Arctangent Algorithm Research Articles

A PGC demodulation scheme based on the Lissajous ellipse fitting for homodyne interferometer without using second harmonic carrier signal

This paper proposes a phase generated carrier (PGC) demodulation scheme for sinusoidal phase modulation interferometer (SPMI) that does not require a second harmonic carrier signal. In this scheme, an in-phase interference signal component with direct current (DC) bias is obtained by low-pass filtering the original interference signal, and a quadrature interference signal component is obtained by conventional means. The Lissajous figure formed by these two quadrature interference signal components is an ellipse whose center deviates from the origin of the I-Q coordinates. By fitting this ellipse, the major and minor axes and the center coordinates can be obtained. These elliptic parameters are then used to eliminate DC bias and normalize the amplitude of the quadrature signal components. Finally, the phase demodulation of the interference signal can be realized by using arctangent algorithm or differential cross multiplication (DCM) algorithm. The results of simulation and experiment have verified the validity and accuracy of the proposed PGC demodulation scheme, and show that it is insensitive to the phase modulation depth C. Compared with conventional PGC demodulation, this scheme requires lower depth of carrier phase modulation. The vibration measurement results indicate that the average signal-to-noise and distortion Ratio (SINAD) and total harmonic distortion (THD) values of demodulation results in the frequency range of 130 Hz to 250 Hz are 34 dB and -33 dB respectively. The proposed PGC demodulation scheme shows good performance in displacement measurement using SPMI.

Research and Hardware Implementation of a Reduced-Latency Quadruple-Precision Floating-Point Arctangent Algorithm

In the field of digital signal processing, such as in navigation and radar, a significant number of high-precision arctangent function calculations are required. Lookup tables, polynomial approximation, and single/double-precision floating-point Coordinate Rotation Digital Computer (CORDIC) algorithms are insufficient to meet the demands of practical applications, where both high precision and low latency are essential. In this paper, based on the concept of trading area for speed, a four-step parallel branch iteration CORDIC algorithm is proposed. Using this improved algorithm, a 128-bit quad-precision floating-point arctangent function is designed, and the hardware circuit implementation of the arctangent algorithm is realized. The results demonstrate that the improved algorithm can achieve 128-bit floating-point arctangent calculations in just 32 cycles, with a maximum error not exceeding 2×10−34 rad. It possesses exceptionally high computational accuracy and efficiency. Furthermore, the hardware area of the arithmetic unit is approximately 0.6317 mm2, and the power consumption is about 40.6483 mW under the TSMC 65 nm process at a working frequency of 500 MHz. This design can be well suited for dedicated CORDIC processor chip applications. The research presented in this paper holds significant value for high-precision and rapid arctangent function calculations in radar, navigation, meteorology, and other fields.

Ameliorated PGC demodulation technique based on the ODR algorithm with insensitivity to phase modulation depth.

For the optical fiber sensing system using phase generated carrier (PGC) technology, it is very important to eliminate the nonlinear effect of phase modulation depth (C) fluctuation on the demodulation results in the actual environment. In this paper, an ameliorated phase generated carrier demodulation technique is presented to calculate the C value and suppress its nonlinear influence on the demodulation results. The value of C is calculated out by the fundamental and third harmonic components with the equation fitted by the orthogonal distance regression algorithm. Then the Bessel recursive formula is used to convert the coefficients of each order of Bessel function contained in demodulation result into C values. Finally, the coefficients in demodulation result are removed by the calculated C values. In the experiment, when the C ranges from 1.0 rad to 3.5 rad, the minimum total harmonic distortion and maximum phase amplitude fluctuation of the ameliorated algorithm are 0.09% and 3.58%, which are far superior to the demodulation results of the traditional arctangent algorithm. The experimental results demonstrate that the proposed method can effectively eliminate the error caused by the fluctuation of the C value, which provides a reference for signal processing in practical applications of fiber-optic interferometric sensors.

Eliminating the influence of modulation depth by a laser interference double detection system.

We propose a method of combining a double optical path detection system with a phase generation carrier arctangent algorithm. The feasibility of eliminating the influence of modulation depth by the double optical path detection vibration measurement system is verified by simulation and experiment. Results show that the average total harmonic distortion of the single frequency signal is 1%, and the signal-to-noise ratio of the speech signal is improved by an average of 5.6dB. Therefore, the double optical path detection system can eliminate the influence of modulation depth, avoid nonlinear distortion, and improve the detection performance of the system.

Reliability Demodulation Algorithm Design for Phase Generated Carrier Signal

Phase-generated carrier (PGC) demodulation technology has been widely used in fiber-optic interferometric sensors system in recent years. Nonlinear errors caused by interference noise and parasitic amplitude modulation (AM) are key factors that affect the reliability of traditional PGC demodulation system based on arctangent (PGC-Arctan) algorithm. In order to enhance the reliable performance of PGC demodulation system under the conditions of low signal-to-noise ratio (SNR) and high parasitic AM interference, an ellipse fitting algorithm based on the Gauss–Newton iteration (GNI) is proposed, which is called the PGC-Arctan-GNI demodulation algorithm. The proposed algorithm uses the Euclidean distance to accurately estimate the geometric parameters of ellipse, which can correct the nonlinear distortion of the demodulated signal. In addition, the PGC demodulation system designed based on GNI algorithm has the advantages of strong antinoise ability, good antiparasitic AM ability, and widely dynamic range of the input signal amplitude. Experimental results show that, compared with the direct demodulation algorithm and the ellipse fitting algorithm based on the least squares (LS) method, the SINAD value of the proposed algorithm is always greater than 30 dB and higher than the other two algorithms under the condition of low SNR. Under the condition that the parasitic AM index <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> changes from 0 to 0.5, the signal-to-noise and distortion ratio (SINAD) curve of the proposed algorithm is more stable than the curves of the other two algorithms, and the fluctuation range of the SINAD value does not exceed 3 dB. This algorithm also has a wider input signal amplitude range than the LS algorithm. Especially, when the amplitude of the measured signal is less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pi$</tex-math></inline-formula> /2, the relative amplitude error of the GNI algorithm is less than 2% and significantly smaller than the LS algorithm. Finally, the demodulation results using real measured data in the actual system show that the frequency and amplitude of the demodulated signal are same as the original signal. At the same time, the SNIAD value of this system reaches 67.40 dB, the spurious-free dynamic range value reaches 68.82 dB, and the total harmonic distortion value reaches -68.40 dB@1 KHz. In summary, the proposed PGC-Arctan-GNI algorithm can eliminate system nonlinear errors caused by the parasitic AM, harmonics, and noise interference by estimating the ellipse parameters, thereby effectively enhancing the stability and reliability of the PGC demodulation system.

Dual-wavelength demodulation technique for interrogating a shortest cavity in multi-cavity fiber-optic Fabry–Pérot sensors

This paper demonstrates, for the first time, a novel demodulation technique that can be applied for interrogating a shortest cavity in multi-cavity Fabry-Pérot (F-P) sensors. In this demodulation technique, using an amplified spontaneous emission (ASE) light source and two optical fiber broadband filters, the interference only occurs in a shortest F-P cavity that is shorter than the half of the coherence length. Using a signal calibration algorithm, two low-coherence interference optical signals with similar coherence lengths were calibrated to obtain two quadrature signals. Then, the change in the cavity length of the shortest F-P cavity was interrogated by the two quadrature signals and the arctangent algorithm. The experimental results show that the demodulation technique successfully extracted 1 kHz and 500 Hz vibration signals with 39.28 µm and 64.84 µm initial cavity lengths, respectively, in a multi-cavity F-P interferometer. The demodulation speed is up to 500 kHz, and the demodulation technique makes it possible for multi-cavity F-P sensors to measure dynamic and static parameters simultaneously. The results show that the demodulation technique has wide application potential in the dynamic measurement of multi-cavity F-P sensors.

A Fabry–Perot Interferometer-Based Fiber Optic Dynamic Displacement Sensor With an Analog in-Phase/Quadrature Generator

In this paper, the design details of a low-cost and high resolution Fiber Optic Dynamic Displacement sensor are discussed. The sensor is designed based on the principle of extrinsic Fiber Optic Fabry-Perot Interferometer (FPI). Higher sensitivity is achieved by using a double pass sensor head configuration. To resolve direction ambiguity, quadrature signals are generated by employing Phase Generated Carrier (PGC) demodulation technique. An Analog In-phase/Quadrature phase (I/Q) generator is designed to perform light intensity compensation, carrier mixing, and filtering operations to obtain the quadrature signals. Arctangent algorithm and phase unwrapping algorithms are used to obtain dynamic displacement from the quadrature signals. The sensor is demonstrated to measure non-contact dynamic displacement at a working distance of 5 cm with an amplitude resolution of 12 nm, a precision of 1.4 nm, and a standard deviation of 2%. The sensor is also able to measure multi-frequency dynamic displacements generated by piezo actuators. Potential applications of the sensor include condition monitoring and predictive maintenance in industries.

Phase Demodulator for the Measurement of Extrinsic Fabry-Perot Interferometric Sensors With Arbitrary Initial Cavity Length

A phase demodulator for fiber optic extrinsic Fabry-Perot interferometric (EFPI) sensors is proposed for dynamic signal (vibration, acoustic, etc.) measurements. Three interferometric signals with phase differences are introduced by three lasers with different wavelengths. The tangent of the desired dynamic signal is extracted from the three interferometric signals. Then, the phase modulation is recovered by the arctangent algorithm and phase unwrapping. For the technique, constant phase differences are not required. Theoretically, the phase demodulator can interrogate EFPIs with arbitrary initial cavity length. The phase demodulator can perform regardless of whether the EFPI is modulated with a phase shift $>{2}\pi $ or not. Dynamic signals are recovered successfully from EFPIs with cavity lengths ranging from $23.065~\mu \text{m}$ to $1094.703~\mu \text{m}$ by the same demodulator. As a passive demodulation method, the proposed demodulator is adapted to the detection of high frequency signals. A sinewave with a frequency of 20 kHz is extracted successfully. The algorithm of the technique is simple, robust, and provides great liberty with respect to choosing different sensors in different applications.

A Novel Demodulation Scheme of Fiber-Optic Interferometric Sensor Based on FM SSB Signal

A novel vibration signal demodulation scheme of fiber-optic interferometric sensor is proposed and demonstrated. A frequency-modulated (FM) single sideband (SSB) light-wave signal is generated by applying an FM microwave signal to a Mach–Zehnder modulator (MZM) for the demodulation of fiber-optic vibration interferometric sensor. A method of 12-sample orthogonal demodulation algorithm (ODA) based on setting the system sampling rate to 12 times of the FM frequency is adopted for the generation of two orthogonal signals for arctangent transformation. The experimental results show that the harmonic distortion is well suppressed compared with the traditional phase generation carrier (PGC) arctangent (PGC-Arctan) algorithm, and a minimum detectable phase shift is achieved about $6.3\times 10^{-5}$ rad/ $\surd $ Hz at 5 kHz.

An improved arctangent algorithm based on phase-locked loop for heterodyne detection system**Project supported by Key Research Program of Frontier Science, Chinese Academy of Sciences (Grant No. QYZDB-SSW-SLH014) and the Yong Scientists Fund of the National Natural Science Foundation of China (Grant No. 61205143).

We present an ameliorated arctangent algorithm based on phase-locked loop for digital Doppler signal processing, utilized within the heterodyne detection system. We define the error gain factor given by the approximation of Taylor expansion by means of a comparison of the measured values and true values. Exact expressions are derived for the amplitude error of two in-phase & quadrature signals and the frequency error of the acousto-optic modulator. Numerical simulation results and experimental results make it clear that the dynamic instability of the intermediate frequency signals leads to cumulative errors, which will spiral upward. An improved arctangent algorithm for the heterodyne detection is proposed to eliminate the cumulative errors and harmonic components. Depending on the narrow-band filter, our experiments were performed to realize the detectable displacement of 20 nm at a detection distance of 20 m. The aim of this paper is the demonstration of the optimized arctangent algorithm as a powerful approach to the demodulation algorithm, which will advance the signal-to-noise ratio and measurement accuracy of the heterodyne detection system.

High stability and low harmonic distortion PGC demodulation technique for interferometric optical fiber sensors

Phase generation carrier (PGC) demodulation technique is widely adopted in the field of interferometric optical fiber sensors, for its high sensitivity, good linearity and large dynamic range. In this paper, a novel single-path differential divide PGC demodulation algorithm (PGC-SDD) is designed and analyzed. Experimental results show that not only the distortion induced by light intensity but also the harmonic distortion induced by the modulation depth are well suppressed by the PGC-SDD algorithm. The signal-to-noise and distortion ratio of sensors using the proposed algorithm achieved a gain of 28.06 dB and 13.13 dB over that using differential-cross-multiplying (PGC-DCM) algorithm and PGC Arctangent (PGC-Arctan) algorithm, respectively. In addition, the PGC-SDD demodulation algorithm can also demodulate non-periodic signals.

Phase Compensation Scheme for Fiber-Optic Interferometric Vibration Demodulation

In order to expand the dynamic range of fiber-optic interferometric demodulation, a novel phase compensation scheme based on the phase generated carrier arctangent algorithm (PGC-arctan) is proposed, which overcomes the limitation of the amplitude range from $- \pi /2$ to $\pi /2$ for arbitrary signals; and a Michelson interferometer measurement system is established to verify the rationality of the compensation algorithm. The phase compensation algorithm is based on analysis of the instantaneous function of the arctangent signal during the phase hopping process, while overcoming the problem of phase winding. Experimental results confirmed the feasibility of the phase compensation scheme as predicted by theoretical and simulation analysis. In particular, the new scheme was shown to be effective in responding to arbitrary large-amplitude and time-dependent vibrational input, with the additional benefit of a strong ability to suppress system and ambient noise.

激光语音检测系统反正切补偿算法

激光语音检测系统反正切补偿算法

High Stability Fiber Optics Sensors with an improved PGC demodulation algorithm

In this paper, an optical fiber interferometric sensor with improved phase generated carrier (PGC) demodulation algorithm is proposed to eliminate the influence of light intensity disturbance. Performance of sensor using proposed algorithm is analyzed and compared with sensor using differential-cross-multiplying (PGC-DCM) algorithm and PGC Arctangent (PGC-Arctan) algorithm, respectively. Experimental results show that distortion, but also harmonic distortion is well suppressed by the improved PGC algorithm. In experiment, the signal-to-noise and distortion ratio of sensor using the proposed algorithm achieve a gain of 16.4 and 10.8 dB over sensor using PGC-DCM and PGC-Arctan algorithm, respectively.

High Dynamic-Range Motion Imaging Based on Linearized Doppler Radar Sensor

Miniaturized Doppler radar sensor (DRS) for noncontact motion detection is a hot topic in the microwave community. Previously, small-scale physiological signals such as human respiration and heartbeat rates are the primary interest of study. In this paper, we propose a comprehensive approach that can be used to improve the demodulation linearity of microwave DRSs, such that detailed time-domain motion information ranging from micro-scale to large scale can be accurately reconstructed. Experiments show that based on a digital-IF receiver architecture, dynamic dc offset tracking, and the extended differentiate and cross-multiply arctangent algorithm, the displacement and velocity of both micrometer-scale vibration of a tuning fork and meter-scale human walking can be accurately recovered. Our work confirms that substantial time-domain motion information is carried by the signals backscattered from moving objects. Retrieval of such information using DRSs can be potentially used in a wide range of healthcare and biomedical applications, such as motion pattern recognition and bio-signal measurements.

Fiber optic extrinsic Fabry–Perot accelerometer using laser emission frequency modulated phase generated carrier demodulation scheme

Dai-Hua WangPing-Gang JiaChongqing UniversityKey Laboratory of Optoelectronic Technology andSystems of the Ministry of Education of ChinaRoom 1407, Main Teaching BuildingChongqing 400044, ChinaandPrecision and Intelligence LaboratoryDepartment of Optoelectronic EngineeringRoom 1407, Main Teaching BuildingChongqing 400044, ChinaE-mail: dhwang@cqu.edu.cnAbstract. The principle of a fiber optic Fabry–Perot (F–P) accelerometer(FOFPA) system using the laser emission frequency modulated phasegenerated carrier (FMPGC) demodulation scheme is first described andexperimentally demonstrated. The F–P cavity, which is constituted byplacing the end face of a gradient-index lens in parallel with the reflectoron the inertial mass, directly translates the inertial mass’s displacementgenerated by the measured acceleration into phase shifts of the interfer-enceoutputfromtheF–Pcavity.AnFMPGCdemodulationschemebasedon the arctangent (Arctan) algorithm is adapted to demodulate the phaseshifts. The sensing model for the FOFPA system using the FMPGC–Arctan demodulation scheme is established and the sensing characteris-tics are theoretically analyzed. On these bases, the FOFPA is designedand fabricated and a prototyping system is built and tested. The resultsindicate that: (1) the nonlinearity of the FOFPA system using theFMPGC–Arctan demodulation scheme is less than 0.58%, (2) the reso-nant frequency, on-axial sensitivity, and resolution are 393 Hz,13.11 rad∕g, and 450 μg∕Hz

Comparison of Hydrophone Calibration by Reciprocity and Heterodyne Interferometer in the Frequency Range 500 kHz to 15 MHz

This paper presents a comparison of high-frequency hydrophone calibration by a heterodyne interferometer and two-transducer reciprocity method respectively. For the calibration using the heterodyne interferometer, a vertical arrangement is applied to avoid the acousto-optical interaction which is a major uncertainty source in previous studies. For the demodulation of the acoustic displacement, a PC-based digital demodulation using arctangent algorithm is applied which has overcome the drawback of analog demodulation and achieved a higher demodulation accuracy. The demodulation is independent of the intensity of the carrier signal and thus can avoid the correction for the frequency response of the photodiode which is another uncertainty source. Comparison results between these two methods showed a good agreement. Owing to the optical measurement is independent of the acoustic field generating by the transducer, it has wider usage compared to the reciprocity method and can be easily extended to higher frequency and higher acoustic power application.

Self-calibrated non-contact fibre-optic Fabry–Perot interferometric vibration displacement sensor system using laser emission frequency modulated phase generated carrier demodulation scheme

In this paper, the principle of a self-calibrated non-contact fibre-optic Fabry–Perot interferometric vibration displacement sensor (FOFPIVDS) system is described and experimentally demonstrated. According to the principle, the Fabry–Perot cavity, constituted by placing the end face of a gradient-index lens in parallel with the measured vibration surface, is used to translate the vibration displacement of the measured vibration surface into phase shifts in the interference signal output from the FOFPIVDS, and the laser emission frequency modulated phase generated carrier (FMPGC) demodulation scheme based on the arctangent (Arctan) algorithm is adapted to demodulate the phase shifts. After eliminating the optical power modulation in the laser emission frequency modulation, the sensing model for the FOFPIVDS system using the FMPGC–Arctan demodulation scheme is established. On these bases, the FOFPIVDS is designed and fabricated, and the prototyping FOFPIVDS system is built and experimentally tested. The research results indicate that the fabricated FOFPIVDS system possesses the characteristics as follows: (1) the sensitivity is only determined by the laser wavelength and the vibration displacement measurement is self-calibrated, (2) the nonlinearity is 0.29% and (3) the resolution is less than 2.06 nm.

Validation of analytical algorithms for the estimation of soil thermal properties using de Vries model

Soil thermal conductivity, and diffusivity together with the damping depth of soil temperature computed using Amplitude decay, Phase shift, Harmonic (amplitude based and phase based), Arctangent, Logarithmic and conduction-convection algorithms were compared with those obtained from de Vries model. The amplitude decay algorithm yielded the most reliable values of the soil thermal properties of all the estimation methods with mean absolute error (MAE), root mean squared error (RMSE) and relative maximum error (RME) of 0.04, 0.05 and 5.63% respectively for soil thermal conductivity. Harmonic algorithm (using the amplitude of the first 4 harmonics) gave values of the soil thermal properties next to the amplitude decay algorithm with MAE, RMSE and RME values 0.41, 0.44 and 47.84% respectively for soil thermal conductivity. Higher error values were associated with the other algorithms. The Arctangent algorithm gave the most deviated values of soil thermal properties with RME of 156.83% for soil thermal conductivity. For soil moisture content between 0.168 and 0.189 (> critical soil moisture content) the values of the soil thermal properties of the loamy sand decreased with increasing soil moisture, while they increased with increasing soil aeration.

用于探测极低频信号的光纤传感器相位生成载波解调方法

A high resolution ultra low frequency phase generated carrier(PGC) demodulation technique for interferometric fiber sensors is proposed. The differential cross multiplication(DCM) algorithm and the Arctangent algorithm of PGC technique are analyzed theoretically and simulated by computer for ultra low frequency demodulation. And the results show that an unnecessary random direct current(DC) drift is induced in the recovered signal of DCM-PGC, but it doesn't exit in that of Arctangent-PGC. To demonstrate the conclusion, an optical fiber interferometer-based ultra low frequency sensing system is set up, and the comparable experiments of DCM-PGC and Arctangent-PGC are carried out. The experimental results show that only Arctangent-PGC can be used for ultra low frequency demodulation, for it avoids the problem of DC drift as DCM-PGC. At last the ultra low frequency demodulation system is achieved by Arctangent-PGC, and the testing results show that the system has the lowest frequency of0.01 Hz, a minimum detectable phase shift of4×10-4 rad/√Hz, a dynamic range of110 dB@1 Hz and a linearity of99.99%.