Choi-Williams Distribution Research Articles

Transverse Vibration Response and Time-Varying Analysis of Axially Moving Beams Based on Choi–Williams Distribution

Cantilever beams with axial extension and contraction functions as time-varying parameter systems have attracted extensive research due to their prevalence in engineering structures. In this paper, the dynamic characteristics of the flat-push bridge mechanism are investigated from the theoretical aspect, firstly, it is simplified into a cantilever beam model with axial motion, and the transverse vibration equations of the cantilever beam are established based on the Euler–Bernoulli beam theory and D’Alembert’s principle, and the transverse displacements of the beam are discretized using the Galerkin truncation method and solved numerically by using the Newmark-[Formula: see text] method for the second-order vibration differential equations with variable coefficients; second-order vibration differential equations were solved numerically by the Newmark-[Formula: see text] method; then the structural examples in the existing literature are verified numerically, and the calculation results are consistent with the literature results, indicating that the transverse vibration equations derived in this paper are solved effectively; then a parametric analysis of the speed of the cantilever beam during axial extension/contraction was done, which showed that the faster the speed of axial motion, the faster the rate of change of the dynamic characteristics of the beam and the more pronounced the vibration; finally, the power signal is decomposed in the time-frequency domain by using the Choi–Williams Distribution (CWD), and the time-frequency characteristics of the system under different speeds are analyzed, and the results show that the use of CWD technique can accurately reflect the system’s transient vibration frequency and vibration energy with the change rule of time and speed, and the axial motion speed of ±0.05[Formula: see text]m/s, the transient frequency of the theoretical prediction and the average relative deviation between the finite element modal frequencies are kept within 5% and the recognition accuracy is high. This analysis provides flexible technical guidance for vibration control in engineering structures.

LPI radar waveform recognition based on semi-supervised model all mean teacher

Low probability of intercept (LPI) radar signal identification plays an important role in electronic warfare, but most existing algorithms are proposed under the condition of sufficient samples, ignoring the problem of a small amount of labeled data in the actual electromagnetic environment. To solve the problem, in this paper, a semi-supervised learning model All Mean Teacher (AMT) based on Mean Teacher (MT) is proposed. First, the LPI radar signal is transformed into Time-frequency images (TFIs) by using the Choi-Williams distribution, and Random Erasing is used for TFIs which improves the generalization ability of the model. Then the Multi-headed Self-Attention Network (MSA-Net) is aimed to extract features, combined with AMT to realize the automatic waveform recognition of radar signals. MSA-Net facilitates feature information propagation by computing contrast costs on TFIs between the student and teacher networks. It solves the problem that TFIs are not easy to train for small amounts of labeled data, improving the accuracy of signal recognition in semi-supervised learning scenarios. Experimental results show that the average recognition accuracy of the proposed method is up to 85.7% at a signal-to-noise ratio of -8 dB.

Radar Signal Recognition Based on Bagging SVM

Radar signal recognition under low signal-to-noise ratio (SNR) conditions is a critical issue in modern electronic reconnaissance systems, which face significant challenges in recognition accuracy due to signal diversity. A novel method for radar signal detection based on the bagging support vector machine (SVM) is proposed in this paper.This method firstly utilizes the Choi–Williams distribution (CWD) and the smooth pseudo Wigner-Ville distribution (SPWVD) to obtain different time–frequency images of radar signals, which effectively leverages CWD’s strong time–frequency aggregation and SPWVD’s robust cross-interference resistance. Moreover, histograms of oriented gradient (HOG) features are extracted from time–frequency images to train multiple SVM classifiers by bootstrap sampling. Finally, the performance of each SVM classifier is aggregated using plurality voting to reduce the risk of model overfitting and improve recognition accuracy. We evaluate the effectiveness of the proposed method using 12 different types of radar signals. The experimental results demonstrate that its overall identification rate reaches around 79% at an SNR of −10 dB, and it improves the recognition rate by 5% compared with a single classifier.

LPI Radar Signal Recognition Based on Feature Enhancement with Deep Metric Learning

Low probability of intercept (LPI) radar signals are widely used in electronic countermeasures due to their low power and large bandwidth. However, they are susceptible to interference from noise, posing challenges for accurate identification. To address this issue, we propose an LPI radar signal recognition method based on feature enhancement with deep metric learning. Specifically, time-domain LPI signals are first transformed into time–frequency images via the Choi–Williams distribution. Then, we propose a feature enhancement network with attention-based dynamic feature extraction blocks to fully extract the fine-grained features in time–frequency images. Meanwhile, we introduce deep metric learning to reduce noise interference and enhance the time–frequency features. Finally, we construct an end-to-end classification network to achieve the signal recognition task. Experimental results demonstrate that our method obtains significantly higher recognition accuracy under a low signal-to-noise ratio compared with other baseline methods. When the signal-to-noise ratio is −10 dB, the successful recognition rate for twelve typical LPI signals reaches 94.38%.

Diff-SwinT: An Integrated Framework of Diffusion Model and Swin Transformer for Radar Jamming Recognition

With the increasing complexity of radar jamming threats, accurate and automatic jamming recognition is essential but remains challenging. Conventional algorithms often suffer from sharply decreased recognition accuracy under low jamming-to-noise ratios (JNR).Artificial intelligence-based jamming signal recognition is currently the main research directions for this issue. This paper proposes a new radar jamming recognition framework called Diff-SwinT. Firstly, the time-frequency representations of jamming signals are generated using Choi-Williams distribution. Then, a diffusion model with U-Net backbone is trained by adding Gaussian noise in the forward process and reconstructing in the reverse process, obtaining an inverse diffusion model with denoising capability. Next, Swin Transformer extracts hierarchical multi-scale features from the denoised time-frequency plots, and the features are fed into linear layers for classification. Experiments show that compared to using Swin Transformer, the proposed framework improves overall accuracy by 15% to 10% at JNR from −16 dB to −8 dB, demonstrating the efficacy of diffusion-based denoising in enhancing model robustness. Compared to VGG-based and feature-fusion-based recognition methods, the proposed framework has over 27% overall accuracy advantage under JNR from −16 dB to −8 dB. This integrated approach significantly enhances intelligent radar jamming recognition capability in complex environments.

LPI Waveform Recognition Using Adaptive Feature Construction and Convolutional Neural Networks

Low probability of intercept (LPI) radar waveform recognition is one of the crucial functions in electronic intelligence systems. Advances in artificial intelligence promote the performance of the LPI waveform recognition with various signal features defined with analytical expressions. However, noisy LPI waveform recognition is still a challenge for traditional approaches even with noise elimination techniques, especially for the heavily contaminated noisy LPI signals. Recently, the adaptive analysis techniques, such as variational mode decomposition (VMD) and empirical mode decomposition (EMD) provide potential methods that explore the inherent features of signals and formulate adaptive features for signal recognition. In this article, we propose an adaptive feature construction framework that utilizes both the adaptive features (via VMD and EMD) and predefined analytical features (via Wigner–Ville distribution, Choi–William distribution, and wavelet analysis) to construct the fusion feature, which is further applied on the convolutional neural networks-based LPI waveform recognition system. The experimental results show that the proposed feature adaptive LPI network-based exploitation (FALPINE) approach achieves higher probability of correct classification than the state-of-the-art works, which demonstrates the superior performance of the proposed approach.

시간-주파수 분석 기법과 딥러닝을 사용한 저피탐 레이다 식별 기술의 정확도와 시간의 Trade-Off

Technology for classifying low probability of intercept (LPI) radar signals with speed and accuracy is critical for cognitive communication research. We used time-frequency analysis (TFA) and deep learning to classify 12 typical LPI radar signals. Traditional methods use the Choi-Williams distribution (CWD), which requires more than 500 times longer TFA generation time than the spectrogram method. In this paper, we show the trade-off relationship between classification accuracy and detection time using a spectrogram, Wigner-Ville distribution (WVD), and CWD as the training datasets. As a result, the CWD model showed higher accuracy than the spectrogram model, but the prediction time was more than 200 times longer. The accuracy difference was only 1 %p for an SNR over −2 dB, but it reached 7.5%p for an SNR of −10 dB. Therefore, a lower SNR shows a distinct trade-off between prediction time and accuracy, depending on the type of TFA.

LPI Radar Signals Modulation Recognition Based on ACDCA-ResNeXt

For low probability of intercept (LPI) radar waveform identification accuracy (ACC) problem at low Signal-to-Noise Ratios (SNRs), an approach based on time-frequency analysis (TFA) and Asymmetric Dilated Convolution Coordinate Attention Residual networks (ACDCA-ResNeXt) is proposed to recognize twelve kinds of LPI radar signals automatically. First, we apply Choi-Williams distribution (CWD), which shows superior performance at low SNRs, to transforming radar signals into time-frequency images (TFI). Then, in order to obtain the high-quality TFIs, a series of image processing techniques, including 2DWiener filtering, image cutting, and image resize, are used to remove the background noise and redundant frequency bands of the TFI and obtain a fixed-size gray scale image containing main morphological features of the TFI. Finally, the TFIs are input into ACDCA-ResNeXt network that can extract and learn deep features to recognize radar waveforms. Furthermore, a fusion loss function, which is composed of a soft-label smoothed cross entropy loss function and a center loss function, improves the generalization capability performance of network and achieves a better clustering effect. Experimental results demonstrate that, for twelve kinds of LPI radar waveforms, the overall recognition ACC of the proposed approach achieves 97.94% when SNR is -8 dB.

Radar signal recognition exploiting information geometry and support vector machine

Aiming at the recognition of low-probability-of-intercept (LPI) radar signals, a support vector machine (SVM)-based algorithm is proposed, where the information geometry theory is utilised to optimise the kernel function of the SVM. Since signals with different modulations have different characteristics in the time-frequency domain, the time-frequency transformation result of the LPI radar signal is considered as an image, referred to as the time-frequency image, and computer vision techniques are utilized to perform recognition. Specifically, the time-frequency images of different LPI radar signals are obtained via the Choi-Williams distribution (CWD) transform, and the AlexNet network, one improved convolutional neural network (CNN), is used to extract time-frequency features. Then, an SVM is adopted to recognise LPI radar signals due to its superiority in addressing the dimension disaster and non-linear inseparability issue. The extracted time-frequency features are fed into the SVM for classification and recognition. Note that the classification performance of SVM depends on the kernel function. Therefore, in the proposed algorithm, information geometry theory is exploited to improve the Gaussian kernel function, and the maximum margin between different categories of samples is further enlarged. As a consequence, the recognition accuracy for LPI radar signals with similar time-frequency images is effectively improved. In addition, the proposed algorithm has better robustness to small samples than other deep learning-based algorithms, since the SVM method minimises the structural risk instead of the empirical risk. Simulation results verify the effectiveness of the proposed algorithm.

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.

Comparison of Time-Frequency Analyzes for a Sleep Staging Application with CNN

Sleep staging is the process of acquiring biological signals during sleep and marking them according to the stages of sleep. The procedure is performed by an experienced physician and takes more time. When this process is automated, the processing load will be reduced and the time required to identify disease will also be reduced. In this paper, 8 different transform methods for automatic sleep-staging based on convolutional neural networks (CNNs) were compared to classify sleep stages using single-channel electroencephalogram (EEG) signals. Five different labels were used to stage the sleep. These are Wake (W), Non Rapid Eye Movement (NonREM)-1 (N1), NonREM-2 (N2), NonREM-3 (N3), and REM (R). The classifications were done end-to-end without any hand-crafted features, ie without requiring any feature engineering. Time-Frequency components obtained by Short Time Fourier Transform, Discrete Wavelet Transform, Discrete Cosine Transform, Hilbert-Huang Transform, Discrete Gabor Transform, Fast Walsh-Hadamard Transform, Choi-Williams Distribution, and Wigner-Willie Distribution were classified with a supervised deep convolutional neural network to perform sleep staging. The discrete Cosine Transform-CNN method (DCT-CNN) showed the highest performance among the methods suggested in this paper with an F1 score of 89% and a value of 0.86 kappa. The findings of this study revealed that the transformation techniques utilized for the most accurate representation of input data are far superior to traditional approaches based on manual feature extraction, which acquires time, frequency, or nonlinear characteristics. The results of this article are expected to be useful to researchers in the development of low-cost, and easily portable devices.

LPI Radar Signal Recognition Based on Dual-Channel CNN and Feature Fusion

The accuracy of low probability of intercept (LPI) radar waveform recognition is an important and challenging problem in electronic warfare. Aiming at the problem of the difficulty in feature extraction and the low recognition rates of the LPI radar signal under a low signal-to-noise ratio, and inspired by the symmetry theory, we propose a new approach for the LPI radar signal recognition method based on a dual-channel convolutional neural network (CNN) and feature fusion. Our new approach contains three main modules: the preprocessing module that converts the LPI radar waveforms into two-dimensional time-frequency images using the Choi–Williams distribution (CWD) transformation and performs image binarization, the feature extraction module that extracts different features obtained from the images, and the recognition module that utilizes a multi-layer perceptron (MLP) network to fuse these features and distinguish the type of LPI radar signals. In the feature extraction module, a two-channel CNN model is proposed that extracts Histogram of Oriented Gradients (HOG) features and deep features from time-frequency images, respectively. Finally, the recognition module recognizes the radar signals using a Softmax classifier based on the fused features from two channels. The experimental results from 12 types of LPI radar signals prove the superiority and robustness of the proposed model. Its overall recognition rate reaches 97% when the signal-to-noise ratio is −6 dB.

Signal Recognition of Polyphase-Coded Radar Signals Based on Multifeature Fusion

For improving recognition accuracy of polyphase-coded radar signals under low signal to noise ratio (SNR), a recognition method based on multifeature and feature selection was proposed. First, feature parameters of polyphase-coded radar signals in time domain, frequency domain, and pseudo-Zernike moments of Choi-Williams distribution (CWD) were extracted, respectively. And then, redundancy features and small-correlation features were removed based on mutual information with greedy idea. Finally, recognition of polyphase-coded radar signals was implemented using support vector machines (SVM). The experimental results showed that the average recognition accuracy of proposed method was over 90% when SNR is 0 dB, and the recognition performance was superior to the existing methods.

СРАВНЕНИЕ ОСНОВНЫХ ЧАСТОТНО-ВРЕМЕННЫХ ПРЕОБРАЗОВАНИЙ СПЕКТРАЛЬНОГО АНАЛИЗА СИГНАЛОВ АКУСТИЧЕСКОЙ ЭМИССИИ

Due to the intensive development of spectroscopic techniques for detecting acoustic emission signals, the problem of providing the best time-frequency resolution through the application of specific time-frequency transformation algorithms comes to the fore. The Short-Time Fourier Transform, the Wavelet Transform, the Smoothed Pseudo Wigner Distribution, the Choi-Williams Distribution, and the Hilbert-Huang Transform are currently the main time-frequency transformations used or integrated into the acoustic emission method. However, today in the literature, there is not enough information that allows evaluating time-frequency transformations regarding the effectiveness of their application to specify the features of discrete and continuous acoustic emission signals. On this basis, the authors carried out an experimental comparison of synthetic and actual model signals to determine the efficiency of specified time-frequency transformations. The synthetic model signals were a chirp signal, ideal sinusoids, and a Dirac delta function. The actual signals were a discrete acoustic emission signal from the Hsu Nelson source decomposed into dispersion modes in the acoustic channel and a continuous acoustic emission signal from the air outflow through a calibrated hole. The analysis shows that only the Fourier transform and the Wavelet transform can define all control features of model signals at the frequency components’ energy gap of about 25 dB. Wigner Distribution, Choi-Williams Distribution, and Hilbert-Huang Transform demonstrated higher time-frequency resolution did not identify frequency components of low energy. Therefore, the authors recommend using them to identify spectral changes in the resonance and discrete signals but in the narrow energy range. The Fourier transform and the Wavelet transform demonstrated the best result to analyze continuous acoustic emission. However, to use the latter, the procedure of selection of the optimal basis function is necessary. The study determined that the Hilbert-Huang transform allows identifying the frequency fluctuations, but it is necessary to develop ways to increase sensitivity and extract basic information from the spectrograms to enhance the validity of its results.

Detecting targets’ longitudinal and angular accelerations based on vortex electromagnetic waves

In this paper, the detections of targets’ longitudinal and angular accelerations, including uniform and variable accelerations, have been investigated, based on vortex electromagnetic (EM) waves carrying orbital angular momentum (OAM), in which the time–frequency analysis approaches are used to process the echo signal. Compared with other methods, the Choi-Williams distribution (CWD) is more suitable for detecting, which has better energy concentration and less cross term. Meanwhile, the multi-OAM modes method has been proposed to decouple the longitudinal and angular accelerations and detect the concrete acceleration coefficients. And the detection errors have been analyzed by transmitting the OAM waves with different mode combinations of l = 1, 3, 5 at 3 GHz. The results indicate that increasing mode intervals and decreasing topological charges can relatively improve detection accuracy in an appropriate sampling frequency range. In addition, the proposed method is also applicable to the case that the transmission modes are impure in practical application. And the directions of the longitudinal and angular accelerations can be acquired by sending OAM with different amounts of power. This work paves the way of detecting 3-dimensional velocity of complex target.

IQ-Data-Based WiFi Signal Classification Algorithm Using the Choi-Williams and Margenau-Hill-Spectrogram Features: A Case in Human Activity Recognition

This paper presents a novel approach that applies WiFi-based IQ data and time–frequency images to classify human activities automatically and accurately. The proposed strategy first uses the Choi–Williams distribution transform and the Margenau–Hill spectrogram transform to obtain the time–frequency images, followed by the offset and principal component analysis (PCA) feature extraction. The offset features were extracted from the IQ data and several spectra with maximum energy values in the time domain, and the PCA features were extracted via the whole images and several image slices on them with rich unit information. Finally, a traditional supervised learning classifier was used to label various activities. With twelve-thousand experimental samples from four categories of WiFi signals, the experimental data validated our proposed method. The results showed that our method was more robust to varying image slices or PCA numbers over the measured dataset. Our method with the random forest (RF) classifier surpassed the method with alternative classifiers on classification performance and finally obtained a 91.78% average sensitivity, 91.74% average precision, 91.73% average F1-score, 97.26% average specificity, and 95.89% average accuracy.

Accurate LPI Radar Waveform Recognition With CWD-TFA for Deep Convolutional Network

Automotive radars, with a widespread emergence in the last decade, have faced various jamming attacks. Utilizing low probability of intercept (LPI) radar waveforms, as one of the essential solutions, demands an accurate waveform recognizer at the intercept receiver. Numerous conventional approaches have been studied for LPI radar waveform recognition, but their performance is inadequate under channel condition deterioration. In this letter, by exploiting deep learning (DL) to capture intrinsic radio characteristics, we propose a convolutional neural network (CNN), namely LPI-Net, for automatic radar waveform recognition. In particular, radar signals are first analyzed by a time-frequency analysis using the Choi-Williams distribution. Subsequently, LPI-Net, primarily consisting of three sophisticated modules, is built to learn the representational features of time-frequency images, in which each module is constructed with a preceding maps collection to gain feature diversity and a skip-connection to maintain informative identity. Simulation results show that LPI-Net achieves the 13-waveform recognition accuracy of over 98% at 0 dB SNR and further performs better than other deep models.

Study for classification and recognition of radar emitter intra-pulse signals based on the energy cumulant of CWD

A novel feature extraction method based on the energy cumulant of Choi-Williams distribution has been proposed, which copes with the complex electromagnetic environment and varied signal styles. The energy cumulant of Choi-Williams distribution has been calculated from the accumulations of each frequency sample value with different time samples. Preceding this procedure, the time frequency distribution via Choi-Williams distribution has been processed through base noise reduction. Henceforth, a simulation analysis based on the discriminability and recognition effect of the proposed features in a low SNR environment has been carried out. In the discriminability experiment, the kernel fuzzy C-means clustering algorithm has been employed for classifying the radar pulse modulated radiation source signals. In the recognition effect analysis, the deep belief network has been proposed to employ the input feature vectors for training to achieve the radar emitter recognition and classification. Simulation results demonstrate that the proposed feature extraction method is feasible and robust in radar emitter classification and recognition even at a low SNR.

A Multipulse Radar Signal Recognition Approach via HRF-Net Deep Learning Models.

In the field of electronic countermeasure, the recognition of radar signals is extremely important. This paper uses GNU Radio and Universal Software Radio Peripherals to generate 10 classes of close-to-real multipulse radar signals, namely, Barker, Chaotic, EQFM, Frank, FSK, LFM, LOFM, OFDM, P1, and P2. In order to obtain the time-frequency image (TFI) of the multipulse radar signal, the signal is Choi–Williams distribution (CWD) transformed. Aiming at the features of the multipulse radar signal TFI, we designed a distinguishing feature fusion extraction module (DFFE) and proposed a new HRF-Net deep learning model based on this module. The model has relatively few parameters and calculations. The experiments were carried out at the signal-to-noise ratio (SNR) of −14 ∼ 4 dB. In the case of −6 dB, the recognition result of HRF-Net reached 99.583% and the recognition result of the network still reached 97.500% under −14 dB. Compared with other methods, HRF-Nets have relatively better generalization and robustness.

A New Modulation Recognition Method Based on Flying Fish Swarm Algorithm

The modulation recognition method based on deep learning plays a significant role in the intelligent communication system. To further improve the recognition rate, especially in the case of small samples with a low signal-to-noise ratio, this paper proposes a new modulation recognition method based on flying fish swarm algorithm. First, Short-Time Fourier Transform, Choi-Williams Distribution, and Cyclic Spectrum are combined to complete multi-channel signal processing. Second, AlexNet, VGGNet, GoogLeNet, and ResNet are transferred to realize feature extraction. Third, the support vector machine classifies the modulations after dimension reduction and feature fusion. Finally, the flying fish swarm is proposed to optimize the signal processing methods, the types of networks, the layers of networks, the dimensions of features, and the parameters of the support vector machine. The method can accurately recognize BPSK, QPSK, OQPSK, 8PSK, 4ASK, QAM16, QAM32, and QAM64. The simulation results show that the average recognition rate of modulation is 94.5% at SNR of 0 dB and 84.7% at SNR of −4 dB. Besides, the proposed modulation recognition method possesses good robustness under low SNR conditions.