Phase Error Reduction Research Articles

Design trades for Rotman lenses

The foundation of a satisfactory Rotman lens design is geometric. The effects of seven design parameters (focal angle, focal ratio, beam angle ratio, maximum beam angle, beam port curve ellipticity, array element spacing, and focal length/ lambda ) on the shape, and on the geometric phase and amplitude errors of a Rotman lens are described. The advantage of beam port shaping to reduce phase error, and of pointing port horns at the opposite apex (instead of normal to the curve) to reduce off-axis beam amplitude asymmetries, are shown numerically. A design procedure for selecting these parameters is given, and a calculation of lens gain is presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An evaluation of selected density-modification methods for protein-structure determination

The effectiveness of density-modification methods in reducing phase errors has been quantitatively analyzed under conditions in which the source and magnitude of phase error could be defined. Progress was gauged by monitoring both the phase errors and electron density. Changes in electron density were monitored in and around the protein density and in the solvent density. The results of this analysis suggest that the improvement possible is more dependent upon the magnitude than on the source of phase error, and that the methods are most effective when the initial phase error is relatively small. As the magnitude of error increases, the improvement gained decreases, and in some cases noise is introduced by the procedure(s).

Diagonally Implicit Runge–Kutta–Nyström Methods for Oscillatory Problems

Implicit Runge–Kutta–Nyström (RKN) methods are constructed for the integration of second-order differential equations possessing an oscillatory solution. Based on a linear homogeneous test model we analyse the phase errors (or dispersion) introduced by these methods and derive so-called dispersion relations. Diagonally implicit RKN methods of relatively low algebraic order are constructed, which have a high order of dispersion (up to 10). Application of these methods to a number of test examples (linear as well as nonlinear) yields a greatly reduced phase error when compared with “conventional” DIRKN methods.

Length error analysis for impedance tube measurements

Length error analysis of the transfer function method and improved standing wave ratio (SWR) method for impedance tube measurements is presented in this letter. The error of measuring the complex reflection coefficients R is caused by errors in measuring the positions of the two microphones (or the two pressure points for the improved SWR method). Results agree with Abom and Boden’s [J. Acoust. Soc. Am. 83, 2429–2438 (1988)] conclusions, i.e., errors will be large if s, the separation distance of the two microphones, is about one-half wavelength and will be small if s is about one-quarter wavelength. Results also show that in order to reduce the amplitude error of R, it is better to locate one microphone near a minimum pressure point and the other near the adjacent maximum pressure point. To reduce the phase error of R for the transfer function method, preference of the two microphone locations depends on the combination of length errors Δl1 and Δl2. Phase error reduction for the improved SWR method requires an accurate measurement of the minimum pressure point.

Modifying sound speed to reduce phase errors in parabolic equations by a Toda flow method

It is well known that the standard parabolic wave equation, in the absence of range dependence, has the correct normal nodes (amplitudes) but incorrect eigenvalues (phase). This is regrettable because the desired information is generally more sensitive to phase errors than to amplitude errors. Several authors have attempted to increase the accuracy of the phases by modifying the sound-speed profile, even though this decreases the accuracy of the amplitudes. Attempts have been made to eliminate the phase errors (accepting the increase in amplitude errors) by a Toda flow method. From a given sound-speed profile, one can easily generate a matrix with the desired eigenvalues. The spectral mapping theorem shows that a certain square root of this matrix would have eigenvalues that are distorted in the opposite way from the parabolic wave equation. The modified sound-speed profile that corresponds to the last matrix is sought: A system of spectrum-preserving differential equations is solved giving the evolution of this matrix into one where the modified sound-speed profile can be recovered.

The Influence of Cycle Slipping on the Error Probability of a PAM Receiver

In this paper, we evaluate the error probability of a PAM receiver, which derives symbol timing from the received waveform by means of a synchronizer. We point out that the conventional way (using the distribution of the synchronizer's modulo-2π- reduced phase error) of evaluating this error probability does not take into account the effect of the synchronizer's cycle slips. We present a correct expression for the error probability, which uses the distribution of the so-called renewal phase. Also, we derive simple but accurate approximations for both the conventional and the correct error probability, which clearly show the influence of the characteristics of the PAM receiver. These approximate expressions indicate that, for a decreasing additive noise level at the receiver input, the conventional error probability becomes vanishingly small, whereas the correct error probability approaches a nonzero value, which is inversely proportional to the mean time between slips. Hence, cycle slipping imposes a (sometimes rather large) lower limit on the attainable error probability.

Explicit Runge–Kutta (–Nyström) Methods with Reduced Phase Errors for Computing Oscillating Solutions

We construct explicit Runge–Kutta (–Nystrom) methods for the integration of first (and second) order differential equations having an oscillatory solution. Special attention is paid to the phase errors (or dispersion) of the dominant components in the numerical oscillations when these methods are applied to a linear, homogeneous test model. RK(N) methods are constructed which are dispersive of orders up to 10, whereas the (algebraic) order of accuracy is only 2 or 3. Application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced.

Vibration Measurement Using Phase-shifting Stroboscopic Holographic Interferometry

A fringe-evaluation system for a vibration measurement is presented, which is based on the phase-shifting method in real-time holographic interferometry. Phase-shifted real-time holographic interferograms are acquired by a TV camera when stroboscopically illuminating the object of interest. An arctangent calculation with phase-shifted fringe patterns gives a phase distribution proportional to the displacement of the object, including the initial phase of the vibration. The effects of the time-integration of the real-time fringe, which involves a reduction in fringe contrast and phase error in the arctangent calculation, are analysed theoretically. Experimental demonstrations are also presented for two cases: vibration which is spatially uniform; and vibration in which the initial phase is non-uniform.

Effects of Upstream-Biased Third-Order Space Correction Terms on Multidimensional Crowley Advection Schemes

The impact of upstream-biased corrections for third-order spatial truncation error on the stability and phase error of the two-dimensional Crowley combined advective scheme with the cross-space term included is analyzed, putting primary emphasis on phase error reduction. The various versions of the Crowley scheme are formally defined, and their stability and phase error characteristics are intercompared using a linear Fourier component analysis patterned after Fromm (1968, 1969). The performances of the schemes under prototype simulation conditions are tested using time-dependent numerical experiments which advect an initially cone-shaped passive scalar distribution in each of three steady nondivergent flows. One such flow is solid rotation, while the other two are diagonal uniform flow and a strongly deformational vortex.

Reduction of phase error in a measuring circuit by shifting the frequency spectrum

Reduction of phase error in a measuring circuit by shifting the frequency spectrum

Electronic instrumentation for heterodyne holographic interferometry

An electronic instrumentation for heterodyne evaluation of holographic interferograms by accurate phase-measurements is described. It allows to achieve an overall interpolation accuracy of 1/1000 of a fringe. The specifications for the electronic circuitry are derived from the conditions in the optical experiment. To improve the SNR, a narrow-band signal-filtering is needed; an AGC circuit has to adapt the gain in the signal channel to various experimental conditions. Special measures for reduction of phase errors due to these circuits are discussed. The phase meter with an accuracy of 0.1 deg is described in detail. It counts continuously the multiples of π’s as obtained when the detectors are crossing fringes in the image plane of a reconstructed interferogram. The total measuring range for phase is given by the capacity of the up-down counter, and thus practically unlimited.

A generalized active compensated noninverting VCVS with reduced phase error and wide bandwidth

A general circuit for the active compensation of the op-amp noninverting VCVS is given. The circuit has the same topology as the Geiger maximally flat magnitude circuit [1]. The proposed design has the advantages of a smaller phase error and a larger bandwidth than the design in [1]. Moreover, it relaxes the restriction on the relizable dc gain obtained using the design in [1]. The proposed modified design includes the recently described, active compensated, voltage follower [2] as a special case.

Computer generated holograms with reduced phase errors

A rigorous elimination of the phase errors in a computer generated hologram is possible by using three apertures in each sampling cell and performing three Fourier transformations. Another way for reducing these errors approximately is the use of a larger object field. Both methods require a considerably higher amountof computation time. We compare their results in relation to their expense and propose a compromise, which also yields better reconstructions, but can be computed as quickly as a simple Lohmann hologram.

The 50 MEZ Transverse Feedback System in the CERN ISR

One of the intensity limits in the ISR is imposed by the transverse coherent instability. The intensity at which the instability occurs may be raised by Landau damping, i.e. increasing the tune spread of the stacked beam. However, due to non linear resonances, the maximum tune spread is limited. A vertical feedback system which detects and damps the first eight modes of the instability has succeeded in increasing the intensity limit by a factor of ~1.6. Recently this factor has once again become insufficient due to the increase in intensity resulting from improvements in the longitudinal density and vacuum system. A new 50 MHz feedback system has now been installed which damps the first 166 modes of the instability. Hence at low chromaticity values the critical intensity is raised by around another factor of two. The electronic gain of this system decreases with frequency in order to limit the blow-up of vertical oscillations due to electronic noise and to reduce phase errors at high frequencies. The 50 MHz system has been used to explore the ISR vacuum limit to intensities of 39 A.

Performance of Phase-Locked Loops in the Presence of Fading Communication Channels

An approach is presented for the analysis of phase-locked loops whose input signal has passed through time-varying channels. The specific channels considered in detail are the Rice-Nakagami, Rayleigh, and lognormal fading channels. Loop performance is characterized in terms of the steady-state probability density function of the reduced phase error process. The basic parameters which characterize performance include the loop signal-to-noise ratio (SNR) and the variance and bandwidth of the fading components introduced by the channel. Particular channel models are used to illustrate the theory for the firstorder loop. The results are also applied to the analysis of the PSK noisy reference problem in the presence of these time-varying channels.

Phase error and stability of second order methods for hyperbolic problems. I

Second order schemes for hyperbolic systems are compared with respect to stability properties and minimization of phase errors. Among the schemes that utilize no points beyond a nine point lattice those with the smallest phase errors are a Strang type splitting scheme, Leapfrog, and Lax-Wendroff (written as a two-step scheme). Because of its optimal permissible time step the time splitting scheme is to be preferred. Both the Burstein and MacCormack schemes are found to be weakly unstable and have larger phase errors while the rotated Richtmyer method has a much larger phase error than these other schemes. It is possible, however, to reduce the phase error by using points beyond the eight nearest neighbors. However, the Richtmyer method together with its generalization by Gourlay and Morris still have very large phase errors. The method of Fromm together with the SHASTA code are nonlinear schemes that do have reduced phase errors. However, because of the smaller permissible time steps coupled with additional complexities it would seem more worthwhile to use higher order schemes once one is willing to use data beyond the nine point rectangular mesh. Numerical experiments, with vector equations, are presented that confirm these results.

The Beam-Programming System of the Bevatron

A brief description is given of the beam-progromming equipment developed for the Bevatron. The system described provides analog-computer-controlled as well as beam-controlled frequency tracking. Equipment is described that permits arbitrary adjustment and control of the radial position of the circulating beam at any energy. Beamintensity control and the automatic reduction of phase errors in the acceleration system are provided. Included also are units that produce long and short beam pulses. Equipment that permits highly multiple operation is also discussed. The integration of these equipments for extended injection is treated.

Multipath phase errors in CW-FM tracking systems

High-precision, cw, missile-tracking systems are subject to multipath errors due to ground reflections and reflections from neighboring objects. To reduce the effects of ground reflection, advantage may be taken of the characteristic of frequency modulation to discriminate against a weaker interfering signal. This paper discusses the magnitude of the phase error reduction to be obtained by use of frequency modulation. It is demonstrated that the phase error can be reduced to any desired value if sufficient band width is utilized in the frequency modulation signal and significant improvement over an amplitude modulation system can be obtained with a wide-band FM system.