Multi-component Linear Frequency Modulation Research Articles

High-resolution multicomponent LFM parameter estimation based on deep learning

This paper addresses the complex challenge of parameter estimation in multi-component Linear Frequency Modulation (LFM) signals by introducing an innovative approach to high-resolution Fractional Fourier Transform (FrFT) parameter estimation, facilitated by convolutional neural networks. Initially, it analyzes the issues of peak shifts and the masking of weaker components due to spectral overlap in the FrFT domain of multi-component LFM signals. Convolutional neural networks are then employed to train and achieve high-resolution representations of FrFT parameters. Specifically, convolutional modules with residual structures are utilized to learn coarse features, while a weighted attention mechanism refines independent features across both channel and spatial dimensions. This approach effectively addresses the challenges posed by spectral peak overlap and frequency shifts in multi-component LFM signals, thereby enhancing the quality of high-resolution parameter estimation. Experimental results demonstrate that the proposed method significantly outperforms traditional methods in processing multi-component LFM signals. Moreover, it exhibits robust detection capabilities for both weak and compact components, thereby underscoring its potential applicability in the field of complex signal processing.

Improved Lv’s Distribution for Noisy Multicomponent LFM Signals Analysis

This paper presents the improved Lv’s distribution (ImLVD) for noisy multicomponent linear frequency-modulated (LFM) signals analysis, which is of significant importance in radar signal processing. Two goals of this paper are (i) to overcome drawbacks of the Lv’s distribution (LVD), and (ii) to study mechanisms of the constant delay introduction. Theoretical comparisons in cross-term suppression, resolution, peak-to-sidelobe level, anti-noise performance and implementation are performed for the maximum likelihood (ML) method, Wigner–Hough transform (WHT), LVD, parameterized centroid frequency–chirp rate distribution (PCFCRD) and ImLVD. Based on theoretical comparisons and illustrative examples, superiorities of the ImLVD are demonstrated and several unclear mechanisms of the introduced constant delay are interpreted. Finally, three numerical examples are given to illustrate that, because of the high cross-term suppression, resolution, peak-to-sidelobe level and anti-noise performance without the non-uniform integration variable, the ImLVD is more suitable for noisy multicomponent LFM signals analysis.

LFM signal parameter estimation in the fractional Fourier domains: Analytical models and a high-performance algorithm

The index of the peak magnitude of the fractional Fourier transform (FrFT) of linear frequency modulated (LFM) signals has been widely used as a powerful chirp-rate estimator. In this paper, we propose analytical approximations for the peak FrFT magnitude (PFM) of mono and multicomponent LFM signals, both for noiseless and noisy cases. In addition, we present a novel coarse-to-fine FrFT-based algorithm designed specifically for chirp-rate estimation of multi-component LFM signals. Our approach entails an initial coarse estimation of the chirp-rates for each component by utilizing our proposed mathematical models. By leveraging these models, we achieve improved performance and a reduced signal-to-noise breakdown threshold. Furthermore, we incorporate a unique and efficient estimate-and-subtract strategy to refine the estimated parameters using our proposed models. Rather than removing the components from the LFM signal, we utilize the derived model to identify and remove peaks in the PFM. This strategy enhances the algorithm’s capability to handle challenging scenarios. Extensive simulation results demonstrate that our proposed algorithm performs very close to the Cramér–Rao lower bound. It effectively eliminates the leakage effect between signal components, avoids error propagation, and maintains an acceptable computational cost compared to other state-of-the-art methods.

Refocusing Swing Ships in SAR Imagery Based on Spatial-Variant Defocusing Property

Synthetic aperture radar (SAR) is an essential tool for maritime surveillance in all weather conditions and at night. Ships are often affected by sea breezes and waves, generating a three-dimensional (3D) swinging motion. The 3D swing ship can thereby become severely defocused in SAR images, making it extremely difficult to recognize them. However, refocusing 3D swing ships in SAR imagery is challenging with traditional approaches due to different phase errors at each scattering point on the ship. In order to solve this problem, a novel method for refocusing swing ships in SAR imagery based on the spatial-variant defocusing property is proposed in this paper. Firstly, the spatial-variant defocusing property of a 3D swing ship is derived according to the SAR imaging mechanism. Secondly, considering the spatial-variant defocusing property, each azimuth line of the SAR 3D swing ship image is modeled as a multi-component linear frequency modulation (MC-LFM) signal. Thirdly, Fractional Autocorrelation (FrAc) is implemented in order to quickly calculate the optimal rotation order set for each azimuth line. Thereafter, Fractional Fourier Transform (FrFT) is performed on the azimuth lines to refocus their linear frequency modulation (LFM) components one by one. Finally, the original azimuth lines are replaced in the SAR image with their focused signals to generate the refocused SAR image. The experimental results from a large amount of simulated data and real Gaofen-3 data show that the proposed algorithm can overcome the spatial-variant defocusing of 3D swing ships. Compared with state-of-the-art algorithms, our approach reduces the image entropy by an order of magnitude, leading to a visible improvement in image quality, which makes it possible to recognize swing ships in SAR images.

An improved signal-dependent QTFD based on iterative regional RGK optimization for multi-component LFM signals

It is quite challenging for the quadratic time-frequency distribution (QTFD) to analyze the multi-component linear frequency modulation (MC-LFM) signals due to its inevitable cross-term issue, particularly in the case of simultaneous strong components and weak components involved in the MC-LFM signals. This paper proposes a novel signal-dependent QTFD based on regional radially Gaussian kernel (RRGK) to improve the time-frequency representations (TFR) of the MC-LFM signals with distinguishing component amplitudes. The model of MC-LFM signals is firstly introduced, and the detrended Radon transform of ambiguity domain (RTA) technique is combined with energy distribution feature in the ambiguity domain analysis for the detection of potential auto-terms concentrated along lines passing through the origin of the Doppler-lag plane. The combined two-step signal detection method can be easily extended to other signal-dependent QTFDs. Then the Doppler-lag distribution area of each detected auto-term is automatically selected with stronger components peeled to restrict the searching area of RGK. Afterwards, a novel iterative regional RGK-based TFD (IRRGD) is developed to resolve strong and weak components under a proper trade-off between auto-term resolution and cross-term suppression in the TFD. Finally, filtering is performed in (t, f) domain to further improve the TFD results. Also, a quantitative indicator is proposed to measure weak signal preservation. Both simulation and experimental results have verified the superiorities of the proposed method over other existing state-of-the-art algorithms.

Three Dimension Airborne SAR Imaging of Rotational Target With Single Antenna and Performance Analysis

For the target with 3-D rotation, the 3-D synthetic aperture radar (SAR) imaging is very important for the feature extraction and classification. To solve this issue, a novel 3-D imaging algorithm through the airborne SAR system with a single antenna is proposed in this article, which has great advantages for the simplification of system structure compared with the traditional interferometric system. The proposed 3-D airborne SAR imaging algorithm can be implemented with three steps: 1) the azimuth signal is modeled as multicomponent linear frequency modulation (LFM) signal due to the relative movement between the target and radar. 2) The scatterer height position can be obtained by estimating the frequency modulation rate (FMR) for the LFM signal. 3) The 3-D airborne SAR image is obtained via the range-Doppler (RD) algorithm. Furthermore, the reconstruction performance under different rotation patterns, including roll, pitch, and yaw, is analyzed. The availability of the presented novel technique is demonstrated by the results of simulated and real experimental data.

A Fast and Accurate Method for Parameter Estimation of Multi-Component LFM Signals

This letter presents a fast and accurate method for parameter estimation of multi-component linear frequency modulated (MC-LFM) signals. We first show that the auto-terms of signal in fractional auto-correlation (FrAc) domain are concentrated along lines passing through the origin with slopes equal to chirp rates. The connection between the FrAc and fractional Fourier transform (FrFT) is exploited to find the chirp rates. We then easily obtain central frequencies from the same FrFT coefficients which are used for the estimation of chirp rates. We finally propose our efficient algorithm to accelerate the precise estimate of the MC-LFM signal parameters. Simulation results validate performance of the approach.

An improved signal-dependent quadratic time-frequency distribution using regional compact kernels for analysis of nonstationary multicomponent LFM signals

The time-frequency analysis of nonstationary multicomponent linear frequency modulation (LFM) signals is still a challenging task due to their cross-term interferences. In this paper, a novel signal-dependent adaptive compact kernel based time-frequency distribution (CKD) named regional adaptive CKD (RACKD) is proposed. The proposed RACKD firstly represents the signal in the Doppler-lag (ν, τ) ambiguity domain, and then applies a signal-dependent regional compact kernel to filter the cross-term interferences, with preserving the auto-terms in the ambiguity domain. By automatically selecting the auto-terms' distribution area which represents the signals in the ambiguity domain, the parameters of the newly designed signal-dependent regional compact kernel can be adaptively set according to the signals to be processed. The proposed RACKD shows a more stable performance in the ability of cross-term rejection and component energy of signals concentration around their respective instantaneous frequency, compared to several existing state-of-the-art methods. Both simulation results and experimental results based on sea-truth data have validated the improved performance of the proposed method. For the simulated signal composed of one weak and two strong LFM components with crossing frequency, the proposed RACKD shows up to 2.6% gain as compared to the signal-dependent multidirectional distribution (MDD) approach.

Frequency estimation of chirp signals based on fractional fourier transform combined with Otsu’s method

The parameters estimation of linear frequency modulation (LFM) signals, which also called chirp signals, have been widely applied in optical communication, LIDAR and many other fields. Fractional Fourier transform (FRFT) is one of the most important methods for estimating the frequency of chirp signals. However, the traditional estimation methods based on FRFT are incapable of high-precision parameter estimation for chirp signals, especially multi-component LFM signals. Therefore, we present a novel parameter estimation method based on the Otsu ratio, i.e., the interclass ratio of impulse signal to noise. An algorithm is used to determine this ratio, and then the frequency parameters of chirp signals are estimated by using the ratio as the fitness function. The parameter estimation of single-component and multi-component chirp signals is simulated. The results show that the proposed method has higher estimation accuracy than the traditional FRFT method.

Parameter estimate of multi-component LFM signals based on GAPCK

In this paper, we propose a fast and robust parameter estimate method for multi-component linear frequency modulated (LFM) signals from a finite number of noisy discrete time observations. First, a new kernel called generalized adjustable parameter correlation kernel (GAPCK) is introduced to avoid the coupling terms between time and lag. Then, Matched Fourier transform (MFT) and Robust Energy accumulation (REA) are utilized to obtain the chirp rate estimate of the received LFM signals. This estimate is used to compensate the time-quadratic phase term of the GAPCK, and then Fast Fourier transform (FFT) along time-axis is performed to estimate the constant coefficient. Moreover, the asymptotic statistical properties of parameter estimates are derived. The proposed method has low computational complexity and favorite performance under low signal-to-noise ratio (SNR) due to the low-order non-linearity of the GAPCK. Finally, simulated and real data are provided to verify the robustness and effectiveness of the proposed method.

Fault diagnosis for rolling bearing based on VMD-FRFT

In order to solve the problems that the fault feature of rolling bearing in early failure period is difficult to extract and the over-decomposition problem arises when applying the variational mode decomposition (VMD), in this paper, the fault diagnosis for rolling bearing based on variational mode decomposition–fractional Fourier transform (VMD-FRFT) is proposed. Firstly, based on the theory of fractional Fourier transform parameter estimation, the definition and derivation of VMD-FRFT are given and the modal component and center frequency estimation of the method are obtained. Secondly, the multicomponent linear frequency modulation (LFM) signal is simulated to verify the non-fragile to the center frequency distribution, the insensitive to the number of modes and the noise robustness of VMD-FRFT. Finally, VMD-FRFT is applied to the fault diagnosis of uniform acceleration rolling bearings, which can provide an effective filtering algorithm for the extraction of fundamental frequency and frequency multiplication of instantaneous frequency. Therefore, the VMD-FRFT proposed in this paper has certain reference significance for the fault diagnosis of rolling bearings.

Sparsely Localized Time-Frequency Energy Distributions for Multi-Component LFM Signals

This letter presents a high resolution method which separates close components of a multi-component linear frequency modulated (LFM) signal and eliminates their Cross-Terms (CTs). We first investigate the energy distribution of the Auto-Terms (ATs) and CTs in ambiguity plane. This reveals that the energy of the CTs of parallel close components is significant around the origin. We propose to mask the samples in which the CTs may have interferences with the ATs. This mask is signal-dependent and its directions are determined using the relationship between the radial slices of ambiguity function (AF) and the fractional Fourier transform (FrFT). Exploiting sparsity in time-frequency (TF) domain and by solving an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization problem, the localized time-frequency distribution (TFD) is extracted from the acquired samples of the AF. Simulation results reveal significant improvements in the efficiency compared to previous works.

Fractional iterative variational mode decomposition and its application in fault diagnosis of rotating machinery

Variational mode decomposition (VMD), a recently developed adaptive mode decomposition technique, has attracted much attention in various fields. However, due to the assumption that the obtained intrinsic mode functions should be band-limited and separable in the Fourier domain, VMD has experienced many obstacles when processing wideband nonstationary signals. In this paper, a new method named fractional iterative variational mode decomposition (FrIVMD) is proposed for the decomposition of a multicomponent linear frequency modulation signal. By accurately estimating the chirp rate of the linear frequency modulation (LFM) component, the original signal is mapped to the fractional Fourier domain by the fractional Fourier transform (FRFT), where the corresponding LFM component is narrowly banded. Then, the conventional VMD is applied to separate the components. Finally, the signal mode in the time domain is obtained by the inverse FRFT. Numerical and real-world vibration signals are employed to validate the effectiveness of the FrIVMD technique. The results prove that the proposed method performs well for noisy signals and even signals containing weak components.

An adaptive method based on fractional empirical wavelet transform and its application in rotating machinery fault diagnosis

Compared with the stable running condition, the vibration signals of rotating machinery during the startup or run-down stage have more abundant information, and these are more sensitive to weak changes in the machine operating state. Therefore, it is of great importance to accurately and efficiently extract state feature information of rotor startup vibration signals. However, non-stationary cases such as amplitude modulation, frequency modulation and a strong background noise do exist in a rotor system with variable speeds, which makes the extraction of feature information more difficult. As a novel analysis method, empirical wavelet transform (EWT) can automatically extract empirical modes of non-stationary signals. Nevertheless, rotor startup vibration signals can be seen as multi-component linear frequency modulation (LFM) signals whose components overlap with each other in the Fourier spectrum, which makes the EWT analysis method no longer valid. Therefore, a new analysis method named ‘fractional empirical wavelet transform (FrEWT)’ is proposed in this paper, which effectively combines the advantages of fractional Fourier transform (FrFT) and EWT. On the one hand, FrFT is very suitable for analyzing LFM signals, which provides each LFM component of the rotor startup vibration signal with compact support and enables the components to be separated from each other in an appropriate fractional Fourier domain. On the other hand, based on the analysis of the EWT, a wavelet filter bank in the fractional Fourier domain is constructed adaptively to extract the fault feature components of rotor startup vibration signals. Finally, the effectiveness of the proposed method is verified by both simulated and experimental data. The analysis results prove that the proposed method demonstrates excellent performance.

Parameter estimation of linear frequency modulation signals based on sampling theorem and fractional broadening.

Linear frequency modulation (LFM) signals are a class of important radar signals, but it is difficult to estimate their parameters in electronic warfare. Fractional Fourier transform (FRFT) is one of the most important methods for estimating the parameters of LFM signals, but the computational efficiency is strongly influenced by the search range and evaluation of the optimum FRFT order. To improve the estimation speed, we present a novel method for the parameter estimation of LFM signals using sampling theorem and fractional broadening. First, sampling theorem is used to calculate the search range of the optimum FRFT transform order. Then, the LFM signals are transformed by FRFT in the search range. Finally, the fractional broadening of the LFM signals is calculated, and the optimum FRFT order is obtained according to the relationship between the fractional broadening and the FRFT order. Experiments are performed to compare the proposed method and the traditional FRFT method. The results show that the proposed method is six times faster than the traditional FRFT method while preserving the accuracy. Moreover, it can quickly and accurately achieve the parameter estimation of multi-component LFM signals in the case of white Gaussian noise with a low signal-to-noise ratio.

Threshold Decision Algorithm Based On The Maximum Interclass Variance For TDCS

Transform domain communication system (TDCS) is a new type of wireless communication system, who can take the initiative to avoid interference. In this paper, Fractional Fourier Transforms (FRFT) is introduced into Transform Domain Communication Systems (FRFT-TDCS), which can effectively target linear frequency modulation (LFM). But the multicomponent linear frequency modulation (MLFM) is difficult to effectively eliminate, when using traditional threshold decision algorithm of TDCS. A novel threshold decision algorithm for FRFT-TDCS is proposed in this paper. Because the order of FRFT has no effect on noise, the threshold decision algorithm by using the idea of maximum interclass variance means minimum misclassification probability. Simulation results show that compared with the traditional hard threshold decision algorithm, the proposed algorithm can effectively eliminate MLFM interference and negative influence of the order of FRFT. Whether bipolar modulation or cyclic shift keying(CSK) modulation, the bit error rate(BER) of this threshold decision algorithm is always better than the traditional threshold decision algorithm; With the increase of signal to noise ratio(SNR), the BER performance of the two threshold decision algorithms increase gradually.

Refining the ambiguity domain characteristics of non-stationary signals for improved time–frequency analysis: Test case of multidirectional and multicomponent piecewise LFM and HFM signals

This paper aims at providing a more accurate description of the ambiguity domain characteristics of a piecewise multicomponent non-stationary signals with focus on piece-wise linear frequency modulated (LFM) (PW-LFM) signal and a mixed LFM and hyperbolic FM (HFM). The main motivation comes from the observed PW-LFM nature of several real life signals. It is essential that the characteristics of these types of signals be taken into account for the design of high resolution Time–Frequency Distributions (TFDs) and therefore to improve the application and interpretation of time–frequency signal analysis and processing. In this paper the ambiguity function (AF) of a general PW-LFM signal is derived exactly and then analyzed to deduce important properties. The precise location and behavior of both auto-terms and cross-terms of a general piecewise LFM signal can be deduced from its AF. Numerical simulations using different types of test signals confirm the analytical derivations. An extension of the PW-LFM test signal is also presented by using HFM signals. For such signals, the exact analytical expression of auto-terms is given as well as the expression of cross-terms between HFM and LFM. The results obtained can be used in future studies to design more advanced quadratic time–frequency distributions (QTFDs) that exhibit improved properties in terms of resolution and accuracy.

A Novel Approach of Slope Detection Combined with Lv’s Distribution for Airborne SAR Imagery of Fast Moving Targets

This paper presents a fast SAR imagery algorithm for Ground Moving Target Imaging (GMTIm) based on the slope detection strategy combined with Time-Frequency Representation (TFR), which is known as Lv’s Distribution (LVD). This fast imagery algorithm focuses on the solution of the ambiguity problems and relevant heavy computing load in SAR imagery. Firstly, according to the relationship between the slope of the range walk trajectory and the cross-track velocity of moving target, a new high-efficiency slope detection strategy based on gradient and level-line angle is presented in the image domain. Then, the Doppler centroid shift induced by cross-track velocity can also be obtained. Secondly, owing to the cross-track velocity estimated before, the Range Walk Migration Correction (RWMC) can be performed to concentrate the echo response of the moving target into a single range cell. Finally, due to the superior performance in representing multi-component Linear Frequency Modulation (LFM) signal, LVD is adopted here to represent the Doppler chirp rate of multiple moving targets in a Doppler Centroid Frequency and Chirp Rate domain (CFCR). The performance of the proposed algorithm is evaluated in terms of superiority and effectiveness using simulations, and the comparison between the proposed algorithm and the other conventional algorithms is also presented.

An Effective LFM Signal Reconstruction Method for Signal Denoising

One of the main challenges in signal denoising is to accurately restore useful signals in low signal-to-noise ratio (SNR) scenarios. In this paper, we investigate the signal denoising problem for multi-component linear frequency modulated (LFM) signals. An effective time-frequency (TF) analysis-based approach is proposed. Compared to the existing approaches, our proposed one can further increase the noise suppressing performance and improve the quality of the reconstructed signal. Experimental results are presented to show that the proposed denoising approach is able to effectively separate the multi-component LFM signal from the strong noise environments.

On the Estimation of LFM Signal Parameters: Analytical Formulation

We propose analytical formulations, approximations, upper and lower bounds for the angle sweep of maximum magnitude of fractional Fourier transform of mono- and multicomponent linear frequency modulated (LFM) signals. We employ a successive coarse-to-fine grid-search algorithm to estimate the chirp rates of multicomponent nonseparable LFM signals. Extensive numerical simulations show the validity of analytical formulations and performance of the proposed estimator. Obtained analytical results may also find themselves other application areas, where nonstationary signals are of interest.