Phase Interpolation Research Articles

High Precision Multifrequency Acoustic Radar with an Unequally Spaced Frequency Array Designed by Simulated Annealing

Multifrequency cw radar with a frequency array is realized by digital signal processing. However, the resolution is limited to the fundamental sampling spacing determined by the aperture width of the frequency array. Furthermore, it has a periodic structure with a usually used equally spaced frequency array which results in the formation of periodic ghosts of an object. These fundamental problems are shown to be solved by the introduction of an unequally spaced frequency array (USFA) and a linear phase interpolation of the reconstructed impulse response between the fundamental sampling points. An amplitude-weighted USFA, which has minimized grating lobes, is designed by the method of simulated annealing from a previously described uniformly weighted USFA. The original level of the grating lobes is as large as -8.93 dB, and the minimized level is -12.5 dB. The resolution is not degraded by the minimization, and a superresolution of 1/60 to 1/300 of the wavelength is maintained.

Optical displacement measurement with GaAs/AlGaAs-based monolithically integrated Michelson interferometers

Two monolithically integrated optical displacement sensors fabricated in the GaAs/AlGaAs material system are reported. These single-chip microsystems are configured as Michelson interferometers and comprise a distributed Bragg reflector (DBR) laser, photodetectors, phase shifters, and waveguide couplers. While the use of a single Michelson interferometer allows measurement of displacement magnitude only, a double Michelson interferometer with two interferometer signals in phase quadrature also permits determination of movement direction. In addition, through the use of two 90/spl deg/ phase-shifted interferometer signals in the latter device, a phase interpolation of 2/spl pi//20 is possible, leading to a displacement resolution in the range of 20 nm. The integration of these complex optical functions could be realized with a relatively simple fabrication process.

A semidigital dual delay-locked loop

This paper describes a dual delay-locked loop architecture which achieves low jitter, unlimited (modulo 2/spl pi/) phase shift, and large operating range. The architecture employs a core loop to generate coarsely spaced clocks, which are then used by a peripheral loop to generate the main system clock through phase interpolation. The design of an experimental prototype in a 0.8-/spl mu/m CMOS technology is described. The prototype achieves an operating range of 80 kHz-400 MHz. At 250 MHz, its peak-to-peak jitter with quiescent supply is 68 ps, and its jitter supply sensitivity is 0.4 ps/mV.

A monolithically integrated double Michelson interferometer for optical displacement measurement with direction determination

A monolithically integrated optical displacement sensor fabricated in the GaAs-AlGaAs material system is reported. The single-chip device consists of a distributed Bragg reflector laser, two photodetectors, two phase modulators, two Y-couplers, and two directional couplers. It is configured as a double Michelson interferometer and allows the determination of both magnitude and direction of a displacement. The detection of two 90/spl deg/ phase-shifted interferometer signals also resulted in on improved phase interpolation of o/20. Despite the relatively simple fabrication process, the integration of rather complex optical functions could be realized.

Distance measurements using two frequency-stabilized Nd:YAG lasers

Two diode-pumped tunable Nd:YAG lasers locked to sub-Doppler transitions of (127)I(2) and (133)Cs(2) are used as a source for two-wavelength interferometry. The synthetic wavelength is highly stable and accurate, owing to the frequency stability of the locked lasers and the precise determination of the frequency difference between Cs(2) and I(2) transitions. The dense spectra of the two molecular absorbers allows selection of synthetic wavelength A over a wide range, between 8.5 mm and more than 1 m, thus enabling distance measurements with a large nonambiguity range. Fringe contrast and phase-shifting methods are used to measure the synthetic phase. An accuracy of 70 µm is achieved for synthetic wavelength Λ ~ 19 mm, corresponding to a phase interpolation accuracy ofΛ/260.

Improvement of Interferometer Accuracy with Self-calibration of Phase Interpolation Effor.

Resolution of interferometers reaches to a few nanometers or less with the phase shifting or other interpolation techniques. However the accuracy of them could not approach to their resolution because any accurate standard for calibration could not be found. The present paper proposes a self-calibration method which can determine the interpolation error between fringes of the interferometer without using any standard. An interferometer of Michelson type has been developed with the phase shifting by a piezo electric actuator. The interpolation error due to the phase shifting actuator has been calibrated by the proposed self-calibration method. The reliability of the calibration result is estimated to be about 0.2 nm from the reproducibility error of the calibration. It has been confirmed that the proposed method can calibrate the interpolation error with the accuracy of the limit given by the resolution or the stability of the interferometer itself.

Accurate measurement of the refractive index of air

An interference refractometer for absolute measurement of the refractive index of air has been developed. It is essentially based on components of a commercial Zeeman type laser interferometer system and uses a differential interferometer and a vacuum chamber with four measurement cells. The resolution and accuracy of interference phase interpolation is improved by an electronic phase meter and a detection technique which compensates for interferometer non-linearity. The system has been designed for an accuracy of a few parts in 10 8. Experimental investigations of individual components and comparison measurements with pure gases, with Edlén's formula and with the BCR interference refractometer confirm this goal.

Modulation analysis of phase-shifted holographic interferograms

A new evaluation technique for phase-shifted holographic interferograms is developed and verified by experiments. In evaluating the interferograms, both the discrete phase distribution and the intensity modulation of interference fringes are calculated and a binary control mask is generated by analysing histograms and tracing local minima of the intensity modulation distribution. The control mask is used to identify valid and invalid areas of the discrete phase distribution in subsequent phase unwrapping and phase interpolation, and to control the path of phase unwrapping. The evaluation of experimentally obtained phase-shifted holographic interferograms shows that this method can be successfully applied for the processing of complex interference patterns, even if the interferograms have fringe discontinuities and shaded areas.

Quadratic phase interpolation for voiced speech synthesis in MBE model

A new phase unwrapping algorithm for phase interpolation in voiced speech synthesis is proposed. Computer simulation results show that the new algorithm outperforms the original phase unwrapping method proposed by Griffin and Lim (1985, 1988) in terms of the number of instances of wrongly unwrapped phase.

Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation

A new method for phase-stepping grating profilometry of complex objects is proposed and verified by experiments. Both, the discrete phase distribution and the modulation depth distribution are calculated. To construct binary control masks the discrete values and the histogram of the modulation intensities are inspected first. Then the blocking lines, which are connecting the pairs of opposite poles, are traced down through the local minimums of the modulation intensities. The control masks are used to identify valid and invalid discrete phase distributions during phase unwrapping and phase interpolation, and to control the path of phase unwrapping. The method could equally well be used in grating profilometry or in interferometry.

On the phase interpolation problem — A brief review and some new results

Reconstruction of a signal from the phase of its Fourier transform is important and useful in a variety of practical applications. In this paper, we give a brief review of the previous results and develop some new ones pertaining to the phase-only reconstruction problem. The review includes conditions under which the signal is uniquely specified by its Fourier transform phase and some algorithms for performing the reconstruction. New results consist of an extension of a previous technique and an explicit formula for reconstruction. A number of potential applications of the reconstruction techniques are briefly mentioned in the concluding section.

Recursive construction of polynomial with prescribed phase on unit circle

An extension of a recent technique for phase interpolation to make it implementable on a computer is described. A computer program has been written an d several examples are presented to demonstrate its usage.

Simultaneous amplitude and phase approximation for fir filters

AbstractAn FIR filter design method with simultaneous amplitude and phase approximation is presented. Two components are proposed in the design method: simultaneous amplitude and phase interpolation in the passband and amplitude interpolation in the stopband. These interpolations are implemented by the Remez exchange algorithm.The polynomials for the interpolations are generated by recurrence formulae and the recurrence coefficients are calculated by a recursive method. the properties of the filters designed with the simultaneous approximation are investigated and compared with those of exact linear and minimum phase filters.

Linear phase decimation and interpolation filters for high-speed applications

A novel hardware realisation for linear phase decimation and interpolation FIR filters, exploiting the symmetry of the coefficients to reduce the number of multipliers, will be described. Combining the multiplications, leads to contra-dataflow or folded filter structures in which simple pipelining is limited, because of the opposite directions of the data flows. It will be shown that these structures can be transformed into new filter structures, which allow conventional pipelining to relax the timing demands for high-speed applications.

ON REVERSE‐TIME MIGRATION*

ABSTRACTMigration is a process whereby events in ‘image space’ are mapped into their correct positions in ‘object space’. The wave equations associated with this mapping may be defined and solved numerically either in image space or in object space. In the former the CMP section, which represents the initial conditions, is extrapolated toward increasing depths, and the migrated data are recovered at zero time. In the latter, the wave‐field extrapolation takes place in the coordinate frame of the depth section, and the CMP data serve as boundary conditions at the surface. Computations begin at the last sample of the record section and continue ‘reverse time’ until time zero.This paper describes a reverse‐time migration (RTM) method and compares its performance with that of an image‐space method based on the idea of phase shift plus interpolation (PSPI). Synthetic zero‐offset sections serve as examples for migration experiments with the RTM and PSPI methods. It is shown that the RTM approach to migration is rather expensive, but its robustness and accuracy are difficult to surpass.

An efficient method for the estimation of the frequency of a single tone in noise from the phases of discrete fourier transforms

The formalism for the phase interpolation estimator (PIE) of the frequency of a single tone in noise is developed. The PIE is a frequency estimator which makes use of the phases of discrete Fourier transforms calculated at a single frequency in the neighbourhood of the tonal frequency. The relationship between the PIE and the maximum likelihood frequency and phase estimators is derived. An analytical expression for the variance of the PIE is obtained, and is seen to be largely insensitive to the choice of frequency at which the Fourier transforms are calculated. Application of the PIE to frequency tracking is discussed.

Comparison of frequency estimators for underwater acoustic data

A number of high‐accuracy frequency estimators have been critically compared using simulated and real passive sonar data. Techniques considered are the standard Prony method, the extended Prony method of Marple, the Kumaresan–Prony algorithm, the Pisarenko algorithm, the maximum likelihood approach, and a phase interpolation method developed by the authors. The latter two techniques appear to offer the most promise for the analysis of sonar data.

Microprocessor-based phase determination for high-resolution optical sensors

A microprocessor-controlled phase measuring system for high-resolution optical sensors is described. An automatic calibration procedure provides drift compensation and long-term accuracy. Using a 12 bit A/D convertor, phase interpolation with a resolution of 0.12° (~1/3000 of a period) is demonstrated.

Migration of seismic data by phase shift plus interpolation

Under the horizontally layered velocity assumption, migration is defined by a set of independent ordinary differential equations in the wavenumber‐frequency domain. The wave components are extrapolated downward by rotating their phases. This paper shows that one can generalize the concepts of the phase‐shift method to media having lateral velocity variations. The wave extrapolation procedure consists of two steps. In the first step, the wave field is extrapolated by the phase‐shift method using ℓ laterally uniform velocity fields. The intermediate result is ℓ reference wave fields. In the second step, the actual wave field is computed by interpolation from the reference wave fields. The phase shift plus interpolation (PSPI) method is unconditionally stable and lends itself conveniently to migration of three‐dimensional data. The performance of the methods is demonstrated on synthetic examples. The PSPI migration results are then compared with those obtained from a finite‐difference method.