Goertzel Algorithm Research Articles

Experimental study and techno-economic evaluation of an active fault detection kit in the prospect of future zero energy building installations

This article presents an active fault detection kit, applicable to low voltage single-phase electrical installations, in the prospect of the Zero Energy Building concept, integrating on-site renewable energy generation and energy storage. This simple, flexible, self-powered and compact kit is capable of detecting faults, such as power theft (meter tampering), unintentional islanding and neutral conductor loss. Its operation is based on harmonic voltage injection, in series with the electrical installation, through a low-power H-bridge inverter and a current transformer, along with the corresponding harmonic current measurement, to estimate the impedance and effectively detect faulty conditions; the fast and robust Goertzel algorithm is utilized. Moreover, it features IoT communication capabilities, employing the ESP32 microcontroller, to exchange data and information with the installation meters. The functionality and effective fault detection of the proposed device are validated, through experimental tests on a custom-developed hardware prototype; IoT connectivity and data uploading are experimentally tested and verified, too. Finally, a sustainability assessment study is performed, using the Life Cycle Cost Analysis tool.

Advances in the Goertzel Filter Bank Channelizer for Cryogenic Sensors Readout

Neutrino mass estimation experiments and cosmic microwave background (CMB) radiation surveys both employ low-temperature detectors (LTD) known as calorimeters and bolometers, respectively. These detectors operate typically between 10 and 300 mK. LTDs multiplexed by means of a microwave superconducting quantum interference device multiplexer (µMUX) demonstrated to be an excellent device for the readout of several detectors in the microwave region. This entails generating a multi-tonal signal and its subsequent readout. A single-tone detection method based on a Goertzel filter bank (GFB) channelizer was used for the readout of the aforementioned signal, implemented in a software-defined radio readout architecture within a field-programmable gate array. The measurements presented here demonstrate remarkable results in validating the suitability of the GFB channelizer for this system.

Goertzel based phasor and frequency measurement for low-cost PMUs

The need for low cost phasor measurement unit (PMU) devices in power system demands less complex measurement algorithms that can be implemented on inexpensive processing platforms. Unlike the widely used DFT, the Goertzel algorithm, having low complexity is not explored much for synchrophasor estimation. An efficient integration of zero crossing detection (ZCD) with Goertzel technique is presented in this work for the design of low cost synchrophasor devices. An extensive analysis has been done in the design of pre-filtering stage, selection of suitable window length, sampling rate, and other parameters to exploit the benefits of proposed algorithm. The metrological characterization of the proposed algorithm is delineated by showing the different test results following the IEEE synchrophasor standards. The proposed method exhibits the lowest computational time in comparison to the state-of-the-art methods with similar low computational complexity. Simulations and experimental results are obtained by using Raspberry Pi 3 model B+ board.

The Goertzel Algorithm for the Extraction of Texture Features

The Goertzel Algorithm for the Extraction of Texture Features

Audio Steganalysis Estimation with the Goertzel Algorithm

Audio steganalysis has been little explored due to its complexity and randomness, which complicate the analysis. Audio files generate marks in the frequency domain; these marks are known as fingerprints and make the files unique. This allows us to differentiate between audio vectors. In this work, the use of the Goertzel algorithm as a steganalyzer in the frequency domain is combined with the proposed sliding window adaptation to allow the analyzed audio vectors to be compared, enabling the differences between the vectors to be identified. We then apply linear prediction to the vectors to detect any modifications in the acoustic signatures. The implemented Goertzel algorithm is computationally less complex than other proposed stegoanalyzers based on convolutional neural networks or other types of classifiers of lower complexity, such as support vector machines (SVD). These methods previously required an extensive audio database to train the network, and thus detect possible stegoaudio through the matches they find. Unlike the proposed Goertzel algorithm, which works individually with the audio vector in question, it locates the difference in tone and generates an alert for the possible stegoaudio. In this work, we apply the classic Goertzel algorithm to detect frequencies that have possibly been modified by insertions or alterations of the audio vectors. The final vectors are plotted to visualize the alteration zones. The obtained results are evaluated qualitatively and quantitatively. To perform a double check of the fingerprint of the audio vectors, we obtain a linear prediction error to establish the percentage of statistical dependence between the processed audio signals. To validate the proposed method, we evaluate the audio quality metrics (AQMs) of the obtained result. Finally, we implement the stegoanalyzer oriented to AQMs to corroborate the obtained results. From the results obtained for the performance of the proposed stegoanalyzer, we demonstrate that we have a success rate of 100%.

A Novel Algorithm Application in Vocal Signals Processing.

This study proposes an innovative application of the Goertzel Algorithm (GA) for the processing of vocal signals in dysphonia evaluation. Compared to the Fast Fourier Transform (FFT) representing the gold standard analysis technique in this context, GA demonstrates higher efficiency in terms of processing time and memory usage, also showing an improved discrimination between healthy and pathological conditions. This suggests that GA-based approaches could enhance the reliability and efficiency of vocal signal analysis, thus supporting physicians in dysphonia research and clinical monitoring.

Phase sensitive detection for embedded sensors

The measurement of some physical properties requires detection of both amplitude and phase of an applied test signal. In portable sensors, this needs to be performed locally with integrated analog circuits or digital processing. Wireless sensor network allows the measurements over wide areas but raises a challenge of using efficient algorithm for phase sensitive detection (PSD) in nodes with constrained processing capacity. It is not straightforward to compare different PSD methods because their performance is closely related to the fine tuning of their parameters such as filter order and cut-off frequency. We propose a methodology to compare several PSD methods subjected to the same response time to assess its ability to provide accurate estimation in the presence of noise. We also determine the computational complexity of the investigated methods in terms of the number of operations required. We investigate both continuous and windowed operation. The methods were tested through simulations and verified using a dedicated embedded system. Our findings demonstrate that Goertzel algorithm rendered the best results, with smaller estimation error and less computation complexity.

Goertzel Algorithm을 통한 주파수 분석 기반의 아크 검출에 대한 연구

Goertzel Algorithm을 통한 주파수 분석 기반의 아크 검출에 대한 연구

An improved active islanding detection method for grid-connected solar inverters with a wide range of load conditions and reactive power

Islanding occurs when a utility-interactive distributed generator (DG) is disconnected from the utility grid and continues to supply power to local loads. This condition has several consequences including damage to the equipment and personnel safety risks. Therefore, it is required by utility interconnection standards for the DG to stop energizing local loads in the case of islanding. In this paper, an active islanding detection method (IDM) based on injecting a disturbance into the phase-locked loop (PLL) of a grid-connected photovoltaic (PV) inverter and monitoring the harmonic components of the point of common coupling (PCC) is proposed. A modified Goertzel algorithm is utilized to detect the injected harmonic which is effective for a wide range of load conditions in terms of load quality factor, resonant frequency, and reactive power. The simulation results in MATLAB/Simulink show that the proposed technique is fast with minimum side effects on power quality features such as output current total harmonic distortion (THD). This method is implemented on a single phase 5 kW grid-tied PV inverter and is tested experimentally. The experimental results verify that the proposed technique is fast with zero non-detection zone (NDZ) which is effective for a variety of load conditions.

Stator ITSC Fault Diagnosis for EMU Induction Traction Motor Based on Goertzel Algorithm and Random Forest

The stator winding insulation system is the most critical and weakest part of the EMU’s (electric multiple unit’s) traction motor. The effective diagnosis for stator ITSC (inter-turn short-circuit) faults can prevent a fault from expanding into phase-to-phase or ground short-circuits. The TCU (traction control unit) controls the traction inverter to output SPWM (sine pulse width modulation) excitation voltage when the traction motor is at a standstill. Three ITSC fault diagnostic conditions are based on different IGBTs’ control logics. The Goertzel algorithm is used to calculate the fundamental current amplitude difference Δi and phase angle difference Δθ of equivalent parallel windings under the three diagnostic conditions. The six parameters under the three diagnostic conditions are used as features to establish an ITSC fault diagnostic model based on the random forest. The proposed method was validated using a simulation experimental platform for the ITSC fault diagnosis of EMU traction motors. The experimental results indicate that the current amplitude features Δi and phase angle features Δθ change obviously with an increase in the ITSC fault extent if the ITSC fault occurs at the equivalent parallel windings. The accuracy of the ITSC fault diagnosis model based on the random forest for ITSC fault detection and location, both in train and test samples, is 100%.

A model predictive Goertzel algorithm based active islanding detection for grid integrated photovoltaic systems

A microgrid is a cluster of several energy sources and interconnected loads which are synchronized to operate either in grid connected or islanded mode. To attain higher efficiency and improved power control the microgrid is integrated with power electronic devices. But various problems arise related to protection among which the major issue to be focused on is islanding detection. Whenever an islanding condition arises the balance between load and generation in the islanded circuit can be violated, thereby leading to abnormal frequencies and voltages. This hazardous state observed in the system can be prevented by early detection of occurrence. In this paper, an active islanding detection with integrated PV system for grid based on Predictive Goertzel algorithm is proposed. The proposed single-phase single stage photovoltaic system injects a small harmonic component of the output of the grid and checks point of common coupling (PCC). The Predictive Goertzel algorithm is a discrete Fourier transform that resolve Non detection Zone (NDZ) and detection time. The highlight of this work is to demonstrate the performance of proposed islanding detection method, a single-stage single-phase grid integrated PV system embraces of 1-Φ full bridge inverter, LCL filter and local load (RLC parallel load) is considered. This algorithm is implemented in MATLAB Simulink environment. From the results obtained, the detection time for real time environment has reduced by 200ms compared to existing algorithm.

A Novel Frequency Estimator for Protection Applications in Active Distribution Networks

Frequency is a useful parameter for ensuring the proper functioning of the power system operation, control, and protection. In this article, a fast and accurate frequency estimator (FE) is developed using a zero crossing and a peak, occurring in the buffered data, unlike the commonly used two zero crossings. Prior to this, the signal is preprocessed using the sliding Goertzel filter (SGF) that mitigates the impact of undesirable features such as harmonics, noise, etc., from the signal. The two FEs, with and without SGF, then work in tandem to form the hybrid estimator and are named as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Z_{C}P_{k}$</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">${{HFE}}$</tex-math></inline-formula> . The estimated frequency is chosen based on the minimum error computed from the two parallel estimators. The performance of the estimator is, then, validated using the simulated signals polluted with harmonics, noise, transients, etc. The article also discusses the protection-oriented applications of the proposed FE in active distribution network (ADN). Moreover, the evaluation results with the real-time field data and the IEEE-13 ADN incorporated with photovoltaic using dSPACE1104 and real-time digital simulator show its capability to estimate the real-time frequency with satisfactory output.

A benchtop induction-based AC magnetometer for a fast characterization of magnetic nanoparticles

In this study, we report a development of a benchtop induction-based AC magnetometer to realize a simple, wideband, and sensitive AC magnetometer for bio-sensing applications and characterization of magnetic nanoparticles (MNPs). We investigate the inductance and parasitic capacitance of six different pickup coil geometries and compare their sensitivity and usable frequency range. In the pickup coil design, the number of turns and coil section separation are varied from 200 to 400 turns, and 1 to 4 sections, respectively. We find that the usable frequency range is greatly affected by the pickup coil’s inductance due to the self-resonance phenomena compared to their parasitic capacitance. A low noise instrument amplifier circuit (AD8429, Analog Devices, USA) was integrated and fabricated on a printed circuit board to amplify the weak signal from the pickup coil. We also implement a generalized Goertzel algorithm to achieve fast signal amplitude and phase extractions at a frequency. The developed magnetometer shows a sensitivity of 10−8 Am2/√Hz at 6 Hz and a frequency range of up to 158 kHz. Using the developed AC magnetometer, we demonstrate the viscosity effect on the frequency response of thermally blocked, single-core nanoparticles (SHP30, Ocean Nanotech, USA) in glycerol solutions. The excitation frequency is swept from 5 Hz to 158 kHz at a field amplitude of 0.55 mTpp within the acquisition time of 5 min (51 points). As a result, the viscosity change is confirmed by the peak shifting in the imaginary magnetization curve towards lower frequency values when the wt/V% of the glycerol solution is increased. The hydrodynamic size and the average anisotropy energy ratio σ are estimated to be 60.6 nm and 25, respectively, from the complex AC magnetization. It can be expected that the developed AC magnetometer can be a valuable tool in providing a fast and reliable assessment of MNPs for bio-sensing applications.

Accurate Goertzel Algorithm: Error Analysis, Validations and Applications

The Horner and Goertzel algorithms are frequently used in polynomial evaluation. Each of them can be less expensive than the other in special cases. In this paper, we present a new compensated algorithm to improve the accuracy of the Goertzel algorithm by using error-free transformations. We derive the forward round-off error bound for our algorithm, which implies that our algorithm yields a full precision accuracy for polynomials that are not too ill-conditioned. A dynamic error estimate in our algorithm is also presented by running round-off error analysis. Moreover, we show the cases in which our algorithms are less expensive than the compensated Horner algorithm for evaluating polynomials. Numerical experiments indicate that our algorithms run faster than the compensated Horner algorithm in those cases while producing the same accurate results, and our algorithm is absolutely stable when the condition number is smaller than 1016. An application is given to illustrate that our algorithm is more accurate than MATLAB’s fft function. The results show that the relative error of our algorithm is from 1015 to 1017, and that of the fft was from 1012 to 1015.

A Novel Phase Difference Measurement Method for Coriolis Mass Flowmeter Based on Correlation Theory

Aiming at the poor precision problem in phase difference measurements with unknown frequencies in engineering practice, a new phase difference measurement method is proposed for Coriolis mass flowmeter based on correlation theories. Firstly, the signal frequency was estimated by using an adaptive notch filter, which was applied to filter the waves and determined the integer period of the sampling signals, and the non-integer period sampling signals needed to be extended. Then, the Hilbert transformation was conducted relative to the extended signals, and the correlation functions of these extended signals with the transformed signals can be computed. Finally, the formula of phase difference can be obtained by utilizing the sinusoidal function. Compared to traditional methods, such as the correlation method, the Hilbert transformation method, and sliding Goertzel algorithm, the proposed method is suitable for both integer period and non-integer period sampling signals, and its accuracy, real-time, and dynamic performance is superior. Simulation and experiment results validate the superiority and effectiveness of the proposed method.

Improved multi-channel interferometric fiber-optic sensor demodulation based on the Goertzel algorithm.

In a multi-channel interferometric fiber-optic sensor system using space-division multiplexing (SDM) and phase-generated-carrier (PGC) demodulation, the phase delay and phase modulation depth fluctuation of each channel will affect the amplitude consistency and harmonic distortion of the demodulation results. In this paper, an improved demodulation scheme based on the Goertzel algorithm is proposed to calculate the multi-channel phase delay and phase modulation depth and to compensate for their fluctuations simultaneously. First, the carrier's 1st to 6th harmonic amplitudes in the interference fringe are extracted using the Goertzel algorithm. Then, the phase delay is calculated using the real and imaginary components of the 1st harmonic amplitude. The phase modulation depth is calculated with a combinatorial operation of the 1st to 6th harmonic amplitudes. In addition, a reference channel is introduced to implement phase delay and modulation depth feedback control. The experimental results demonstrate that the improved scheme can effectively suppress the harmonic distortion and improve the amplitude consistency of multi-channel interferometric fiber-optic sensors with low resource consumption.

FPGA-Based EO-PLL With Repetitive Control for Highly Linear Laser Frequency Tuning in FMCW LIDAR Applications

We developed a digital electro-optical phase-locked-loop (EO-PLL) utilizing a repetitive control, allowing the use of very nonlinear tunable lasers for frequency-modulated continuous-wave (FMCW) light detection and ranging (LIDAR) applications. This approach allows reducing the phase error of continuously repeated frequency chirps to virtually zero by using information of previous chirps and additionally guaranties sub-second settling times. With this method, much higher nonlinearities in the laser frequency modulation slope <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γ</i> are tolerable than with common real-time analog PLLs. Furthermore, we applied a Goertzel algorithm generalized to non-integer multiples of the fundamental frequency to achieve a mean measurement precision of 5.41 µm and 16.35 µm at a distance from the interferometer baseline of 3 m and 10.3 m, respectively.

Comparative Study of AC Signal Analysis Methods for Impedance Spectroscopy Implementation in Embedded Systems

For embedded impedance spectroscopy, a suitable method for analyzing AC signals needs to be carefully chosen to overcome limited processing capability and memory availability. This paper compares various methods, including the fast Fourier transform (FFT), the FFT with barycenter correction, the FFT with windowing, the Goertzel filter, the discrete-time Fourier transform (DTFT), and sine fitting using linear or nonlinear least squares, and cross-correlation, for analyzing AC signals in terms of speed, memory requirements, amplitude measurement accuracy, and phase measurement accuracy. These methods are implemented in reference systems with and without hardware acceleration for validation. The investigation results show that the Goertzel algorithm has the best overall performance when hardware acceleration is excluded or in the case of memory constraints. In implementations with hardware acceleration, the FFT with barycentre correction stands out. The linear sine fitting method provides the most accurate amplitude and phase determinations at the expense of speed and memory requirements.

Методика определения частотных характеристик микрофонов в незаглушенном лабораторном помещении, основанная на применении широкополосного сигнала возбуждения

The paper introduces a method for determining the frequency response of microphones using a broadband excitation signal. The method is based on time deiay spectrometry and uses the recursive filter of the Goertzel algorithm and additional sliding complex weighted averaging under conditions burdened by sound reverberation. A combination of field and pressure measurements was used to minimize the effect of reflections during measurements. Relying on the algorithm, we developed software for post-processing the measurement results when determining the frequency response of microphones in an unmuffled laboratory room. Furthermore, we assembled a research measuring complex and experimentally tested the developed method and software determining the methodical error in the frequency response of microphones using a broadband excitation signal. The developed method and software are recommended for determining the frequency response of measuring microphones by the field in unmuffled reverberating laboratory rooms without auxiliary equipment, such as a damped chamber and a small-volume chamber, provided the practical recommendations given in the research are taken into consideration

Joint Vital Signs and Position Estimation of Multiple Persons Using SIMO Radar

Respiration rate monitoring using ultra-wideband (UWB) radar is preferred because it provides contactless measurement without restricting the person’s privacy. This study considers a novel non-contact-based solution using a single-input multiple-output (SIMO) UWB impulse radar. In the proposed system, the collected radar data are converted to several narrow-band signals using the generalized Goertzel algorithm (GGA), which are used as the input of the designed phased arrays for position estimation. In this context, we introduce the incoherent signal subspace methods (ISSM) for the direction of arrivals (DOAs) and distance evaluation. Meanwhile, a beam focusing approach is used to determine each individual and estimate their breathing rate automatically based on a linearly constrained minimum variance (LCMV) beamformer. The experimental results prove that the proposed algorithm can achieve high estimation accuracy in a variety of test environments, with an error of 2%, 5%, and 2% for DOA, distance, and respiration rate, respectively.