Non-Gaussian noise Progress report
Probability density functions for quantization noise, continuous wave interference, impulse noise, and atmospheric noise
Tuncer, K. F.↗
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Estimation of the state of a nonlinear process in the presence of nongaussian noise and disturbances.
Optimum estimate of nonlinear process in presence of non-Gaussian noise and disturbances, deriving algorithm for correcting approximation
OPTIMIZATION↗
Edge detection for synthetic aperture radar and other noisy images
The development is examined of a new edge detector which is shown to perform adequately in the non-Gaussian multiplicative noise environment which characterizes radar images. This edge detector operates over larger local neighborhood and is less susceptible to noise than previous edge detectors and is therefore more suitable for radar. In addition, a radar image noise model is employed for the design of this new operator. This edge detector is unique in that it is assumed that every local area belongs to either the class of local areas not containing edges or to the class of local areas containing edges. Each pixel's local neighborhood is then assigned to one of these two classes using a statistical hypothesis (a likelihood ratio) test. It is demonstrated that this algorithm is useful for detecting edges in radar images.
Frost, V. S.↗
Fast frequency acquisition via adaptive least squares algorithm
A new least squares algorithm is proposed and investigated for fast frequency and phase acquisition of sinusoids in the presence of noise. This algorithm is a special case of more general, adaptive parameter-estimation techniques. The advantages of the algorithms are their conceptual simplicity, flexibility and applicability to general situations. For example, the frequency to be acquired can be time varying, and the noise can be non-Gaussian, nonstationary and colored. As the proposed algorithm can be made recursive in the number of observations, it is not necessary to have a priori knowledge of the received signal-to-noise ratio or to specify the measurement time. This would be required for batch processing techniques, such as the fast Fourier transform (FFT). The proposed algorithm improves the frequency estimate on a recursive basis as more and more observations are obtained. When the algorithm is applied in real time, it has the extra advantage that the observations need not be stored. The algorithm also yields a real time confidence measure as to the accuracy of the estimator.
Kumar, Rajendra↗
Adaptive line enhancers for fast acquisition
Three adaptive line enhancer (ALE) algorithms and architectures - namely, conventional ALE, ALE with double filtering, and ALE with coherent accumulation - are investigated for fast carrier acquisition in the time domain. The advantages of these algorithms are their simplicity, flexibility, robustness, and applicability to general situations including the Earth-to-space uplink carrier acquisition and tracking of the spacecraft. In the acquisition mode, these algorithms act as bandpass filters; hence, the carrier-to-noise ratio (CNR) is improved for fast acquisition. In the tracking mode, these algorithms simply act as lowpass filters to improve signal-to-noise ratio; hence, better tracking performance is obtained. It is not necessary to have a priori knowledge of the received signal parameters, such as CNR, Doppler, and carrier sweeping rate. The implementation of these algorithms is in the time domain (as opposed to the frequency domain, such as the fast Fourier transform (FFT)). The carrier frequency estimation can be updated in real time at each time sample (as opposed to the batch processing of the FFT). The carrier frequency to be acquired can be time varying, and the noise can be non-Gaussian, nonstationary, and colored.
Yeh, H.-G.↗
An investigation of error characteristics and coding performance
The performance of forward error correcting coding schemes on errors anticipated for the Earth Observation System (EOS) Ku-band downlink are studied. The EOS transmits picture frame data to the ground via the Telemetry Data Relay Satellite System (TDRSS) to a ground-based receiver at White Sands. Due to unintentional RF interference from other systems operating in the Ku band, the noise at the receiver is non-Gaussian which may result in non-random errors output by the demodulator. That is, the downlink channel cannot be modeled by a simple memoryless Gaussian-noise channel. From previous experience, it is believed that those errors are bursty. The research proceeded by developing a computer based simulation, called Communication Link Error ANalysis (CLEAN), to model the downlink errors, forward error correcting schemes, and interleavers used with TDRSS. To date, the bulk of CLEAN was written, documented, debugged, and verified. The procedures for utilizing CLEAN to investigate code performance were established and are discussed.
Ebel, William J.↗
Statistics and topology of the COBE differential microwave radiometer first-year sky maps
We use statistical and topological quantities to test the Cosmic Background Explorer (COBE) Differential Microwave Radiometer (DMR) first-year sky maps against the hypothesis that the observed temperature fluctuations reflect Gaussian initial density perturbations with random phases. Recent papers discuss specific quantities as discriminators between Gaussian and non-Gaussian behavior, but the treatment of instrumental noise on the data is largely ignored. The presence of noise in the data biases many statistical quantities in a manner dependent on both the noise properties and the unknown cosmic microwave background temperature field. Appropriate weighting schemes can minimize this effect, but it cannot be completely eliminated. Analytic expressions are presented for these biases, and Monte Carlo simulations are used to assess the best strategy for determining cosmologically interesting information from noisy data. The genus is a robust discriminator that can be used to estimate the power-law quadrupole-normalized amplitude, Q(sub rms-PS), independently of the two-point correlation function. The genus of the DMR data is consistent with Gaussian initial fluctuations with Q(sub rms-PS) = (15.7 +/- 2.2) - (6.6 +/- 0.3)(n - 1) micro-K, where n is the power-law index. Fitting the rms temperature variations at various smoothing angles gives Q(sub rms-PS) = 13.2 +/- 2.5 micro-K and n = 1.7(sup (+0.3) sub (-0.6)). While consistent with Gaussian fluctuations, the first year data are only sufficient to rule out strongly non-Gaussian distributions of fluctuations.
Smoot, G. F.↗
Three-Point Correlations in the COBE DMR 2 Year Anisotropy Maps
We compute the three-point temperature correlation function of the COBE Differential Microwave Radiometer (DMR) 2 year sky maps to search for evidence of non-Gaussian temperature fluctuations. We detect three-point correlations in our sky with a substantially higher signal-to-noise ratio than from the first-year data. However, the magnitude of the signal is consistent with the level of cosmic variance expected from Gaussian fluctuations, even when the low-order multipole moments, up to l = 9, are filtered from the data. These results do not strongly constrain most existing models of structure formation, but the absence of intrinsic three-point correlations on large angular scales is an important consistency test for such models.
Hinshaw, G.↗
Bispectral analysis of equatorial spread F density irregularities
Bispectral analysis has been applied to density irregularities at frequencies 5-30 Hz observed with a sounding rocket launched from Peru in March 1983. Unlike the power spectrum, the bispectrum contains statistical information about the phase relations between the Fourier components which make up the waveform. In the case of spread F data from 475 km the 5-30 Hz portion of the spectrum displays overall enhanced bicoherence relative to that of the background instrumental noise and to that expected due to statistical considerations, implying that the observed f exp -2.5 power law spectrum has a significant non-Gaussian component. This is consistent with previous qualitative analyses. The bicoherence has also been calculated for simulated equatorial spread F density irregularities in approximately the same wavelength regime, and the resulting bispectrum has some features in common with that of the rocket data. The implications of this analysis for equatorial spread F are discussed, and some future investigations are suggested.
Labelle, J.↗
Limits on three-point correlations in the COBE DMR first-year anisotropy maps
We compute the three-point temperature correlation function of the Cosmic Background Explorer (COBE) Differential Microwave Radiometer (DMR) first-year sky maps to search for non-Gaussian temperature fluctuations. The level of fluctuations seen in the computed correlation function are too large to be attributable solely to instrument noise. However the fluctuations are consistent with the level expected to result from a superposition of istrument noise and sky signal arising from a Gaussian power-law model of initial fluctuations, with a quadrupole normalized amplitude of 17 micro K and a power-law spectral index n = 1. We place limits on the amplitude of intrinsic three-point correlations with a variety of predicted functional forms.
Hinshaw, G.↗
Non-Gaussian microwave background fluctuations from nonlinear gravitational effects
Whether the statistics of primordial fluctuations for structure formation are Gaussian or otherwise may be determined if the Cosmic Background Explorer (COBE) Satellite makes a detection of the cosmic microwave-background temperature anisotropy delta T(sub CMB)/T(sub CMB). Non-Gaussian fluctuations may be generated in the chaotic inflationary model if two scalar fields interact nonlinearly with gravity. Theoretical contour maps are calculated for the resulting Sachs-Wolfe temperature fluctuations at large angular scales (greater than 3 degrees). In the long-wavelength approximation, one can confidently determine the nonlinear evolution of quantum noise with gravity during the inflationary epoch because: (1) different spatial points are no longer in causal contact; and (2) quantum gravity corrections are typically small-- it is sufficient to model the system using classical random fields. If the potential for two scalar fields V(phi sub 1, phi sub 2) possesses a sharp feature, then non-Gaussian fluctuations may arise. An explicit model is given where cold spots in delta T(sub CMB)/T(sub CMB) maps are suppressed as compared to the Gaussian case. The fluctuations are essentially scale-invariant.
Salopek, D. S.↗
Non-Gaussian approach for parametric random vibration of non-linear structures
The dynamic response of a nonlinear, single degree of freedom structural system subjected to a physically white noise parametric excitation is investigated. The Ito stochastic calculus is employed to derive a general differential equation for the moments of the response coordinates. The differential equations of moments of any order are found to be coupled with higher order moments. A non-Gaussian closure scheme is developed to truncate the moment equations up to fourth order. The statistical of the stationary response are computed numerically and compared with analytical solutions predicted by a Gaussian closure scheme and the stochastic averaging method. It is found that the computed results exhibit the jump phenomenon which is typical of the characteristics of deterministic nonlinear systems. In addition, the numerical algorithm leads to multiple solutions all of which give positive mean squares. However, two of these solutions are found to violate the properties of high order moments. One solution preserves the moments properties and demonstrates that the system achieves a stationary response.
Ibrahim, R. A.↗
Topology of microwave background fluctuations - Theory
Topological measures are used to characterize the microwave background temperature fluctuations produced by 'standard' scenarios (Gaussian) and by cosmic strings (non-Gaussian). Three topological quantities: total area of the excursion regions, total length, and total curvature (genus) of the isotemperature contours, are studied for simulated Gaussian microwave background anisotropy maps and then compared with those of the non-Gaussian anisotropy pattern produced by cosmic strings. In general, the temperature gradient field shows the non-Gaussian behavior of the string map more distinctively than the temperature field for all topology measures. The total contour length and the genus are found to be more sensitive to the existence of a stringy pattern than the usual temperature histogram. Situations when instrumental noise is superposed on the map, are considered to find the critical signal-to-noise ratio for which strings can be detected.
Gott, J. Richard, III↗
Improving the performance of soft decision Viterbi decoding in a non-Gaussian environment through non-linear quantization
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.
Mcgregor, D. N.↗
Stochastic inflation lattice simulations - Ultra-large scale structure of the universe
Non-Gaussian fluctuations for structure formation may arise in inflation from the nonlinear interaction of long wavelength gravitational and scalar fields. Long wavelength fields have spatial gradients, a (exp -1), small compared to the Hubble radius, and they are described in terms of classical random fields that are fed by short wavelength quantum noise. Lattice Langevin calculations are given for a toy model with a scalar field interacting with an exponential potential where one can obtain exact analytic solutions of the Fokker-Planck equation. For single scalar field models that are consistent with current microwave background fluctuations, the fluctuations are Gaussian. However, for scales much larger than our observable Universe, one expects large metric fluctuations that are non-Gaussian. This example illuminates non-Gaussian models involving multiple scalar fields which are consistent with current microwave background limits.
Salopek, D. S.↗
The topology of large-scale structure. II - Nonlinear evolution of Gaussian models
The evolution of non-Gaussian behavior in the large-scale universe from Gaussian initial conditions is studied. Topology measures developed in previous papers are applied to the smoothed initial, final, and biased matter distributions of cold dark matter, white noise, and massive neutrino simulations. When the smoothing length is approximately twice the mass correlation length or larger, the evolved models look like the initial conditions, suggesting that random phase hypotheses in cosmology can be tested with adequate data sets. When a smaller smoothing length is used, nonlinear effects are recovered, so nonlinear effects on topology can be detected in redshift surveys after smoothing at the mean intergalaxy separation. Hot dark matter models develop manifestly non-Gaussian behavior attributable to phase correlations, with a topology reminiscent of bubble or sheet distributions. Cold dark matter models remain Gaussian, and biasing does not disguise this.
Melott, Adrian L.↗
Comments on the statistical analysis of excess variance in the COBE differential microwave radiometer maps
Cosmic anisotrophy produces an excess variance sq sigma(sub sky) in the Delta maps produced by the Differential Microwave Radiometer (DMR) on cosmic background explorer (COBE) that is over and above the instrument noise. After smoothing to an effective resolution of 10 deg, this excess sigma(sub sky)(10 deg), provides an estimate for the amplitude of the primordial density perturbation power spectrum with a cosmic uncertainty of only 12%. We employ detailed Monte Carlo techniques to express the amplitude derived from this statistic in terms of the universal root mean square (rms) quadrupole amplitude, (Q sq/RMS)(exp 0.5). The effects of monopole and dipole subtraction and the non-Gaussian shape of the DMR beam cause the derived (Q sq/RMS)(exp 0.5) to be 5%-10% larger than would be derived using simplified analytic approximations. We also investigate the properties of two other map statistics: the actual quadrupole and the Boughn-Cottingham statistic. Both the sigma(sub sky)(10 deg) statistic and the Boughn-Cottingham statistic are consistent with the (Q sq/RMS)(exp 0.5) = 17 +/- 5 micro K reported by Smoot et al. (1992) and Wright et al. (1992).
Wright, E. L.↗
An efficient system for reliably transmitting image and video data over low bit rate noisy channels
This research project is intended to develop an efficient system for reliably transmitting image and video data over low bit rate noisy channels. The basic ideas behind the proposed approach are the following: employ statistical-based image modeling to facilitate pre- and post-processing and error detection, use spare redundancy that the source compression did not remove to add robustness, and implement coded modulation to improve bandwidth efficiency and noise rejection. Over the last six months, progress has been made on various aspects of the project. Through our studies of the integrated system, a list-based iterative Trellis decoder has been developed. The decoder accepts feedback from a post-processor which can detect channel errors in the reconstructed image. The error detection is based on the Huber Markov random field image model for the compressed image. The compression scheme used here is that of JPEG (Joint Photographic Experts Group). Experiments were performed and the results are quite encouraging. The principal ideas here are extendable to other compression techniques. In addition, research was also performed on unequal error protection channel coding, subband vector quantization as a means of source coding, and post processing for reducing coding artifacts. Our studies on unequal error protection (UEP) coding for image transmission focused on examining the properties of the UEP capabilities of convolutional codes. The investigation of subband vector quantization employed a wavelet transform with special emphasis on exploiting interband redundancy. The outcome of this investigation included the development of three algorithms for subband vector quantization. The reduction of transform coding artifacts was studied with the aid of a non-Gaussian Markov random field model. This results in improved image decompression. These studies are summarized and the technical papers included in the appendices.
Costello, Daniel J., Jr.↗