spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

legendre multi galerkin methods for fredholm integral equations with weakly singular kernel and the corresponding eigenvalue problem abstract in this paper we consider legendre multi galerkin methods to solve fredholm integral equations of the second kind with weakly singular kernel and the corresponding eigenvalue problem we obtain the convergence rates for the approximated solution and iterated solution in weakly singular fredholm integral equations of the second kind in both l 2 and infinity norm we also establish error bounds of approximated eigenelements with exact eigenelements in the eigenvalue problem of a compact integral operator with weakly singular kernel in both infinity and l 2 norm numerical examples are presented to prove the theoretical estimates

dithered quantizers a theory of overall quantization noise for nonsubtractive dither was originally developed in unpublished work by j n wright and by t j stockham and subsequently expanded by l k brinton s p lipshitz j vanderkooy and r a wannamaker it is suggested that since these latter results are not as well known as the original results misunderstanding persists in the literature new proofs of the properties of quantizer dither both subtractive and nonsubtractive are provided the new proofs are based on elementary fourier series and rice s characteristic function method and do not require the use of generalized functions impulse trains of dirac delta functions and sampling theorem arguments the goal is to provide a unified derivation and presentation of the two forms of dithered quantizer noise based on elementary fourier techniques

the generalized lasso for sub gaussian measurements with dithered quantization in the problem of structured signal recovery from high dimensional linear observations it is commonly assumed that full precision measurements are available under this assumption the recovery performance of the popular generalized lasso g lasso is by now well established in this paper we extend these types of results to the practically relevant settings with quantized measurements we study two extremes of the quantization schemes namely uniform and one bit quantization the former imposes no limit on the number of quantization bits while the second only allows for one bit in the presence of a uniform dithering signal and when measurement vectors are sub gaussian we show that the same algorithm i e the g lasso has favorable recovery guarantees for both uniform and one bit quantization schemes our theoretical results shed light on the appropriate choice of the range of values of the dithering signal and accurately capture the error dependence on the problem parameters for example our error analysis shows that the g lasso with one bit uniformly dithered measurements leads to only a logarithmic rate loss compared to the full precision measurements

recovering the electrical conductivity of the soil via a linear integral model abstract electromagnetic induction techniques are often used to ascertain the presence of conductive substances in the subsoil the device we are interested in generates an electromagnetic field and measures the secondary field produced by eddy currents induced by the primary field we analyze a linear model which reproduces the device readings consisting of a system of two fredholm linear integral equations of the first kind after studying the continuous problem in various function spaces we describe three collocation methods to transform the integral equations into a system of linear equations which is solved by a regularization technique numerical experiments show the performance of the different discretizations and of a particular approach to deal with the integral truncation error we also verify the influence on the solution of regularization matrices based on the discretization of the first and second derivative

noise enhanced parameter estimation this paper investigates the phenomenon of noise enhanced systems for a general parameter estimation problem when the estimator is fixed and known the estimation performance before and after the addition of noise are evaluated performance comparisons are made between the original estimators and noise enhanced estimators based on different criteria the form of the optimal noise probability density function pdf is determined the results are further extended to the general case where the noise is introduced to the system via a transformation for the case where the estimator is fixed and unknown approaches are also proposed to find the optimum noise finally two illustrative examples are presented where the performance comparison is made between the optimal noise modified estimator and gaussian noise modified estimator

generating dithering noise for maximum likelihood estimation from quantized data the quantization theorem i qt i implies that the likelihood function can be reconstructed from quantized sensor observations given that appropriate dithering noise is added before quantization we present constructive algorithms to generate such dithering noise the application to maximum likelihood estimation mle is studied in particular in short dithering has the same role for amplitude quantization as an anti alias filter has for sampling in that it enables perfect reconstruction of the dithered but unquantized signal s likelihood function without dithering the likelihood function suffers from a kind of aliasing expressed as a counterpart to poisson s summation formula which makes the exact mle intractable to compute with dithering it is demonstrated that standard mle algorithms can be re used on a smoothed likelihood function of the original signal and statistically efficiency is obtained the implication of dithering to the cramer rao lower bound crlb is studied and illustrative examples are provided

superconvergence of legendre projection methods for the eigenvalue problem of a compact integral operator in this paper we consider the galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the legendre polynomials of degree n we prove that the error bounds for eigenvalues are of the order o n 2 r and the gap between the spectral subspaces are of the orders o n r in l 2 norm and o n 1 2 r in the infinity norm where r denotes the smoothness of the kernel by iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence o n 2 r in both l 2 norm and infinity norm we illustrate our results with numerical examples

decentralized estimation in an inhomogeneous sensing environment we consider decentralized estimation of a noise corrupted deterministic parameter by a bandwidth constrained sensor network with a fusion center the sensor noises are assumed to be additive zero mean spatially uncorrelated but otherwise unknown and possibly different across sensors due to varying sensor quality and inhomogeneous sensing environment the classical best linear unbiased estimator blue linearly combines the real valued sensor observations to minimize the mean square error mse unfortunately such a scheme cannot be implemented in a practical bandwidth constrained sensor network due to its requirement to transmit real valued messages in this paper we construct a decentralized estimation scheme des where each sensor compresses its observation to a small number of bits with length proportional to the logarithm of its local signal to noise ratio snr the resulting compressed bits from different sensors are then collected and combined by the fusion center to estimate the unknown parameter the proposed des is universal in the sense that each sensor compression scheme requires only the knowledge of local snr rather than the noise probability distribution functions pdf while the final fusion step is also independent of the local noise pdfs we show that the mse of the proposed des is within a constant factor of 25 8 of that achieved by the classical centralized blue estimator

bandwidth constrained distributed estimation for wireless sensor networks part i gaussian case we study deterministic mean location parameter estimation when only quantized versions of the original observations are available due to bandwidth constraints when the dynamic range of the parameter is small or comparable with the noise variance we introduce a class of maximum likelihood estimators that require transmitting just one bit per sensor to achieve an estimation variance close to that of the clairvoyant sample mean estimator when the dynamic range is comparable or larger than the noise standard deviation we show that an optimum quantization step exists to achieve the best possible variance for a given bandwidth constraint we will also establish that in certain cases the sample mean estimator formed by quantized observations is preferable for complexity reasons we finally touch upon algorithm implementation issues and guarantee that all the numerical maximizations required by the proposed estimators are concave

sequential decentralized parameter estimation under randomly observed fisher information we consider the problem of decentralized scalar parameter estimation using wireless sensor networks with gaussian noise specifically we propose a novel framework based on level triggered sampling a non uniform sampling strategy and sequential estimation the proposed estimator can be used as an asymptotically optimal fixed sample size decentralized estimator when the observed fisher information i e fisher information without expectation is deterministic as an alternative to the one shot estimators commonly found in the literature it can also be used as an asymptotically optimal sequential decentralized estimator when the observed fisher information is random we show that the optimal centralized estimator under gaussian noise which is the maximum likelihood estimator is characterized by two processes namely the observed fisher information ut and the observed correlation vt it is noted that vt is always random even when ut is not in the proposed scheme each sensor computes its local random processes and sends a single bit to the fusion center fc whenever the local random processes passes certain predefined levels the fc upon receiving a bit from a sensor updates its approximation to the corresponding global random process and accordingly its estimate the sequential estimation process terminates when ut or the approximation to it reaches a target value we provide an asymptotic analysis for the proposed estimator and the one based on conventional uniform in time sampling under both deterministic and random ut and determine the conditions under which they are asymptotically optimal consistent and asymptotically unbiased analytical results together with simulation results demonstrate the superiority of the proposed estimator based on level triggered sampling over the traditional decentralized estimator based on uniform sampling

discrete multi projection methods for eigen problems of compact integral operators abstract in this paper a discrete multi projection method is developed for solving the eigenvalue problem of a compact integral operator with a smooth kernel we propose a theoretical framework for analysis of the convergence of these methods the theory is then applied to establish super convergence results of the corresponding discrete galerkin method collocation method and their iterated solutions numerical examples are presented to illustrate the theoretical estimates for the error of these methods

signal parameter estimation using 1 bit dithered quantization motivated by the estimation of spatio temporal events with cheap simple sensors we consider the problem of estimation of a parameter thetas of a signal s x thetas corrupted by noise assuming that only 1 bit precision dithered quantized samples are available an estimate that does not require the knowledge of the dither signal and the noise distribution is proposed and it is analyzed in detail under variety of nonidealities the consistency and asymptotic normality of the estimate is established for deterministic and random sampling imprecise knowledge of sampling locations gaussian and non gaussian noise with possibly infinite variance a wide class of dither distributions and under erroneous transmission of the binary observations via binary symmetric channels bscs it is also shown that if approximation to the log likelihood equation in the full precision case yields a good estimate then there is a corresponding good estimate based on 1 bit dithered samples the proposed estimate requires no more computation than the maximum likelihood estimate for the full precision case and suffers only a logarithmic rate loss compared to the full precision case when uniform dithering is used it is shown that uniform dithering leads to the best rate among a broad class of dither distributions a condition under which no dithering leads to a better estimate is also given

bandwidth constrained distributed estimation for wireless sensor networks part ii unknown probability density function wireless sensor networks wsns deployed to perform surveillance and monitoring tasks have to operate under stringent energy and bandwidth limitations these motivate well distributed estimation scenarios where sensors quantize and transmit only one or a few bits per observation for use in forming parameter estimators of interest in a companion paper we developed algorithms and studied interesting tradeoffs that emerge even in the simplest distributed setup of estimating a scalar location parameter in the presence of zero mean additive white gaussian noise of known variance herein we derive distributed estimators based on binary observations along with their fundamental error variance limits for more pragmatic signal models i known univariate but generally non gaussian noise probability density functions pdfs ii known noise pdfs with a finite number of unknown parameters iii completely unknown noise pdfs and iv practical generalizations to multivariate and possibly correlated pdfs estimators utilizing either independent or colored binary observations are developed and analyzed corroborating simulations present comparisons with the clairvoyant sample mean estimator based on unquantized sensor observations and include a motivating application entailing distributed parameter estimation where a wsn is used for habitat monitoring

the reconstruction of analog signals from the sign of their noisy samples the reconstruction of a continuous time signal s t from the sign of its deliberately contaminated samples is considered sequential generally nonlinear estimates of s t are established and their performance is studied error bounds and convergence rates are derived the signal s t need not be band limited the convergence rates obtained here are faster than those obtained previously for nonsequential estimates the degradation in the reconstruction of the signal due to transmission over a noisy channel is also investigated and bounds on the additional error are obtained

legendre multi projection methods for solving eigenvalue problems for a compact integral operator in this paper we consider the m galerkin and m collocation methods for solving the eigenvalue problem for a compact integral operator with smooth kernels using legendre polynomial bases we obtain error bounds for the eigenvalues and the gap between the spectral subspaces in both legendre m galerkin and legendre m collocation methods we also obtain superconvergence results for the eigenvalues and iterated eigenvectors in both l 2 and infinity norm we illustrate our results with numerical examples

estimation from quantized gaussian measurements when and how to use dither subtractive dither is a powerful method for removing the signal dependence of quantization noise for coarsely quantized signals however estimation from dithered measurements often naively applies the sample mean or midrange even when the total noise is not well described with a gaussian or uniform distribution we show that the generalized gaussian distribution approximately describes subtractively dithered quantized samples of a gaussian signal furthermore a generalized gaussian fit leads to simple estimators based on order statistics that match the performance of more complicated maximum likelihood estimators requiring iterative solvers the order statistics based estimators outperform both the sample mean and midrange for nontrivial sums of gaussian and uniform noise additional analysis of the generalized gaussian approximation yields rules of thumb for determining when and how to apply dither to quantized measurements specifically we find subtractive dither to be beneficial when the ratio between the gaussian standard deviation and quantization interval length is roughly less than one third when that ratio is also greater than 0 822 k text 0 930 for the number of measurements k text 20 estimators we present are more efficient than the midrange

on the limits of communication with low precision analog to digital conversion at the receiver as communication systems scale up in speed and bandwidth the cost and power consumption of high precision e g 8 12 bits analog to digital conversion adc becomes the limiting factor in modern transceiver architectures based on digital signal processing in this work we explore the impact of lowering the precision of the adc on the performance of the communication link specifically we evaluate the communication limits imposed by low precision adc e g 1 3 bits for transmission over the real discrete time additive white gaussian noise awgn channel under an average power constraint on the input for an adc with k quantization bins i e a precision of log2 k bits we show that the input distribution need not have any more than k 1 mass points to achieve the channel capacity for 2 bin 1 bit symmetric quantization this result is tightened to show that binary antipodal signaling is optimum for any signal to noise ratio snr for multi bit quantization a dual formulation of the channel capacity problem is used to obtain tight upper bounds on the capacity the cutting plane algorithm is employed to compute the capacity numerically and the results obtained are used to make the following encouraging observations a up to a moderately high snr of 20 db 2 3 bit quantization results in only 10 20 reduction of spectral efficiency compared to unquantized observations b standard equiprobable pulse amplitude modulated input with quantizer thresholds set to implement maximum likelihood hard decisions is asymptotically optimum at high snr and works well at low to moderate snrs as well

bayesian parameter estimation using single bit dithered quantization the bayesian parameter estimation problem using a single bit dithered quantizer is considered this problem arises e g for channel estimation under low precision analog to digital conversion adc at the receiver based on the bayesian cramer rao lower bound crlb bounds on the mean squared error are derived that hold for all dither strategies with strictly causal adaptive processing of the quantizer output sequence in particular any estimator using the binary quantizer output sequence is asymptotically in the sequence length at least 10 log10 2 1 96 db worse than the minimum mean squared error estimator using continuous observations for any dither strategy moreover dither strategies are designed that are shown by simulation to closely approach the derived lower bounds

effects of sampling and quantization on single tone frequency estimation the effects of sampling and quantization on frequency estimation for a single sinusoid are investigated the cramer rao bound for 1 bit quantization is derived and compared with the limit of infinite quantization it is found that 1 bit quantization gives a slightly worse performance however with a dramatic increase of variance at certain frequencies this can be avoided by using four times oversampling the effect of sampling when using nonideal antialiasing lowpass filters is therefore investigated through derivation of the cramer rao lower bounds finally fast estimators for 1 bit quantization in particular correlation based estimators are derived and their performance is investigated the paper is concluded with simulation results for four times oversampled 1 bit quantization