Residual Phase Error Research Articles

Hardware simulation of real-time wavelength corrected phase projection

We demonstrate the real-time signal processing operation of a dispersion-free phase projection algorithm intended for atmospheric correction of multi-aperture optical phased arrays. It uses interferometric phase measurements at multiple sensing wavelengths, offset by 50 GHz, to compute a phase correction at a third, remote wavelength. This is useful where phase sensing cannot be implemented at the wavelength of interest, enabling interferometric level control from wavelength offset targeting beacons or guidestars. The digital signal processing implementation we demonstrate has a residual temporal phase error of 4×10−4rad/Hz while being capable of 100 MHz throughput with 0.53 µs latency, making it a viable approach for either feedback or feed-forward atmospheric correction in segmented piston-phase control systems.

RIS-NOMA communications over Nakagami-m fading with imperfect successive interference cancellation

Considering imperfect successive interference cancellation (SIC) for non-orthogonal multiple access (NOMA) communications, this work studies the cooperative reconfigurable intelligent surface (RIS)- and relay-assisted system under Nakagami-m fading. We focus on the performance comparison for such cooperative schemes, under different channel conditions and system parameters. In addition, we analyze the minimum required RIS elements number of the RIS-assisted scheme to achieve the same performance of the relay-assisted scheme with given signal-to-noise ratio (SNR). The cases of the optimal continuous phase shift and discrete phase shift designs of RIS are also discussed. We make a comparison between them, aiming to study the impact of the residual phase errors on performance. Specially, we compare the active RIS and passive RIS with the same system power constraint. Simulation results demonstrating the reliability of the analysis, validate that the relay-assisted scheme is superior to that of RIS-assisted one when the RIS elements number is small and transmitted power is lower. The results also confirm that the deployment of RIS should consider the actual situation of the application scenario.

Coherent beam combining of two all-PM thulium-doped fiber chirped pulse amplifiers

In this paper, we report a coherent beam combining (CBC) system that involves two thulium-doped all-polarization maintaining (PM) fiber chirped pulse amplifiers. Through phase-locking the two channels via a fiber stretcher by using the stochastic parallel gradient descent (SPGD) algorithm, a maximum average power of 265 W is obtained, with a CBC efficiency of 81% and a residual phase error of λ/17. After de-chirping by a pair of diffraction gratings, the duration of the combined laser pulse is compressed to 690 fs. Taking into account the compression efficiency of 90% and the main peak energy proportion of 91%, the corresponding peak power is calculated to be 4 MW. The laser noise characteristics before and after CBC are examined, and the results indicate that the CBC would degrade the low frequency relative intensity noise (RIN), of which the integration is 1.74% in [100 Hz, 2 MHz] at the maximum combined output power. In addition, the effects of the nonlinear spectrum broadening during chirped pulse amplification on the CBC efficiency are also investigated, showing that a higher extent of pulse stretching is effective in alleviating the spectrum broadening and realizing a higher output power with decent combining efficiency.Graphical

Phase external calibration in interferometric microwave radiometer

The receiver error of the interferometric microwave radiometer can be calibrated by injecting the correlated noise into the receiver’s calibration port. However, in the antenna part without the internal calibration link, the residual phase error can be calibrated by an external noise source in an anechoic chamber and the significance of phase external calibration [1][2] is also verified by imaging the anechoic chamber.

High dynamic range 3D shape reconstruction: Combination of adaptive fringe intensity, iterative exposure time adjustment and least quadratic curve fitting

For three-dimensional (3D) reconstruction of specular surfaces based on fringe projection profilometry (FPP), a hybrid approach combining adaptive fringe projection, iterative exposure time adjustment and least quadratic curve fitting is proposed. First, we propose an adaptive projection fringe intensity generation method in the camera image. Second, we build a new pixel matching between the camera and projector, and correct the pixel matching on the specular surfaces based on linear interpolation. Third, we propose an iterative exposure time adjustment method based on saturated pixel statistics. Fourth, we fuse the multiple results to obtain the final 3D reconstructed surface. Finally, we adopt a least quadratic curve fitting algorithm to fit the residual phase errors. Experiments confirmed the validity of the proposed method.

A Phase Correction Model for Fourier Transform Spectroscopy

In Fourier transform spectroscopy (FTS), the conventional Mertz method is commonly used to correct phase errors of recovered spectra, but it performs poorly in correcting nonlinear phase errors. This paper proposes a phase correlation method–all-pass filter (PCM-APF) model to correct phase errors. In this model, the proposed improved phase correlation method can correct linear phase errors, and all-pass filters are applied to correct the residual nonlinear phase errors. The optimization algorithm for the digital all-pass filters employs an improved algorithm which combines the subtraction-average-based optimizer (SABO) and the golden sine algorithm (Gold-SA). The proposed PCM-APF model demonstrates high correction precision, and the optimization algorithm for the filters converges faster than traditional intelligent optimization algorithms.

Spatial diversity processing mechanism based on the distributed underwater acoustic communication system.

To address the problem of unreliable single-link underwater acoustic communication caused by large signal delays and strong multipath effects in shallow water environments, this paper proposes a distributed underwater acoustic diversity communication system (DUA-DCS). DUA-DCS employs a maneuverable distributed cross-medium buoy network to form multiple distributed, non-coherent, and parallel communication links. In the uplink, a receiving diversity processing mechanism of joint decision feedback equalizer embedded phase-locked loop and maximum signal-to-interference ratio combining (DFE-PLL-MSIRC) is proposed to achieve waveform-level diversity combining of underwater signals. A phase-locked loop module is embedded in each branch of the decision feedback equalizer to eliminate the residual frequency and phase errors after Doppler compensation. Meanwhile, the combining coefficients are determined based on the maximum signal-to-interference ratio criterion, taking into account the residual inter-symbol interference after equalization, resulting in efficient and accurate computation. Additionally, the combined decision values are fed back to the feedback filters in each branch to ensure more accurate feedback output. Simulation and lake experiment results demonstrate that, compared to the single-link communication system, DFE-PLL-MSIRC can achieve a diversity gain of more than 5.2 dB and obtain about 3 dB more diversity gain than the comparison algorithm. And the BER of DFE-PLL-MSIRC can be reduced by at least one order of magnitude, which is lower by at least 0.6 order of magnitude compared to the comparison algorithm. In the downlink, a transmitting diversity processing mechanism of complex orthogonal space-time block coding (COSTBC) is proposed. By utilizing a newly designed generalized complex orthogonal transmission matrix, complete transmission diversity can be achieved at the coding rate of 3/4. Compared to the single-link communication system, the system can achieve a diversity gain of more than 6 dB.

Interactive Errors Analysis and Scale Factor Nonlinearity Reduction Methods for Lissajous Frequency Modulated MEMS Gyroscope.

Although the Lissajous frequency modulated (LFM) mode can improve the long-term and temperature stability of the scale factor (SF) for mode mismatch MEMS gyroscopes, its SF nonlinearity poses a significant limitation for full-scale accuracy maintenance. This paper examines the interaction effects among stiffness coupling, system phase delay, readout demodulation phase shift, and velocity amplitude mismatch within the control process. Based on the completion of frequency difference control and demodulation phase matching, we clarify that the remaining stiffness coupling and residual system phase error are the primary factors influencing SF nonlinearity. Furthermore, SF nonlinearity is reduced through error compensation. On one hand, this paper suppresses stiffness coupling through the observation of the instantaneous frequency difference and the application of the quadrature voltage. On the other hand, system phase error is compensated by observing the amplitude control force and tuning the reference in the Phase-Locked Loops (PLLs). Subsequent simulations of these methods demonstrated a remarkable 97% reduction in SF nonlinearity within the measurement range of ±500°/s. In addition, an observed rule dictates that maintaining a sufficiently large frequency split effectively constrains the SF nonlinearity.

The performance analysis of the laser heterodyne ultra-high frequency detection based on the optical phase-locked loop

The performance of the laser heterodyne ultra-high frequency detection based on the optical phase-locked loop is analyzed through theoretical calculations and numerical simulations, considering the residual phase error and carrier stability. Appling the Wiener–Khinchin theorem, a general expression of the residual phase variance with essential affecting factors is derived. Furthermore, we define the error gain factor based on the ultra-high frequency carrier stability, which consists of the amplitude deviation and frequency deviation. Then the model of the demodulation output with the amplitude deviation and frequency deviation is presented. According to numerical simulation results, we acquired the variation curve of the residual phase variance with the loop bandwidth and the performance of the carrier signal stability. For the residual phase variance expression, the minimum variance corresponding to the optimal parameters can be obtained through the numerical simulation. The simulation experiment proves that the numerical results are consistent with the theoretical analysis. Our model for evaluating the influence of the carrier stability on the heterodyne ultra-high frequency detection can provide powerful support for improving the demodulation accuracy. The quantitative analysis of the paper will be available for theoretical basis and guidance for laser heterodyne detection based on optical phase-locked loops.

A Quasi-Coherent Detection Method Based on Radon–Fourier Transform Using Multi-Frequency-Based Passive Bistatic Radar

Passive bistatic radar (PBR)-based moving target detection (MTD) has benefited greatly from multi-frequency (MF) integration, which can effectively improve the detection capability of weak targets. However, with the increase in the coherent processing interval (CPI) and carrier-frequency separation, Doppler spread will appear in the range-Doppler maps (RDMs) over different frequency bands, which severely limits the processing gain of MF integration. In this paper, a novel MTD algorithm is proposed to achieve both long-time integration and quasi-coherent MF integration. More specifically, the proposed method consists of two main steps, where a modified Radon–Fourier transform (RFT), termed as MF-based RFT (MF-RFT), is, firstly, used to eliminate the Doppler spread via designing a sequential of MF-based Doppler filter banks. Following the MF-RFT, a phase-compensation-based method is also developed to further remove the residual phase errors. This method involves formulating an optimization problem based on the minimum-entropy criterion and employing a particle swarm optimization (PSO) algorithm to solve it, after which quasi-coherent MF integration can be achieved with robustness. Both numerical results and field test results based on digital video broadcasting-satellite (DVB-S) signals demonstrate that the proposed algorithm outperforms the existing methods in the scenario of weak MTD.

Experimental study of the impact of carrying a telecom signal on LOCSET-based coherent beam combining.

We report on what is, to our knowledge, one of the first realizations of a CBC (coherent beam combining)-based laser emitter carrying a 10.66 Gb/s telecom signal in free-space optics, within the laboratory environment. Two telecom modulations have been tested: NRZ (non-return-to-zero, in amplitude) and DPSK (differential phase-shift keying, in phase). The modulated signal is split and amplified in three fiber amplifiers, delivering up to 3 W each. CBC of data amplified signals is achieved with residual phase errors well below < λ/60 RMS, using a phase-tagging technique (LOCSET). A first analysis of the influence of various parameters (such as phase-tagging modulation depth, optical path difference, number of channels, amplifier power) on the locking and data transmission quality is investigated. The study shows that the phase-tagging modulation depth and optical path difference are the main critical issues when carrying data on a CBC signal.

Fully-Metallic Additively Manufactured Monolithic Double-Ridged Waveguide Rotman Lens in the K/Ka-Band.

This paper reports on the design and experimental validation of a fully-metallic double-ridged waveguide 10 × 10 Rotman lens additively manufactured as a single part. The wide band operation of this quasi-optical beamformer enables us to cover the uplink and downlink frequencies allocated to satellite communications in the K/Ka-band, from 17.3 GHz to 30 GHz. The feeding port design was adjusted to enable vertical printing, thus minimizing the use of supporting structures. A prototype was manufactured and tested. The reported results indicate losses in the range of 0.5 dB in the lower-frequency band and 0.8 dB in the upper-frequency band, including the waveguide transitions added for test purposes. The measured reflection and coupling coefficients remain below -11.5 dB over the operating band. The standard deviation of the residual phase error across the array ports is below 5° in simulation and below 10° in measurements. Array factors synthesized using the scattering parameters confirm the good stability of the beamforming functionality over the wide frequency band analyzed. This monolithic design is a promising step toward more integrated antenna systems, such as a compact dual-stack configuration for planar array design.

Improving Forest Canopy Height Estimation Using a Semi-Empirical Approach to Overcome TomoSAR Phase Errors

Forest canopy height is an important forest indicator parameter. Synthetic aperture radar tomography (TomoSAR) is an effective method to characterize forest canopy height and describe forest 3D structure; however, the residual phase error of TomoSAR affects the focus of the relative reflectance and can lead to errors in forest canopy height estimation. Therefore, this paper proposes a semi-empirical method to overcome the residual phase effects on forest canopy height estimation. In this study, we used airborne multi-baseline UAVSAR data to estimate forest canopy height via TomoSAR techniques and applied a semi-empirical method to improve forest canopy height estimation without phase calibration to mitigate the effects of phase error. The process is divided into three stages: the first step uses a semi-empirical method to initially determine the optimal relative reflectance loss threshold (K) by excluding the inverse extremes; in the second and third steps, the percentile height was used to gradually reduce the height interval between the upper and lower envelopes to minimize overestimation of extreme values and the lower vegetation. When the root mean square error (RMSE) was minimized, the percentile combinations were determined between the inversion results and a LiDAR dataset of the area. The results show that the canopy height estimation results are not satisfactory when relying solely on the K value to estimate the height difference between the envelope at the top of the forest and the ground; the best result was obtained when K = 0.4, but the corresponding R2 value was only 0.13, and the RMSE was 15.23 m. In our proposed method, the K value is determined as 0.3 by excluding the extreme values of the inversion result in the initial step—the corresponding R2 and RMSE values were 0.59 and 10.73 m, respectively, representing an RMSE decrease of 29.54% relative to the initial K value. After two steps of correction overestimation, the inversion accuracy was significantly improved with an R2 value of 0.65 and an RMSE of 9.69 m, corresponding to an RMSE decrease of 36.38%. Overall, the findings of the study represent an important reference for optimizing future spaceborne TomoSAR forest canopy height estimates.

Mathematical modelling and simulation of a wideband PLL for application in frequency synthesizers

In this paper the mathematical modelling of the Voltage Controlled Oscillator, Phase Frequency Detector, Low Pass Filter, Charge pump for designing a PLL circuit is proposed. Based on the modelling schematics are designed and proposed for the application in frequency synthesizer. The main design feature of this modelling is that VCO tunning switches are controlled digitally. The behaviour of PLL under its region of operation is simulated. The effect of temperature variations and insufficient bandgap overlapping on the output frequency of PLL is also presented and discussed. To avoid the current mismatching and residual phase errors the charge pump is modelled as to calibrate by its own circuit. Various simulations are done for the performance analysis of wideband PLL. The proposed design will have its application in frequency synthesizers suitable for multi band transceivers.

Joint 2-D Sparse ISAR Imaging and Autofocusing by Using 2-D-IADIANet

Compressive sensing (CS) based methods have been widely used for sparse inverse synthetic aperture radar (ISAR) imaging. However, many CS-based methods are sensitive to the selection of model parameters, and the residual phase error of the echo also causes trouble for imaging and autofocusing. To address these problems, a novel deep learning approach, named as 2D-IADIANet, is proposed to achieve 2D sparse ISAR imaging with 2D phase error estimation in this paper. First, a 2D ISAR sparse echo model with 2D phase error into account is established, and a 2D-ADMM frame-work-based method, dubbed as 2D-IADIA, is presented to solve this compound reconstruction problem. Second, 2D-IADIA is further unfolded and mapped into a deep network form by integrating with 2D phase error compensation network. Moreover, all adjustable parameters can be learned adaptively by training the network through back propagation algorithm in complex domain directly. At last, experiment results verify that the well-learned 2D-IADIANet, which is only trained by a small amount of simulation samples, can also be generalized to measured data application. Especially, owing to the good performance of the network, the proposal has a superior reconstruction performance than 2D-IADIA under the low 2D sample rate and/or signal-to-noise ratio scenarios.

Deep learning phase error correction for cerebrovascular 4D flow MRI

Background phase errors in 4D Flow MRI may negatively impact blood flow quantification. In this study, we assessed their impact on cerebrovascular flow volume measurements, evaluated the benefit of manual image-based correction, and assessed the potential of a convolutional neural network (CNN), a form of deep learning, to directly infer the correction vector field. With IRB waiver of informed consent, we retrospectively identified 96 MRI exams from 48 patients who underwent cerebrovascular 4D Flow MRI from October 2015 to 2020. Flow measurements of the anterior, posterior, and venous circulation were performed to assess inflow-outflow error and the benefit of manual image-based phase error correction. A CNN was then trained to directly infer the phase-error correction field, without segmentation, from 4D Flow volumes to automate correction, reserving from 23 exams for testing. Statistical analyses included Spearman correlation, Bland–Altman, Wilcoxon-signed rank (WSR) and F-tests. Prior to correction, there was strong correlation between inflow and outflow (ρ = 0.833–0.947) measurements with the largest discrepancy in the venous circulation. Manual phase error correction improved inflow-outflow correlation (ρ = 0.945–0.981) and decreased variance (p < 0.001, F-test). Fully automated CNN correction was non-inferior to manual correction with no significant differences in correlation (ρ = 0.971 vs ρ = 0.982) or bias (p = 0.82, Wilcoxon-Signed Rank test) of inflow and outflow measurements. Residual background phase error can impair inflow-outflow consistency of cerebrovascular flow volume measurements. A CNN can be used to directly infer the phase-error vector field to fully automate phase error correction.

Highly coherent phase-lock of an 8.1 μm quantum cascade laser to a turn-key mid-IR frequency comb

A continuous-wave Fabry–Pérot quantum cascade laser (QCL) emitting at 8.1 μm operating in the single mode regime has been coherently phase locked to a turn-key low-noise commercial mid-Infrared (mid-IR) optical frequency comb. The stability of the comb used as a reference is transferred to the QCL resulting in an integrated residual phase error of 0.4 rad. The laser linewidth is narrowed by more than two orders of magnitude reaching sub-kHz level at 1 ms observation time, limited by the spectral purity of the mid-IR comb. Our experiment is an important step toward the development of both powerful and metrology-grade QCLs and fully stabilized QCL frequency comb and opens perspectives for precision measurements and frequency metrology in the mid-IR.

Online Expectation-Maximization Based Frequency and Phase Consensus in Distributed Phased Arrays

Distributed phased arrays are comprised of separate, smaller antenna systems that coordinate with each other to support coherent beamforming towards a destination. However, due to the frequency drift and phase jitter of the oscillators, as well as the frequency and phase estimation errors induced at the nodes, there exists decoherence that degrades the beamforming process. A decentralized frequency and phase consensus (DFPC) algorithm was proposed in prior work for undirected networks in which the nodes locally share their frequencies and phases with their neighbors to reach synchronization. Kalman filtering (KF) was also integrated with DFPC (KF-DFPC) to lower the total residual phase error upon convergence. Since these DFPC-based algorithms rely on the average consensus protocol, they do not converge for directed networks. In this paper, we propose a push-sum based frequency and phase consensus (PsFPC) algorithm for the directed networks. The residual phase error of PsFPC is theoretically derived as well. Kalman filtering is also integrated with PsFPC and the resulting KF-PsFPC algorithm shows a significant reduction in the residual phase error upon convergence. KF assumes that the model parameters, i.e., the measurement noise and innovation noise covariance matrices, are known. Since they may not be known in practice, we develop an online expectation maximization (EM) based algorithm that iteratively computes the maximum likelihood (ML) estimate of the unknown matrices in an online manner. EM is integrated with KF-PsFPC to propose the EM-KF-PsFPC algorithm. Simulation results are included where the performance of the PsFPC-based algorithms is analyzed for different distributed phased arrays and is compared to other algorithms.

Spectrally dispersed kernel phase interferometry with SCExAO/CHARIS: proof of concept and calibration strategies

Kernel phase interferometry (KPI) is a data processing technique that allows for the detection of asymmetries (such as companions or disks) in high-Strehl images, close to and within the classical diffraction limit. We show that KPI can successfully be applied to hyperspectral image cubes generated from integral field spectrographs (IFSs). We demonstrate this technique of spectrally-dispersed kernel phase by recovering a known binary with the SCExAO/CHARIS IFS in high-resolution K-band mode. We also explore a spectral differential imaging (SDI) calibration strategy that takes advantage of the information available in images from multiple wavelength bins. Such calibrations have the potential to mitigate high-order, residual systematic kernel phase errors, which currently limit the achievable contrast of KPI. The SDI calibration presented here is applicable to searches for line emission or sharp absorption features, and is a promising avenue toward achieving photon-noise-limited kernel phase observations. The high angular resolution and spectral coverage provided by dispersed kernel phase offers novel opportunities for science observations which would have been challenging to achieve otherwise.

Frequency and Phase Synchronization in Distributed Antenna Arrays Based on Consensus Averaging and Kalman Filtering

A decentralized approach for joint frequency and phase synchronization in distributed antenna arrays is presented. The nodes in the array share their frequencies and phases with their neighboring nodes to align these parameters across the array. Our signal model includes the frequency drifts and phase jitters of the local oscillators as well as the frequency and phase estimation errors at the nodes and models them using practical statistics. A decentralized frequency and phase consensus (DFPC) algorithm is proposed which uses an average consensus method in which each node in the array iteratively updates its frequency and phase by computing an average of the frequencies and phases of their neighboring nodes. Simulation results show that upon convergence the DFPC algorithm can align the frequencies and phases of all the nodes up to a residual phase error that is governed by the oscillators and the estimation errors. To reduce this residual phase error and thus improve the synchronization between the nodes, a Kalman filter based decentralized frequency and phase consensus (KF-DFPC) algorithm is presented. The total residual phase error at the convergence of the KF-DFPC and DFPC algorithms is derived theoretically. The synchronization performances of these algorithms are compared to each other in light of this theoretical residual phase error by varying the duration of the signals, connectivity of the nodes, the number of nodes in the array, and signal to noise ratio of the transmitted signals. Simulation results demonstrate that the proposed KF-DFPC algorithm converges in fewer iterations than other algorithms including DFPC. Furthermore, for shorter intervals between local information broadcasts, the KF-DFPC algorithm significantly outperforms the DFPC algorithm in reducing the residual total phase error, irrespective of the signal to noise ratio of the received signals.