Systematic Phase Errors Research Articles

An analysis of systematic phase errors due to nonlinearity in fringe scanning systems

In a fringe scanning interferometer, if the digitized intensity data do not change sinusoidally due to, for example, nonlinearity of the detector or to the fringe shape itself, systematic errors will occur in determining the phases in an interferogram. Unless this nonlinearity is compensated for, the three-bucket or four-bucket method will be unable to determine the phases accurately. This problem is discussed and some examples are shown of the necessary number of sampling steps to reduce under-sampling errors.

Stable nonlinear methods for sensor array processing;

Most nonlinear high-resolution bearing estimators are unstable in the presence of correlated noise, system phase errors, and multipath arrivals because they inadvertently place too much emphasis on unstable eigenvectors of the cross-sensor correlation matrix. For moderately correlated noise there will be sufficiently many eigenvectors to resolve and localize discrete sources. A is given (the stable nonlinear method or SNLM) whereby reweighting of the eigenvectors is achieved implicitly, without actual calculation of the eigenvectors. This SNLM is compared with Capon's maximum likelihood (MLM) in simulations of correlated noise, partially correlated signals, and phase errors, and is shown to provide good stability in the cases considered.

Systematic errors in the behaviour of cyclones in the ECMWF operational models

Systematic errors in the forecasts (up to 10 days) of cyclones over the north Atlantic and Europe by the ECMWF operational grid point model (N48) for the 1981–82 winter seasons are described, and some corresponding results for the ECMWF operational spectral model (T63) for the 1983–84 winter season are shown. Statistics for the cyclone tracks, central pressure, deepening rates, etc., are given. The forecasts from the grid point model tend to displace the cyclone tracks towards the south, especially after day 4, except near Newfoundland where a northward displacement is found. The corresponding errors are reduced in the spectral model. A systematic phase error is evident in the grid point model with the waves being too slow, usually in the developing stages of cyclones. This error is also apparent for the fast moving cyclones in the spectral model. For the grid point model, the forecast deepening and filling rates are less than those observed. However, the deepening rate is more realistic in the spectral model, though the filling rate is still underestimated. For both models the horizontal scale of the cyclones is too small during the first 2 or 3 days, especially in the forecasts started from several days prior to the day on which the cyclone was generated; during the later stages of cyclone development, the scale is too large. In comparing these errors to similar errors documented for other numerical models, some hypotheses are discussed about their likely causes. DOI: 10.1111/j.1600-0870.1985.tb00429.x

Determination of the optical constants of solids in the visible by dispersive Fourier transform spectroscopy

A Michelson interferometer has been developed for determining the optical constants of solids in the visible and UV by DFTS. A simple measuring procedure has been developed for use in amplitude reflection spectroscopy which eliminates systematic phase errors from most sources, including non-flatness of the specimen surface. The performance of the instrument is demonstrated with measurements in the visible region of the optical constants of thin Cu films vacuum deposited on fused quartz substrates.

Digital fibre transmission using optical homodyne detection

Homodyne detection of a 140 Mbit/s PSK signal transmitted over 30 km of cabled fibre has been achieved using a balanced optical phase-locked-loop receiver. System phase error was dominated by the laser frequency instability and not the cabled fibre.

Impact of beam steering errors on shifted sideband and phase shift beamforming techniques

The Shifted Sideband Beamformer (SSB) is a digital, time-domain beamformer intended for bandpass applications. Computational efficiency results from the digital processing of the complex envelope of the sensor data at a rate proportional to the bandwidth of the data. In a previous paper, the spatial response of the SSB was examined and compared to that of a conventional bandpass digital beamformer by modeling the beam steering quantization error as a uniformly distributed random variable. This paper examines, in detail, the specific case of a line array of uniformly spaced sensors where beam steering quantization is shown to produce systematic time-delay errors for both the conventional beamformer and SSB implementations. Expressions are presented for predicting the directions of grating side lobes in the spatial response pattern as a result of these errors. Also, the spatial response of the Phase Shift Beamformer (PSB), which is a limiting case of the SSB and intended for narrow-band applications, is examined. The systematic phase errors of the PSB for a planar array are shown to produce errors in the beam pointing direction for frequencies other than band center. Examples demonstrate this effect for the special case of a line array. Finally, it is shown that the SSB can be interpreted as time-domain beamforming with partial beam outputs formed from subclusters of adjacent sensors with a PSB.

Forecast Intercomparisons from Three Numerical Weather Prediction Models

A forecast intercomparison study using six different initial states was carried out with forecasts made by the 2.5°, six-layer, second-generation National Center for Atmospheric Research (NCAR) general circulation model (GCM); the 4° × 5°, nine-layer Goddard Institute for Space Studies (GISS) GCM; and the 4° (at 50°N), six-layer National Meteorological Center (NMC) operational model. The initial conditions for geopotential and velocity were obtained from the NMC operational analyses during December 1972 and January 1973 for the Northern Hemisphere. The operational NMC analyses were used for verification of the forecast 1000 and 500 mb heights. Forecast projections to either three or five days were examined using conventional skill scores, wavenumber analysis of longitude-time plots constructed from the forecasts, and maps of the actual difference (forecast minus verification). The forecast errors were defined for each model and the common errors in all three forecasts were identified. The GISS and NMC models proved to be nearly equal in skill with the GISS forecasts being slightly better beyond two days. The present version of the NCAR model appears to be significantly less skillful, apparently due to initialization sensitivity, systematic bias and damping of the small scales. All three models tend to produce similar errors in the phase speeds of all waves. Smaller scale waves not only show systematic phase and amplitude errors, but also fail to develop and decay at the proper times. Phase and amplitude errors in the planetary wave structure are comparable in magnitude to errors made at higher wavenumbers. There was also a distinct tendency for errors in each model to be highly correlated with one another. The NCAR model produces a large mean error and systematically damps the smaller scale waves during the forecasts. The GISS model exhibited the best skill in the planetary-wave structure but the phase speeds of shorter waves were, on the average, slower than those in the other models.

A Double-Loop Tracking System

A nonlinear analysis which can be used to assess certain statistical characteristics of double-loop tracking systems is presented. It takes into account the mutual coupling effects of the loops in the system. Two approaches are taken to obtain steady-state probability density functions (pdf's) of the system phase errors, φ 1 and φ 2 From these pdf's, important system performance statistics, e.g., the phaseerror variances, can be calculated, thus illustrating the application and usefulness of the analysis. The analysis is applied to a satellite transponder as an example.

Nonlinear analysis of generalized tracking systems

This paper sets forth a rather general analysis pertaining to the performance and synthesis of generalized tracking systems. The analysis is based upon the theory of continuous Markov processes, in particular, the Fokker-Planck equation. We point out the interconnection between the theory of continuous Markov processes and Maxwell's wave equations by interpreting the charge density as a transition probability density function (pdf). These topics presently go under the name of probabilistic potential theory. Although the theory is valid for (N+1)-order tracking systems with an arbitrary, memoryless, periodic nonlinearity, we study in detail the case of greatest practical interest, viz., a second-order tracking system with sinusoidal nonlinearity. In general we show that the transition pdf p(y, t|y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> , t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ) is the solution to an (N+1)-dimensional Fokker-Planck equation. The vector (y, t)=(φ, y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> ,..., y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> , t) is Markov and φ represents the system phase error. According to the theory the transition pdf's {p(φ, t|φ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> , t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ), P(y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> |y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k0</inf> , t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ); k=1,..., N} of the state variables satisfy a set of second-order partial differential equations which represent equations of flow taking place in each direction of (N+1)-space. Each equation, and solution, is characterized by a potential function U <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , t); which is related to the nonlinear restoring force h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , t)=-∇U <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , t); k=0, 1,..., N. In turn the potential functions are completely determined by the set of conditional expectations {E(y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , t|φ), E(g(φ), t|y); k=1, 2,..., N}. It is conjectured that the potential functions represent the projections of the system Lyapunov function which characterizes system stability. This paper explores these relationships in detail.

A study of tracking accuracy in monopulse phased arrays

The requirements for high-data rates in modem weapons systems favor the use of phased arrays for monopulse tracking antennas. A discussion is presented of the tracking errors produced by the phase and amplitude errors in the excitation function of a phased array. Both systematic and random phase errors are considered. The effects of both scan and aperture correlation of the random errors is discussed. Even when off-boresight tracking is performed, boresight errors were found to be the predominant tracking error.