Quadratic Chirp Rate 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 a Maneuvering Target Based on Parameter Estimation of Multicomponent Cubic Phase Signals

In inverse synthetic aperture radar (ISAR) imaging for a uniformly moving rigid-body target, a finely focused ISAR image can be obtained by using the conventional range-Doppler algorithm. However, the ISAR image quality may significantly deteriorate when the time-vary Doppler phases in virtue of target maneuvering motions are present, such as an airplane with nonuniformly rotation and a ship with fluctuation. This has become a challenging task, especially under nonhigh signal-to-noise ratio (SNR) environment. In this article, a novel ISAR imaging algorithm for a maneuvering target with moderate reflection intensity is proposed. After motion compensation, the radar echo signal in a range cell is modeled as a multicomponent cubic phase signal (CPS), in which the chirp rate and the quadratic chirp rate are two important physical quantities that may determine the target ISAR focusing quality. Based on a symmetrical instantaneous autocorrelation function, the received CPSs are transformed into the time and lag-time plane, and then a 2-D coherent integration can be realized after the generalized time-scaled transform and 1-D maximization. This forms a high-quality ISAR image. The effectiveness and superiority of the proposed algorithm are validated by the ISAR imaging results of simulated and real measured data.

Imaging and Relocation for Extended Ground Moving Targets in Multichannel SAR-GMTI Systems

In a multichannel synthetic aperture radar (SAR) system, because of the target uncooperative motion, a ground moving target (GMT) is usually smeared, distorted, and shifted in an SAR image. In this article, a novel approach for multichannel SAR-GMT indication (GMTI) processing is proposed. The main innovations of this method are that a GMT can be well refocused and relocated since the target high-order Doppler parameters can be precisely estimated based on a high-order polynomial phase signal (PPS) model, and the target statistical amplitude and phase information is jointly applied to improve the radial velocity estimation robustness. Compared with the current SAR-GMTI algorithms, the improvements of this method over the existing methods are: 1) the topography interferometric phase can be effectively compensated by applying an adaptive 2-D spectrum filtering technique via iterative processing; 2) a GMT can be well imaged since the target Doppler chirp rate and the quadratic chirp rate can be well estimated via the 2-D coherent integration in the time–frequency plane; and 3) a GMT can be precisely relocated into its original position by applying the generalized amplitude and phase weighting technique. Real-measured SAR data processing results are presented to validate the effectiveness and feasibility of the proposed method.

Fast ISAR imaging method for complex manoeuvring target based on local polynomial transform‐fast chirp Fourier transform

In ISAR imaging, echo signals of complex manoeuvring targets have to be modelled as multi-component cubic phase signals (m-CPS). In this condition, the Doppler diffusion will seriously affect the quality of ISAR imaging. Herein, a complex manoeuvring target imaging method based on local polynomial transform-fast chirp Fourier transform (LPT-FCFT) is proposed. In this method, the instantaneous frequency and the quadratic chirp rate of the echo signal are estimated by local polynomial transformation, then the chirp rate and the amplitude are estimated by fast chirp Fourier transform, and finally the high-resolution ISAR image of the complex manoeuvring target is obtained by CLEAN technology. Compared with the traditional method, the proposed method has the advantages of less computational complexity, clearer imaging results and better anti-noise performance. Simulation results indicate the effectiveness and superiority of the proposed method.

ISAR Imaging of Nonuniformly Rotating Targets With Low SNR Based on Third Order Autocorrelation Function

This paper focuses on the inverse synthetic aperture radar (ISAR) imaging of nonuniformly rotating targets under low signal-to-noise ratio (SNR) environment, and an effective ISAR imaging algorithm based on a novel parameter estimation method is proposed. For nonuniformly rotating targets, the received signals in a range cell can be modeled as multicomponent cubic phase signals (CPSs) after the migration compensation and translation-induced phase error correction. For the CPS model, the chirp rate (CR) and quadratic chirp rate (QCR) will induce image defocus. To estimate the CR and QCR of CPS, a novel parameter method is presented. The basic idea of this method is to design a third order autocorrelation function (TOAF) which enable coherent accumulation of the signal energy in both of the time and lag-time domain. Compared with several state-of-the-art algorithms, the proposed method has a significant anti-noise performance improvement, while offering a proper computational complexity. The effectiveness and the superiority of the proposed ISAR imaging algorithm have been demonstrated through several simulation results.

An Approach for Parameter Estimation of Maneuvering Targets With Nonlinear Motions

To compensate the range cell migration and Doppler frequency spread of a maneuvering target during the integration time, this paper proposes a new parameter estimation method based on sub-band dual-frequency conjugate (SDFC)–Radon chirp rate–quadratic chirp rate algorithm. In this method, SDFC process is first applied to reduce the equivalent frequency and Radon chirp rate–quadratic chirp rate algorithm is then performed to obtain the estimates. Both simulated and real-data results demonstrate the effectiveness of the proposed algorithm.

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.

An ISAR Imaging Algorithm for Nonuniformly Rotating Targets With Low SNR Based on Modified Bilinear Parameter Estimation of Cubic Phase Signal

For nonuniformly rotating target in the low signal-to-noise ratio (SNR) environment, the inverse synthetic aperture radar (ISAR) imaging is a challenging task due to the Doppler spread induced by time-varying rotation. In this paper, an effective modified bilinear parameter estimation algorithm is proposed and applied to reconstruct the ISAR image of nonuniformly rotating targets with low SNR. First, azimuth echoes of a range cell are modeled as multicomponent cubic phase signals (multi-CPSs) after range alignment and phase adjustment. Then, to estimate the parameter of the CPS (the chirp rate and the quadratic chirp rate, which are identified as causes of the target image defocus), a novel integrated parametric cubic phase function (IPCPF) and a reversing Wigner–Ville distribution processing are developed. Compared to other four representative algorithms, the bilinearity of the IPCPF guarantees a higher antinoise performance and a better suppression on cross terms. Moreover, by utilizing the nonuniform fast Fourier transform and the generalized scaled Fourier transform, the brute-force searching is eliminated and the computational cost is reduced. Finally, a new cross-range scaling method based on a regression analysis is proposed for the ISAR imaging. With the synthetic data and the real radar data, several simulation examples and ISAR imaging results demonstrate the effectiveness and the superiority of the proposed ISAR imaging algorithm.

Quadratic Frequency Modulation Signals Parameter Estimation Based on Two-Dimensional Product Modified Parameterized Chirp Rate-Quadratic Chirp Rate Distribution.

In an inverse synthetic aperture radar (ISAR) imaging system for targets with complex motion, the azimuth echo signals of the target are always modeled as multicomponent quadratic frequency modulation (QFM) signals. The chirp rate (CR) and quadratic chirp rate (QCR) estimation of QFM signals is very important to solve the ISAR image defocus problem. For multicomponent QFM (multi-QFM) signals, the conventional QR and QCR estimation algorithms suffer from the cross-term and poor anti-noise ability. This paper proposes a novel estimation algorithm called a two-dimensional product modified parameterized chirp rate-quadratic chirp rate distribution (2D-PMPCRD) for QFM signals parameter estimation. The 2D-PMPCRD employs a multi-scale parametric symmetric self-correlation function and modified nonuniform fast Fourier transform-Fast Fourier transform to transform the signals into the chirp rate-quadratic chirp rate (CR-QCR) domains. It can greatly suppress the cross-terms while strengthening the auto-terms by multiplying different CR-QCR domains with different scale factors. Compared with high order ambiguity function-integrated cubic phase function and modified Lv’s distribution, the simulation results verify that the 2D-PMPCRD acquires higher anti-noise performance and obtains better cross-terms suppression performance for multi-QFM signals with reasonable computation cost.

Parameterized two-dimensional representation for multicomponent cubic phase signals and its application in ISAR imaging of fluctuating ship

The cubic phase signal (CPS) can be determined by two physical quantities, the chirp rate and quadratic chirp rate. With this characteristic, this paper proposes the parameterized two-dimensional representation (PTDR), which can be seen as an extension of the chirp rate-quadratic chirp rate distribution (CRQCRD). The goal of the PTDR is to accurately estimate parameters of multicomponent CPSs under low signal-to-noise ratios (SNRs). The PTDR is simple and can be implemented by using the complex multiplication, fast Fourier transform (FFT) and nonuniform FFT (NUFFT). In this paper, the computational cost, resolution, cross term suppression and anti-noise performance are analyzed for the PTDR. With several synthetic examples, comparisons with other five algorithms are performed and demonstrate that the PTDR can achieve higher resolution, cross term suppression and anti-noise performance with a moderate computational cost. Finally, the PTDR is employed to reconstruct inverse synthetic aperture radar (ISAR) image of the fluctuating ship and the real radar data simulations validate the effectiveness of the PTDR-based ISAR imaging algorithm.

Analysis of Maneuvering Targets with Complex Motions by Two-Dimensional Product Modified Lv's Distribution for Quadratic Frequency Modulation Signals.

For targets with complex motion, such as ships fluctuating with oceanic waves and high maneuvering airplanes, azimuth echo signals can be modeled as multicomponent quadratic frequency modulation (QFM) signals after migration compensation and phase adjustment. For the QFM signal model, the chirp rate (CR) and the quadratic chirp rate (QCR) are two important physical quantities, which need to be estimated. For multicomponent QFM signals, the cross terms create a challenge for detection, which needs to be addressed. In this paper, by employing a novel multi-scale parametric symmetric self-correlation function (PSSF) and modified scaled Fourier transform (mSFT), an effective parameter estimation algorithm is proposed—referred to as the Two-Dimensional product modified Lv’s distribution (2D-PMLVD)—for QFM signals. The 2D-PMLVD is simple and can be easily implemented by using fast Fourier transform (FFT) and complex multiplication. These measures are analyzed in the paper, including the principle, the cross term, anti-noise performance, and computational complexity. Compared to the other three representative methods, the 2D-PMLVD can achieve better anti-noise performance. The 2D-PMLVD, which is free of searching and has no identifiability problems, is more suitable for multicomponent situations. Through several simulations and analyses, the effectiveness of the proposed estimation algorithm is verified.

Radar high-speed maneuvering target detection based on joint second-order keystone transform and modified integrated cubic phase function

Long-time coherent integration is an effective means to improve the radar detection ability of high-speed maneuvering targets with jerk motion. However, the range migration (RM) and Doppler frequency migration (DFM) have a great impact on the integration performance. To overcome these problems, a unique method, called the second-order keystone transform modified integrated cubic phase function (SKT-MICPF), is proposed. In this method, the velocity compensation and SKT are jointly employed to correct the RM. After the RM correction, the azimuth echoes of a range cell where a target is located can be modeled as a cubic phase signal (CPS), whose chirp rate (CR) and quadratic CR are related to the target’s radial acceleration and jerk, respectively. Thereafter, an effective parameters’ estimation algorithm for CPS, called MICPF, is proposed and applied to compensate the DFM. After that, coherent integration and target detection are accomplished via the fast Fourier transform and constant false alarm rate technique, successively. Compared with the improved axis rotation discrete chirp Fourier transform, the SKT-MICPF achieves close detection performance, but greatly reduces the computational complexity. The results of simulation and real radar data demonstrate the validity of the proposed algorithm.

Inverse synthetic aperture radar imaging of targets with complex motion based on the local polynomial ambiguity function

In inverse synthetic aperture radar (ISAR) imaging of targets with complex motion, the azimuth echoes have to be modeled as multicomponent cubic phase signals (CPSs) after motion compensation. For the CPS model, the chirp rate and the quadratic chirp rate deteriorate the ISAR image quality due to the Doppler frequency shift; thus, an effective parameter estimation algorithm is required. This paper focuses on a parameter estimation algorithm for multicomponent CPSs based on the local polynomial ambiguity function (LPAF), which is simple and can be easily implemented via the complex multiplication and fast Fourier transform. Compared with the existing parameter estimation algorithm for CPS, the proposed algorithm can achieve a better compromise between performance and computational complexity. Then, the high-quality ISAR image can be obtained by the proposed LPAF-based ISAR imaging algorithm. The results of the simulated data demonstrate the effectiveness of the proposed algorithm.

Errata: Inverse synthetic aperture radar imaging of targets with complex motions based on modified chirp rate–quadratic chirp rate distribution for cubic phase signal

The <i>Journal of Applied Remote Sensing</i> (JARS) is an online journal that optimizes the communication of concepts, information, and progress within the remote sensing community to improve the societal benefit for monitoring and management of natural disasters, weather forecasting, agricultural and urban land-use planning, environmental quality monitoring, ecological restoration, and numerous other commercial and scientific applications.

Inverse synthetic aperture radar imaging of targets with complex motions based on modified chirp rate–quadratic chirp rate distribution for cubic phase signal

For targets with complex motions, the time-varying Doppler frequency deteriorates inverse synthetic aperture radar (ISAR) images. After range alignment and phase adjustment, azimuth echoes in a range cell can be modeled as multicomponent cubic phase signals (CPSs). The chirp rate and the quadratic chirp rate of the CPS are identified as the causes of the time-varying Doppler frequency; thus, it is necessary to estimate these two parameters correctly to obtain a well-focused ISAR image. The parameter-estimation algorithm based on the modified chirp rate-quadratic chirp rate distribution (M-CRQCRD) is proposed for the CPS and applied to the ISAR imaging of targets with complex motions. The computational cost of M-CRQCRD is low, because it can be implemented by the fast Fourier transform (FFT) and the nonuniform FFT easily. Compared to two representative parameter-estimation algorithms, the M-CRQCRD can acquire a higher antinoise performance due to the introduction of an optimal lag-time. Through simulations and analyses for the synthetic radar data, the effectiveness of the M-CRQCRD and the imaging algorithm based on the M-CRQCRD are verified.

ISAR Imaging of Targets With Complex Motions Based on Modified Lv’s Distribution for Cubic Phase Signal

For targets with complex motions, such as highly maneuvering airplanes and ships fluctuating with oceanic waves, the Doppler frequencies of scatterers are actually time-varying and azimuth echoes of a range cell have to be modeled as multicomponent cubic phase signals (CPSs) after the range alignment and the phase adjustment. In inverse synthetic aperture radar (ISAR) imaging based on the CPS model, the chirp rate and the quadratic chirp rate are identified as causes of the image defocus. In this paper, by employing a novel parametric symmetric self-correlation function and the keystone transform, an effective estimation algorithm, known as the modified Lv’s distribution (MLVD), is presented for the CPS and applied to ISAR imaging of targets with complex motions. The MLVD is simple and can be easily implemented using the complex multiplication, the fast-Fourier transform (FFT), and the inverse FFT (IFFT). The implementation, the cross-term, the antinoise performance, and the computational cost are analyzed for the MLVD. Compared to three representative estimation algorithms for the CPS, the MLVD can acquire a higher antinoise performance and eliminate the brute-force searching without the interpolation. Through simulations and analyses for synthetic models and the real radar data, we verify the effectiveness of the MLVD and the corresponding ISAR imaging algorithm.

ISAR Imaging of Nonuniformly Rotating Target Based on a Fast Parameter Estimation Algorithm of Cubic Phase Signal

In inverse synthetic aperture radar (ISAR) imaging of nonuniformly rotating targets, such as highly maneuvering airplanes and ships fluctuating with oceanic waves, azimuth echoes have to be modeled as cubic phase signals (CPSs) after the range migration compensation and the translational-induced phase error correction. For the CPS model, the chirp rate and the quadratic chirp rate, which deteriorate the azimuth focusing quality due to the Doppler frequency shift, need to be estimated with a parameter estimation algorithm. In this paper, by employing the proposed generalized scaled Fourier transform (GSCFT) and the nonuniform fast Fourier transform (NUFFT), a fast parameter estimation algorithm is presented and utilized in the ISAR imaging of the nonuniformly rotating target. Compared to the scaled Fourier transform-based algorithm, advantages of the fast parameter estimation algorithm include the following: 1) the computational cost is lower due to the utilization of the NUFFT, and 2) the GSCFT has a wider applicability in ISAR imaging applications. The CPS model and the algorithm implementation are verified with the real radar data of a ship target. In addition, the cross-term, which plays an important role in correlation algorithms, is analyzed for the fast parameter estimation algorithm. Through simulations of the synthetic data and the real radar data, we verify the effectiveness of the fast parameter estimation algorithm and the corresponding ISAR imaging algorithm.

ISAR Imaging for Fluctuating Ships Based on a Fast Bilinear Parameter Estimation Algorithm

For inverse synthetic aperture radar (ISAR) imaging of ships fluctuating with oceanic waves, azimuth echoes of a range cell have to be modeled as cubic phase signals (CPSs) after the range alignment and the phase adjustment. In ISAR imaging based on the CPS model, the chirp rate and the quadratic chirp rate are identified as causes of the target image defocus and need to be estimated with an effective algorithm. In this paper, a fast bilinear parameter estimation algorithm is proposed and applied in ISAR imaging for fluctuating ships by employing the cubic phase bilinear function, the nonuniform fast Fourier transform (NUFFT), and the parameter space switching method. Compared to two existing representative parameter estimation algorithms for the CPS, the advantages of this proposed estimation algorithm are: 1) the computational cost is lower due to the NUFFT and the parameter space switching method and 2) the bilinearity and the energy accumulation operation guarantee a higher anti-noise performance and a better suppression on cross-terms. Through simulations on synthetic models and the real radar data, we verify the effectiveness of this fast bilinear parameter estimation algorithm and the corresponding ISAR imaging algorithm.

ISAR Imaging of Targets With Complex Motion Based on the Chirp Rate–Quadratic Chirp Rate Distribution

In inverse synthetic aperture radar (ISAR) imaging of targets with complex motion such as fluctuating ships with oceanic waves and high maneuvering airplanes, the azimuth echo signals can be modeled as cubic phase signals (CPSs) after the migration compensation. The chirp rate (CR) and the quadratic chirp rate (QCR) are two important physical quantities of the CPS, which deteriorate the azimuth focusing quality due to the Doppler frequency shift. With these two quantities, other parameters can be estimated by using the fast Fourier transform (FFT). Therefore, the CPS can be uniquely determined by both CR and QCR. In this paper, based on the proposed generalized keystone transform and the parametric instantaneous autocorrelation function, a novel distribution of the CPS, known as the CR–QCR distribution (CRQCRD), is presented and applied in a newly proposed ISAR imaging algorithm for targets with complex motion. The CRQCRD is simple and only requires the FFT and the nonuniform FFT (NUFFT). Owing to the application of the NUFFT, the computational cost is saved, and the searching procedure is unnecessary for the nonuniformly spaced signal. Compared to other four representative methods for CPSs, the CRQCRD, which can acquire higher antinoise performance and no error propagation, is searching-free and more suitable for the situation of multitargets. Several simulation examples, analyses of the antinoise performance, and ISAR images validate the effectiveness of the CRQCRD and the corresponding ISAR imaging algorithm.