linear and nonlinear algorithms for identification in h sub infinity with error bounds a linear algorithm and a nonlinear algorithm for the problem of system identification in h sub infinity posed by helmicki et al 1990 for discrete time systems are presented the authors derive some error bounds for the linear algorithm which indicate that it is not robustly convergent however the worst case identification error is shown to grow as log n where n is the model order a robustly convergent nonlinear algorithm is derived and bounds on the worst case identification error in the h sub infinity norm are obtained

sample adaptive product quantization asymptotic analysis and examples vector quantization vq is an efficient data compression technique for low bit rate applications however the major disadvantage of vq is that its encoding complexity increases dramatically with bit rate and vector dimension even though one can use a modified vq such as the tree structured vq to reduce the encoding complexity it is practically infeasible to implement such a vq at a high bit rate or for large vector dimensions because of the huge memory requirement for its codebook and for the very large training sequence requirement to overcome this difficulty a structurally constrained vq called the sample adaptive product quantizer sapq has recently been proposed we extensively study the sapq that is based on scalar quantizers in order to exploit the simplicity of scalar quantization through an asymptotic distortion result we discuss the achievable performance and the relationship between distortion and encoding complexity we illustrate that even when sapq is based on scalar quantizers it can provide vq level performance we also provide numerical results that show a 2 3 db improvement over the lloyd max 1982 1960 quantizers for data rates above 4 b point

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

a pyramid vector quantizer the geometric properties of a memoryless laplacian source are presented and used to establish a source coding theorem motivated by this geometric structure a pyramid vector quantizer pvq is developed for arbitrary vector dimension the pvq is based on the cubic lattice points that lie on the surface of an l dimensional pyramid and has simple encoding and decoding algorithms a product code version of the pvq is developed and generalized to apply to a variety of sources analytical expressions are derived for the pvq mean square error mse and simulation results are presented for pvq encoding of several memoryless sources for large rate and dimension pvq encoding of memoryless laplacian gamma and gaussian sources provides rose improvements of 5 64 8 40 and 2 39 db respectively over the corresponding optimum scalar quantizer although suboptimum in a rate distortion sense because the pvq can encode large dimensional vectors it offers significant reduction in rose distortion compared with the optimum lloyd max scalar quantizer and provides an attractive alternative to currently available vector quantizers

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

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

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

upper bounds to the asymptotic performance of block quantizers upper bounds to the asymptotic performance of block quantizers with difference distortion measures are derived in many eases these upper bounds approach known lower bounds as the block length of the quantizer approaches infinity a condition for the optimal point density function of the output levels is derived it is shown to particularize to a known result of gersho the behavior of the bounds for large block lengths is investigated

structure selection for bounded parameter models consistency conditions and selection criterion strong consistency conditions for structure selection in bounded parameter models are studied a certain robust selection criterion based on the volume of the exact parameter bounding polytope is proposed for linear regression models the effectiveness of the polytope volume criterion is demonstrated on a model nonlinear in its variables but linear in its parameters its strong consistency is proved for a large class of noise distributions the usual assumptions on the noise namely independence constant variance or martingale difference properties need not be made but asymptotic independence is assumed

quantization based on a novel sample adaptive product quantizer sapq in this paper we propose a novel feedforward adaptive quantization scheme called the sample adaptive product quantizer sapq this is a structurally constrained vector quantizer that uses unions of product codebooks sapq is based on a concept of adaptive quantization to the varying samples of the source and is very different from traditional adaptation techniques for nonstationary sources sapq quantizes each source sample using a sequence of quantizers even when using scalar quantization in sapq we can achieve performance comparable to vector quantization with the complexity still close to that of scalar quantization we also show that important lattice based vector quantizers can be constructed using scalar quantization in sapq we mathematically analyze sapq and propose a algorithm to implement it we numerically study sapq for independent and identically distributed gaussian and laplacian sources through our numerical study we find that sapq using scalar quantizers achieves typical gains of 13 db in distortion over the lloyd max quantizer we also show that sapq can he used in conjunction with vector quantizers to further improve the gains

on the value of information in system identification bounded noise case assuming instantaneous bounds on the noise system parameter identification is formulated as membership set estimation problem sequential algorithms are constructed to estimate the membership sets of the parameters which are consistent with the measurements and the noise constraints the important new feature of the proposed algorithms is their ability to ignore redundant data the efficient data extraction property of the new algorithms is achieved with small computational effort and with improved performance when compared to the least square algorithm the convergence properties and the notion of identifiability in the set theoretic context are also studied

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

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

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

quantization dimension for condensation systems let be the attracting measure of a condensation system associated with a self similar measure we determine the upper and lower quantization dimension of under the strong separation condition

norm estimates related to self commulators abstract for a given self adjoint matrix h with h 1 and tr h 0 we consider the number h which is defined to be the minimum of t 2 for those t satisfying t t h we show that 1 h 2 and that h is close to 2 if h is suitably chosen

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

optimal asymptotic identification under bounded disturbances the intrinsic limitation of worst case identification of linear time invariant systems using data corrupted by bounded disturbances when the unknown plant is known to belong to a given model set is studied this is done by analyzing the optimal worst case asymptotic error achievable by performing experiments using any bounded input and estimating the plant using any identification algorithm it is shown that under some topological conditions on the model set there is an identification algorithm which is asymptotically optimal for any input and the optimal asymptotic error is characterized as a function of the inputs these results which hold for any error metric and disturbance norm are applied to three specific identification problems identification of stable systems in the l sub 1 norm identification of stable rational systems in the h sub infinity norm and identification of unstable rational systems in the gap metric for each of these problems the general characterization of optimal asymptotic error is used to find near optimal inputs to minimize the 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

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

set membership identification of systems with parametric and nonparametric uncertainty a method for parameter set estimation in which the system model is assumed to contain both parametric and nonparametric uncertainty is presented in the disturbance free case the parameter set estimate is guaranteed to contain the parameter set of the true plant in the presence of stochastic disturbances the parameter set estimate obtained from finite data records is shown to have the property that it contains the true plant parameter set with probability one as the data length tends to infinity

the root lattices as low bit rate vector quantizers vector quantizers based on the a and d lattices are explored using the mean square error mse fidelity criterion for independent identically distributed sources with normal laplacian and gamma probability density functions code books are studied for uniform coding rates of up to 2 b sample and two and four dimensions the expected mse performance of the quantizers is better than that of optimal nonuniform lloyd max quantizers while the design algorithm presented offers only modest improvements in mse performance for a normally distributed source improvement gains of up to several decibels can be achieved when coding laplacian and gamma sources

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

controller design for plants with structured uncertainty this paper addresses the problem of designing feedback controllers to achieve good performance in the presence of structured plant uncertainty and bounded but unknown disturbances a general formulation for the performance robustness problem is presented and exact computable conditions are furnished these conditions are then utilized for synthesizing robust controllers which involves solving l1 optimization problems these solutions are computed using the duality theory of lagrange multipliers approximations and computational issues are discussed

parameter identification in the presence of non parametric dynamic uncertainty abstract this paper provides identification laws which yield monotonic parameter error reduction in the absence of external noise inputs but in the presence of non parametric dynamic uncertainty unmodelled dynamics this paper considers two non parametric uncertainty sets motivated by robust control the weighted ball in h and the weighted ball in the induced l2 norm the non parametric uncertainty produces signals which corrupt the signals used in identification this paper shows that the corruption is non conservatively bounded in an l2 sense this l2 bound is then compared with the signal energy with the result used to enable disable the parameter adjustment of a gradient least squares adjustment mechanism it is proved that the resulting parameter adjustment yields non increasing parameter errors with strictly decreasing parameter errors whenever the parameter errors are distinguishable from zero given the system input and output histories

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

extremality of numerical radii of matrix commutators and jordan products abstract it is known that the numerical radius of the commutator resp jordan product a b b a resp a b b a of two n by n matrices a and b is related to those of a and b by w a b b a 4 w a w b in this paper we give complete characterizations of a and b for which the equality is attained the matrices involved can be considered as elaborate generalizations of the equality attaining a 0 a 0 0 and b 0 0 b 0

vector quantization analysis of spl sigma spl delta modulation when considering a class of finite dimensional input signals such as bandlimited signals in the discrete fourier sense within a finite time window o t of observation we show that a single loop spl sigma spl delta modulator behaves like the encoder of a vector quantizer the intrinsic behavior of the modulator can be studied by analyzing the partition it generates in the input space we show that this partition yields a particular structure that we call the hyperplane wave structure as one consequence to this analysis it can be proved for the considered class of bandlimited signals that the mean squared error mse of reconstruction cannot asymptotically decrease with the oversampling ratio r faster than o r sup 4 regardless of the type of reconstruction used we then generalize this analysis to n loop spl sigma spl delta modulators and show that the mse cannot decrease faster than o r sup 2n 2

a simple class of asymptotically optimal quantizers a simple class of quantizers is introduced which are asymptotically optimal as the number of quantization levels increases to infinity with respect to a mean r th power absolute error distortion measure these asymptotically optimal quantizers are very easy to compute their performance is evaluated for several distributions and compares favorably with the performance of the optimal quantizers in all cases for which the latter have been computed in addition their asymptotic robustness is studied under location scale and shape mismatch for several families of distributions

mean squared error analysis of quantizers with error feedback quantization is a fundamental process in digital signal processing delta sigma modulators are often utilized for quantization which can be easily implemented with static uniform quantizers and error feedback filters in this paper we analyze the mean squared quantization error of the quantizer with error feedback including the delta sigma modulators first we study the quantizer with an ideal optimal error feedback filter that minimizes the mean squared error mse of quantization we show that the amplitude response of the optimal error feedback filter can be parameterized by one parameter this parameterization enables us to find the optimal error feedback filter numerically second the relationship between the number of bits used for the quantizer and the achievable mse is clarified by using the optimal error feedback filter this makes it possible to investigate the efficiency of the quantizer with the optimal error feedback filter in terms of mse then ideal optimal error feedback filters are approximated by practically implementable filters using the yule walker method and the linear matrix inequality based method numerical examples are provided for demonstrating our analysis and synthesis

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

lower bound on the mean squared error in oversampled quantization of periodic signals using vector quantization analysis oversampled analog to digital conversion is a technique which permits high conversion resolution using coarse quantization classically by lowpass filtering the quantized oversampled signal it is possible to reduce the quantization error power in proportion to the oversampling ratio r in other words the reconstruction mean squared error mse is in spl oscr r sup 1 it was recently found that this error reduction is not optimal under certain conditions it was shown on periodic bandlimited signals that an upper bound on the mse of optimal reconstruction is in spl oscr r sup 2 instead of spl oscr r sup 1 in the present paper we prove on the same type of signals that the order spl oscr r sup 2 is the theoretical limit of reconstruction as an mse lower bound the proof is based on a vector quantization approach with an analysis of partition cell density

performance of sigma delta quantizations in finite frames in this paper we consider sigma delta sd quantization of geometrically uniform gu finite frames in the first part we prove that under some conditions the variant i and ii permutation modulation pm codes first introduced by slepian 1968 1965 can belong to the class of gu frames then we focus essentially on a subclass of a gu frame namely cyclic geometrically uniform cgu frame family of frames containing finite harmonic frames both in bbc m and bbr m for first and second order sd quantizers we establish that the reconstruction minimum squares error mse behaves as 1 r 2 where r denotes the frame redundancy this result is shown to be true both under the deterministic quantization model used in benedetto 2004 benedetto 2006 and yilmaz 2001 as well as under the widely used additive white quantization noise assumption for the widely used l th order noise shaping filter g z 1 z 1 l we show that the mse behaves as 1 r 2 irrespectively of the filter order l more importantly we prove also that in the case of tight and normalized cgu frame when the frame length is too large compared to the filter order and under some conditions on the quantizer the reconstruction mse can decay as fast as o 1 r 2 l 1 finally it is shown that degrees of freedom in cgu frames when compared to harmonic frames result in a smaller mse albeit mse prop 1 r 2 in both cases

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

condition for the numerical range to contain an elliptic disc for an n by n matrix a and an elliptic disc e in the plane we show that the sum of the number of common supporting lines and the number of common intersection points to e and the numerical range w a of a should be at least 2n 1 in order to guarantee that e be contained in w a this generalizes previous results of anderson and thompson as an application our result is used to verify a special case of the poncelet property conjecture 2003 elsevier science inc all rights reserved

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

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

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

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

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

a framework for robust parametric set membership identification this paper proposes a new framework for studying robust parametric set membership identification the authors derive some new results on the fundamental limitations of algorithms in this framework given a particular model structure the new idea is to quantify uncertainty only with respect to the finite dimensional parametric part of the model and not the fixed size unmodeled dynamics thus the measure of uncertainty is different from the measures used in previous robust identification work where system norms are used to quantify uncertainty as an example the results are used to assess the fidelity of a certain approximate robust parametric set membership identification algorithm

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