Quantization Error Model Research Articles

A Joint Model Extraction and Data Detection Framework for IRS-NOMA System

This paper addresses the problem of data detection for intelligent reflecting surface (IRS) assisted non-orthogonal multiple access (NOMA) systems. Also, the phase noise, analog-to-digital converter (ADC) quantization error and imperfect channel estimation are taken into consideration. At present, IRS is designed to be deployed in a wide range of scenarios. However, the quantization error model and IRS phase error model developed for limited scenarios may not be well adapted to the complex and changing environment, which brings a new challenge for the application of IRS. Therefore, this paper investigates a new hybrid method for joint noise model extraction, device activity detection and data detection in the case of unknown noise models, to solve the aforementioned IRS problem. Firstly, the Gaussian mixture model (GMM) is used to describe the model of quantization noise and phase noise. Thus the problem of model extraction is transformed into the calculation of the key parameters of GMM. Secondly, an improved expectation maximization algorithm is used to calculate the quantization noise model parameters. Finally, utilizing the property of the structure of the problem, a message-passing based algorithm is investigated. Simulation results show the merits of the proposed hybrid algorithm.

A hybrid spatial model based on identified conditions for 3D pore pressure estimation

Accurate spatial modeling of pore pressure helps to provide extensive and continuous information regarding the insightful downhole geological environment for safe drilling plans, performance analysis, and efficient reservoir modeling. Considering the varying drilling depth and pore pressure of study wells, a hybrid spatial modeling method is developed to estimate the pore pressure of an area. The method comprises three parts. In the first part, an appropriate number of pressure conditions is determined adaptively based on quantization error modeling (QEM), and similar pressure conditions are identified based on fuzzy c-means (FCM) method. In the second part, a kriging interpolation (Kriging) algorithm is introduced to produce profiles of different pressure subspaces. Afterward, random forest (RF) submodels are built separately according to the pressure profiles. In the last part, all of these submodels coalesced into one whole model. Verification of the proposed method is carried out on real case studies using field data from Songliao basin. The results show that the proposed method achieves the best prediction accuracy than the other five well-known methods for six wells, especially for well XS3, the mean absolute error and mean absolute percentage error of the proposed method achieve 0.717 and 6.063, which are at least 31% less than those of other five well-known methods.

Medium-precision ADC quantization error modeling for MIMO-OFDM

Analog-to-digital converter (ADC) is an indispensable application for wireless communication systems, but the model of the quantization error induced by ADC is still an open interdisciplinary problem. This paper proposes a general framework of quantization error model for multiple-input multiple-output (MIMO) system employing orthogonal frequency-division multiplexing (OFDM) with medium-precision ADCs. The Gaussian mixture model (GMM) is used to represent the error caused by ADCs, and then an optimization structure is constructed. In the proposed optimization method, with the key parameters of the error distribution obtained by the expectation-maximization (EM) algorithm, the generalized approximate message passing (GAMP) algorithm can detect the data accurately. The simulation demonstrates that the proposed quantization error calculation algorithm can improve the performance of data detection under imperfect channel state information (CSI).

Modeling and analysis of pixel quantization error of binocular vision system with unequal focal length

Current Analysis of pixel quantization error has following problems in the binocular vision system. Most literatures are based on the ideal binocular vision system with equal focal length. Moreover, the precision of pixel quantization is relatively low. In order to solve above problems, an analysis method of pixel quantization error for binocular vision system with unequal focal length is proposed. Firstly, a pixel quantization error model is established and a mathematical expression is presented to characterize the pixel quantization error model. Secondly, in the case of equal and unequal focal lengths, measurement error of object point is discussed respectively. Finally, simulation experiments were conducted and the effect of some parameters on measurement error was analyzed. Reasonability and efficiency of proposed methods are verified.

Estimating the number of clusters in a ranking data context

This study introduces two methods for estimating the number of clusters specially designed to identify the number of groups in a finite population of objects or items ranked by several judges under the assumption that these judges belong to a homogeneous population. The proposed methods are both based on a hierarchical version of the classical Plackett–Luce model in which the number of clusters is set as an additional parameter. These methods do not require continuous score data to be available or restrict the number of clusters to be greater than one or less than the total number of objects, thereby enabling their application in a wide range of scenarios. The results of a large simulation study suggest that the proposed methods outperform well-established methodologies (Calinski & Harabasz, gap, Hartigan, Krzanowski & Lai, jump, and silhouette) as well as some recently proposed approaches (instability, quantization error modeling, slope, and utility). They realize the highest percentages of correct estimates of the number of clusters and the smallest errors compared with these well-established methodologies. We illustrate the proposed methods by analyzing a ranking dataset obtained from Formula One motor racing.

Double-Talk Robust Multichannel Acoustic Echo Cancellation Using Least-Squares MIMO Adaptive Filtering: Transversal, Array, and Lattice Forms

In this paper, we address the problem of noise-robust multiple-input multiple-output (MIMO) adaptive filtering that is optimal in least-squares sense with application to multichannel acoustic echo cancellation. We formulate the problem as minimization of a multichannel least-squares cost function that incorporates near-end speech and noise statistics resulting in a novel noise-robust framework for MIMO adaptive filtering. Although the issue of numerical stability has been widely explored in the context of recursive least-squares (RLS) filtering, a rigorous mathematical treatment of the MIMO case in the context of numerically stable noise-robust multichannel echo cancellation remains absent. Guided by quantization-error modeling, we resolve the issue of numerical instability in our noise-robust scheme by utilizing transversal RLS filtering of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Type 2</i> . Thereafter, an explicit derivation of its inverse QR-decomposition (IQRD) counterpart based on Givens rotations is presented. We also derive computationally efficient lattice forms for our noise-robust RLS Type-2 and IQRD algorithms. It is highlighted that propagation of angle-normalized errors occurs <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">naturally</i> within the numerically stable least-squares lattice (LSL). Thus, our approach combines the four sought after attributes in a multichannel echo cancellation scheme, i.e., computational efficiency, numerical stability, fast convergence and tracking, and robustness against noise. We analyze our formulations using simulations in terms of convergence, re-convergence, robustness in the presence of double-talk, and numerical stability.

Quantization Bias for Digital Correlators

In radio interferometry, the quantization process introduces a bias in the magnitude and phase of the measured correlations which translates into errors in the measurement of source brightness and position in the sky, affecting both the system calibration and image reconstruction. In this paper, we investigate the biasing effect of quantization in the measured correlation between complex-valued inputs with a circularly symmetric Gaussian probability density function (PDF), which is the typical case for radio astronomy applications. We start by calculating the correlation between the input and quantization error and its effect on the quantized variance, first in the case of a real-valued quantizer with a zero mean Gaussian input and then in the case of a complex-valued quantizer with a circularly symmetric Gaussian input. We demonstrate that this input-error correlation is always negative for a quantizer with an odd number of levels, while for an even number of levels, this correlation is positive in the low signal level regime. In both cases, there is an optimal interval for the input signal level for which this input-error correlation is very weak and the model of additive uncorrelated quantization noise provides a very accurate approximation. We determine the conditions under which the magnitude and phase of the measured correlation have negligible bias with respect to the unquantized values: we demonstrate that the magnitude bias is negligible only if both unquantized inputs are optimally quantized (i.e. when the uncorrelated quantization error model is valid), while the phase bias is negligible when (1) at least one of the inputs is optimally quantized, or when (2) the correlation coefficient between the unquantized inputs is small. Finally, we determine the implications of these results for radio interferometry.

Quantization and Colored Noises Error Modeling for Inertial Sensors for GPS/INS Integration

In this paper, modeling approaches for quantization and colored noises have been proposed. To accommodate the quantization noise, a modified inertial navigation system (INS) error dynamics is developed in this paper, and the quantization noise is incorporated into the modified INS error dynamics as augmenting driving noise. The three kinds of colored noises are modeled by using an equivalent differential equation driven by a unit white noise, and a technique is developed in this paper to augment the Kalman Filter of GPS/INS integration using this equivalent differential equation. Experimental test results show that the proposed stochastic error modeling approaches for quantization and colored noises significantly improves the accuracies of the estimated inertial drifts and the navigation solutions.

Sensitivity-Based Pole and Input-Output Errors of Linear Filters as Indicators of the Implementation Deterioration in Fixed-Point Context

Input-output or poles sensitivity is widely used to evaluate the resilience of a filter realization to coefficients quantization in an FWL implementation process. However, these measures do not exactly consider the various implementation schemes and are not accurate in general case. This paper generalizes the classical transfer function sensitivity and pole sensitivity measure, by taking into consideration the exact fixed-point representation of the coefficients. Working in the general framework of the specialized implicit descriptor representation, it shows how a statistical quantization error model may be used in order to define stochastic sensitivity measures that are definitely pertinent and normalized. The general framework of MIMO filters and controllers is considered. All the results are illustrated through an example.

Fixed-Point Signal Processing

This book is intended to fill the gap between the ideal digital signal processing (DSP) that is widely taught, and the limited precision implementation skills that are commonly required in fixed-point processors and field programmable gate arrays (FPGAs). These skills are often neglected at the university level, particularly for undergraduates. We have attempted to create a resource both for a DSP elective course and for the practicing engineer with a need to understand fixed-point implementation. Although we assume a background in DSP, Chapter 2 contains a review of basic theory and Chapter 3 reviews random processes to support the noise model of quantization error. Chapter 4 details the binary arithmetic that underlies fixed-point processors and then introduces fractional format for binary numbers. Chapter 5 covers the noise model for quantization error and the effects of coefficient quantization in filters. Because of the numerical sensitivity of IIR filters, they are used extensively as an example system in both Chapters 5 and 6. Fortunately, the principles of dealing with limited precision can be applied to a wide variety of numerically sensitive systems, not just IIR filters. Chapter 6 discusses the problems of product roundoff error and various methods of scaling to avoid overflow. Chapter 7 discusses limit cycle effects and a few common methods for minimizing them. There are a number of simple exercises integrated into the text to allow you to test your understanding. Answers to the exercises are included in the footnotes.

Tracking Performance Limitations in a Linear Feedback System with Remote Sensors

Abstract This paper studies tracking performance limitations for a networked feedback control system. In the system, the plant is a linear time invariant (LTI) SISO system and the measurement signal is received from a remote site through a network. The reference signal in the tracking problem is a step signal. The tracking performance is measured by an integral square error between the output of the plant and the reference signal. To transmit the measurement signal through a network, this signal is quantized and then certain information which the original signal possesses could be lost. The major issue which we study in this paper is: How does a logarithmic quantization law constrain the best attainable tracking performance of the feedback system? Here the quantization error is modeled as a product of the original signal and a bounded nonlinear function. An upper bound of the best attainable tracking performance of the system is presented in terms of the quantization error model and the characteristics of the plant. It is also found that, if the nonlinear function in the quantization error model is an H ∞ norm-bounded uncertainty, this upper bound is the tracking performance limit of the feedback system under the worst uncertainty. In the case where the quantizer and network are not used in the system, the upper bound is equal to the tracking performance limit of the LTI system.

Recent advances in DFT codes based quantized frame expansions for erasure channels

Overcomplete frame expansions have been introduced recently as signal representations resilient to erasures on packet networks such as the Internet and wireless networks. This paper is dedicated to a broad analysis of a class of frames derived from the generator matrices of DFT codes. In contrast to earlier presentations, which focused on the analysis of only low-pass DFT codes, this analysis aims at comparing the performance of all BCH codes and MDS negacyclic codes derived from the DFT codes. It is proven that all BCH codes, and MDS negacyclic codes derived from the BCH codes, are equivalent in the sense that they produce the same mean square error averaged over all erasure patterns of a given length under the same quantization error model. On the way to this result, the special property of the frames associated with the BCH codes, and the reconstruction equivalence between the frame-theoretic approach and the coding-theoretic approach, are proved in general settings. The paper also discusses optimal packetization of BCH codevectors and the performance of non-BCH codes that have the same erasure recovery capacity as BCH codes.

Analytical Modeling of Sensor Quantization in Strapdown Inertial Navigation Error Equations

Although generally not considered a major contributor to system inaccuracy, inertial sensor quantization error, if not properly modeled, can lead to erroneously large estimates of its impact on inertial navigation system performance. Analytical methods are described for modeling inertial sensor quantization in strapdown inertial system error parameter propagation and measurement equations. Error propagation equations are derived in classical differential error state dynamic and discrete difference format. It is shown how the attitude, velocity, and sensor error parameters in these equations must be modified to enable proper sensor quantization error modeling as white noise and to account for differences in attitude, velocity, and position update frequencies. The discrete difference error equation form is used to develop values for attitude/velocity measurement noise covariances and for spectral densities of white quantization noise terms in the differential error propagation equations. A general discussion is included of how quantization white noise spectral densities should be computed for the differential error propagation equations for compatibility with two-speed attitude, velocity, and position updating algorithms. Validity limits for white noise modeling approximations and methods for reducing quantization noise are discussed. Numerical examples are provided.

Quantized Frame Expansions with Erasures

Frames have been used to capture significant signal characteristics, provide numerical stability of reconstruction, and enhance resilience to additive noise. This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of frames are sometimes themselves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness to losses in packet networks such as the Internet. With a simple model for quantization error, it is shown that a normalized frame minimizes mean-squared error if and only if it is tight. With one coefficient erased, a tight frame is again optimal among normalized frames, both in average and worst-case scenarios. For more erasures, a general analysis indicates some optimal designs. Being left with a tight frame after erasures minimizes distortion, but considering also the transmission rate and possible erasure events complicates optimizations greatly.

JPEG standard uniform quantization error modeling with applications to sequential and progressive operation modes

The <i>Journal of Electronic Imaging</i> (JEI), copublished bimonthly with the Society for Imaging Science and Technology, publishes peer-reviewed papers that cover research and applications in all areas of electronic imaging science and technology.

Validity of uniform quantization error model for sinusoidal signals without and with dither

A technique is presented for determining the probability density function (PDF) and variance of the quantization error of a sinusoidal signal applied to a uniform quantizer, namely, an ideal analog-digital converter. The results are the basis for determining the validity of the uniform quantization noise model for this class of signals. When dither is used it tends to decorrelate quantization error and the input signal. The effect of added uniform dither on the PDF of the quantizer input is investigated, as well as the PDF of the quantization error corresponding to the dithered sinusoid. The results prove the validity of the uniform quantization model in this case. >

Nondetour phase digital holograms: an analysis: errata

An expression for the required special bandwidth of a computer generated hologram is desired using result from the theory of frequency modulation. The expression is the same as a rule of thumb first presented by Lee. A simple quantization error model is presented for one type of nondetour phase class. It is shown that this hologram may be used to achieve a near optimum simulation of the quantization problem for random phase images.

Nondetour phase digital holograms: an analysis

An expression for the required special bandwidth of a computer generated hologram is desired using results from the theory of frequency modulation. The expression is the same as a rule of thumb first presented by Lee. A simple quantization error model is presented for one type of nondetour phase class. It is shown that this hologram may be used to achieve a near optimum simulation of the quantization problem for random phase images.