Quadratic Frequency Modulation Research Articles

Convolutional Neural Networks for Local Component Number Estimation from Time–Frequency Distributions of Multicomponent Nonstationary Signals

Frequency-modulated (FM) signals, prevalent across various applied disciplines, exhibit time-dependent frequencies and a multicomponent nature necessitating the utilization of time-frequency methods. Accurately determining the number of components in such signals is crucial for various applications reliant on this metric. However, this poses a challenge, particularly amidst interfering components of varying amplitudes in noisy environments. While the localized Rényi entropy (LRE) method is effective for component counting, its accuracy significantly diminishes when analyzing signals with intersecting components, components that deviate from the time axis, and components with different amplitudes. This paper addresses these limitations and proposes a convolutional neural network-based (CNN) approach for determining the local number of components using a time–frequency distribution of a signal as input. A comprehensive training set comprising single and multicomponent linear and quadratic FM components with diverse time and frequency supports has been constructed, emphasizing special cases of noisy signals with intersecting components and differing amplitudes. The results demonstrate that the estimated component numbers outperform those obtained using the LRE method for considered noisy multicomponent synthetic signals. Furthermore, we validate the efficacy of the proposed CNN approach on real-world gravitational and electroencephalogram signals, underscoring its robustness and applicability across different signal types and conditions.

A novel automatic generation control method with hybrid sampling for multi-area interconnected girds

Introduction: The emerging “net-zero carbon” police will accelerate the large-scale penetration of renewable energies in the power grid, which would bring strong random disturbances due to the unpredictable power output. It would affect the coordinated control performance of the distributed grids.Method: From the quadratic frequency modulation perspective, this paper proposes a fast Q-learning-based automatic generation control (AGC) algorithm, which combines full sampling with full expectation for multi-area coordination. A parameter σ is used to balance the state between the full sampling update and only the expectation update so as to improve the convergence accuracy. Meanwhile, fast Q-learning is incorporated by replacing the historical estimation function with the current state estimation function to accelerate the convergence speed.Results: Simulations on the IEEE two-region load frequency control model and Hubei power grid model in China have been performed to validate that the proposed algorithm can achieve optimal multi-area coordination and improve the control performance of frequency deviations caused by the strong random disturbances.Discussion: The proposed Q-learning-based AGC method outperforms the convergence accuracy, speed, and control performance compared with other reinforcement learning algorithms.

When FrFT meets quadratic frequency modulation functions—A novel tool for nonstationary signals and time-varying systems

The fractional Fourier transform (FrFT) is a general form of the Fourier transform, which has provided a theoretical framework for linear frequency modulation signal processing. The quadratic frequency modulation (QFM) function is a generalization of the linear frequency modulation function, which balances the precision and complexity of nonstationary signals in practical applications. However, there is no framework for analyzing and processing QFM signals, which limits the applications of QFM signal model. The QFMFrFT, a generalization of the FrFT, is developed in this study to tackle this issue. The impact of the QFMFrFT on the time-frequency distribution explains its usefulness in sampling, denoising, and filter design of QFM signals. Two one-place statistical functions are defined for QFM-type stochastic signals, of which the input-output relationships of a linear time-varying system are discussed. Using these relations, a linear time-varying matched filter is designed. Two types of generalized convolution operators are designed associated with the QFMFrFT, which allow a multiplicative realization of a time-varying system. The QFM signal is demonstrated as a singular function of this type of linear time-varying system, which inspires a multi-carrier technique in spread-spectrum communications. Simulations verify the applications of the QFMFrFT in parameter estimation, matched filter design, and communications.

Instantaneous Frequency and Chirp Rate Estimation for Noisy Quadratic FM Signals by CNN

Deep learning and machine learning are widely employed in various domains. In this paper, Artificial Neural Network (ANN) and Convolution Neural Network (CNN) are used to estimate the Instantons Frequency (IF), Linear Chirp Rate (LCR), and Quadratic Chirp Rate (QCR) for Quadratic Frequency Modulated (QFM) signals under Additive White Gaussian (AWG) noise and Additive Symmetric alpha Stable (ASαS) noise. SαS distributions are impulsive noise disturbances except for a few circumstances, lack a closed-form Probability Density Function (PDF), and an infinite second-order statistic. Geometric SNR (GSNR) is used to determine the impulsiveness of mixture noise for Gaussian and SαS noise. ANN is a machine learning classifier with few layers that reduce FE, LCRE, and QCRE complexity and achieve high accuracy. CNN is a deep learning classifier that is built with multiple layers of FE, LCRE, and QCRE. CNN is more accurate than ANN when dealing with large amounts of data and determining optimal features. The results reveal that SαS noise is substantially more damaging to FE, LCRE, and QCRE than Gaussian noise, even when the magnitude is modest, and it is less damaging when alpha is greater than one. After training DCNN for FE, LCRE, and QCRE estimation of QFM signals. The 2D-CNN model accuracy achieved 98.7603 and 1D-CNN is 75.8678 for ten epochs. ANN model accuracy achieved 37.5 for 1000 epochs. The accuracy of TFD (spectrogram & pspectrum) for frequency estimation of QFM signals was 38.4254 by spectrogram and 38.6746 by pspectrum.

Reduced-Complexity Estimation of FM Instantaneous Parameters via Deep-Learning

Signal frequency estimation is a fundamental problem in signal processing. Deep learning is a fundamental method to solve this problem. This paper used five deep learning methods and three datasets including different singles Single Tone (ST), Linear- Frequency-Modulated (LFM), and Quadratic-Frequency-Modulated (QFM). This signal is affected by Additive White Gaussian (AWG) noise and Additive Symmetric alpha Stable (SαS) noise. Geometric SNR (GSNR) is used to determine the impulsiveness of noise in a Gaussian and SαS noise mixture. Deep learning methods are Gated Recurrent Unit (GRU), Long Short-Term Memory (LSTM), Bi-Direction Long Short-Term Memory (BiLSTM), and Convolution Neural Network (1D-CNN & 2D-CNN). When compared to a deep learning classifier with few layers to get on high accuracy and complexity reduces for Instantaneous Frequency (IF) estimation, Linear Chirp Rate (LCR) estimation, and Quadratic Chirp Rate (QCR) estimation. IF estimation of ST signals, IF and LCR estimation of LFM signals, and IF, LCR, and QCR estimation of QFM signals. The accuracy of the ST dataset in GRU is 58.09, LSTM is 46.61, BiLSTM is 45.95, 1D-CNN is 51.48, and 2D-CNN is 54.13. The accuracy of the LFM dataset in GRU is 82.89, LSTM is 66.28, BiLSTM is 20%, 1D-CNN is 74.79, and 2D-CNN is 98.26. The accuracy of the QFM dataset in GRU is 78.76, LSTM is 67.8, BiLSTM is 69.91, 1D-CNN is 75.8, and 2D-CNN is 98.2. The results show that 2D-CNN is better than other methods for parameter estimation in LFM signals and QFM signals, and the GRU is better for parameter estimation in ST signals.

Parametric Instantaneous Frequency Estimation via PWSR With Adaptive QFM Dictionary

In this letter, an effective parametric IF estimation method based on piece-wise sparse representation (PWSR) is proposed to enhance the precision of conventional time-frequency distribution (TFD)-based estimators. Firstly, the signal is divided into a series of short-time segments according to the piece-wise quadratic frequency modulation (QFM) model. Then, the corresponding QFM parameters of each segment are accurately estimated by solving a sparse representation (SR) problem. Moreover, a construction scheme enabling the QFM dictionary to vary adaptively with the analyzed short-time signal is also provided. Finally, the IF of each segment is reconstructed according to the SR solution individually and combined together to generate the complete IF estimates. A radar imaging example verifies that the proposed method achieves a notable improvement in estimation accuracy as compared to existing TFD-based approaches.

Dynamic intelligent measurement of multiple chirped signals of different types based on the optical computing STFT and the YOLOv3 neural network.

We propose a simultaneous measurement system for multiple signals of different types which combines the optical computing short-time Fourier transform (STFT) and You Only Look Once (YOLOv3) neural network. Through the system, the analytical expressions of multiple broadband signals of different types can be obtained in real time with high-frequency resolution. Experimentally, the accuracy of the signal type in the detection results can almost reach 100%. Additionally, the parameter measurement errors for the bandwidth (BW), pulse width (PW), center frequency (CF), and time of arrival (TOA) of each linear frequency-modulated (LFM) or quadratic frequency-modulated (QFM) signal are within ±30 MHz, ±20 ns, ±15 MHz, and ±20 ns, respectively. The frequency resolution can reach 60 MHz. Factors affecting the performance of the measurement system, such as the quantity of the signal and the number of the category, are discussed.

A robust method for instantaneous frequency estimation via inverse spectral decomposition

Instantaneous frequency is an important seismic attribute, which can indicate thin beds and lithofacies boundaries. However, instantaneous frequency attribute is susceptible to noise when it is obtained by the traditional Hilbert transform (HT) method. We propose a robust method for instantaneous frequency estimation. The method first obtains the time-frequency distribution of the seismic signal by inverse spectral decomposition (ISD) and then calculates the analytic signal through the window HT transform. Inverse spectral decomposition achieves high-resolution time-frequency distribution by adding sparse constraint to the corresponding inverse problem, and therefore the noise can be suppressed. The algorithm we choose to solve the mix ℓ2 − ℓ1 problem is the fast iterative shrinkage-thresholding algorithm (FISTA). Compared with the traditional iterative least squares (IRLS) algorithm, FISTA can achieve a better computational efficiency. We perform the method on a quadratic frequency modulation (QFM) signal, a synthetic data based on a wedge model and field data sets to demonstrate its performance, compared with the HT method and the time-frequency adaptive filtering method.

Detection and Estimation Algorithm for Marine Target With Micromotion Based on Adaptive Sparse Modified-LV’s Transform

Due to the complex marine environment and high-order frequency modulation (FM) on radar echo from the micromotion of the target, the effective and robust detection of a marine target with micromotion under heavy sea clutters’ background is a challenging task. In this article, we propose a novel detection and estimation algorithm based on adaptive sparse modified-Lv’s transform (ASMLVT). First, the micro-Doppler (m-D) characteristics of marine targets are employed and modeled as quadratic frequency-modulated (QFM) signals. Second, we modify the 2-D robust sparse Fourier transform (2-D-RSFT) and make it adaptive to the sea clutters’ background, namely, 2-D adaptive sparse Fourier transform (2-D-ASFT). Then, we substitute the 2-D Fast Fourier transform (2-D-FFT) operation with 2-D-ASFT in the modified-Lv’s transform (MLVT). The proposed algorithm can not only achieve good energy accumulation and accurate parametric estimation for marine targets with micromotion but is also robust to the heavy sea clutters and can greatly reduce false alarms. Besides, it has a good cross-term suppression ability to detect multitargets. Experiments with simulated and real radar datasets show that the proposed algorithm can effectively detect and estimate multitargets with micromotion under heavy sea clutter and low signal-to-clutter ratio (SCR) background.

Detection of LFM Radar Signals and Chirp Rate Estimation Based on Time-Frequency Rate Distribution.

Linear frequency-modulated (LFM) signals are the most significant example of waveform used in low probability of intercept (LPI) radars, synthetic aperture radars and modern communication systems. Thus, interception and parameter estimation of the signals is one of the challenges in Electronic Support (ES) systems. The methods, which are widely used to accomplish this task are mainly based on transformations from time to time-frequency domain, which concentrate the energy of signals along an instantaneous frequency (IF) line. The most popular examples of such transforms are the short time Fourier transform (STFT) and Wigner-Ville distribution (WVD). However, for LFM waveforms, methods that concentrate signal energy along a line in the time-frequency rate domain may allow to obtain better detection and estimation performance. This type of transformation can be obtained using the cubic phase (CP) function (CPF). In the paper, the detection of LFM waveform and its chirp rate (CR) parameter estimation based on the extended forms of the standard CPF is proposed. The CPF was originally introduced for instantaneous frequency rate (IFR) estimation for quadratic frequency modulated (QFM) signals i.e., cubic phase signals. Summation or multiplication operations on time cross-sections of the CPF allow to formulate the extended forms of the CPF. Based on these forms, detection test statistics and the estimation procedure of LFM signal parameters have been proposed. The widely known estimation methods assure satisfying accuracy for high SNR levels, but for low SNRs the reliable estimation is a challenge. The proposed approach based on joint analysis of detection and estimation characteristics allows to increase the reliability of chirp rate estimates for low SNRs. The results of Monte-Carlo simulation investigations on LFM signal detection and chirp rate estimation evaluated by the mean squared error (MSE) obtained by the proposed methods with comparisons to the Cramer-Rao lower bound (CRLB) are presented.

InISAR Imaging for Maneuvering Target Based on the Quadratic Frequency Modulated Signal Model With Time-Varying Amplitude

Interferometric inverse synthetic aperture radar (InISAR) has proven to be an effective tool for 3-D imaging of noncooperative targets. The traditional InISAR imaging algorithms are almost entirely based on the assumption of a constant amplitude polynomial phase signal (PPS) model. However, target’s maneuverability tends to cause the echo signal to exhibit the characteristic of time-varying amplitude (TVA) in practice. To remedy this problem, a novel InISAR imaging algorithm for maneuvering targets based on quadratic frequency modulated (QFM) signal model with TVA is presented. First, the echo of each range cell is modeled as a multicomponent TVA-QFM signal. Then, through the matrix derivation, the parameter estimation problem of this signal is converted to a convex optimization problem. Subsequently, with the help of the scaled Fourier transform (SCFT), an efficient iterative update approach based on alternative direction method of multipliers (ADMM) framework is proposed to solve this optimization problem. Furthermore, associated with the range instantaneous Doppler (RID) method and multichannel interference technology, 2-D ISAR images and 3-D space shape of the target can be generated. Finally, some simulation results are provided to evaluate the effectiveness and robustness of the proposed algorithm.

Instantaneous frequency extraction of highly nonstationary optical interferometric signal using Reassigned Smoothed Pseudo Wigner Ville Distribution

Instantaneous frequency (IF) of optical interferometric signal often carries information on various physical parameters such as velocity, strain and displacement. We propose smoothed Pseudo Wigner-Ville distribution method for extracting IF of highly non-stationary polynomial frequency modulated (FM) fringe signal. Cohen class signal representation is followed and time and frequency resolutions are independently optimized using separable kernel. Reassignment technique is further applied for sharpening of the spectrogram with these optimized kernels that improves accuracy in IF significantly. Quadratic FM along with strong nonlinear IF based fringe signal is considered for showing the efficacy of proposed method. Results are compared with that of existing Polynomial Wigner Ville Distribution method and Continuous Wavelet Transform (CWT) based approach and found to be more accurate in proposed technique. Finally fringe signal generated by varying the length of one of the legs of Michelson interferometer has been used to validate the proposed method experimentally.

Parameter Estimation of Polynomial-Phase Signals

Introduction . Polynomial phase signals frequently appear in radar, sonar, communication and technical applications. Therefore, estimation of polynomial phase coefficients of such signals is an urgent problem in signal theory. Currently, a large number of estimation algorithms have been proposed. The best way is the maximum likelihood (ML) method. However, its implementation is associated with a multidimensional retrieval, which makes the method unsuitable for practical implementation. A number of alternative strategies have been developed to circumvent the ML difficulties. These strategies are very close to optimal. Among them one can single out the HAF-algorithm based on the computation of the High order Ambiguity Function and the CPF algorithm, which uses the computation of the Cubic Phase Function and produces very accurate estimates for signals with the quadratic frequency modulation. However, both algorithms have obvious drawbacks. The HAF algorithm pro-duces a large number of combinatorial noise components. The CPF algorithm is limited in its implementation to the third order polynomial signals and does not use fast algorithms, such as the Fast Fourier Transform. Aim. Synthesis of an estimation algorithm that produces a small number of noise combinatorial components and uses the Fast Fourier Transform computation algorithms to find coefficient estimates of an arbitrary order phase polynomial. Materials and methods. In the paper a concept of a decisive function was introduced. It was calculated so that its phase contained only a first-order monomial with a coefficient equal to the highest coefficient of the signal phase polynomial. Results. A new estimation algorithm was proposed able to use Fast Fourier Transform computation algorithms to find estimates. Each polynomial coefficient was estimated on the basis of a unified procedure, which reduced the number of combinatorial noise components in an estimate search. Conclusions. The synthesized algorithm gives asymptotically efficient estimates for lower signal-to-noise ratios in comparison with the HAF-algorithm.

Quadratic FM Signal Detection and Parameter Estimation Using Coherently Integrated Trilinear Autocorrelation Function

This paper proposes an integrated time-frequency rate distribution (TFRD) technique, i.e., coherently integrated trilinear autocorrelation function (CITAF), for the detection and parameter estimation of quadratic frequency modulated (QFM) signal embedded in white Gaussian noise. The basic idea is to design a trilinear autocorrelation function (TAF) enabling a coherent integration of the signal energy in both of the time and lag-time domains. Theoretical analysis and numerical simulations show that this new method has a lower signal to noise ratio (SNR) threshold and higher estimation accuracy in comparison with several state-of-the-art algorithms. Furthermore, the coherently integrated nonuniform TAF (CINTAF) is briefly introduced.

A novel method of dynamic monitoring and parameter estimation for rock-fall based on multichannel SAR

Rock-fall incidents have the characteristics of multiple, sudden and random, which threaten the surrounding buildings and traffic lines. This paper studied the monitoring and prediction of rock falls with a new synthetic aperture radar (SAR) technology. One of the most important factors in rock-fall target detection is the signal-to-clutter-plus noise ratio (SCNR) after clutter suppression, which should be maximized before rock-fall target detection. Through analysing the remainder characteristics of rock-fall targets after clutter suppression, a method of removing the quadratic frequency-modulated (FM) component associated with the azimuth velocity and along-track velocity of a rock-fall target is proposed. After removing the quadratic FM component using dechirp technology, focus is placed on the remainder signal of a rock-fall target in a Doppler domain image, and the SCNR of the signal is maximized. This method provides effective rock-fall target detection. To resolve the contradiction between the computational accuracy and the accuracy of the along-track velocity, this paper adopts gradual approach technology. Finally, by analysing the focused signal characteristics, a method of rock-fall target parameter estimation is proposed. The effectiveness of the presented method is demonstrated by both theoretic analysis and simulations.

ISAR Imaging of Maneuvering Target Based on the Estimation of Time Varying Amplitude With Gaussian Window

Due to the maneuverability of target’s motion in the inverse synthetic aperture radar (ISAR) imaging, the Doppler frequency of the received signal has the nonlinear time-varying characteristic. Under the circumstances, it is accurate enough to model the received signal of each range bin as a multicomponent quadratic frequency-modulated (QFM) signal. In this paper, the scaled Fourier transform (SCFT) is applied to obtain the phase parameters of the QFM signal. Owing to the phenomenon of the scatterers’ migration through resolution cell (MTRC) and the change of attitude angle, the amplitude of the received signal is no longer constant but time-varying. Thus, a novel and precise method using the adaptive Gaussian window is presented to estimate the time varying amplitude of the received signal. After that, the corresponding ISAR imaging algorithm is presented consequently. The validity of the proposed method is demonstrated by the simulated and real-data ISAR results.

Parameter estimation of quadratic frequency modulated signal based on three‐dimensional scaled Fourier transform

In this study, a novel algorithm based on three-dimensional scaled Fourier transform (TDSFT) is proposed for parameter estimation of a quadratic frequency modulated (QFM) signal. To solve the problem of signal-to-noise ratio loss caused by constant delay, a variable-delay instantaneous autocorrelation function is first proposed. After that, two steps TDSFT and addition operation are performed to achieve three-dimensional decoupling and coherent integration of signal energy. The analysis shows that the proposed algorithm can be efficiently implemented by a fast Fourier transform without any searching procedure. Moreover, it can deal with multi-component QFM signals with satisfactory suppression of cross terms. Comparisons with three other popular methods, chirp rate-quadratic chirp rate distribution, generalized scaled Fourier transform and generalized decoupling technique demonstrate that the proposed algorithm has superior anti-noise performance due to the coherent integration of all signal energies. Extensive numerical simulations validate the effectiveness of the TDSFT-based algorithm.

Beam-forming of broadband QFM signals using generalized time–frequency transform

ABSTRACTDetection and localisation of contacts using sensor array play a significant role in the area of array signal processing. In this paper, a new beam-forming analysis for broad-banded quadratic frequency modulated (QFM) signals using generalised time–frequency transform (GTFT) is presented by extending the conventional fractional Fourier transform (FrFT). As FrFT is constrained to the analysis of linear chirps, detection using GTFT is found to be the appropriate selection for higher-order chirps with the suitable choice of the kernel. The parameters of the QFM signal is extracted using the polynomial chirplet transform time–frequency distribution method. Subsequently, the direction of arrival (DoA) estimation algorithms viz. conventional beam-former (CBF), minimum variance distortionless response (MVDR), multiple signal classification (MUSIC) and minimum norm (MN) are re-cast using the quadratic steering vector derived for QFM chirps and hence renamed as GTFT-CBF, GTFT-MVDR, GTFT-MUSIC and GTFT-MN, respectively. The efficacy of the algorithm is validated through various signals including practical sonar array data. It is seen from the theoretical analysis as well as simulations that GTFT-MUSIC excels other DoA (bearing) estimation techniques. The GTFT analysis on back propagated signals using sonar array aided in effective underwater channel modelling.

Quadratic Frequency Modulation Signals Parameter Estimation Based on Product High Order Ambiguity Function-Modified Integrated Cubic Phase Function

In inverse synthetic aperture radar (ISAR) imaging system for targets with complex motion, such as ships fluctuating with oceanic waves and high maneuvering airplanes, the multi-component quadratic frequency modulation (QFM) signals are more suitable model for azimuth echo signals. The quadratic chirp rate (QCR) and chirp rate (CR) cause the ISAR imaging defocus. Thus, it is important to estimate QCR and CR of multi-component QFM signals in ISAR imaging system. The conventional QFM signal parameter estimation algorithms suffer from the cross-term problem. To solve this problem, this paper proposes the product high order ambiguity function-modified integrated cubic phase function (PHAF-MICPF). The PHAF-MICPF employs phase differentiation operation with multi-scale factors and modified coherently integrated cubic phase function (MICPF) to transform the multi-component QFM signals into the time-quadratic chirp rate (T-QCR) domains. The cross-term suppression ability of the PHAF-MICPF is improved by multiplying different T-QCR domains that are related to different scale factors. Besides, the multiplication operation can improve the anti-noise performance and solve the identifiability problem. Compared with high order ambiguity function-integrated cubic phase function (HAF-ICPF), the simulation results verify that the PHAF-MICPF acquires better cross-term suppression ability, better anti-noise performance and solves the identifiability problem.

QFM Signals Parameters Estimation Based on Double Scale Two Dimensional Frequency Distribution

This paper proposes a novel estimation method, named double scale two dimensional frequency distribution (DSTFD), to estimate the parameters of quadratic frequency modulated (QFM) signals. In the DSTFD, by using a novel parametric instantaneous self-correlation function and the idea of the keystone transform, the QFM signals are transformed into two-dimensional frequency domain, and the QFM signals are detected by searching peaks. To reduce computational cost, a double scale (DS) estimation strategy, which consists of coarse estimation and fine estimation, is proposed. The DS estimation strategy can be implemented by using the Chirp Z-transform and an improved transform named local scaled Fourier transform (LSFT). The LSFT only consists of complex multiplications, fast Fourier transform (FFT), and inverse FFT operations. The implementation, anti-noise performance, and computational cost are analyzed for the proposed method. Through simulations and analyses, the results demonstrate that the DSTFD outperforms other compared algorithms.