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Nonlinear estimation with quantized measurements and application to pulse code modulation, data compression, and predictive quantization
Nonlinear estimation with quantized measurements, applying algorithms to PCM, predictive quantization and data compression
A study is made of the effects of quantization of the radar returns transmitted from aircraft or spacecraft employing a synthetic aperture radar system. The study is based on the output images obtained after one-bit, two-bit, and eight-bit quantizations and comparing the results to ground truth. In this way the degradation resulting from data or bandwidth reduction is determined. Quantization is evaluated in terms of crater scene, number of looks, and transmission error rate. It is found that two-bit quantization of raw radar data from homogeneous scenes processed to 32 looks yields nearly all the details of the original. One-bit quantization of raw radar data from homogeneous scenes processed to 32 looks yields a good visual representation of the scene but some fine detail is lost and the absolute reflectivity level is not reliable. Image quality is observed to improve with more looks and video and intermediate frequency quantization are not distinguishable even for one-bit quantizations. Image quality is not influenced by bit error rates less than about 2 to the -7th power.
The quantizer problem is first considered. The conditions under which the uniform white sequence model for the quantizer error is valid are established independent of the sampling rate. An equivalent spectral density is defined for the quantizer error resulting in an effective SNR value. This effective SNR may be used to determine quantized performance from infinitely fine quantized results. Attention is given to sampling rate considerations. Sampling rate characteristics of the digital phase-locked loop (DPLL) structure are investigated for the infinitely fine quantized system. The predicted phase error variance equation is examined as a function of the sampling rate. Simulation results are presented and a method is described which enables the minimum required sampling rate to be determined from the predicted phase error variance equations.
If the VCO of a phase-locked receiver is to be replaced by a digitally programmed synthesizer, the phase error signal must be sampled and quantized. Effects of quantizing after the loop filter (frequency quantization) or before (phase error quantization) are investigated. Constant Doppler or Doppler rate noiseless inputs are assumed. The main result gives the phase jitter due to frequency quantization for a Doppler-rate input. By itself, however, frequency quantization is impractical because it makes the loop dynamic range too small.
The amount of quantization noise generated in a digital-to-analog converter is dependent on the number of bits or quantization levels used to digitize the analog signal in the analog-to-digital converter. The minimum number of quantization levels and the minimum sample rate were derived for a digital voice channel. A sample rate of 6000 samples per second and lowpass filters with a 3 db cutoff of 2400 Hz are required for 100 percent sentence intelligibility. Consonant sounds are the first speech components to be degraded by quantization noise. A compression amplifier can be used to increase the weighting of the consonant sound amplitudes in the analog-to-digital converter. An expansion network must be installed at the output of the digital-to-analog converter to restore the original weighting of the consonant sounds. This technique results in 100 percent sentence intelligibility for a sample rate of 5000 samples per second, eight quantization levels, and lowpass filters with a 3 db cutoff of 2000 Hz.
The well-known algorithm of Max is used to determine the minimum distortion quantizers for normal, two-sided exponential, and specialized two-sided gamma input distributions and for mean-square, magnitude, and relative magnitude error distortion criteria. The optimum equally-spaced and unequally-spaced quantizers are found, with the resulting quantizer distortion and entropy. The quantizers, and the quantizers with entropy coding, are compared to the rate distortion bounds for mean-square and magnitude error.
In quantum time (QT) schemes, time is promoted to a degree of freedom, allowing Lorentz covariance to be made explicit for single particles. We ask whether this can be lifted to QFT so that Lorentz covariance becomes manifest at the Hilbert-space level, rather than being hidden as in the standard canonical formulation. We address this question by proposing a second-quantized approach in which the elementary particle is the QT particle itself, leading naturally to the notion of spacetime field algebras and of quantum action. We show, however, that a naive many-body construction runs into inconsistencies. To pinpoint their origin we introduce a classical counterpart of the second-quantized formalism, spacetime classical mechanics (SCM), and prove a no-go theorem: Dirac quantization of SCM collapses back to standard QFT and therefore hides covariance. We circumvent this problem by presenting a quantum-action-based quantization that yields a spacetime version of quantum mechanics (SQM), making covariance manifest for (interacting) QFTs. Finally, we show that this resolution is tied to a genuine spacetime generalization of the notion of a quantum state, required by causality and closely connected to recent “states over time” proposals and, in dS/CFT–motivated settings, to microscopic notions of timelike entanglement and emergent time.
We consider optimum uniform data quantization for noisy channels. We present a general formulation for natural encoding that results in simple expressions for the mean-square error. Specifically, we show that the optimum location of the center of the quantizer is at the mean of the distribution for all error rates. The optimum levels for quantization and the corresponding mean-square error are presented for Gaussian and uniform data. For the latter the width of the optimum quantizer for noisy channels is shown to be smaller than the entire range of probability distribution.
Typically an analog signal from a space system is sampled, quantized by Analog-to-Digital (A/D) conversion, merged into a bit stream, communicated to a ground station, received by the ground station, and processed by the ground station to extract useful information for dissemination to the users. The cost of each of these steps is reduced as the number of quantization steps is reduced in the A/D converter. The number of quantization steps should be as small as possible without losing the required information content. This report deals specifically with the accuracy of averages as a function of the number of quantized samples used to compute the averages with the noise on the analog signal as a parameter. For example, the success of the Visible Infrared Spin Scan Radiometer (VISSR) Atmospheric Sounder (VAS) Demonstration depends upon temporally averaging multiple samples in an effort to reduce noise to a sufficiently low level such that temperature profile sounding is made possible. A tutorial description of this process is presented.
The performance of Viterbi decoding in a non-Gaussian environment is investigated using a nonlinear quantization strategy. The channel model consists of a convolutionally encoded BPSK signal transmitted to a satellite where it is corrupted with additive white Gaussian noise and pulsed radio frequency interference (RFI). The resultant signal is then passed through a satellite nonlinearity and transmitted to a ground station where it is coherently detected. Interleaving is assumed in order to make the channel memoryless. The presence of RFI makes the channel statistics non-Gaussian, leading to a nonlinear log-likelihood function. A near optimum quantization scheme is found by maximizing a channel parameter, or by matching the quantizer to the log-likelihood function in a mean square error sense. Bit error rate performance improvement is achieved by using such nonlinear quantization.
Design of optimum quantizers for quantizer-input message signal and quantizer-input message signal contaminated by noise
Some neutron stars known as magnetars possess very strong magnetic fields, with surface fields as large as 10 15 G and internal fields that are possibly stronger. Recent observations of the radio pulsar GLEAM-X J1627 suggest it may have a surface field as strong as 10 16 G. In the presence of a strong magnetic field, Leinson and Perez showed and Baiko and Yakovlev confirmed that the energy levels of electrons and protons are quantized and the direct Urca process allows neutron stars to cool rapidly, even at low density. For the case of magnetic fields 𝐵 ≳ 10 16 G, we find features in the emissivity due to energy quantization that are not captured by the frequently employed quasiclassical approximation where energy levels are treated as nearly continuous. Resonances can result in amplification of the neutrino emissivity at specific densities compared to a calculation that neglects quantization, particularly at low temperature. These effects are not important for the thermal evolution of an entire neutron star, but may be relevant for phenomena that depend on behavior at specific densities. Here, we present a fully relativistic calculation of the direct Urca rate in a strong magnetic field using the standard V-A weak Lagrangian incorporating mean field nuclear effects and discuss approaches to the numerical challenge the modified wave functions present and a new semianalytic approximation. These tools are also applicable to calculating neutrino opacities in strong magnetic fields in the ejecta of binary neutron star mergers. We calculate the opacities for neutrinos capturing on free nucleons at subsaturation densities and temperatures exceeding an MeV. We find an enhancement to capture processes of the lowest energy neutrinos by an order of magnitude or more due to suppression of electron Pauli blocking in the case of capture on neutrons, and from the effect of the nucleon magnetic moments in the case of capture on protons.
The existence of quantizer-induced limit cycles in digital control systems is a well-known phenomenon. This paper reports the results of an investigation into the discrete describing function technique which is applicable not only to the quantizer nonlinearity but also to any nonlinearity in an otherwise linear discrete system. First a general expression is developed for the discrete describing function that is applicable to any nonlinearity. The describing function is then obtained for the quantizer nonlinearity. Finally, the use of the discrete describing function is illustrated by an example of a digital control system.
Quantized vortices can occur around nodal points in wavefunctions. The derivation depends only on the wavefunction being single valued, continuous, and having continuous first derivatives. Since the derivation does not depend upon the dynamical equations, the quantized vortices are expected to occur for many types of waves such as electromagnetic and acoustic. Such vortices have appeared in the calculations of the H + H2 molecular collisions and play a role in the chemical kinetics. In a companion paper, it is shown that quantized vortices occur when optical waves are internally reflected from the face of a prism or particle beams are reflected from potential energy barriers.
In this paper a technique is developed for choosing programing forms and bit configurations for digital filters that minimize the quantization errors. The technique applies to digital filters operating in fixed-point arithmetic in either open-loop or closed-loop systems, and is implemented by a digital computer program that is based on a digital simulation of the system. As an output the program gives the programing form required for minimum quantization errors, the total bit configuration required in the filter, and the location of the binary decimal point at each quantizer within the filter.
Some theoretical and experimental results on optimal sampling and quantization of synthetic aperture radar (SAR) signals are presented. It includes a description of a derived theoretical relationship between the pixel signal to noise ratio of processed SAR images and the number of quantization bits per sampled signal, assuming homogeneous extended targets. With this relationship known, a solution may be realized for the problem of optimal allocation of a fixed data bit-volume (for specified surface area and resolution criterion) between the number of samples and the number of bits per sample. The results indicate that to achieve the best possible image quality for a fixed bit rate and a given resolution criterion, one should quantize individual samples coarsely and thereby maximize the number of multiple looks. The theoretical results are then compared with simulation results obtained by processing aircraft SAR data.
Qualitative arguments are adduced which indicate that the apparently chaotic dynamics on the Henon-Heiles (1964) surface display sufficient regularity on a short to intermediate (but not long) time scale to allow the use of standard EBK quantization techniques. This takes advantage of the remnants of manifold structure implied. A complete uniform semiclassical quantization is performed using the time independent technique of the Birkhoff-Gustavson normal form, which was recently introduced in the context of semiclassical quantization by Swimm and Delos (1977, 1979).