Phase Ambiguity Problem Research Articles

RESOLVING PHASE AMBIGUITY IN THE INVERSE PROBLEM OF REFLECTION-ONLY MEASUREMENT METHODS

We have applied the phase unwrapping technique to resolve the phase ambiguity problem arising from complex expressions of scattering parameters, for re∞ection-only measurement conflgurations, since, at some instances, only one side of the sample under test is accessible for electromagnetic measurements. We considered two difierent measurement conflgurations for testing the applicability of the phase unwrapping technique as: 1) two identical samples with difierent lengths ∞ushed by a short-circuit termination and 2) one sample shorted by a varying short-circuit termination. For each measurement conflguration, the underlying expressions for the re∞ection scattering parameters are derived. For both cases, we evaluated the suitability of the phase unwrapping technique by considering a highly-dispersive medium (distilled water) as our test sample. We note that continuity of the real part of the complex wavelength is a key issue in the unwrapping technique for (one-port) re∞ection-only measurements.

Robust Chinese Remainder Theorem Ranging Method Based on Dual-Frequency Measurements

The Chinese remainder theorem (CRT) is an effective tool to solve the phase ambiguity problem in phase-based range estimation. However, existing methods suffer from problems such as requiring special measuring frequency, low spectrum efficiency, noise sensitivity, etc. To overcome these problems, this paper presents a CRT ranging method using two “adjacent” frequencies. As a result, all the frequencies in the given frequency span can be used, which greatly increases spectrum utilization. Moreover, since the same distance is measured by different frequency pairs, statistical processing can be performed on the results, which further improves estimation accuracy. Simulations verify the validity of the proposed method.

Resolving Phase Ambiguity in the Inverse Problem of Transmission /Reflection Measurement Methods

Inherent to transmission/reflection measurement methods and posed by the multiple-valued logarithm function of the complex transmission coefficient, the phase ambiguity problem is solved by the phase wrapping technique. Here extended and generalized, the proposed technique relies on properly adding to the phase of the complex logarithmic function a stepwise function built in from the resonance frequencies at which the phase of the transmission coefficient reaches ±π. In a concrete example the method is illustrated by correctly retrieving from complex scattering parameters the constitutive parameters of a highly-dispersive medium (distilled water) over the 0–250 GHz frequency range. Implication of a mathematically negative wavelength is also discussed.

Application of least-square estimation in white-light scanning interferometry

White-light scanning interferometry (WLSI) is a powerful tool for investigating the profile of a test object that contains sharp steps. Due to the light source used in WLSI system, it is able to overcome phase ambiguity problem, which is often encountered in monochromatic interferometry. In this paper, a new algorithm based on least-square estimation using short-time Fourier transform (STFT) is proposed to measure the profile of a test object. STFT is used to extract the peak position of the coherence envelope of a white-light interference signal and retrieve the corresponding phase values simultaneously at first. A complex phasor (CP) method is introduced to further reduce the phase noise. Then, the phase values at several positions around are utilized to achieve a more accurate peak position based on least-square line fitting. Both simulated and experimental results show that the proposed algorithm is able to accurately measure the profile of a test object.

Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry

White-light scanning interferometry (WLSI) has been widely used in micro-profile measurement such as Micro-Electro-Mechanical Systems (MEMS) and Computer Generated Hologram (CGH) diffractive elements. It does not contain phase ambiguity problem which is often encountered in monochromatic wavelength interferometry. This paper presents an algorithm based on windowed Fourier transform (WFT) to extract the phase of a white-light interferogram and compensates for the difference in zero optical path difference (ZOPD) position in WLSI. With the WFT technique, the center wavelength of a white-light source and the phase of a white-light interferogram could be retrieved simultaneously. The effect of noise, scanning interval of a piezoelectric transducer (PZT) and the window size of WFT are also analyzed. Both simulated and experimental results show that the proposed algorithm has good noise immunity and is able to accurately measure the micro-profile of a specimen.

Dynamic phase measurement in shearography by clustering method and Fourier filtering

Quantitative phase extraction is a key step in optical measurement. While phase shifting technique is widely employed for static or semi-static phase measurement, it requires several images with known phase shifts at each deformed stage, thus is not suitable for dynamic phase measurement. Fourier transform offer a solution to extract phase information from a single fringe pattern. However, a high frequency spatial carrier which is sometimes not easy to generate is required to solve the phase ambiguity problem. In this paper, we aim to propose an ideal solution for dynamic phase measurement. Four images with known phase shift are captured at the reference stage to analyze the initial phase information. After the object starts continuous deformation, only one image is captured at each deformed stage. A clustering phase extraction method is then applied for deformation phase extraction utilizing the phase clustering effect within a small region. This method works well for speckle image with low and medium fringe density. When the fringe density is high, especially in the case of shearographic fringe, information insufficiency inherent with merely one deformed speckle image often results in poor quality wrapped phase map with plenty of phase residues, which make phase unwrapping a difficult task. In the light of this limitation, a Fourier transform based phase filtering method is proposed for fringe frequency analysis and adaptive filtering, and effectively removes most of the phase residues to reconstruct a high quality wrapped phase map. Several real experiments based on shearography are presented. Comparison between the proposed solution and standard phase evaluation methods is also given. The results demonstrate the effectiveness of the proposed integrated dynamic phase extraction method.

Dual Phase-shifting Moiré Projection with Tunable High Contrast Fringes for Three-Dimensional Microscopic Surface Profilometry

A new dual phase-shifting moiré projection method is proposed for microscopic three-dimensional surface profilometry. One of the most difficult technological challenges encountered in employing moiré triangulation projection for automatic optical inspection (AOI) is to detect high contrast projected fringes from an object's surface having vast variation of surface reflectance. Surface reflectance from different surface regions could vary significantly due to diverse surface characteristics, such as texture and roughness, in which returning light may be arbitrary combination of specular reflection, diffusion and scattering. Undesired measurement results are common when deformed moiré fringes cannot be satisfactorily detected by an imaging detector. To resolve this, the proposed method generates a moiré fringe formed by employing a combination of the deformed fringe detected from the object's surface and a virtual sinusoidal grating self-generated by the computer. A spatial averaging approach is applied to phase-shifted moiré interferograms formed between one deformed fringe and virtual fringes having an equal shifted phase, such as π/2. High-contrast phase-shifted interferograms can be generated for more accurate 3-D surface reconstruction via phase wrapping and unwrapping. One great advantage of the method is its tunable control over the moiré fringe period for reaching more measurable step heights without encountering phase ambiguity problem. From the experimental analysis using standard targets and industrial PCB, it is verified that the method can significantly improve measurement accuracy with a measuring standard deviation less than 0.14% of the overall measuring depth range when measuring a low reflected industrial object surface.

Surface profile inspection of a moving object by using dual-frequency Fourier transform profilometry

The 2π phase ambiguity caused by surface isolations and large height step can be solved by dual-frequency projection grating profilometry. However, in the Fourier transform profilometry (FTP) of a moving object, only one single deformed fringe pattern can be obtained. In order to introduce the dual-frequency technique into the FTP of moving object, a novel experimental system is designed to capture two fringe patterns with different frequency at the same time. A grating structure comprising two regions with different frequencies is projected upon the surface of the detected object. Two line-scan CCD cameras are used to capture the surface images encoded by the two kinds of patterns, respectively. By getting the corresponding image intensity at the same point of the object surface in the two acquired images, the dual-frequency technique is applied to extract the real phase without phase ambiguity. The surface profile of a specimen with a large height step is measured to prove the feasibility of the proposed method. The experimental results show that the proposed method can solve the 2π phase ambiguity problem successfully in the surface profile inspection of a moving object.

An efficient ranging method based on Chinese remainder theorem for RIPS measurement

The radio interferometric positioning system (RIPS) measures the phase difference of the interference signal to provide high accuracy and at the same time maintain simple hardware configuration for wireless sensor networks. However, it suffers from phase ambiguity problem because of the periodicity of phase, which makes it hard to determine the actual distance difference only from a single phase measurement. To solve the problem, RIPS makes multiple measurements at different frequencies so as to determine the distance difference from multiple phases. However, this is a computationally intensive searching process and not suitable for energy-constrained wireless sensor nodes. In this paper, we introduce the Chinese remainder theorem (CRT) to RIPS to solve the phase ambiguity problem. Meanwhile, we utilize some properties of the coefficients in the CRT algorithm to avoid the over-sensitivity of the traditional CRT, which increases the robustness of the algorithm. We apply this robust CRT algorithm to the ranging process which calculates the distance difference directly from a closed-form equation and therefore reduces the response time and the energy consumption of the ranging procedure.

Phase ambiguity diminishing space-time trellis codes

Phase ambiguities affect any joint decoding and channel estimation algorithm over fading channels, making codes that mitigate this problem of considerable interest. This work presents a design that transforms a Space-Time Trellis Code (STTC), prone to carrier phase ambiguities, into a Phase Ambiguity Diminishing STTC (PAD-STTC) that is robust to such impairment. A PAD-STTC can be considered as a Multiple STTC (MSTTC) with special constellation mappings. The design criteria for these mappings are developed in this work by trading among the PAD property, diversity, and coding gain. Two PAD-STTC structures termed PAD1-STTC and PAD2-STTC are proposed, where PAD2-STTC increases the number of states over the constituent STTC. Computer simulation results indicate that PAD-STTCs can solve the phase ambiguity problem and provide a larger diversity gain than the constituent STTC over high Doppler fading channels. Moreover, with the same constituent STTC, the PAD2-STTC outperforms the PAD1-STTC at the expense of increased complexity.

Compensation of Frequency Offset for Differentially Encoded 16- and 64-QAM in the Presence of Laser Phase Noise

The compensation of frequency offset for differentially encoded 16- and 64-ary quadrature amplitude modulation (QAM) in the presence of laser phase noise is investigated. Differential encoding is employed to solve the four-fold phase ambiguity problem in a nondata-aided transmission system with square QAM constellations. Simulation results show that frequency offset and phase noise can successfully be compensated using a second-order digital filter loop for the square QAM constellations.

Short pulse multi-frequency phase-based time delay estimation

An approach for time delay estimation, based on phase difference detection, is presented. A multiple-frequency short continuous wave pulse is used to solve the well-known phase ambiguity problem when the maximum distance exceeds a full wavelength. Within an unambiguous range defined with the lowest frequency difference between components, the corresponding phase difference is unique and any distance within this range can be determined. Phase differences between higher frequency components are used to achieve a finer resolution. The concept will be presented and the effectiveness of the approach will be investigated through theoretical and practical examples. The method will be validated using underwater acoustic measurements, simulating noisy environments, demonstrating resolutions better than a 50th of a wavelength, even in the presence of high levels (-5 dB) of additive Gaussian noise. Furthermore, the algorithm is simple to use and can be easily implemented, being based on phase detection using the discrete Fourier transform.

Isoclinic Ambiguity Unwrapping of Circular Ring under Diametric Compression

Isoclinic and isochromatic parameters in photoelastic analysis are formulated by arc tangent function of several phase shifting frames but for isochromatic calculation it is, in addition, in terms of isoclinic data also. Thus, any isoclinic phase ambiguity would also result in the phase inconsistencies of isochromatic calculation and cause unwrapping problems and difficulties. Many methods had been proposed to treat these kinds of isochromatic fringes but lots of treatments were needed for the correct retrieval. In this work, the isoclinic ambiguity problem is investigated and solved directly by a novel regional phase unwrapping technique. Once the isoclinic phase ambiguity problems are solved, a correct isoclinic result can be obtained. Then, substituting this result into the isochromatic formation yields an ambiguity free isochromatic phase map, which can be easily restored by any phase unwrapping algorithm. A stress frozen sample - circular ring under diametric compression verifies its effectiveness. Usually happened is that this kind of sample is with ambiguous isoclinic data which would cause phase ambiguities of the isochromatic formulation if not treated well first.

A gaussian sum approach to blind carrier phase estimation and data detection in turbo coded transmissions

We present a joint phase estimation and decoding method for convolutional turbo codes in the presence of strong phase noise. In order to overcome the problem of phase ambiguity and cycle slips, a combined state-space model for the time varying phase and the component convolutional codes is introduced. The proposed algorithm uses a Gaussian sum approach to approximate the multimodal a posteriori probability density function (pdf) of the phase in a blind context. We compare our method to the well known alternative consisting in discretizing the phase.Monte-Carlo simulations for the turbo code used in the DVB-RCS standard show that the performances of the proposed scheme are close to decoding with perfect knowledge of the phase.

Phase/Frequency Tracking in a GNSS Software Receiver

In the paper, a software-based technique for the tracking of frequency and phase of Global Navigation Satellite System (GNSS) signals is proposed. It is shown that a software-based receiver with precomputed, zero-phase carrier replicas for phase tracking is subject to phase jump in phase tracking and ambiguity problems in frequency estimation. A phase/frequency tracking architecture is proposed in which the frequency of the incoming signal is estimated based on a lead-lag structure which is similar to a traditional delay locked loop and the phase, after de-rotation, is estimated using a frequency-aiding phase locked loop. The effects of thermal noise and resolution-induced noise are analyzed and verified using simulation. The significance of frequency aiding in enhancing the phase tracking response is also described.

Unified approach for rough phase measurement without phase unwrapping by changing the sensitivity factor

Phase evaluation is extraordinarily important in optical, acoustic and radar techniques where coherent signals are employed as information carriers. In most of the cases, phase information are obtained from an inverse trigonometric function and wrapped into −π to π. A phase unwrapping process is thus required to obtain the final unwrapped phase which represents the ultimate physical quantity to be measured. One-dimensional phase unwrapping is easily achieved by adding or subtracting an integer multiple of 2π to the wrapped phase to establish a smooth phase map. Two-dimensional phase unwrapping, however, is quite troublesome and an elegant unwrapping routing should be chosen in most of the cases to deal with phase residues caused by noise and other error sources. It would be valuable if two-dimensional phase unwrapping can be avoided and the physical quantity obtained directly. In the past, researchers have proposed other methods such as the multiple wavelengths approach which incorporates information from multiple wavelengths to eliminate the need for phase unwrapping. In this study, we extend the multiple wavelengths approach by varying the sensitivity factor, which is more convenient and cost-effective, to achieve the aim of requiring no phase unwrapping. Furthermore, an elegant phase derivative approach is used to solve the phase ambiguity problem in the multiple wavelengths method. Both simulation results and real experiment data of shadow moiré and shearography demonstrate the usefulness of this method, especially for discontinuous surface profile measurements such as steps. Advantages and disadvantages for the proposed method are also discussed in this paper.

Low-frequency vibration measurement by temporal analysis of projected fringe patterns

Fringe projection is a whole field, non-contacting optical technique allowing direct measurement of surface contour of an object. In many cases, 3D surface profiling is required for vibrating objects or for objects with continuously changing profile, where conventional phase shifting techniques are not applicable. Carrier-based spatial Fourier, window Fourier analyses, or wavelet analysis are normally used to retrieve instantaneous phase maps in dynamic measurement from one fringe pattern, but these techniques require high-quality surface, such as uniform reflectivity, continuity of the surface and a regular shape. Otherwise, large errors will be generated during the process. In this study, a grating is projected on the surface of a low-frequency vibrating object and a sequence of fringe patterns is captured by a high-speed camera. The fringe patterns are processed point-by-point along time axis by different algorithms. To avoid the phase ambiguity problem, a temporal carrier is introduced in the experiment by shifting the grating with a constant speed. The results show that high-quality instantaneous surface profile and the kinematic parameters of vibration, such as displacement, velocity and acceleration can be obtained precisely.

Vibration measurement of a miniature component by high-speed image-plane digital holographic microscopy

Measuring deformation of vibrating specimens whose dimensions are in the submillimeter range introduces a number of difficulties using laser interferometry. Normal interferometry is not suitable because of a phase ambiguity problem. In addition, the noise effect is much more serious in the measurement of small objects because a high-magnification lens is used. We present a method for full-field measurement of displacement, velocity, and acceleration of a vibrating miniature object based on image-plane digital holographic microscopy. A miniature cantilever beam is excited by a piezoelectric transducer stage with a sinusoidal configuration. A sequence of digital holograms is captured using a high-speed digital holographic microscope. Windowed Fourier analysis is applied in the spatial and spatiotemporal domains to extract the displacement, velocity and acceleration. The result shows that a combination of image-plane digital holographic microscopy and windowed Fourier analyses can be used to study vibration without encountering a phase ambiguity problem, and one can obtain instantaneous kinematic parameters on each point.

Reliable method for material characterisation using quasi-optical free-space measurement in W-band

An extension to the Nicolson–Ross–Weir (NRW) algorithm to retrieve complex permittivity and permeability from both reflection and transmission coefficients is presented. This new numerical retrieval approach solves the phase ambiguity problem associated with the NRW algorithm. A free-space quasi-optical two-port measurement setup is employed to obtain the scattering parameters of different samples in the W-band frequency range. The results of different samples with various thicknesses are presented. The advantages and limitations of the method are also discussed.

Maximum accuracy evaluation scheme for wireless saw delay-line sensors

This paper describes an evaluation scheme that prevents phase ambiguity of surface acoustic wave (SAW) delay-line sensors. Although it is well-known that phase evaluation yields accuracies of 150~1500 times higher than time-delay evaluation, the problem of phase ambiguity has prevented phase evaluation of sensors operating over a range larger than 2 pi. This paper addresses this unsolved problem with a complete strategy. Furthermore, the existence of an optimum choice of the relative reflector positions on the sensor is shown. The presented relations enable the design of maximum accuracy SAW delay-line sensors.