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

quasi cohen macaulay properties of local homomorphisms abstract for a large class of local homomorphisms r s including those of finite g dimension studied by avramov and foxby proc london math soc 75 1997 241 270 we assign a new numerical invariant called the quasi cohen macaulay defect of and a local homomorphism is called quasi cohen macaulay if it is of finite g dimension and has trivial quasi cohen macaulay defect we show among other things the following g dim spec spec g dim

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

grassmann algebra and fermions at finite temperature for any d dimensional self interacting fermionic model all coefficients in the high temperature expansion of its grand canonical partition function can be put in terms of multivariable grassmann integrals a new approach to calculate such coefficients based on direct exploitation of the grassmannian nature of fermionic operators is presented we apply the method to the soluble hatsugai kohmoto model reobtaining well known results

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

canonical elements in local cohomology modules and the direct summand conjecture one of the objectives of this paper is to show that the usual homological consequences of the existence of big cohen macaulay henceforth c m modules e g the new intersection conjecture of peskine szpiro and roberts and the evans griffith syzygy conjecture follow from the direct summand conjecture when the residual characteristic of the local ring is positive this gives a new and substantially more elementary proof of the standard homological conjectures in case the characteristic of the ring itself is positive and reduces the general case of all these conjectures to one rather down to earth conjecture of course this places the direct summand conjecture in a position of central importance so that it now merits an allout attack some partial results on this problem are given in section 6 the conjecture asserts that a regular noetherian ring r is a direct summand as an r module of every module finite extension ring s 3 r the other main objective of this paper is to formulate and develop a theory of certain canonical elements in the local cohomology of special modules of syzygies neither the modules of syzygies nor the induced maps between them are canonical but the identification between the canonical elements is independent of the choices in particular a canonical element qr e hz syz k is associated see section 3 for details with each ndimensional local ring r m k rings are commutative associative with identity local ring means noetherian ring with a unique maximal ideal the conjecture that qr is nonzero for all local rings r turns out to be equivalent to the direct summand conjecture for a given r an infinite family of cases of the latter implies the former

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

unidimensional generalized hubbard model in the high temperature limit abstract we explore the properties of the non commutative grassmann algebra to study the unidimensional generalized hubbard model obtaining the analytical expressions for the first three terms in the high temperature expansion of its grand canonical partition function with no restrictions to the constant parameters of the model we obtain corrections to known results in the case of half filled band with hopping constants t 0 and x 0

conformal invariance and the operator content of the xxz model with arbitrary spin the authors are concerned with the critical properties of the anisotropic heisenberg chain or xxz model with arbitrary integer or half integer spin the eigenspectra of these hamiltonians with periodic boundaries are calculated for finite chains by solving numerically their associated bethe ansatz equations the resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified for lattices with an even and odd number of sites the results for spin 1 and spin 3 2 indicate that the conformal anomaly c for these models in the gapless regime has the value c 3s 1 s independent of the anisotropy and the exponents vary continuously with the anisotropy as in the eight vertex model the operator content of these models indicate that the underlying field theory governing these critical spin s models is described by composite fields formed by the product of gaussian and z n fields with n 2s finally some of the irrelevant operators which produce the leading finite size corrections of the eigenenergies are identified

semi dualizing complexes and their auslander categories let r be a commutative noetherian ring we study r modules and complexes of such with excellent duality properties while their common properties are strong enough to admit a rich theory we count among them such potentially diverse objects as dualizing complexes for r on one side and on the other the ring itself in several ways these two examples constitute the extremes and their well understood properties serve as guidelines for our study however also the employment in recent studies of ring homomorphisms of complexes lying between these extremes is incentive

remarks on the convergence of the β expansion of and s 1 ising models abstract in a recent work j math phys 43 2002 1390 we derived analytical expressions for the coefficients in the high temperature expansion of the helmholtz free energy of periodic one dimensional chains in the cumulant method for arbitrary order in in order to do so an auxiliary function was defined the purpose of the present paper is to show that a poor convergence of the high temperature expansions of thermodynamic quantities are signaled by the existence of non physical singularity ies in the auxiliary function our discussion is based on the exactly solvable spin 1 and spin 1 2 ising models whose respective auxiliary functions have algebraic expressions valid for 0

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

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

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

homology over local homomorphisms the notions of betti numbers and of bass numbers of a finite module n over a local ring r are extended to modules that are only assumed to be finite over s for some local homomorphism r s various techniques are developed to study the new invariants and to establish their basic properties in some cases they are computed in closed form applications go in several directions one is to identify new classes of finite r modules whose classical betti numbers or bass numbers have extremal growth another is to transfer ring theoretical properties between r and s in situations where s may have infinite flat dimension over r a third is to obtain criteria for a ring equipped with a contracting endomorphism such as the frobenius endomorphism to be regular or complete intersection these results represent broad generalizations of kunz s characterization of regularity in prime characteristic

cohen macaulay properties of ring homomorphisms abstract numerical invariants which measure the cohen macaulay character of homomorphisms r s of noetherian rings are introduced and studied comprehensive results are obtained for homomorphisms which are locally of finite flat dimension they provide a point of view from which a variety of phenomena receive a unified treatment the conceptual clarification and technical versatility of this approach leads among other things to a determination of those homomorphisms which preserve the cohen macaulay character of the rings to the discovery of new classes of homomorphisms with remarkable stability properties and to solutions of some problems on flat homomorphisms raised by grothendieck

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

on the series expansion of the general spin s ising chain abstract thermodynamical properties of spin s ising chains can nowadays be easily obtained using numerical calculation however from a mathematical point of view its exact solution for arbitrary spin is still a challenge only limiting cases have been solved exactly such as the infinite spin limit and lowest spin values the present article addresses this issue using the high temperature series expansion we obtain a new analytical series expansion of the partition function for the ising chain in the absence of magnetic field our general results cover all spins from 1 2 to infinite in the high temperature region up to order 40 kt 1 in order to extend our results to finite temperature we employ the method presented in the work of ref bernu and misguich phys rev b63 2001 134409 we also present a matrix formulation of our series expansion and relate it to the transfer matrix technique

construction of modules with finite homological dimensions a new homological dimension called g dimension is defined for every finitely generated module m over a local noetherian ring r it is modeled on the ci dimension of avramov gasharov and peeva and has parallel properties in particular a ring r is gorenstein if and only if every finitely generated r module has finite g dimension the g dimension lies between the ci dimension and the g dimension of auslander and bridger this relation belongs to a longer sequence of inequalities where a strict inequality in any place implies equalities to its right and left over general local rings we construct classes of modules that show that a strict inequality can occur at almost every place in the sequence

thermodynamics of the spin 1 2 heisenberg ising chain at high temperatures a rigorous approach this work develops a rigorous setting allowing one to prove several features related to the behaviour of the heisenberg ising or xxz spin 1 2 chain at finite temperature t within the quantum inverse scattering method the physically pertinent observables at finite t such as the per site free energy or the correlation length have been argued to admit integral representations whose integrands are expressed in terms of solutions to auxiliary non linear integral equations the derivation of such representations was based on numerous conjectures the possibility to exchange the infinite volume and the infinite trotter number limits the existence of a real non degenerate maximal in modulus eigenvalue of the quantum transfer matrix the existence and uniqueness of solutions to the auxiliary non linear integral equations as well as the possibility to take the infinite trotter number limit on their level we rigorously prove all these conjectures for temperatures large enough as a by product of our analysis we obtain the large t asymptotic expansion for a subset of sub dominant eigenvalues of the quantum transfer matrix and thus of the associated correlation lengths this result was never obtained previously not even on heuristic grounds

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

bass series of local ring homomorphisms of finite flat dimension nontrivial relations between bass numbers of local commutative rings are established in case there exists a local homomorphism r s which makes s into an r module of finite flat dimension in particular it is shown that an inequality r i depth r s i depth s holds for all i z this is a consequence of an equality involving the bass series i r m t i z r i m t i of a complex m of r modules which has bounded above and finite type homology and the bass series of the complex of s modules m r s where denotes the derived tensor product it is prove that there is an equa lity of formal laurent series i s m rs t i r m t i f t where f is the fiber of considered as a homomorphism of commutative differential graded rings

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

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

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

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

high temperature expansion for a chain model we consider an arbitrary translationally invariant chain model with nearest neighbors interaction and satisfying periodic boundary condition the approach developed here allows a thermodynamic description of the chain model directly in terms of grand potential per site this thermodynamic function is derived from an auxiliary function constructed only from open connected subchains in order to exemplify its application and how this approach works we consider the heisenberg xxz model we obtain the coefficients of the high temperature expansion of the free energy per site of the model up to third order

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

locally complete intersection homomorphisms and a conjecture of quillen on the vanishing of cotangent homology classical deflnitions of locally complete intersection l c i homomor phisms of commutative rings are limited to maps that are essentially of flnite type or at the concept introduced in this paper is meaningful for homo morphisms ri s of commutative noetherian rings it is deflned in terms of the structure of in a formal neighborhood of each point of specs w e characterize the l c i property by difierent conditions on the vanishing of the andre quillen homology of the r algebra s one of these descriptions estab lishes a very general form of a conjecture of quillen that was open even for homomorphisms of flnite type if s has a flnite resolution by at r modules and the cotangent complex l sjr is quasi isomorphic to a bounded complex of at s modules then is l c i the proof uses a mixture of methods from commutative algebra difierential graded homological algebra and homotopy theory the l c i property is shown to be stable under a variety of operations including composition decomposition at base change localization and com pletion the present framework allows for the results to be stated in proper generality many of them are new even with classical assumptions for in stance the stability of l c i homomorphisms under decomposition settles an open case in fulton s treatment of orientations of morphisms of schemes