One-bit Quantization Research Articles

Performance Limit for Distributed Estimation Systems With Identical One-Bit Quantizers

Full precision Crame?r-Rao lower bound (CRLB) where no quantization is assumed is often employed to evaluate and compare distributed estimation performance even though the sensor observations are quantized before any further processing. However, as it completely disregards quantization and often does not exist when the sensor observation noise is bounded, full precision CRLB is often too optimistic or not applicable. In this work, we determine the performance limit of a distributed estimation system with identical one-bit quantizers in terms of the metric minimax CRLB. The performance limit that a distributed estimation scheme with identical quantizers can achieve is found as well as the set of optimal noise distribution functions and quantizers. Compared to the full precision CRLB, the performance limit is shown to be a much tighter bound when the parameter range is relatively large and reveals the important role of the quantization system.

Distributed estimation over fading channels using one-bit quantization

The problem of distributed estimation of an unknown parameter in noise is considered. Sensor observations are compressed using a one-bit quantizer and then transmitted to a fusion center over fading channels. We propose a mean estimator which requires only the knowledge of the mean of the channel gain and a sign estimator where the signs of the received signals are used for parameter estimation. Our analysis shows that these two estimators are not only computationally efficient, but also achieve comparable performance of maximum likelihood estimation. A robust iterative algorithm is proposed to address the design issue of local quantization thresholds.

ANALYSIS AND DESIGN OF ONE-BIT NOISE-SHAPING QUANTIZER USING VARIABLE STRUCTURE CONTROL APPROACH

This paper offers a straightforward analysis of a one-bit noise-shaping quantizer based on variable structure control methods. The method uses the concept of equivalent control to access the bound on the state signals when the sliding condition is reached. For a stable loop filter, the global stability can be proved to exist by using the Lyanpunov stability theory. For an unstable loop filter, this paper proposes a state constraint approach to ensure that the sliding condition is achieved. Simulation examples and an experimental circuit constructed using off-the-shelf electronic components are given to demonstrate the correctness of the theoretical analysis.

Multimode Fiber as Random Code Generator— Application to Massively Parallel MIMO Transmission

We propose a novel multiple-input multiple-output (MIMO) scheme over multimode fiber, acting as a distributed random code generator fed by spatial codes, using silicon photonics in the transmitter and maximum-likelihood (ML) electronic detection in the receiver, providing an alternative to coarse wavelength division multiplexing (CWDM) for implementation of ultrahigh speed parallel transmission over short-range optical interconnects. The optical MIMO system utilizes mutually coherent transmission and conventional direct detection with one-bit quantization, facilitating cost-effective application to 100 Gb/s links over < 50 m.

Robust Filter Design for Sigma-Delta Modulators with One-Bit Quantizer and Analog Mismatch

A method for improving the signal-to-noise ratio (SNR) of single-stage sigma-delta modulators (SDMs) with one-bit quantizer is presented. It is well known that uncertainties and noises are sources of error which cause performance degradation. Specifically, the modulators are naturally subject to analog mismatch in capacitor values and finite amplifier gain, which are manifested as parametric uncertainties of the SDMs, whereas the quantization error due to the coarse quantizer is modeled as a bounded-peak additive noise. Robust stability margin optimization and suppression of the output swing of the non-ideal integrators are also considered in the integral design. Based on a frequently used linear model, we propose a new architecture for the SDMs. The associated digital filter is determined by robust control theory. For numerical illustration, comparisons between the proposed SDM and the conventional one are made. It is shown that the proposed SDM has an effective resolution of 16 bits.

Noise-Improved Bayesian Estimation With Arrays of One-Bit Quantizers

A noisy input signal is observed by means of a parallel array of one-bit threshold quantizers, in which all the quantizer outputs are added to produce the array output. This parsimonious signal representation is used to implement an optimal Bayesian estimation from the output of the array. Such conditions can be relevant for fast real-time processing in large-scale sensor networks. We demonstrate that, for input signals of arbitrary amplitude, the performance in the estimation can be improved by the addition of independent noises onto the thresholds in the array. These results constitute a novel instance of the phenomenon of suprathreshold stochastic resonance in arrays, by which nonlinear transmission or processing of signals with arbitrary amplitude can be improved through cooperative coupling with noise.

Fisher information and noise-aided power estimation from one-bit quantizers

We investigate power estimation on a random noise from measurements taken by one-bit quantizers, with an efficacy assessed by the Fisher information. In isolated quantizers, an optimal tuning of the quantization threshold exists to maximize the estimation efficacy. When the quantizers are assembled in parallel arrays, no specific tuning of the quantization threshold is any longer required. Instead, addition of noise in the array can be employed as a means of enhancing the estimation efficacy. This is interpreted as a form of stochastic resonance or improvement by noise, applied to parametric estimation on a noise, which is shown improvable by adding more noise.

Adaptive Delta-Sigma Modulation for Enhanced Input Dynamic Range

An adaptive delta-sigma modulator of 1st order with one-bit quantization is presented. Adaptation is instantaneous and based on an exponential law. The feedback signal is a multibit discrete-level signal generated by a digital-to-analog converter (DAC). Compared to a nonadaptive delta-sigma modulator of 1st order, the input dynamic range is significantly enhanced. The gain in dynamic range is 6 dB per bit defining the feedback amplitude. The influence of nonideal DAC performance is discussed. It is demonstrated that an implementation of the system is realistic with standard CMOS technology. To relax the requirements to the one-bit quantizer, the quantizer input signal is amplified adaptively (Q-Switching).

A low-power correlation-derivative CMOS VLSI circuit for bearing estimation

We present a CMOS integrated circuit (IC) for bearing estimation in the low-audio range that performs a correlation derivative approach in a 0.35-/spl mu/m technology. The IC calculates the bearing angle of a sound source with a mean variance of one degree in a 360/spl deg/ range using four microphones: one pair is used to produce the indication and the other to define the quadrant. An adaptive algorithm decides which pair to use depending on the direction of the incoming signal, in such a way to obtain the best estimate. The IC contains two blocks with 104 stages each. Every stage has a delay unit, a block to reduce the clock speed, and a 10-bit UP/DN counter. The IC measures 2 mm by 2.4 mm, and dissipates 600 /spl mu/W at 3.3 V and 200 kHz. It is purely digital and uses a one-bit quantization of the input signals.

One‐bit sigma‐delta quantization with exponential accuracy

AbstractOne‐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 λ → ∞, 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−λ 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 π‐bandlimited signals is at most O(2−.07λ). © 2003 Wiley Periodicals, Inc.

Asymptotic MSE law of nth-order ΣΔ modulators

The mean squared error (MSE) of an nth-order /spl Sigma//spl Delta/ modulator is classically understood to decay with the oversampling ratio M at the rate /spl alpha//M/sup 2n+1/. In this paper, we demonstrate by experimental evidence and basic analytical explanations that the accurate general MSE law is in fact in (/spl beta//M/sup 2n/)+(/spl gamma//M/sup 2n+1/) instead. The coefficient /spl beta/ is in general not zero, except when the input is rigorously constant, or when using the ideal standard nonoverloaded n-bit n-loop modulator. The term /spl beta//M/sup 2n/ is dominant starting from low values of M when using a one-bit quantizer.

Perfect spline approximation

Our study of perfect spline approximation reveals: (i) it is closely related to ΣΔ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is h r−1 . This rate is optimal in the sense that a function can be approximated with a better rate if and only if it is a polynomial of degree < r. The uniqueness of best approximation is studied, too. Along the way, we also give a result on an extremal problem, that is, among all perfect splines with integer knots on R , (multiples of) Euler splines have the smallest possible norms.

Adaptive sigma-delta modulation with one-bit quantization

A method for improving the signal-to-noise ratio (SNR) of sigma-delta modulators with one-bit quantization is presented. The two-level feedback signal of a standard sigma-delta modulator is replaced by a multilevel signal, which is a superposition of two parts. One part s(n) represents a rough estimate of the instantaneous amplitude of the input signal (prediction signal), and the other y/sub b/(n) is the sign of the quantizer output, multiplied with constant b. Compared to a nonadaptive modulator, the amplitude of y/sub b/(n) is reduced. Therefore, less noise power is introduced in the quantizer, and the SNR is considerably enhanced. Signal s(n) is derived numerically from the quantizer output y/sub 0/(n) according to a particular adaptation algorithm. Except for the DC-level of s(n), sequence y/sub 0/(n) contains the full digital information of the modulator input signal. From y/sub 0/(n), a digital multilevel sequence w/sub 0/(n) can be calculated, which represents the digital modulator output. The price paid for the improved SNR is a moderate slew rate limitation of the input signal. The approach is basically suited for a wide class of sigma-delta modulators. Here, simulation results and an example for a practical implementation of an adaptive sigma-delta modulator of first order are presented.

Third-order SigmaDelta modulator feedback coefficients for maximum dynamic range

A new analysis of the third-order ΣΔ modulator is based on parameterization of the comparator gain and the fact that the one-bit quantizer delivers constant output power. It is shown that a wrong choice of feedback coefficients reduces the dynamic range considerably. The new analysis derives a set of feedback coefficients to give the best large input signal handling capabilities. The quantization noise and its spectrum are evaluated in order to find the signal-to-noise dynamic range for each case. Optimum feedback coefficients are then chosen to give maximum signal-to-quantization noise ratio. A comparison with non-ideal coefficients is also shown. The choice of non-ideal coefficients may reduce the circuit complexity.

A monolithic 4-bit 2-Gsps resonant tunneling analog-to-digital converter

The combination of resonant-tunneling diodes and heterostructure field-effect transistors provides a versatile technology for implementing microwave digital and mixed-signal applications. Here we demonstrate and characterize the first monolithic flash analog-to-digital converter (ADC) in this technology. The first-pass ADC achieved 2.7 effective bits at 2 gigasamples per second (Gsps) for a 220-MHz input signal. The one-bit quantizer achieved a single-tone spurious free dynamic range greater than 40 dB at 2 Gsps for a 220-MHz single-tone input with dithering.

Spectral shaping of circuit errors in digital-to-analog converters

Recently, various multibit noise-shaping digital-to-analog converters (DACs) have been proposed that use digital signal processing techniques to cause the DAC noise arising from analog component mismatches to be spectrally shaped. Such DACs have the potential to significantly increase the present precision limits of /spl Delta//spl Sigma/ data converters by eliminating the need for one-bit quantization in delta-sigma modulators. This paper extends the practicality of the noise-shaping DAC approach by presenting a general noise-shaping DAC architecture along with two special-case configurations that achieve first- and second-order noise-shaping, respectively. The second-order DAC configuration, in particular, is the least complex of those currently known to the author. Additionally, the paper provides a rigorous explanation of the apparent paradox of how the DAC noise can be spectrally shaped even though the sources of the DAC noise-the errors introduced by the analog circuitry-are not known to the noise-shaping algorithm.

Analysis of ΔΣ modulators with zero mean stochastic inputs

In this paper, a new framework for the analysis of /spl Delta//spl Sigma/ modulators with stochastic inputs is proposed. The framework is based on assuming that the input to the one-hit quantizer is a Gaussian random process with zero mean and is thus able to interrelate the autocorrelation and cross-correlation of different signals of the modulator. Two main equations describing the behavior of two different /spl Delta//spl Sigma/ topologies are derived. These two equations are generally nonlinear and can be of arbitrary order, hence approximations are used to study some interesting cases. First, the nonlinear equations are linearized and solved analytically for first-order modulator with white inputs and numerically for colored inputs both with and without dithering. Also, a numerical iterative approach is used for second and fourth order modulators with white and colored inputs. In these cases, the variance of the one-bit quantizer input is found as a function of modulator input power. Next, the variance of the one-bit quantizer input is calculated when large amplitude oscillations are present assuming a large amplitude limit cycle to have a sinusoidal autocorrelation. Finally, an attempt is made to estimate the modulator's critical input power level beyond which these large amplitude limit cycles start. >

NONLINEAR DYNAMICS OF A CLASS OF ANALOG-TO-DIGITAL CONVERTERS

Oversampled sigma-delta modulators are finding widespread use in audio and other signal processing applications, due to their simple structure and robustness to circuit imperfections. Exact analyses of the system are complicated by the presence of a discontinuous nonlinear element—a one-bit quantizer. In this paper, we study the dynamics of the one-dimensional mapping which models the behavior of the single-loop modulator. This mapping has a discontinuity at the origin and constant slope at all other points. With slope one, the dynamics in the region of interest reduce to those of the rotation of the circle. With slope less than one, almost all system inputs give rise to globally asymptotically stable periodic orbits. We emphasize the case with slope greater than one, and explain the structure of the resultant bifurcation diagram. A symbolic dynamics based study allows us to explain the self-similarity of the dynamics and the nature of chaos in the system.

Effect of phase errors on field patterns generated by an ultrasound phased-array hyperthermia applicator

The effect of phase quantization errors and Gaussian distributed random phase errors on field patterns synthesized by a rectangular ultrasound phased array hyperthermia applicator is studied. The parameters defined show that, over the range of four-bit to one-bit quantization, the simulated, patterns degrade with increasing phase errors. However, the overall shape and position of the foci remain unchanged. Simulation results show that, even with one-bit phase quantization, or with a Gaussian distributed random phase error of standard deviation of about 52 degrees , the applicator can still produce useful, if not particularly clean, power deposition patterns. Thus, the power deposition patterns are remarkably stable tolerating quite large phase error levels. This suggests that the power deposition patterns inside the body may be relatively insensitive to phase errors due to tissue inhomogeneities.

One-bit quantization of computer-stored spectra

The behavior of reconstructed spectra after application of one-bit quantization in comparison to the usual absorption-type spectra in modern computerized Fourier-transform spectroscopy is re-investigated. By means of a “synthetic” and an experimental 13C-F.t.-n.m.r. spectrum, the results of some computer manipulations are illustrated. These manipulations enable the inherent distortions in the reconstructed spectra to be reduced by half.