Cross-term Suppression Research Articles

Dynamic mode decomposition-based technique for cross-term suppression in the Wigner-Ville distribution

This paper presents a new method for time-frequency representation (TFR) using dynamic mode decomposition (DMD) and Wigner-Ville distribution (WVD), which is termed as DMD-WVD. The proposed method helps in removing cross-term in WVD-based TFR. In the suggested method, the DMD decomposes the multi-component signal into a set of modes where each mode is considered as mono-component signal. The analytic modes of these obtained mono-component signals are computed using the Hilbert transform. The WVD is computed for each analytic mode and added together to obtain cross-term free TFR based on the WVD. The effectiveness of the proposed method for TFR is evaluated using Rényi entropy (RE). Experimental results for synthetic signals namely, multi-component amplitude modulated signal, multi-component linear frequency modulated (LFM) signal, multi-component nonlinear frequency modulated (NLFM) signal, multi-component signal consisting of LFM and NLFM mono-component signal, multi-component signal consisting of sinusoidal and quadratic frequency modulated mono-component signals, and synthetic mechanical bearing fault signal and natural signals namely, electroencephalogram (EEG) and bat echolocation signals are presented in order to show the effectiveness of the proposed method for TFR. It is clear from the results that the proposed method suppresses cross-term effectively as compared to the other existing methods namely, smoothed pseudo WVD (SPWVD), empirical mode decomposition (EMD)-WVD, EMD-SPWVD, variational mode decomposition (VMD)-WVD, VMD-SPWVD, and DMD-SPWVD.

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.

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.

Noise-Robust ISAR Translational Motion Compensation via HLPT-GSCFT

Translational motion compensation is a prerequisite of inverse synthetic aperture radar (ISAR) imaging. Translational motion compensation for datasets with low signal-to-noise ratio (SNR) is important but challenging. In this work, we proposed a noise-robust translational motion compensation method based on high-order local polynomial transform–generalized scaled Fourier transform (HLPT-GSCFT). We first model the translational motion as a fourth-order polynomial according to order-of-magnitude analysis, and then design HLPT-GSCFT for translation parameter estimation and parametric translational motion compensation. Specifically, HLPT is designed to estimate the acceleration and third-order acceleration of the translational motion and GSCFT is introduced to estimate the second-order acceleration. Both HLPT and GSCFT have a strong ability for cross-term suppression. In addition, we use a minimum weighted entropy algorithm to estimate the velocity of the translational motion, which can improve the noise robustness of the parameter estimation. Experimental results based on a measured dataset prove that the proposed method is effective and noise-robust.

Tunable hyperbolic Cohen-class kernel for cross-term diminishing in time–frequency distributions

Nonstationary signals are found in many engineering applications. The analysis of nonstationary signals trough bi-dimensional functions allows tracing their time and frequency evolution. In this regard, energy-distribution representations allocate the analyzed-signal power over two description variables, time and frequency, and offer a large number of mathematical properties, such as an ideal infinite resolution in both domains. Unfortunately, energy-distribution representations generate cross terms for multi-component nonstationary signals, which might mislead the analysis interpretation. Cross-term effects can be lessened by applying an appropriate smoothing kernel; they distort the shape of auto terms at the same time. In this work, a novel tunable signal-independent hyperbolic kernel function is introduced for reducing cross-term effects, preserving valuable properties, during the analysis of multi-component nonstationary signals. Obtained results from computer-model and real-life cases of study validate and demonstrate the performance superiority of the proposed kernel function, in terms of cross-term suppression and time–frequency resolution, against the well-known and widely used the Choi-Williams distribution and the cone-shape distribution.

A Time–Frequency Representation Approach of Undersampled Signals With Multiple Periodic FM Components

Periodic frequency-modulated (FM) interference signals, commonly employed by jammers and radars, may lead to the performance degradation or even failure of global navigation satellite systems and wireless communications. Time–frequency (TF) representations play an essential role in FM signal analysis and instantaneous frequency (IF) estimation, which is the basis of many nonstationary interference suppression algorithms. This article provides a TF representation approach for signals with multiple periodic FM components in the presence of sample loss. The proposed approach utilizes the periodicity and the sparsity of signal components in the TF domain to suppress cross-terms and artifacts. We reconstruct TF distributions of each component by time-averaging Wigner–Ville distribution. Then, the residual artifacts are eliminated by a low-complexity sparse-based adaptive directional TF filtering method. Simulation results confirm that the proposed approach provides an outstanding suppression of cross-terms and artifacts. The proposed TF representation approach can effectively improve the accuracy of the IF estimation of each signal component.

Time-frequency readability enhancement of compact support kernel-based distributions using image post-processing: Application to instantaneous frequency estimation of M-ary frequency shift keying signals

Time-frequency distributions (TFDs) based on time-lag kernels with compact support (KCS) have proved their high performance in terms of resolution and crossterms suppression. However, as for all kernel-based quadratic TFDs, these distributions suffer from spreading out signal terms. This is due to the unavoidable smoothing effects of the kernel in the ambiguity domain. The main objective of this paper is to improve concentration, interference rejection and so time-frequency localization of this representation class. The latter has the advantage of being tuned using a single parameter while external windows are no longer needed. The KCS-TFDs, referred to as KCSDs, are first optimized using a selection of the most used objective performance measures in the literature. Important signal features are extracted as well through analysis of time slice plots. The obtained TF diagrams are then enhanced using a specific method that includes two-dimensional Wiener filter, automatic binarization and morphological image processing techniques. The enhanced plots are compared to those obtained from the original TFDs using several tests on real-life and multicomponent frequency modulated (FM) signals including the noise effects. Moreover, a comparative study involving a selection of the best-performing reassignment time-frequency distributions is provided. The obtained results show a significant improvement of concentration, time-frequency localization of the autoterms as well as interference and noise suppression. As viable applications, the proposed approach is used first to instantaneous frequency (IF) estimation of several synthetic and real-life M-ary frequency shift keying (MFSK) signals. It is shown that the IF estimator from the enhanced plots performs better than smoothed pseudo Wigner-Ville distribution (SPWVD) and reassignment post-processing-based TFDs in terms of mainlobe width (MLW) and variance, respectively, even at low signal-to-noise ratio (SNR). On the other hand, time-frequency characterization of continuous wave linear frequency modulation (CW-LFM) and pulse linear FM (PLFM) radar signals is also investigated.

A Data-Driven High-Resolution Time-Frequency Distribution

The design of high-resolution and cross-term (CT) free time-frequency distributions (TFDs) has been an open problem. Classical kernel based methods are limited by the trade-off between resolution and CT suppression, even under optimally derived parameters. To break the current limitation, we propose a data-driven model directly based on Wigner-Ville distribution (WVD). The proposed data-driven high-resolution TFD (DH-TFD) includes several stacked multi-channel convolutional kernels. Specifically, convolutional layers with skipping operators are utilized to learn coarse features, while a weighted block is employed to refine these features independently in both channel and spatial dimensions. By doing so, CTs can be effectively eliminated while maintaining a high resolution. Numerical experiments on both synthetic and real-world data confirm the superiority of the proposed DH-TFD in simultaneously extracting and representing a target signal over state-of-the-art methods.

An Efficient Cross-Terms Suppression Method in Time–Frequency Domain Reflectometry for Cable Defect Localization

Time–frequency domain reflectometry (TFDR) is a highly sensitive method to locate the defects of cables. However, there are cross terms when a multicomponent signal is converted to the time–frequency domain by the conventional Wigner–Ville distribution (WVD), which will greatly interfere with the localization results. To solve this problem, pseudo Wigner–Ville distribution (PWVD) with Gaussian-rectangular window is proposed in this article, which can significantly reduce the amplitude of the cross terms without changing the energy distribution in the time–frequency domain. The simulation results show that the normalized amplitude of the cross term has been greatly reduced with the method proposed in this article. Due to the large number of independent variables in this algorithm and large storage requirements for calculation, a simplification method based on dimensionality reduction is proposed to improve the computational efficiency. Experiments were also carried out to verify the effectiveness of this algorithm. The amplitudes of the localization curve at the defects based on PWVD are much larger than the results via conventional WVD. Besides, the numerator of the cross correlation function between the incident signal and the reflected signal is found to be related to the degree of the defect. The method is expected to significantly improve diagnosis accuracy in cable defect localization and assessment in the future.

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

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

Cross-Term Suppression Method for Time-Frequency Spectrum of Engineering Blasting Signals

Abstract Engineering blasting vibration signals are affected by the test environment and contain noise and trend components, which leads to significant cross-term in the time-frequency (TF) distribution. To address these problems, a combined algorithm of matching pursuit (MP) and smooth pseudo-Wigner-Ville distribution (SPWVD) is proposed to suppress the TF spectrum cross-term and extract the signal features. An overcomplete atom library is constructed for signal feature matching according to sparse signal representation theory. Atomic reconstruction of the subsignal is achieved by MP decomposition, and the SPWVD distribution of the reconstructed subsignal is calculated and superimposed, yielding a more accurate TF expression of the blasting signal. The results show that the combined MP-SPWVD method can accurately suppress cross-term and improve the TF identification and feature analysis ability, which verifies the accuracy of its application in signal TF analysis. The combined algorithm is thus suitable for TF feature extraction of engineering blasting signals.

Crossterm-free time-frequency representation exploiting deep convolutional neural network

Bilinear time-frequency (TF) analyses provide high-resolution time-varying frequency characterization of nonstationary signals. However, because of their bilinear natures, such TF representations (TFRs) suffer from crossterms. TF kernels, which amount to low-pass weighting or masking in the ambiguity function domain, are commonly used to reduce crossterms. However, existing fixed and adaptive kernels do not guarantee effective crossterm suppression and autoterm preservation, particularly for signals with overlapping autoterms and crossterms in the ambiguity function. In this paper, we develop a new method that offers high-resolution TFRs of nonstationary signals with desired autoterm preservation and crossterm mitigation capabilities, especially for signals with slowly time-varying instantaneous frequencies. The proposed method exploits a convolutional autoencoder network which is trained to construct crossterm-free TFRs. For the signals being considered, the proposed technique with properly trained networks offers the capability to outperform state-of-the-art TF analysis algorithms based on adaptive kernels and compressive sensing techniques.

A Novel GNSS Interference Detection Method Based on Smoothed Pseudo-Wigner-Hough Transform.

This paper develops novel Global Navigation Satellite System (GNSS) interference detection methods based on the Hough transform. These methods are realized by incorporating the Hough transform into three Time-Frequency distributions: Wigner–Ville distribution, pseudo -Wigner–Ville distribution and smoothed pseudo-Wigner–Ville distribution. This process results in the corresponding Wigner–Hough transform, pseudo-Wigner–Hough transform and smoothed pseudo-Wigner–Hough transform, which are used in GNSS interference detection to search for local Hough-transformed energy peak in a small limited area within the parameter space. The developed GNSS interference detection methods incorporate a novel concept of zero Hough-transformed energy distribution percentage to analyze the properties of energy concentration and cross-term suppression. The methods are tested with real GPS L1-C/A data collected in the presence of sweep interference. The test results show that the developed methods can deal with the cross-term problem with improved interference detection performance. In particular, the GNSS interference detection performance obtained with the smoothed pseudo-Wigner–Hough transform method is at least double that of the Wigner–Hough transform-based approach; the smoothed pseudo-Wigner–Hough transform-based GNSS interference detection method is improved at least 20% over the pseudo-Wigner–Hough transform-based technique in terms of the zero Hough-transformed energy percentage criteria. Therefore, the proposed smoothed pseudo-Wigner–Hough transform-based method is recommended in the interference detection for GNSS receivers, particularly in challenging electromagnetic environments.

Internal Combustion Engine Fault Diagnosis Method Based MICEEMD-PWVD

As to multi-signals, the figure of time-frequency distribution becomes difficult to understand due to cross-terms in the Wigner-Ville distribution (WVD). At present time, cross-terms suppression of time-frequency analysis is one of the top interests in signal processing, and more and more research had been put on this topic, but the present research on cross-terms suppression exist conflicting problem that these methods couldn’t suppress the cross-terms while holding high time-frequency resolution. A novel method of cross-terms suppression in the WVD based on Complementary Ensemble Empirical Mode Decomposition (CEEMD) was proposed to solve the above problem. Firstly, multi-component signals were decomposed into intrinsic mode functions (IMFs) which were single component signal by CEEMD; Secondly, the Mutual Information values of IMFS and original signal were calculated to judge and delete the CEEMD false components; Thirdly, these Wigner-Ville distributions of true component of IMFs were computed; Finally, WVDs of each single component signal were added linearly to reconstruct the WVD of original signal. Test on synthetic data shows the effectiveness of the proposed method, which can suppress the cross-terms while holding high time-frequency resolution. Apply MICEEMD - PWVD in internal combustion engine valve clearance fault diagnosis, useTD-2DPCA method to extract MICEEMD-PWVD time-frequency image feature, then classify the characteristic parameters with the nearest neighbor classifier. The results show that the IC Engine Fault Diagnosis Method Based MICEEMD-PWVD and TD-2DPCA can accurately diagnose IC engine valve fault.

ISAR Imaging for Low-Earth-Orbit Target Based on Coherent Integrated Smoothed Generalized Cubic Phase Function

In the high-resolution inverse synthetic aperture radar (ISAR) imaging of low-earth-orbit space targets, the received signal from one range bin can be modeled as a multicomponent cubic phase signal (CPS) after motion compensation. The chirp rates and the quadratic chirp rates of the multicomponent CPS need to be estimated with parameter estimation algorithms to obtain the focused ISAR image. In this article, a new parametric range-instantaneous-Doppler ISAR imaging method is proposed, which introduces a new algorithm for parameter estimation of multicomponent CPSs. The cross terms of the generalized cubic phase function (GCPF) are analyzed. By eliminating the cross terms in the ambiguity function domain, the coherent integrated smoothed GCPF (CISGCPF) is proposed to estimate the quadratic chirp rates. Numerical examples are provided to verify the accuracy of the parameter estimation and the effectiveness of the cross-term suppression for CISGCPF in processing multicomponent signals. Simulated and real data imaging results demonstrate the performance of the CISGCPF-based ISAR imaging method. Comparisons with the existing algorithms show that the proposed method is effective in reconstructing focused images with less fake scatterers.

Computationally efficient TDOA, FDOA and differential Doppler rate estimation algorithm for passive emitter localization

This paper addresses the joint time difference of arrival (TDOA), frequency difference of arrival (FDOA) and differential Doppler rate estimation problem in passive emitter localization, considering the range and Doppler frequency migrations during the observation time. A computationally efficient estimation algorithm via second-order keystone transform and Lv's distribution is proposed. This method can effectively remove two types of migrations and accomplish the parameter estimation. Cross-terms suppression ability of the proposed method is also analyzed and its characteristic indicates the applicability in the scenario of multi-targets. The whole estimation step can be fast implemented by complex multiplications, fast Fourier transform (FFT) and inverse FFT operations without any searching procedure. Extensive numerical experiments validate that with a much lower computational cost, the proposed method could achieve comparable estimation performance with the maximum likelihood estimator and outperforms the other representative algorithms.

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.

Computationally efficient TDOA and FDOA estimation algorithm in passive emitter localisation

This study addresses the joint time difference of arrival (TDOA) and frequency difference of arrival (FDOA) estimation problem in passive emitter localisation, involving range migration during the observation time. A computational efficient estimation algorithm based on scaled Fourier transformation is proposed. This method can effectively remove the RM and accomplish the parameter estimation. Cross-terms suppression ability of the proposed method is also analysed and its characteristic indicates the applicability in the scenario of multi-targets. The whole estimation step can be fast implemented by complex multiplications, fast Fourier transformation (FFT) and inverse FFT without any searching process. Compared with the ideal maximum likelihood estimator via extensive numerical experiments, the proposed method can achieve comparable estimation performance and have gained more than ten thousand times reduction in the computational cost, which helps practical application.

A fast non-searching algorithm for radar maneuvering target detection

The detection of maneuvering targets is a challenging work for radar system. During a long integration time, the target’s complex motion will cause severe range migration and Doppler frequency migration, which greatly degrade the energy integration performance. In order to obtain a well-focused result of the maneuvering target, this paper proposes a novel and fast algorithm without any parameter searching process. In our algorithm, a parameter separation (PS) operation is conducted to isolate the target’s acceleration from other motion parameters. Then, the second-order keystone transform (SOKT) is utilized to correct range curvature and relocate the target’s energy into one range cell. Lastly, the target’s energy is integrated along the slow time axis via performing the non-uniform sparse Fourier transform (NUSFT), which is a fast implementation of Fourier transform (FT) for the input signal with unequally sampled points and frequency domain sparse representation. According to the simulation results, the proposed PS–SOKT–NUSFT method can realize multiple targets detection due to its good ability to suppress cross terms. In addition, the runtime of PS–SOKT–NUSFT (106.23 s) is much shorter than the PS–acceleration search (AS) method (2168.67 s).

A Matching Pursuit-Based Method for Cross-Term Suppression in WVD and its Application to the ENPEMF

The earth’s natural pulse electromagnetic field (ENPEMF) signal, which is released by the instantaneous disturbance of the earth’s natural changing magnetic field, contains a large amount of information about the changing geological structures and their kinetic principles. The analysis based on the time-frequency (TF) representation can provide significant interpretations for the ENPEMF signal. Wigner–Ville distribution (WVD) is one of the typical TF representations and has a high degree of TF concentration but is accompanied by severe cross-term interference. Hence, we propose a new TF representation called matching pursuit-based double WVD (MP-DWVD) to suppress cross-term interference, and synthetic signals are used to demonstrate the high TF concentration of the proposed TF analysis method. Furthermore, we apply MP-DWVD to explore the TF characteristics of the ENPEMF signal collected during the Lushan Ms7.0 earthquake. The results show that MP-DWVD can reveal accurate TF distributions of the ENPEMF signal before and after the earthquake and extract anomaly TF components, which have the potential to reveal electromagnetic anomaly information contained in the ENPEMF signal.