Quadrature Error Research Articles

Two formulae with nodes related to zeros of Bessel functions for semi-infinite integrals: extending Gauss–Jacobi-type rules

For approximating integrals $$\varvec{\int _0^{\infty }\!\!} \varvec{x}^{\varvec{\alpha }}\varvec{f(x)dx}$$ ( $$\varvec{\alpha >-1}$$ ) over a semi-infinite interval $$\varvec{[0,\infty )}$$ with a given function $$\varvec{f(x)}$$ , two formulae, one of them new and another associated with an existing formula, are presented. They are constructed in a limiting process to a semi-infinite interval $$\varvec{[0,\infty ]}$$ with a linear transformation for a well-known approximation method, the Gauss–Jacobi (GJ) rule and its family rules: the Gauss–Jacobi–Radau (GR) and Gauss–Jacobi–Lobatto (GL) rules on a finite interval. This procedure was used in constructing our previous limit Clenshaw–Curtis-type formulae. The limit GJ formula (LGJ) constructed in this way uses as nodes zeros of the Bessel function $$\varvec{\varvec{J_{\alpha }(x)}}$$ squared after multiplied by a positive constant a and the limit GR (LGR) (and limit GL (LGL)) formula those with zeros of $$\varvec{J_{\alpha +1}(x)}$$ . The LGJ formula is also shown to be derived from the formula developed by Frappier and Olivier (Math. Comp. 60:303–316, 1993) for an integral on $$\varvec{[0,\infty )}$$ . The LGR and LGL formulae give the same and new formula. We show that for a function f(z) analytic on a domain in the complex plane z and satisfying some appropriate conditions, there exists a constant $${d>0}$$ such that the errors of both formulae are $$\varvec{O(e^{-2d/a})}$$ as $$\varvec{a\rightarrow +0}$$ . The average of the LGJ and LGR formulae gives smaller quadrature errors than each formula. Numerical examples confirm these behaviors and show that the LGJ and LGR formulae give asymptotically the same quadrature errors of opposite sign. Consequently, the LGR formula behaves like an anti-LGJ formula in the same way as the Lobatto rule for integrals on $$\varvec{[-1,1]}$$ behaving like the anti-Gauss rule.

Weighted averaged Gaussian quadrature rules for modified Chebyshev measures

This paper is concerned with the approximation of integrals of a real-valued integrand over the interval [−1,1] by Gauss quadrature. The averaged and optimal averaged quadrature rules ([13,21]) provide a convenient method for approximating the error in the Gauss quadrature. However, they are applicable to all integrands that are continuous on the interval [−1,1] only if their nodes are internal, i.e. if they belong to this interval.We discuss two approaches to determine averaged quadrature rules with nodes in [−1,1]: (i) truncating the Jacobi matrix associated with the optimal averaged rule, and (ii) weighting the optimal averaged quadrature rule. We consider Chebyshev measures of the first, second, and third kinds that are modified by a linear over linear rational factor, and discuss the internality of averaged, optimal averaged, and truncated optimal averaged quadrature rules. Moreover, we show that the weighting yields internal averaged rules if a weighting parameter is properly chosen, and we provide bounds for this parameter that guarantee internality. Finally, we illustrate that the weighted averaged rules give more accurate estimates of the quadrature error than the truncated optimal averaged rules.

A locally corrected multiblob method with hydrodynamically matched grids for the Stokes mobility problem

Inexpensive numerical methods are key to enabling simulations of systems of a large number of particles of different shapes in Stokes flow and several approximate methods have been introduced for this purpose. We study the accuracy of the multiblob method for solving the Stokes mobility problem in free space, where the 3D geometry of a particle surface is discretised with spherical blobs and the pair-wise interaction between blobs is described by the RPY-tensor. The paper aims to investigate and improve on the magnitude of the error in the solution velocities of the Stokes mobility problem using a combination of two different techniques: an optimally chosen grid of blobs and a pair-correction inspired by Stokesian dynamics. Different optimisation strategies to determine a grid with a given number of blobs are presented with the aim of matching the hydrodynamic response of a single accurately described ideal particle, alone in the fluid. It is essential to obtain small errors in this self-interaction, as they determine the basic error level in a system of well-separated particles. With an optimised grid, reasonable accuracy can be obtained even with coarse blob-resolutions of the particle surfaces. The error in the self-interaction is however sensitive to the exact choice of grid parameters and simply hand-picking a suitable geometry of blobs can lead to errors several orders of magnitude larger in size. The pair-correction is local and cheap to apply, and reduces the error for moderately separated particles and particles in close proximity. Two different types of geometries are considered: spheres and axisymmetric rods with smooth caps. The error in solutions to mobility problems is quantified for particles of varying inter-particle distances for systems containing a few particles, comparing to an accurate solution based on a second kind BIE-formulation where the quadrature error is controlled by employing quadrature by expansion (QBX).

Reducing the Sensitivity of Fabrication Error in the MEMS Butterfly Gyroscope by Adopting an Improved Vibration Beam

Microelectromechanical system (MEMS) butterfly gyroscope is one of the most potential high-precision <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ir=textit{x}$ </tex-math></inline-formula> / <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$textit{y}$ </tex-math></inline-formula> -axis gyroscopes, whose vibration beam as the critical component has a large effect on its performance. This article researched the influence of the spindle azimuth on the vibration beam by building a mathematical model, which found the larger and more robust spindle azimuth to process variation benefited the sensitivity and quadrature error. A novel butterfly gyroscope with a concave vibration beam was proposed. And the effect of its parameters on spindle azimuth was analyzed theoretically, which helped to propose a method for designing the most optimal concave vibration beam. In the presence of the same fabrication error, the robustness of the spindle azimuth to process variation was improved about double theoretically compared to the former vibration beam. The fabrication process and the corresponding control system were designed and conducted. Experiments appeared that the butterfly gyroscope with a concave vibration beam was nearly half in the quadrature error ratio (avg 0.869) than the former (avg 1.669). More importantly, the distribution of the quadrature error in the novel butterfly gyroscopes was more concentrated. Their standard deviation was calculated as 0.292, which was nearly half of that of the oblique beam (0.579). It meant the concave beam appeared to have better robustness to process variation.

Multi-coefficient eigenmode operation—breaking through 10°/h open-loop bias instability in wideband aluminum nitride piezoelectric BAW gyroscopes

In this paper, a modification to the eigenmode operation of resonant gyroscopes is introduced. The multi-coefficient eigenmode operation can improve cross-mode isolation due to electrode misalignments and imperfections, which is one of the causes of residual quadrature errors in conventional eigenmode operations. A 1400 µm annulus aluminum nitride (AlN) on a silicon bulk acoustic wave (BAW) resonator with gyroscopic in-plane bending modes at 2.98 MHz achieves a nearly 60 dB cross-mode isolation when operated as a gyroscope using a multi-coefficient eigenmode architecture. The as-born frequency mismatches in multiple devices are compensated by physical laser trimming. The demonstrated AlN piezoelectric BAW gyroscope shows a large open-loop bandwidth of 150 Hz and a high scale factor of 9.5 nA/°/s on a test board with a vacuum chamber. The measured angle random walk is 0.145°/√h, and the bias instability is 8.6°/h, showing significant improvement compared to the previous eigenmode AlN BAW gyroscope. The results from this paper prove that with multi-coefficient eigenmode operations, piezoelectric AlN BAW gyroscopes can achieve a noise performance comparable to that of their capacitive counterpart while having the unique advantage of a large open-loop bandwidth and not requiring large DC polarization voltages.

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of [formula omitted]-functions of matrices via quadrature

In this article, we propose an algorithm for approximating the action of φ−functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the φ−functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.

A High-Frequency and Low-Jitter DLL With Quadrature Error and Duty-Cycle Corrections Based on Asynchronous Sampling

This paper presents a quadrature error and duty-cycle correction circuit based on an asynchronous sampling technique. Sampling the multi-phased clocks provides their phase and duty-cycle information, which are then utilized to compensate for the quadrature and duty-cycle errors. The intrinsic down-conversion operation of the proposed sampling approach enhances the correction accuracy by mitigating circuit mismatch effects. The quadrature-error corrector (QEC) and duty-cycle corrector (DCC) circuits receive control signals generated from the asynchronous phase detectors and correct the clock outputs accordingly. Combined with a delay-locked loop (DLL), the proposed design is fabricated in a 40-nm CMOS process, consumes 7.2 mW, and occupies 0.023 mm. It achieves 0.837-ps RMS jitter and 1.68∘ maximum phase error from multiple chip samples.

Design and Experiment for N = 3 Wineglass Mode Metal Cylindrical Resonator Gyroscope Closed-Loop System

This paper studies a kind of gyro structure of N = 3 Wineglass Mode Metal Cylindrical Resonator Gyroscope (WMMCRG). Compared with traditional Cylindrical Vibrating Gyroscope (CVG), the designed structure has higher scale factor and lower frequency split. This paper provides a more specific processing method and the parameters of resonator materials. A closed-loop controlling system with low error and low noise is designed for WMMCRG. The system is composed of three independent closed-loop systems: drive closed-loop, sensing closed-loop, and quadrature error correction closed-loop. Through the test of the high-precision turntable, under the premise of the same material and processing technology, the bias instability, bias stability, zero bias, Angular Random Walk (ARW), and frequency split of WMMCRG is 1.974°/h, 10.869°/h, 10.3323°/s, 16 (°)/√h, 0.02 Hz, respectively.

A dual-mass fully decoupled MEMS gyroscope with optimized structural design for minimizing mechanical quadrature coupling

Mechanical quadrature coupling is one of the main sources of zero-rate output in microelectromechanical system (MEMS) gyroscope, which limits the further improvement of gyroscope performance. This paper reports a dual-mass MEMS tuning fork gyroscope based on fully decoupled structure. The theoretical analysis and finite element method (FEM) simulation are performed to identify the sources of the mechanical quadrature error in detail and to find ways to optimize the structural design of MEMS gyroscope. The results show that reducing the quadrature coupling stiffness of local sensing components is a key factor in minimizing the overall gyroscope quadrature error. By deliberately optimization of the geometry of the sensing components including sense frame and the drive springs closed to the sense frame, the quadrature induced deformation of sense electrode can be reduced from 160 nm to 5.1 nm with a driving displacement of 1 μm based on FEM simulation. The equivalent angular rate output caused by quadrature coupling of each sense electrode can be reduced from 3960 °/s to 60 °/s. The optimized gyroscope operates in split-mode with open-loop readout. The measured scale factor of gyroscope is 90.46 LSB/(°/s) with a non-linearity of 810 ppm in the full scale range of ±300 °/s. The measurement result shows a bias instability of 8.9 °/h and an angular random walk (ARW) of 0.74°/h at room temperature, respectively. The proposed approach of structural improvements can be used to reduce the quadrature error of the gyroscope in processing tolerances.

Common-Mode Driven Synchronous Filtering of the Powerline Interference in ECG

Powerline interference (PLI) is a major disturbing factor in ground-free biopotential acquisition systems. PLI produces both common-mode and differential input voltages. The first is suppressed by a high common-mode rejection ratio of bioamplifiers. However, the differential PLI component evoked by the imbalance of electrode impedances is amplified together with the diagnostic differential biosignal. Therefore, PLI filtering is always demanded and commonly managed by analog or digital band-rejection filters. In electrocardiography (ECG), PLI filters are not ideal, inducing QRS and ST distortions as a transient reaction to steep slopes, or PLI remains when its amplitude varies and PLI frequency deviates from the notch. This study aims to minimize the filter errors in wide deviation ranges of PLI amplitudes and frequencies, introducing a novel biopotential readout circuit with a software PLI demodulator–remodulator concept for synchronous processing of both differential-mode and common-mode signals. A closed-loop digital synchronous filtering (SF) algorithm is designed to subtract a PLI estimation from the differential-mode input in real time. The PLI estimation branch connected to the SF output includes four stages: (i) prefilter and QRS limiter; (ii) quadrature demodulator of the output PLI using a common-mode driven reference; (iii) two servo loops for low-pass filtering and the integration of in-phase and quadrature errors; (iv) quadrature remodulator for synthesis of the estimated PLI using the common-mode signal as a carrier frequency. A simulation study of artificially generated PLI sinusoids with frequency deviations (48–52 Hz, slew rate 0.01–0.1 Hz/s) and amplitude deviations (root mean square (r.m.s.) 50–1000 μV, slew rate 10–200 μV/s) is conducted for the optimization of SF servo loop settings with artificial signals from the CTS-ECG calibration database (10 s, 1 lead) as well as for the SF algorithm test with 40 low-noise recordings from the Physionet PTB Diagnostic ECG database (10 s, 12 leads) and CTS-ECG analytical database (10 s, 8 leads). The statistical study for the PLI frequencies (48–52 Hz, slew rate ≤ 0.1 Hz/s) and amplitudes (≤1000 μV r.m.s., slew rate ≤ 40 μV/s) show that maximal SF errors do not exceed 15 μV for any record and any lead, which satisfies the standard requirements for a peak ringing noise of &lt; 25 μV. The signal-to-noise ratio improvement reaches 57–60 dB. SF is shown to be robust against phase shifts between differential- and common-mode PLI. Although validated for ECG signals, the presented SF algorithm is generalizable to different biopotential acquisition settings via surface electrodes (electroencephalogram, electromyogram, electrooculogram, etc.) and can benefit many diagnostic and therapeutic medical devices.

A New First Order Expansion Formula with a Reduced Remainder

This paper is devoted to a new first order Taylor-like formula, where the corresponding remainder is strongly reduced in comparison with the usual one, which appears in the classical Taylor’s formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at n+1 equally spaced points between the two points, where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange P1- interpolation error estimate and the trapezoidal quadrature error, in order to assess the gain of the accuracy we obtain using this new Taylor-like formula.

Suppressing Quadrature Error and Harmonics in Resolver Signals via Disturbance-Compensated PLL

The aim of this study was to obtain accurate angular positions and velocities from resolver signals; resolver-to-digital conversion (RDC) often adopts a phase-locked loop (PLL) as a demodulation algorithm. However, resolver signals often come with quadrature errors and harmonics, which lead to a severe reduction in PLL accuracy. The conventional PLL does not consider the impact of the quadrature error, and the bandwidth of the PLL is much larger than the fundamental frequency of resolver signals for pursuing a low dynamic error. These reasons render the retention of resolver harmonics in the demodulation results. In this paper, a disturbance-compensated PLL (DC-PLL) is proposed, which consists of a phase detector for suppressing quadrature error and harmonics (SQEH-PD) and a second-order observer. Firstly, since the quadrature error does not change with the angle velocity, the pre-estimated quadrature error is used in the SQEH-PD to compensate for the quadrature error in resolver signals. Secondly, although the frequency of the harmonics changes with the velocity, the amplitudes of the harmonics do not change. Therefore, the pre-estimated amplitudes of harmonics and estimated angular position are used in the SQEH-PD to compensate for the harmonics in resolver signals. Thirdly, a second-order observer is designed to estimate the angular position and velocity by regulating the phase detector error. Compared with the conventional PLL, the proposed DC-PLL has a stronger anti-disturbance ability against the quadrature error and harmonics by configurating the phase detector error and the estimated position error, which have a linear relation. Simulation and experimental results prove the effectiveness of the proposed method.

Modeling and Analysis of a MEMS Vibrating Ring Gyroscope Subject to Imperfections

We present a new mathematical model for a vibrating ring gyroscope (VRG) in the presence of imperfections, namely, structural defects and material anisotropy. As a novelty, we calculate the mode shapes of the internal suspension structure to enable a more accurate and modular analysis of the VRG’s mass and stiffness distributions. Solving the associated eigenvalue problem shows that imperfections result in the frequency split between the gyroscope’s operating mode shapes, rotating their orientation with respect to the nominal drive and sense axes. We then use perturbation analysis to solve the VRG’s equations of motion and analyze the quadrature error that arises from frequency/damping mismatch between the mode shapes. We use our model to detail the various effects of the etching-related undercuts, structural uncertainties, and Young’s modulus anisotropy–in the form of suitable space-dependent functions–on the mode shapes and the quadrature error for the first time. The results reveal that rings are robust against imperfection, while the straight beams used in the suspension system are most likely responsible for the frequency split and quadrature error. For example, 50 nm (0.5%) width variation in a beam that connects the VRG’s suspension to an anchored internal structure leads to 4700°/s quadrature error. To validate our modeling, using the experimental data from a fabricated 59 kHz VRG, we provide rigorous, comparative simulations against the finite element method (FEM). [2022-0035]

Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball

We consider the numerical integrationINTd(f)=∫Bdf(x)wμ(x)dx for the weighted Sobolev classes BWp,μr and the weighted Besov classes BBτr(Lp,μ) in the randomized case setting, where wμ,μ≥0, is the classical Jacobi weight on the ball Bd, 1≤p≤∞, r>(d+2μ)/p, and 0<τ≤∞. For the above two classes, we obtain the orders of the optimal quadrature errors in the randomized case setting are n−r/d−1/2+(1/p−1/2)+. Compared to the orders n−r/d of the optimal quadrature errors in the deterministic case setting, randomness can effectively improve the order of convergence when p>1.

A Quadrature Error Corrector for Aperiodic, Quarter-rate Data Strobe Signals in HBM3 Interfaces

This paper proposes a quadrature error corrector (QEC) for the aperiodic, quarter-rate data strobe (DQS) signals in the high bandwidth memory generation 3 (HBM3) interfaces. Unlike the existing approaches, which are applicable to periodic clock signals only, the proposed QEC can adjust the phase spacings among four aperiodic, quarter-rate DQS signals to be equal to 1/4 of the operating clock period. Its pulse-width detector measures each phase spacing by charging a capacitor with a fixed current during the difference period and comparing its voltage against the voltage of a four-times larger capacitor charged for the whole clock period. The post-layout simulation results of a prototype QEC designed in a 40-nm CMOS process show that the QEC can correct the phase errors ranging -43.2~43.2° with errors less than 5.01° for both seamless and burst modes. The QEC consumes the maximum power of 2.42mW when operating at 1.6-GHz and 1.1-V supply, and occupies the active area of 0.01 mm².

Method for Estimating and Compensating the Phase Imbalance of Quadrature Signal Components

Currently, methods of direct modulation using complex signals are widely used. A complex signal consists of in-phase I (In-phase) and quadrature Q (Quadrature) components. When a signal passes through a communication channel and a receiving path, mismatches of the signal components occur as a result of interference. The mismatch, in turn, leads to increase in bit error rate (BER) during signal demodulation. The quality of the received signal is expressed in terms of bit error rate. The article considers phase imbalance of the quadrature components of a complex signal. Phase imbalance occurs in the receive path and depends on the quality of the receiver's local oscillators, on the operating temperature and the difference in propagation time of the I and Q components. The article shows an algorithm for estimating the phase imbalance of the quadrature signal components for digital modulation methods. Examples of signal constellation distortions in case of phase imbalance of quadrature signal components are considered. The phase imbalance estimation is based on the modulation constellation method for measuring signal parameters. Formulas for calculating the angle of phase error and the magnitude of quadrature error are given. Formulas for compensating the phase imbalance are also given, taking into account the calculated quadrature error. A mathematical model of the transmitter, communication channel and receiver has been developed to study the method for estimating and compensating for phase imbalance. The mathematical model is built in the Matlab software environment and is an m-script software model. With the help of the mathematical model, method for estimating and compensating for phase imbalance has been studied. In the course of the study, relationships of error probability due to the mismatch of the quadrature components of the signal were obtained. The noise immunity of the receiver paths with and without compensation for the phase mismatch of the quadrature signal components is compared. Based on the results of the study, diagrams of the error probability and the phase mismatch of the receiver's local oscillators were obtained. The study shows that phase mismatch of the receiver local oscillators at a fixed signal-to-noise ratio leads to an increase in the probability of received bit errors. But when applying the phase imbalance compensation method, the error probability remains fixed as the phase mismatch of receiver local oscillators increases at a fixed signal-to-noise ratio.

Treatment of near-incompressibility and volumetric locking in higher order material point methods

We propose a novel projection method to treat near-incompressibility and volumetric locking in small- and large-deformation elasticity and plasticity within the context of higher order material point methods. The material point method is well known to exhibit volumetric locking due to the presence of large numbers of material points per element that are used to decrease the quadrature error. Although there has been considerable research on the treatment of near-incompressibility in the traditional material point method, the issue has not been studied in depth for higher order material point methods. Using the B¯ and F¯ methods as our point of departure we develop an appropriate projection technique for material point methods that use higher order shape functions for the background discretization. The approach is based on the projection of the dilatational part of the appropriate strain rate measure onto a lower dimensional approximation space, according to the traditional B¯ and F¯ techniques, but tailored to the material point method. The presented numerical examples exhibit reduced stress oscillations and are free of volumetric locking and hourglassing phenomena.

The Effect of the Bending Beam Width Variations on the Discrepancy of the Resulting Quadrature Errors in MEMS Gyroscopes.

In this paper, we develop a new approach in order to understand the origin of the quadrature error in MEMS gyroscopes. As the width of the flexure springs is a critical parameter in the MEMS design, it is necessary to investigate the impact of the width variations on the stiffness coupling, which can generate a quadrature signal. To do so, we developed a method to determine the evolution of the stiffness matrix of the gyroscope springs with respect to the variation of the bending beams width of the springs through finite element analysis (FEA). Then, a statistical analysis permits the computation of the first two statistical moments of the quadrature error for a given beam width defect. It turns out that even small silicon etching defects can generate high quadrature level with up to a root mean square (RMS) value of for a bending beam width defect of 0.9%. Moreover, the quadrature error obtained through simulations has the same order of magnitude as the ones measured on the gyroscopes. This result constitutes a great help for designing MEMS gyroscopes, as the consideration of the bending beams width defects is needed in order to avoid high quadrature error.

The novel approach to the closed-form average bit error rate calculation for the Nakagami-m fading channel

The research presents a procedure of the closed-form average bit error rate evaluation for wireless communication systems in the presence of multipath fading. A generalization of the classical moment generating function is applied, and a connection between the Hankel-type contour-integral representation of the Marcum Q-function and Gauss Q-function is obtained and applied to the analytic derivation of the quadrature amplitude modulated signal average bit error rate in the presence of Nakagami-m fading. The correctness of the obtained solution was verified, and its computational efficiency (in terms of accuracy and time gain), compared with the prevailing approximation, was demonstrated. The proposed methodology can be extended to a wide variety of composite channels.

Online Compensation of Phase Delay Error Based on P-F Characteristic for MEMS Vibratory Gyroscopes.

In this paper, an online compensation method of phase delay error based on a Phase-Frequency (P-F) characteristic has been proposed for MEMS Coriolis Vibratory Gyroscopes (CVGs). At first, the influences of phase delay were investigated in the drive and sense mode. The frequency response was acquired in the digital control system by collecting the demodulation value of drive displacement, which verified the existence and influence of the phase delay. In addition, based on the P-F characteristic, that is, when the phase shift of the nonresonant drive force through the resonator is almost 0° or 180°, the phase delay of the gyroscope is measured online by injecting a nonresonant reference signal into the drive-mode dynamics. After that, the phase delay is self-corrected by adjusting the demodulation phase angle without affecting the normal operation of the gyroscopes. The approach was validated with an MEMS dual-mass vibratory gyroscope under double-loop force-to-rebalance (in-phase FTR and quadrature FTR) closed-loop detection mode and implemented with FPGA. The measurement results showed that this scheme can detect and compensate phase delay to effectively eliminate the effect of the quadrature error. This technique reduces the zero rate output (ZRO) from −0.71°/s to −0.21°/s and bias stability (BS) from 23.30°/h to 4.49°/h, respectively. The temperature sensitivity of bias output from −20 °C to 40 °C has reached 0.003 °/s/°C.