Doppler Frequency Shift Research Articles

Self-adaptive retro-reflective Doppler cloak based on planar space-time modulated metasurfaces

As engineered electromagnetic covers based on (space-) time modulated metamaterials and metasurfaces, Doppler cloaks are able to compensate the Doppler effect induced by the motion of a scatterer, making it appear as if it were at rest to a detecting radar system. Perfect Doppler compensation can be theoretically always achieved for any relative velocity and motion direction of the cloaked scatterer with respect to the detecting system. However, the motion can be still detected from the cross section variation of the Doppler cloaked scatterer, especially under oblique incidence illumination. The challenge is, therefore, to have a proper Doppler compensation and maintain the amount of scattered energy toward the detection system as much constant as possible with respect to the illumination angle. In this Letter, we propose the design of self-adaptive retro-reflective planar Doppler cloak composed of a pair of space-time modulated metasurfaces: the first metasurface focuses the incident field in a specific location on the second metasurface, which is designed for enabling retro-reflection and Doppler frequency shift compensation. Here, the self-adaptive Doppler cloak is applied to a metallic planar reflector, moving toward its normal direction, and illuminated by an oblique plane wave. We demonstrate that the proposed Doppler cloak can perform frequency conversion and simultaneously maintain the radar cross section of the reflector as much stable as possible within an angular range of about 60° centered at the normal direction. The self-adaptive Doppler cloak may enhance the undetectability of cloaked moving objects.

Rotational Doppler shift tripling via third-harmonic generation of spatially structured light in a quasi-periodically poled crystal

The rotational Doppler effect—the angular version of the Doppler effect—shows a frequency shift phenomenon happening when light passes through a rotating object and its angular momentum is changed. The rotational Doppler effect has been widely studied with various light waves based on different angular momentum transform mechanisms and successfully utilized to detect the rotation of objects. However, these studies are mostly limited to linear optics. As the rotational Doppler effect is closely related to the frequency degree of freedom and frequency conversion always happens in nonlinear optics, it is of great significance to explore the rotational Doppler effect in nonlinear optics. Although nonlinear rotational Doppler phenomena have been observed in second-harmonic generation with circularly polarized Gaussian fundamental light and a rotating nonlinear crystal, there is still the open challenge of how to realize Doppler frequency shift conversion (similar to conventional nonlinear frequency conversion), such as doubling or tripling. Here we report the experiment observation of rotational Doppler shift conversion in nonlinear optics. We demonstrate the tripling of a rotational Doppler frequency shift in third-harmonic generation of a spatially structured fundamental wave. Frequency shifts can be extracted from the intensity beating signals with the interference of two components of vector fields with opposite topological charges and the measured modulation frequency of the beating signal for the generated third harmonic. Our results will excite more research on nonlinear optics with spatially structured light, and our method may create opportunities for precision measurement of frequency shifts.

A Lightweight and Drift-Free Fusion Strategy for Drone Autonomous and Safe Navigation

Self-localization and state estimation are crucial capabilities for agile drone autonomous navigation. This article presents a lightweight and drift-free vision-IMU-GNSS tightly coupled multisensor fusion (LDMF) strategy for drones’ autonomous and safe navigation. The drone is carried out with a front-facing camera to create visual geometric constraints and generate a 3D environmental map. Ulteriorly, a GNSS receiver with multiple constellations support is used to continuously provide pseudo-range, Doppler frequency shift and UTC time pulse signals to the drone navigation system. The proposed multisensor fusion strategy leverages the Kanade–Lucas algorithm to track multiple visual features in each input image. The local graph solution is bounded in a restricted sliding window, which can immensely predigest the computational complexity in factor graph optimization procedures. The drone navigation system can achieve camera-rate performance on a small companion computer. We thoroughly experimented with the LDMF system in both simulated and real-world environments, and the results demonstrate dramatic advantages over the state-of-the-art sensor fusion strategies.

Authentication for Satellite Communication Systems Using Physical Characteristics

Satellite communication networks have gained a lot of attention recently as a solution to mitigate the limitations of terrestrial networks such as stability and coverage. However, integrating satellite and terrestrial networks makes the system more vulnerable to spoofing attacks. Thus, robust and effective authentication is required. Physical layer authentication (PLA) has emerged as an alternative paradigm that uses physical characteristics to achieve authentication. In this paper, PLA is proposed for low earth orbit (LEO) satellites using the Doppler frequency shift (DS) and received power (RP) characteristics. Hypothesis testing using a threshold or machine learning (ML) is considered to discriminate between legitimate and illegitimate satellites. For ML, a one-class classification support vector machine (OCC-SVM) is employed which uses training data from only legitimate users. The performance is evaluated using real satellite data from the system tool kit (STK). Results are presented which show that the authentication rate (AR) with DS is higher than with RP at low elevation angles for both schemes, but is higher with RP at high elevation angles. Further, the ML authentication scheme provides a higher AR than the threshold scheme for a small percentage of the training data considered as outliers, but at larger percentages the OR threshold scheme is better.

OTFS-based efficient handover authentication scheme with privacy-preserving for high mobility scenarios

Handover authentication in high mobility scenarios is characterized by frequent and short-term parallel execution. Moreover, the penetration loss and Doppler frequency shift caused by high speed also lead to the deterioration of network link quality. Therefore, high mobility scenarios require handover schemes with less handover overhead. However, some existing schemes that meet this requirement cannot provide strong security guarantees, while some schemes that can provide strong security guarantees have large handover overheads. To solve this dilemma, we propose a privacy-preserving handover authentication scheme that can provide strong security guarantees with less computational cost. Based on Orthogonal Time Frequency Space (OTFS) link and Key Encapsulation Mechanism (KEM), we establish the shared key between protocol entities in the initial authentication phase, thereby reducing the overhead in the handover phase. Our proposed scheme can achieve mutual authentication and key agreement among the user equipment, relay node, and authentication server. We demonstrate that our proposed scheme can achieve user anonymity, unlinkability, perfect forward secrecy, and resistance to various attacks through security analysis including the Tamarin. The performance evaluation results show that our scheme has a small computational cost compared with other schemes and can also provide a strong guarantee of security properties.

Retrieval of 2-D Ocean Total Surface Current From Sentinel-1 Doppler Frequency Shift Using 2-D Kalman Filtering

Synthetic aperture radar (SAR) has all-weather, all-day, large-area observation capability and can measure ocean total surface currents with Doppler frequency shift. However, conventional SAR can only obtain the radial velocity of the ocean total surface current. In this letter, we propose a method for inverting ocean total surface current vectors using a two-dimensional (2D) Kalman filter and a priori information about ocean current fields from Ocean Surface Current Analyses Real-time (OSCAR). We establish a complete filtering process including filter creation, filtering path selection and fusion of filtering results. We construct the Kalman filter according to the motion characteristics of ocean currents and transform the ocean surface current vector inversion problem into a filtering problem. The introduction of two-dimensional filtering can better adapt to the directionality of ocean currents. Furthermore, we design the filtering path to obtain smoother filtering results. In order to synthesize the filtering results of different filtering paths, we use the Covariance Intersection (CI) fusion method in distributed fusion to obtain the final ocean current vector inversion results. The accuracy of the inversion is verified by the measured data. The Root Mean Square Error (RMSE) of the proposed method with the mean Doppler error of 5.43Hz is 0.182 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m/s</i> in eastward speed and 0.141 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m/s</i> in northward speed. The reduction is 0.009 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m/s</i> and 0.029 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m/s</i> against OSCAR.

Two-Stage Time-Varying Vibration Compensation for Coherent LiDAR Based on the Adaptive Differential Evolution Method

Coherent LiDAR, which is based on triangular frequency modulated continuous wave (T-FMCW), is a promising technology for high-resolution imaging, because it can achieve long-range and high-precision measurements using low power. However, owing to the short wavelength of laser, the system is extremely sensitive to time-varying vibration. Most vibration compensation methods perform well in the scenes that contain strong scatterers using enough data. However, they lose validity in scenes without strong scatterers and sufficient observation time. To address this problem, we develop a second-order vibration error model for coherent LiDAR that divides the Doppler frequency shift caused by time-varying vibrations into quadratic and primary errors. Based on the vibration error model, we propose a two-stage time-varying vibration compensation method using the adaptive differential evolution (ADE) algorithm. In the first stage, we iteratively determine the optimal quadratic vibration coefficients that minimize the spectral entropy of the compensated dechirp signals using the ADE algorithm. Subsequently, we compensate for the quadratic errors using the estimated quadratic vibration coefficients. In the second stage, we use the frequency spectrum cross-correlation method to estimate the primary vibration coefficients and compensate for the primary errors. We perform experiments in the scene containing a point target and the scenes containing multi-component targets and distributed targets without strong scatterers. The results indicate that the proposed method can effectively compensate for time-varying vibration errors using only one-period T-FMCW data in different scenes. Further, the results of the proposed method are more accurate and stable than the Doppler frequency shift method.

A Closed-Form Solution for 3D Source Localization Using Angles and Doppler Shifted Frequencies

A Closed-Form Solution for 3D Source Localization Using Angles and Doppler Shifted Frequencies

Waveform Design and DoA-DoD Estimation of OFDM-LFM Signal Based on SDFnT for MIMO Radar

In comparison to orthogonal frequency division multiplexing (OFDM), the OFDM linear frequency modulation (OFDM-LFM) signal has similar time-bandwidth product, lower peak-to-average power ratio, and higher sensitivity to Doppler frequency shift, making it a promising candidate waveform for multiple-in multiple-out (MIMO) radar applications. However, the OFDM-LFM signal is unsatisfactory in practical applications due to the challenges in waveform design and its non-stationary property, which significantly degrades the performance of current high-resolution subspace techniques for direction of departure (DoD) and direction of arrival (DoA) angle estimation. In this paper, we first proposed a novel OFDM-LFM waveform design method based on the scale discrete Fresnel transform (SDFnT), which efficiently generates orthogonal LFM signals with unlimited number of subcarriers and can be convieniently implemented using the fast Fourier transform (FFT). Then, for angle estimation in OFDM-LFM MIMO radars, we presented a new scale discrete Fresnel transform multiple signal classification (SDFnT-MUSIC) method. By converting the steering vector into the chirp domain and making it non-time-vary, high resolution angle estimation for OFDM-LFM wideband signals can be achieved. Simulation results shows that the proposed waveform design method requires less computational time than conventional method based based on the fractional Fourier transform (FrFT), whereas the proposed DoA-DoD estimation method outperform the FrFT-MUSIC method in terms of computational complexity and target angle estimate performance.

Reducing Bias for Multistatic Localization of a Moving Object by Transmitter At Unknown Position

Multistatic localization plays an important role in object localization. In this paper, we address the problem of multistatic localization of a moving object in the absence of transmitter position when the transmitter is not synchronized with the receivers. We propose to jointly estimate the unknowns, including the object position and velocity, the transmitter position, and the clock and frequency offsets, using both time delay and Doppler frequency shift measurements. To this end, we first formulate a constrained weighted least squares (CWLS) minimization problem, whose solution is found to have significant bias caused by approximations in transforming the model equations and solving the problem. To reduce the bias, we further formulate a non-convex bias reduced CWLS (BR-CWLS) problem by imposing a quadratic constraint, which is constructed by considering the second-order noise and errors of the matrices and vectors involved in the CWLS problem. One particular aspect of the proposed BR-CWLS method is that the errors in approximating the weighting matrix are taken into consideration for bias reduction. The non-convex BR-CWLS problem is solved by applying the semidefinite relaxation technique to relax it as a convex semidefinite program. In addition, we show through the mean square error (MSE) analysis that the performance of the BR-CWLS solution can approach the Cramer-Rao lower bound performance when the noise is not significant. We also derive the theoretical bias expression for evaluating the amount of bias. Simulation results demonstrate the good performance of the proposed method in terms of both MSE and bias.

Parameter Extraction of Accelerated Motion Targets Based on Vortex Electromagnetic Wave Radar

Vortex electromagnetic (EM) wave radar can obtain more accurate rotation parameters for target classification and recognition. However, the existing rotation parameter extraction methods are difficult to obtain the acceleration and initial phase of composite moving target in the case of multiple scattering points. In this paper, the estimation error of accelerated motion target features in a situation with numerous scattering points is examined for the first time, along with a method for extracting accelerated motion target parameters based on vortex electromagnetic wave radar. Firstly, the relationship between the characteristics of non derivable points (NDPs) in the rotating Doppler frequency shift and the characteristics of the target’s spin motion is analyzed, the method of estimating the number of scattering points is studied, and the formula for extracting the spin parameters is derived. After that, the time-frequency analysis of the echo signal is combined to extract the target translational motion characteristics. Simulation results show that the proposed method is effective.

МОДЕЛЮВАННЯ БАГАТОПРОМЕНЕВОГО КАНАЛУ ЗВ'ЯЗКУ

In the article, the simulation of a multipath communication channel with two bands or subbands (intervals) of frequencies, the system function of which is expressed in a closed form through elliptic functions. A multipath communication channel is a linear system with time various parameters. Multipath leads to the fact that there is always a certain number of errors. They characterize the lower limit of the probability of errors, regardless of the applied methods of error-correcting coding and an unlimited increase in the energy potential of the radio link. Mathematical simulation of a multipath communication channel provides the joint consideration of time delays and Doppler frequency shift of the received signal. The correlation between the transfer function 퐻 (휔,푡) and the system function 푆 ̃(휏,휔 ̃) is shown. Іn the article at the first time, the mathematical theory of elliptic functions and elliptic integrals is applied to the definition of a system function, which in known literature sources is determined using the mathematical methods of statistical analysis. In article is shown the application an approach to mobile radio systems, which are operating in several frequency bands, such as a combined mobile digital troposcatter-radiorelay station or a digital troposcatter station. Their work is carried out in two frequency bands or two frequency subbands (transmit / receive). Therefore, in this case it becomes very difficult to create a mathematical model of a multipath communication channel with the definition of system functions. This situation is complicated if a frequency adaptation system is used. At the same time, several frequency intervals are used to combat fading and interferences, in the simplest case – two intervals. The article describes the approach to the representation of system functions through elliptic functions. This allows to have close form of the transfer function for the multipath communication channel. For the considered case of two-frequency intervals, the final expression is written in terms of theta-functions and the complete elliptic integral of the first kind.

A Simple Photonic System for DFS and AOA Simultaneous Measurement

A simple photonics-based dual-channel system is proposed to simultaneously measure the Doppler frequency shift (DFS) and angle of arrival (AOA) of microwave signals. The system applies two parallel push–pull Mach–Zehnder modulators (MZMs) for carrier suppression dual-sideband (CS-DSB) modulation. The introduction of the reference signal results in a DFS measurement without direction ambiguity. The DFS can be determined by measuring the frequency of the down-converted intermediate frequency (IF) signal, and the AOA can be calculated by comparing the phase shift of the two channels. A proof-of-concept experiment shows that the DFS measurement error is less than 0.4 Hz during ±100 kHz, and the AOA measurement error is within 1.5° in a range of 0–70°.

Physical effects of geocosmic storms. Part 3

A review of the results of observing the effects of the most interesting ionospheric and magnetic storms that took place in 2018–2019 is presented. Ionospheric and magnetic storms are considered as components of geocosmic storms. In accordance with the systemic paradigm of L.F. Chernogor, a geocosmic storm is a complex of processes caused by a solar storm, reduced to synergistically interacting storms in a magnetic field (magnetic storm), ionosphere (ionospheric storm), atmosphere (atmospheric storm) and electric field atmospheric-ionospheric-magnetospheric origin (electric storm). Using the analysis of specific magnetic and ionospheric storms as an example, it is shown that, in addition to general patterns, each storm has its own individual characteristics. The expediency of a comprehensive study of the manifestations of each new geocosmic storm has been confirmed. A wide variety of ionospheric and magnetic storms have been confirmed. A weak magnetic storm is not necessarily accompanied by a weak ionospheric storm, while a strong one is accompanied by a strong one. An ionospheric storm can be single-phase (positive or negative) or multi-phase, when the phases alternate. Storms can last from several hours to many days. Storms are accompanied by both aperiodic and quasi-periodic variations. The latter are caused by atmospheric gravity waves (periods 10–180 min) and infrasound (period less than 5 min). Of all the components of a geocosmic storm, except for academic ones, ionospheric storms are of the greatest practical interest. It is these storms that limit the potential characteristics of radio systems for various purposes (telecommunications, radio navigation, radar, remote radio sounding, radio astronomy). It is shown that ionospheric storms are accompanied by the appearance of multipath, broadening of the Doppler spectra (and even their collapse), significant aperiodic and quasi-periodic variations in the Doppler frequency shift, the amplitude of radio signals, and the trajectories of radio waves in the decameter range. In particular, the height of reflection of radio waves changed during ionospheric storms by 50–180 km. The electron concentration during moderate negative ionospheric storms decreased up to 2 times, and during moderate positive ionospheric storms it increased up to 3 times. Under the action of atmospheric gravity waves, the relative amplitude of quasi-periodic variations in the electron concentration reached 50%, and due to low-frequency infrasound, it did not exceed 0.7%.

Ku-band multichannel frequency comb Doppler reflectometer on the GAMMA 10/potential control and divertor simulating experiment (PDX) tandem mirror.

A Ku-band (12-18GHz) multichannel Doppler reflectometer (DR) has been developed in the GAMMA 10/potential control and divertor simulating experiment (PDX) tandem mirror device to improve the applicability of DR measurement for simultaneous monitoring of velocity of electron density turbulence at different locations. Our previous single-channel DR circuit has been replaced by the multichannel microwave system using a nonlinear transmission line based comb generator with heterodyne technique. The multichannel DR system has been installed in the central cell of GAMMA 10/PDX. Initial results of application to GAMMA 10/PDX plasma are presented, showing Doppler frequency shifts during an additional ion cyclotron resonance frequency heating and gas-puffing experiment.

Small Satellite Orbit Determination Using Single Pass Doppler Measurements

The Doppler measurements of the telemetry radio signals nanosatellite CubeBel-1 for a single pass over the Belarusian State University ground station were carried out. Two methods for orbit determination of a small satellite (SS) are considered. The first method is based on the SGP4 model and requires additional information from the North American Aerospace Defense Command (NORAD) two-line elements (TLE) catalog of the satellite orbital parameters. An unknown SS is identified using the NORAD TLE catalog based on a probabilistic estimation of the elevation angle and the Doppler frequency shift of receiving telemetry signals. The first method was allowed to determine the nanosatellite CubeBel-1 unambiguously. The second method is based on processing experimental measurements of the Doppler frequency of the telemetry radio signals and the Keplerian circular motion model for SS. It does not require additional information from the NORAD database of satellite orbital parameters. The second method was allowed only to calculate the average state vector unknown satellite. Finally, the corrected state vector in the geocentric inertial coordinate system was obtained based on the differential correction method.

A fast TDOA and VDOA estimation method of the wideband frequency hopping signal

In the mobile multi-station passive positioning scenario, the carrier frequency hopping of the wideband frequency hopping communication signal will produce a changing Doppler frequency shift. Therefore, it is not possible to directly estimate the Doppler frequency difference of the observed signal containing multiple pulses. In this case, only the time difference and frequency difference of the single-hop signal can be estimated, but the accuracy is low. The existing methods often need the carrier frequency of each FH pulse signal as a priori information, and the high computational complexity of the two-dimensional search method of the time difference and frequency difference is difficult to overcome. Aiming at the above problems, a fast and high-precision estimation algorithm for the time difference of arrival and velocity difference of wideband frequency hopping signals is proposed in this paper. Firstly, the wideband frequency hopping signal is segmented in the time domain with multiple pulses as the unit, the signal model of time-varying time difference is established, and the time difference of each short-time segment is estimated by the generalized cross-correlation algorithm. The estimated values of the initial time difference and Doppler velocity difference can be worked out by least squares fitting of the time difference sequence obtained from the single pulse estimation. Finally, taking the Link16 data link signal as an example, the simulation experiment of the proposed method is conducted, which verifies the correctness and performance superiority of the algorithm.

Development of Verification Device for Multitarget Radar Velocimeter Based on Echo Signal Simulation Technology

Speeding is one of the leading causes of traffic crashes worldwide. Radar velocimeter is widely used in the capture monitoring of road overspeed violations, which can effectively reduce the probability of traffic accidents and protect people’s life and property safety to the greatest extent. As a new type of radar velocimeter, multitarget radar velocimeter (MTRV) can monitor the speed of more than two vehicles at the same time. However, the verification method and device of MTRV’s performance need to be studied. In order to solve the problem of performance verification for MTRV, a verification device based on echo signal simulation technology is developed in this paper. The measurement mechanism of MTRV with different performance including velocity, distance, and angle is first introduced. Then, a verification method based on the echo signal simulation technology is proposed. The verification device can receive the emission signal of MTRV and process the signal by echo simulation technology, including target generation, Doppler frequency shift, time delay, and angel control, and targets are simulated with nominal velocity, distance, and angle value. The processed echo signal with simulated nominal parameter values is reflected to the MTRV. After the echo signal is received and processed by MTRV, the measurement values of simulated velocity, distance, and angle for targets are obtained. Comparing the measured values of the MTRV with the simulated nominal values of the verification device, the measurement error of MTRV is obtained. The verification device of MTRV is realized to verify the accuracy and reliability of the MTRV measurement results. The simulated velocity range of the verification device is up to (-300~300) km/h, and the simulated distance range of the verification device is up to (10~45) m when the simulated incident angle range was within the range of (-60~60)°. The simulation target generation for the two targets of the device is also verified. And the maximum permissible error (MPE) of the simulated velocity was ±0.05 km/h, the MPE of simulated distance is ±0.3 m, and the MPE of simulated angle is ±0.2°. Finally, the verification and uncertainty evaluation results of the MTRV sample validated the effectiveness and feasibility of the proposed verification method and the developed verification device of MTRV.

Motion parameter estimation of AUV based on underwater acoustic Doppler frequency measured by single hydrophone

This paper describes the use of a single hydrophone to estimate the motion parameters of an autonomous underwater vehicle (AUV) from the underwater acoustic signal excited by its propulsion motor. First, the frequency range of the hydroacoustic signal radiated by the AUV motor is determined, and a detection and recognition model is designed. In the case of uniform linear motion of the AUV, the geometric relationship between the Doppler frequency shift curve of the sound source is derived and the motion model of the sound source and sound line propagation is established. An estimation algorithm for the motion parameters of multiple AUVs based on data from a single hydrophone is derived. Then, for Doppler underwater acoustic signals disturbed by independent identically distributed noise with an arbitrary probability distribution, a cumulative phase difference power amplification instantaneous frequency estimation method is proposed. This method is based on the sum of multiple logarithmic functions. Finally, the effectiveness and accuracy of the algorithm in estimating the motion parameters of multiple AUVs are verified through simulations and experiments.

RESEARCH OF SYNCHRONIZATION CIRCUITS FOR DIGITAL COMMUNICATION SYSTEMS

In a digital communication system, the transmitter and receiver have several generators for modulation and demodulation; increasing and decreasing the sampling frequency; synchronization of symbols and bit streams. The causes of phase and carrier frequency errors are the instability of the frequency of the local generators of the transmitter and receiver; presence of Doppler frequency shift; signal propagation delay from the transmitter to the receiver. Synchronization circuits of modern digital communication systems are built on the basis of phase-locked loop (PLL). The purpose of the work is: research of various characteristics of the digital PLL (locking time; established error; transient behavior) for various types of input action; study of the bit error rate of a coherent digital communication system. The PLL consists of the following components: a phase detector that generates a signal that varies in proportion to the phase difference between the input signal and a locally generated sinusoid; controlled generator that generates an output signal whose phase and frequency depends on the input signal; loop filter, which removes unwanted high-frequency components in the output signal of the phase detector and forms a signal that controls the NCO. During adaptation, the PLL has some transient process that depends, in particular, on three factors: the presence of a zero-phase error is determined by the PLL contour filter; the determined bandwidth of the circuit and the initial deviation between the input and reference frequencies affect the PLL adaptation time; the attenuation coefficient of the PLL affects the adaptation behavior: the speed and magnitude of emissions. The study of the output signal of the linear PLL model for damping factor , and , and we will use an input signal of the step function type, linear voltage changes and hyperbola. type 1, type 2, and type 3 PLLs can adapt to a zero-error step input. If the input signal is a linearly varying voltage, PLL types 2 and 3 can adapt with zero phase error, while type 1 adapts with a residual phase error. For hyperbola input, only type 3 PLL can fully adapt: type 2 adapts with residual error, while type 1 cannot adapt. The damping factor should be in the range of . When changing the input signal frequency from 3.55 MHz to 3.72 MHz, the capture time of the type 2 PLL changes from about 150 μs to about 600 μs, that is, when the frequency increases by 170 kHz, the capture time increases almost four times. The error that has occurred is affected by the choice of the contour filter and the features of the synchronization.