Quadratic Frequency-modulated Signal Research Articles

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.

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.

Ground Maneuvering Targets Imaging for Synthetic Aperture Radar Based on Second-Order Keystone Transform and High-Order Motion Parameter Estimation

With the developments of the synthetic aperture radar (SAR) system, the long dwell time provides high resolution but at the same time it produces large range migration (RM) and Doppler frequency migration, which brings a new challenge for ground moving target imaging. To address those problems, a ground moving target refocusing method based on a novel parameter estimation algorithm, namely second-order Keystone transform (SOKT)-Radon-cubic phase function-Fourier transform (RCFT), is proposed in this work. First, the SOKT is applied to correct the range curvature. After that, the Hough transform is utilized to estimate the trajectory slope for the range walk correction and the across-track velocity acquirement. Finally, a novel coherently integrated parameter estimation method, termed as RCFT, is developed in this work to transform the azimuth quadratic frequency-modulated signal into a peak point in two-dimensional (2-D) time-frequency plane. In the process, the second- and third-order motion parameters are simultaneously obtained, which eliminates the error propagation effect. Due to 2-D coherent integration realization and its bilinear function feature, the proposed RCFT method is capable of obtaining accurate parameter estimates in low signal-to-noise ratio scenario with a relatively low computational burden. Compared with the conventional SAR imaging methods using the second-order phase model, the proposed method produces better imaging quality since the third-order Doppler frequency migration is effectively eliminated. The performance of the proposed method is demonstrated by both the simulated and real data.

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.

An Efficient ISAR Imaging Method for Non-Uniformly Rotating Targets Based on Multiple Geometry-Aided Parameters Estimation

The inverse synthetic aperture radar (ISAR) imaging for non-uniformly rotating target plays an important role in many applications. However, it always faces a serious challenge, i.e., the time-varying Doppler frequency induced by the target complex motions will degrade the quality of ISAR image seriously. Several algorithms based on motion parameter estimation have been proposed in recent years. However, most of them always suffer from the heavy computational burden and considerable propagation errors, which deteriorate the performances of algorithms and limit their practical applications. To tackle this problem, an efficient ISAR imaging algorithm is proposed for non-uniformly rotating targets based on geometry-aided parameters estimation in this paper. After the motion compensation, the ISAR echoes are modeled as the quadratic frequency modulated signal first, then is transformed into time-chirp distribution (TCD) plane and the least square Radon transform (RT) based on TCD (LSRT) to estimate the derivative of chirp rate efficiently is proposed. To improve the estimation performance and robustness, the pre-processing and the weighted least square RT TCD methods are both developed. Similarly, the weighted LSRT based on time-frequency distribution is also proposed to estimate the chirp rate. The proposed multiple geometry-aided method can eliminate the effect of propagation error and reduce the computational complexity simultaneously. The simulated data and real-measured data results are both presented to validate the proposed method.

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.

Quantitative SNR analysis of QFM signals in the LPFT domain with Gaussian windows

The purpose of this paper is to present a quantitative SNR analysis of quadratic frequency modulated (QFM) signals. This analysis is located in the continuous-time local polynomial Fourier transform (LPFT) domain using a Gaussian window function based on the definition of 3 dB signal-to-noise ratio (SNR). First, the maximum value of the local polynomial periodogram (LPP), and the 3 dB bandwidth in the LPFT domain for a QFM signal is derived, respectively. Then, based on these results, the 3 dB SNR of a QFM signal with Gaussian window function is given in the LPFT domain with one novel idea highlighted: the relationship among standard SNR, parameters of QFM signals and Gaussian window function is clear, and the potential application is demonstrated in the parameter estimation of a QFM signal using the LPFT. Moreover, the 3 dB SNR in the LPFT domain is compared with that in the linear canonical transform (LCT) domain. The validity of theoretical derivations is confirmed via simulation results. It is shown that, in terms of SNR, QFM signals in the LPFT domain can achieve a significantly better performance than those in the LCT domain.

Ground Moving Target Refocusing in SAR Imagery Using Scaled GHAF

In this paper, a new method is proposed to refocus a ground moving target in synthetic aperture radar imagery. In this method, range migration is compensated in the 2-D frequency domain, which can easily be implemented by using the complex multiplications, the fast Fourier transform (FFT), and the inverse FFT operations. Then, the received target signal in a range gate is characterized as a quadratic frequency-modulated (QFM) signal. Finally, a novel parameter estimation method, i.e., scaled generalized high-order ambiguity function (HAF), is proposed to transform the target signal into a signal on 2-D time–frequency plane and realize the 2-D coherent integration, where the peak position accurately determines the second- and third-order parameters of a QFM signal. Compared with our previously proposed generalized Hough-HAF method, the proposed method can obtain a better target focusing performance, since it can eliminate the incoherent operations in both range and azimuth directions. In addition, the proposed method is computationally efficient, since it is free of searching in the whole target focusing procedure. Both simulated and real data processing results are provided to validate the effectiveness of the proposed algorithm.

Parameter estimation of QFM signal based on MPKF

In order to reconstruct the quadratic frequency modulated (QFM) signal, such as radar signal due to the complexity. A novel method for the parameter estimation of multi-component QFM signal based on a modified product kernel function (MPKF) is proposed in this paper. The method has fewer external cross-terms and light computational burden because of non-iterative operations. Moreover, it is robust against additive noise. Finally, simulated and real data results confirm the proposed method.

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

Inverse synthetic aperture radar (ISAR) imaging of maneuvering target plays an important role in the field of national defense and surveillance. In this case, the conventional range Doppler algorithm is unsuitable to generate a well-focused ISAR image due to the time-varying property of the Doppler frequency during the integrated imaging interval. Furthermore, the dihedral or trihedral components of the imaging target or the phenomena of migration through resolution cell for the target's complex motion will induce the time-varying performance of the amplitude for the received signal. In this paper, the received signal in a range bin is depicted as multicomponent quadratic frequency-modulated signal with time-varying amplitude, and the generalized cubic phase function is used to estimate the parameters of it. Associated with the range instantaneous Doppler technique, the images quality can be enhanced distinctly compared with the conventional constant amplitude signal model. Results of simulated and read data demonstrate the effectiveness of the novel algorithm proposed in this paper.

An ISAR Imaging Algorithm for Maneuvering Targets With Low SNR Based on Parameter Estimation of Multicomponent Quadratic FM Signals and Nonuniform FFT

The inverse synthetic aperture radar (ISAR) imaging for maneuvering targets has always been a challenging task due to the time-varying Doppler parameter, especially in the low signal-to-noise ratio (SNR) environment. In this paper, a novel ISAR imaging algorithm for maneuvering targets in the low SNR environment based on the parameter estimation approach is presented. First, the received signals of the ISAR in a range bin are modeled as a multicomponent quadratic frequency-modulated (QFM) signal after the migration compensation. Then, to estimate the parameters of the QFM signal, two cubic phase functions (CPFs), namely coherently integrated generalized CPF (CIGCPF) and coherently integrated CPF (CICPF), are developed. The CIGCPF and CICPF are simple and only require the fast Fourier transform (FFT) and the nonuniform FFT (NUFFT). Due to the usage of the NUFFT, the computational cost is reduced, and the searching procedure is unnecessary for the nonuniformly spaced signal. Thanks to the coherent integrations and NUFFT, the CIGCPF and CICPF, which demonstrate the excellent noise-tolerant performance and reduce the error propagation effect, are efficient and suitable for the multicomponent QFM signals in the low SNR environment. Finally, several simulation examples, analyses of the noise-tolerant performance of the proposed method, and ISAR imaging results of the simulated data demonstrate the effectiveness of the proposed method.

Imaging of high‐speed manoeuvering target via improved version of product high‐order ambiguity function

This study presents a novel algorithm for radar imaging of high-speed manoeuvering target. It is based on the fact that the azimuth part of the echo is modelled as the superposition of multiple quadratic frequency modulated (QFM) signals, and the range-Doppler algorithm is inappropriate to generate a focused radar image in this case. In this study, the improved version of product high-order ambiguity function (IPHAF) is proposed to estimate the parameters of multiple QFM signals. This estimator requires only one-dimensional (1D) maximisations without the scaling operation compared with the traditional product high-order ambiguity function. The radar image quality can be improved by the IPHAF technique, and the experimental results validate the effectiveness of the method presented in this study.

Algorithm based on the linear canonical transform for QFM signal parameters estimation

In this study, a novel algorithm based on the linear canonical transform (LCT) is proposed for parameters estimation of a quadratic frequency modulated (QFM) signal. First, a new kind of generalised LCT (GLCT) is defined and the GLCT of the QFM signal will generate an impulse. The third-order phase coefficient of the QFM signal can be estimated in accordance to the position information of such impulse. After compensating off the third-order phase coefficient, the QFM signal can be approximated to linear frequency modulated (LFM) signal and the second-order, first-order phase coefficient and the amplitude can be estimated by algorithms for estimating the LFM signal. The proposed algorithm does not suffer a heavy computational burden because it only requires one dimensionality maximisation and the LCT has fast numerical algorithm as well. Moreover, the proposed algorithm has accurate estimation and low signal to noise ratio (SNR) threshold. Meanwhile, high-output SNR of the proposed algorithm can be gotten with a small number of sampling points. Comparisons with existing algorithms verify that the proposed algorithm has a good performance in the aspects of computational complexity, accuracy and output SNR in some cases.

Parameter estimation method for multiple quadratic frequency‐modulated signals

Generalised cubic phase function (GCPF)-based estimator is efficient in estimating the third-order coefficient for mono-component quadratic frequency-modulated (QFM) signal. However, it suffers from cross-terms and spurious peaks when dealing with multi-component QFM signals. To efficiently estimate the third-order coefficients of multi-component QFM, a novel coherently integrated GCPF (CIGCPF) algorithm is proposed to enhance the auto-terms and suppress the cross-terms and spurious peaks. Comparisons with state-of-the-art algorithms indicate that CIGCPF not only solves the identification problem for multi-component QFM signals, but also can acquire higher anti-noise performance.

Sea clutter suppression and micromotion marine target detection via radon‐linear canonical ambiguity function

The micro-Doppler (m-D) signature of a marine target is used for detection within sea clutter in this study. The marine target usually has translational and rotational motions, and the corresponding radar returns can be modelled as a quadratic frequency modulated signal over a period of time. To compensate the range walk and Doppler migration of the micromotion target simultaneously, a novel long-time coherent integration detection algorithm, that is, Radon-linear canonical ambiguity function (RLCAF), is proposed. It has been proved that the m-D signal can be well matched and accumulated in the RLCAF domain using the long-time instantaneous autocorrelation function and associated with the parameterisation of the linear canonical transform. Furthermore, to suppress sea clutter and improve signal-to-clutter ratio, the authors propose a RLCAF spectrum subtraction method using the different properties of RLCAF representations between the target and sea clutter. Finally, experiments with real radar dataset indicate that the proposed method can achieve better integration and detection performance of a marine target with micromotion in case of high sea state.

Inverse Synthetic Aperture Radar Imaging of Nonuniformly Rotating Target Based on the Parameters Estimation of Multicomponent Quadratic Frequency-Modulated Signals

Inverse synthetic aperture radar (ISAR) is very useful in radar signature applications. In ISAR imaging of nonuniformly rotating target, the radar echo signal in a range cell can be modeled as multicomponent quadratic frequency-modulated (QFM) signals. The high-quality radar images can be obtained by the parameters estimation of QFM signals combined with the range-instantaneous-Doppler technique. In this paper, a novel algorithm for the parameters estimation of QFM signal via the modified cubic phase function (MCPF) is presented. This algorithm estimates the quadratic phase coefficient at first, and the other parameters can be estimated by the traditional CPF and the fast Fourier transform. The implementation of MCPF algorithm requires only 1-D maximizations, and the comparison with some existing algorithms is discussed. The ISAR imaging results of simulated and raw data validate the effectiveness of the novel algorithm proposed in this paper.

Radon-Linear Canonical Ambiguity Function-Based Detection and Estimation Method for Marine Target With Micromotion

Robust and effective detection of a marine target is a challenging task due to the complex sea environment and target's motion. A long-time coherent integration technique is one of the most useful methods for the improvement of radar detection ability, whereas it would easily run into the across range unit (ARU) and Doppler frequency migration (DFM) effects resulting distributed energy in the time and frequency domain. In this paper, the micro-Doppler (m-D) signature of a marine target is employed for detection and modeled as a quadratic frequency-modulated signal. Furthermore, a novel long-time coherent integration method, i.e., Radon-linear canonical ambiguity function (RLCAF), is proposed to detect and estimate the m-D signal without the ARU and DFM effects. The observation values of a micromotion target are first extracted by searching along the moving trajectory. Then these values are carried out with the long-time instantaneous autocorrelation function for reduction of the signal order, and well matched and accumulated in the RLCAF domain using extra three degrees of freedom. It can be verified that the proposed RLCAF can be regarded as a generalization of the popular ambiguity function, fractional Fourier transform, fractional ambiguity function, and Radon-linear canonical transform. Experiments with simulated and real radar data sets indicate that the RLCAF can achieve higher integration gain and detection probability of a marine target in a low signal-to-clutter ratio environment.

ISAR Imaging of a Ship Target Based on Parameter Estimation of Multicomponent Quadratic Frequency-Modulated Signals

High-resolution inverse synthetic aperture radar (ISAR) imaging of a ship target is a challenging task because of fluctuation with the ocean waves. The images obtained with a standard range-Doppler algorithm are usually blurred. Consequently, the range-instantaneous-Doppler (RID) technique should be used to improve the image quality. In this paper, the received signal in a range cell is modeled as a multicomponent quadratic frequency-modulated (QFM) signal after range compression and motion compensation, and then a new RID ISAR imaging algorithm is proposed that introduces a new method for estimating parameters of the QFM signal. By defining a new function and using the scaled Fourier transform (SCFT) with respect to the time axis, the coherent integration of auto-terms can be realized via the subsequent Fourier transformation with respect to the lag-time axis, and a peak can be obtained in the 2-D frequency plane, which is appropriate for parameter estimation of the QFM signal to reconstruct RID images. The proposed algorithm is accurate and fast since the defined function has moderate order nonlinearity and the SCFT can be performed via chirp z-transform. Experiments demonstrate the performance of the new algorithm. Comparisons with existing algorithms are also given, which show that the proposed algorithm can efficiently produce a focused image with less fake scatterers.