer moreover the proposed constrained quantizer renders the reconstruction error nearly white in the second part of the paper we investigate whether uncorrelated reconstruction error requires random coding to achieve asymptotic optimality we show that for a gaussian source the optimal vector quantizer of asymptotically high dimension whose quantization error is uncorrelated with the source is indeed random thus random encoding in this setting of rate distortion theory is not merely a tool to characterize performance bounds but a required property of quantizers that approach such bounds

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

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

semisystolic incremental realization of fir digital filters using ternary arithmetic a fully incremental realization of finite impulse response fir filters is presented in which ternary incremental representations are used for all the signals in the structure and the impulse response delta modulation dm is used for this purpose allowing a considerable simplification of the arithmetic involved in the filtering operations simulation showed that the proposed dm adder has lower mean square error mse and reduced processing delay with respect to existing dm adders while demonstrating the feasibility of the proposed modular incremental realization for fir filters

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

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

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

the approximate solution of a class of fredholm integral equations with a weakly singular kernel a method for finding the numerical solution of a weakly singular fredholm integral equation of the second kind is presented the taylor series is used to remove singularity and legendre polynomials are used as a basis furthermore the legendre function of the second kind is used to remove singularity in the cauchy type integral equation the integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the cauchy principal value sense without using numerical quadratures four examples are given to show the accuracy of the method

the fixed point property for mappings admitting a center abstract we introduce a class of nonlinear continuous mappings in banach spaces which allow us to characterize the banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping later on we give an application to the existence of zeroes for certain kinds of accretive operators

a theory of nonsubtractive dither a detailed mathematical investigation of multibit quantizing systems using nonsubtractive dither is presented it is shown that by the use of dither having a suitably chosen probability density function moments of the total error can be made independent of the system input signal but that statistical independence of the error and the input signals is not achievable similarly it is demonstrated that values of the total error signal cannot generally be rendered statistically independent of one another but that their joint moments can be controlled and that in particular the error sequence can be rendered spectrally white the properties of some practical dither signals are explored and recommendations are made for dithering in audio video and measurement applications the paper collects all of the important results on the subject of nonsubtractive dithering and introduces important new ones with the goal of alleviating persistent and widespread misunderstandings regarding the technique

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

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

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

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

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

comb based decimation filters for sigma delta a d converters novel schemes and comparisons this paper proposes several novel decimation filtering schemes tailored to the design of sigmadelta analog to digital a d converters on the one hand it extends the class of modified sinc filters introduced by lo presti in order to obtain decimation filters capable of decoding high order sigmadelta modulators say up to the fifth order on the other hand it introduces the class of sharpened modified comb filters smcf aimed at increasing the rejection of the sigmadelta quantization noise around the folding bands and reducing the passband drops of the designed decimation filters with respect to the classic comb structures design criteria leading to optimized structures are described the paper compares classic comb filters with modified and smcf in terms of performance metrics widely adopted for comb filters finally it derives the maximum sigmadelta modulator order decodable by any proposed filter cell in the class of the smcfs

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

lax pair darboux transformation and nth order rogue wave solutions for a 2 1 dimensional heisenberg ferromagnetic spin chain equation abstract a new 2 1 dimensional heisenberg ferromagnetic spin chain equation is investigated which can be used to describe magnetic soliton excitations in two dimensional space fields and a time field the lax pair of the equation is first constructed based on the lax pair initial seed solution and darboux transformation the analytic first second and third order rogue wave solutions are obtained and a general expression of the n th order n 3 rogue wave solutions is presented the impacts of the system parameters on the rogue waves are demonstrated through numerical visualization method

weakly singular functional integral equation in infinite dimensional banach spaces the existence of l p w e e to be a banach space and w be a bounded closed subset in the euclidean space r n of a weakly singular functional integral equation has been established the notation of measure of noncompactness and monch s fixed point theorem are the main tools in our analysis

statistics of the binary quantizer error in single loop sigma delta modulation with white gaussian input representations and statistical properties of the process e defined by e n 1 e n n are given here u u b sign u m and n n 0 is gaussian white noise the process e represents the binary quantizer error in a model for single loop sigma delta modulation the innovations variables are found and the existence and uniqueness of an invariant probability measure ergodicity properties as well as the existence of the exponential moment with respect to the invariant probability are proved using markov process theory we consider also e as a random perturbation for small values of the variance of n of the orbits of sn 1 sn here sn has the uniform invariant distribution on the interval m h m b analytical approximations to the structure of the power spectrum of e are obtained using a linear prediction in terms of the innovations variables and the perturbation approach

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

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

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

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

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

a digital requantizer with shaped requantization noise that remains well behaved after nonlinear distortion a major problem in oversampling digital to analog converters and fractional n frequency synthesizers which are ubiquitous in modern communication systems is that the noise they introduce contains spurious tones the spurious tones are the result of digitally generated quantized signals passing through nonlinear analog components this paper presents a new method of digital requantization called successive requantization special cases of which avoids the spurious tone generation problem sufficient conditions are derived that ensure certain statistical properties of the quantization noise including the absence of spurious tones after nonlinear distortion a practical example is presented and shown to satisfy these conditions

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

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

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

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

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

extracting state information form a quantized output record abstract we consider the problem of manipulating the input to a discrete time state space linear system with the goal of obtaining information at each time about the system s current state from a record of past quantized measurements of the system s output we find that if the system is not excessively unstable there exist feedback control strategies that allow one to make an asymtotically perfect determination of the current stage based on the output records that result even if the system is too unstable to apply such strategies there are feedback control laws that make the system s output record more informative about the system s state evolution than one might expect in deriving these control laws we regard quantized measurements of real numbers more as partial observations than as strict approximations and employ techniques from information theory and the theory of markov chains with countable state spaces

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

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

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

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

analytic study on a 2 1 dimensional nonlinear schrodinger equation in the heisenberg ferromagnetism in this paper a 2 1 dimensional nonlinear schrodinger equation for a 2 1 dimensional heisenberg ferromagnetic spin chain with the bilinear and anisotropic interactions is investigated via the hirota method and symbolic computation bilinear forms and multi soliton solutions are derived the one two and three solitons are analyzed graphically and we find the amplitudes and widths of the two and three solitons keep invariant after each interaction the bell shape one soliton as well as parallel crossed two and three solitons are respectively observed through the asymptotic analysis expressions which denote the two solitons before and after the interactions are obtained and interactions between the two solitons are proved to be elastic

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

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

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

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

on a bifurcation structure mimicking period adding in this work we investigate a piecewise linear discontinuous scalar map defined on three partitions this map is specifically constructed in such a way that it shows a recently discovered bifurcation scenario in its pure form owing to its structure on the one hand and the similarities to the nested period adding scenario on the other hand we denoted the new bifurcation scenario as nested period incrementing bifurcation scenario the new bifurcation scenario occurs in several physical and electronical systems but usually not isolated which makes the description complicated by isolating the scenario and using a suitable symbolic description for the asymptotically stable periodic orbits we derive a set of rules in the space of symbolic sequences that explain the structure of the stable periodic domain in the parameter space entirely hence the presented work is a necessary step for the understanding of the more complicated bifurcation scenarios mentioned above

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

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

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

active probing for information in control systems with quantized state measurements a minimum entropy approach in this paper the effect of state quantization in scaler discrete time linear control systems is studied by analyzing the system as a partially observed stochastic system the problem of optimal state information gathering and filtering is investigated using information theoretic measures and formulating the state estimation problem as an entropy optimization problem the active probing effect of the feedback control is thoroughly studied optimal feedback controls which minimize various types of entropy costs are determined and it is shown that this problem is equivalent to an optimal control problem for a controlled markov chain

noise aided processing revisiting ditheringin a spl sigma spl delta quantizer in this paper we show that a particular spl sigma spl delta quantizer which is a strongly nonlinear dynamical system can provide noise enhanced processing effects in some sense in particular we will show that quantizing a sine which means adding a small amount of noise in the input can improve the quantization process in particular the output local signal to noise ratio snr exhibits a maximum when plotted against the noise amplitude

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

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

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

exterior differential expression of the 1 1 dimensional nonlinear evolution equation with lax integrability abstract a 1 1 dimensional nonlinear evolution equation with lax integrability is investigated in this paper with the aid of differential forms and exterior differentials equivalent definition of the lax integrability is given relation between the lax integrability and complete integrability is discussed geometric interpretation of the lax equation is presented gauge transformation and darboux transformation are restated in terms of the differential forms and exterior differentials if i s is a darboux transformation between two lax equations the system that the matrix s should satisfy is given where is the spectral parameter and i is the identity matrix the completely integrable condition of that system is obtained according to the frobenius theorem and it is shown that s h h 1 satisfies that completely integrable condition with h and as two matrices

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