Multi-component Linear Frequency Modulated Signals 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.

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.

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.

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.

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.

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.

Parameterized Centroid Frequency-Chirp Rate Distribution for LFM Signal Analysis and Mechanisms of Constant Delay Introduction

This paper presents an extension of the integrated time-chirp rate analysis technique, known as the parameterized centroid frequency-chirp rate distribution (PCFCRD), for noisy multicomponent linear frequency modulated signals analysis. The PCFCRD is based on a newly defined correlation function and its auto term accumulation is two-dimensionally coherent. The principle, cross term characteristic, implementation, properties, parameter selection criterion, antinoise performance, and estimation accuracy are analyzed for the PCFCRD. In this paper, comparisons with the coherently integrated cubic phase function, maximum likelihood method, and Lv's distribution are performed. With mathematic analyses and numerical simulations, we demonstrate that the PCFCRD outperforms these three representative counterparts. Further, by combining with analyses of the PCFCRD, several unclear mechanisms of the constant delay introduction are mathematically interpreted, especially quantitative influences on the chirp rate resolution and theoretical antinoise performance.

Performances Analysis of Coherently Integrated CPF for LFM Signal Under Low SNR and Its Application to Ground Moving Target Imaging

The detection and parameters estimation of linear frequency-modulated (LFM) signal are important for modern radar applications, but they are also challenged by the fact that echo signal is often of low signal-to-noise ratio (SNR) due to reasons of long imaging distance and/or limited transmitted power, and the target of small size and/or hidden characteristics. To enhance the SNR, in our previous work, a novel coherently integrated cubic phase function (CICPF) was recently developed for the parameters estimation of the multicomponent LFM signal. In the CICPF, the auto-terms are coherently integrated to enhance the performance in the case of low SNR and also to suppress the cross-terms and spurious peaks. In this paper, as an extension of our previous work, the theoretical performance analyses including several important properties and the fast implementation are provided. Furthermore, the asymptotic mean squared error of a CICPF-based estimator as well as the output SNR of a CICPF-based detector are theoretically derived in closed-forms. From the performance point of view, the proposed CICPF attains the Cramer-Rao bound at low input SNR. The complexity analysis also indicates that the CICPF with the nonuniform fast Fourier transform is computationally efficient without needing the interpolation operation and parameter search. Numerical studies of the CICPF confirm the theoretical analysis and demonstrate superior performance of the proposed approach compared with other state-of-the-art approaches, especially under the low-SNR condition. Finally, the proposed CICPF is applied for the ground moving target imaging in synthetic aperture radar. Results using simulated and experimental data demonstrate that it provides an effective means to obtain well-focused image for ground moving targets.

PCFCRD for multicomponent LFM signals analysis

A novel extension of the cubic phase function (CPF), known as the parameterised centroid frequency-chirp rate distribution (PCFCRD), is proposed for multicomponent linear frequency modulated signals analysis. The PCFCRD is coherent and introduces a constant delay to reduce the noise correlation. Mathematic analysis and numerical simulations demonstrate that, compared with the coherent integrated CPF, the PCFCRD can obtain a higher anti-noise performance.

A Keystone-Based Cosine Transform

This paper introduces a new time-frequency transform, referred to as the keystone-based cosine transform (KCT), to process real-valued multi-component linear frequency-modulated signals. Theoretical derivation shows that the KCT is a bilinear-based transform, but has a property of asymptotic linearity with negligible cross-terms. It also has high signal energy concentration. Experimental simulations show that compared with fractional cosine transform, the KCT can provide better performance on the distribution concentration and noisy signal detection.

Robust multicomponent LFM signals synthesis algorithm based on masked ambiguity function

Wigner distribution (WD) based multiple components linear frequency modulation (LFM) signal synthesis method (SSM) is often adversely affected by cross-terms. Masked WD (MWD) is one of the widely used cross-term suppression techniques in practice due to its simplicity and efficiency. However, the cross-terms are hardly masked out when auto-terms and cross-terms are overlapped (Case I) or the components are very close to each other in the time–frequency (TF) plane (Case II). To solve these problems, we present a robust ambiguity function (AF) based approach for multicomponent signals synthesis. This algorithm consists of two stages. First, a SSM from the AF is proposed according to matrix rearrangement and eigenvalue decomposition. However, the existence of cross-term makes the signal synthesis entirely erroneous. To settle this issue, we present a masked AF (MAF) algorithm based on Radon and its inverse transforms in the second stage. Applying the presented algorithm, multicomponent signals can be synthesized efficiently even in Case I and Case II. Simulation results demonstrate the effectiveness and feasibility of the proposed algorithm.

Coherently integrated cubic phase function for multiple LFM signals analysis

The cubic phase function (CPF) based estimator is efficient in estimating the parameters for mono-component linear frequency-modulated (LFM) signals. However, it suffers from cross-terms and spurious peaks when dealing with multi-component LFM signals. Aimed at this identifiability problem, a coherently integrated CPF (CICPF) algorithm is proposed to enhance the auto-terms and suppress spurious peaks. Comparisons with several existing algorithms are made, which show that the CICPF not only solve the identifiability problem for multi-component LFM signals, but also acquires high anti-noise performance.

Simultaneous measurement of in-plane and out-of-plane displacements using pseudo-Wigner-Hough transform

A new method based on pseudo-Wigner-Hough transform is proposed for the simultaneous measurement of the in-plane and out-of-plane displacements using digital holographic moiré. Multiple interference phases corresponding to the in-plane and out-of-plane displacement components are retrieved from a single moiré fringe pattern. The segmentation of the interference field allows us to approximate it with a multicomponent linear frequency modulated signal. The proposed method accurately and simultaneously estimates all the phase parameters of the signal components without the use of any signal separation techniques. Simulation and experimental results demonstrate the efficacy of the proposed method and its robustness against the variations in object beam intensity.

Fast Parameters Estimation of LFM Signals via Radon-Wigner Transform

The line integral of Radon-Wigner Transform (RWT) can transform the line in Wigner-Ville distribution into a point in the plane, which can be used to estimate the chirp rate and initial frequency. The calculation is costly because of two-dimensional (2-D) searching in the plane. Therefore, how to improve calculation velocity is the key of RWT method. Multi-resolution fast parameters estimation algorithm for Linear Frequency Modulation (LFM) signals via RWT is offered in this paper. The analysis results indicate that RWT is similar to Radon-ambiguity Transform (RAT) in calculation load. Performance of the parameters estimation is studied based on Residual Expansion Coefficient (REC) and Frequency Deviation Coefficient (FDC). Simulation results show that the algorithm has reached the Cramer-rao Bound (CRB) even on low SNR. The method has prospective benefits of parameters estimation for multi-component LFM signals.