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Research on sparsity of frequency modulated signal in fractional Fourier transform domain
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摘要
调频信号广泛应用于声纳、雷达、激光和新兴光学交叉研究领域，其紧致性(稀疏性)是调频信号采样、去噪、压缩等研究中面临的共性基础问题。本文致力于研究调频信号在分数傅里叶变换域的稀疏性，提出了一种最大奇异值法来估计调频信号的紧致分数傅里叶变换域。该方法利用调频信号幅度谱的最大奇异值来度量其紧致域，并应用鲸鱼优化算法来搜寻紧致域，有效改善了现有方法的不足。与MNM和MACF方法相比，本文方法给出了调频信号在分数傅里叶变换域更加稀疏的表征，具有更少的重要振幅数。最后，给出了该方法在调频信号滤波中的初步应用。
Abstract
Frequency modulated (FM) signal is extensively applied in sonar, radar, laser and emerging optical cross-research, its sparsity is a common basic issue for the sampling, denoising and compression of FM signal. This paper mainly studies the sparsity of FM signal in the fractional Fourier transform (FRFT) domain, and a maximum singular value method (MSVM) is proposed to estimate the compact FRFT domain of FM signal. This method uses the maximum singular value of amplitude spectrum of FM signal to measure the compact domain, and WOA is used to search the compact domain, which effectively improves the shortcomings of the existing methods. Compared with MNM and MACF, this method gives a sparser representation of FM signal in the FRFT domain, which has less number of significant amplitudes. Finally, the primary application of this method in the FM signal filtering is given.
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Overview
Overview:Frequency modulated (FM) signal is a typical non-stationary signal, which is widely used in sonar, radar, laser and other traditional fields. In recent years, it has been applied to the new field of optical intersection. Its sparsity is a common basic problem in the FM signal processing. Fractional Fourier transform (FRFT) uses the orthogonal chirp function to decompose signal and is unaffected by the cross terms, and thus is very suitable for analyzing and processing FM signal. Due to the advantages of FRFT in the FM signal processing, FRFT is also applied to explore the sparsity of FM signal. FRFT can represent the signal from any fractional domain between the time domain and the frequency domain. Therefore, there is at least one optimal fractional Fourier transform domain, which makes the FM signal have best sparsity in this optimal domain. This optimal domain is named as compact fractional Fourier transform domain. In the process of finding the compact fractional Fourier transform domain, the measurement and searching of optimal domain are two key points. On the basis of the above advantages, this paper is devoted to studying the sparsity of FM signal in fractional Fourier transform domain, and a sparse representation method of FM signal based on FRFT and singular value decomposition is proposed, called as maximum singular value method (MSVM). On the one hand, the maximum singular value of amplitude spectrum in FRFT domain is taken as the measurement of optimal domain, which makes MSVM has better sparsity and noise robustness. Since singular value decomposition can map high-dimensional data space to a relatively low-dimensional data space, and thus singular value decomposition effectively reduces the dimension of data processing. The larger the singular value of the amplitude spectrum, the better the sparsity of the FM signal in the corresponding fractional Fourier transform domain. Moreover, the singular value decomposition is a kind of decomposition method which can be applied to any matrix, and has a wider applicability. On the other hand, whale optimization algorithm is used to search optimal domain. Whale optimization algorithm is a new heuristic bionic algorithm, which imitates the behavior of humpback whales in searching, seizing and foraging. Because whale optimization algorithm is flexible and has no gradient limitation. It can effectively avoid falling into the local optimum, and effectively improve the shortcomings of the coarse-to-fine grid search and traversal search methods, and is not influenced by the search step size. The quantitative index is the number of significant amplitudes (NSA), the less NSA means better sparsity. By the simulation, compared with MACF and MNM, MSVM has less NSA in the compact fractional Fourier transform domain. It is concluded that the MSVM can give better sparsity of FM signal in the compact fractional Fourier transform domain. In the end, this paper presents the application of MSVM in the filter of linear FM signal, which basically achieves the filtering of noise and the maintenance of signal behavior.
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图 1 双分量线性调频信号在时域和紧致分数傅里叶变换域中的表征
Figure 1. Representation of bi-component LFM signal in the time domain and compact FRFT domain
图 2 三种方法给出的线性调频信号在紧致分数傅里叶变换域中的表征
Figure 2. Representation of bi-component LFM signal in the compact FRFT domain for three methods
图 3 三种方法得到的二次调频信号在紧致分数傅里叶变换域中的表征
Figure 3. Representation of QFM signal in the compact FRFT domain for three methods
图 4 单分量线性调频信号的滤波。(a)原始信号；(b)加噪信号(SNR=5 dB)；(c)加噪信号在紧致分数傅里叶变换域的幅度谱；(d)滤波之后的信号
Figure 4. Filter of single component LFM signal. (a) Original signal; (b) Noisy signal (SNR=5 dB); (c) Amplitude spectrum of the noisy signal in the compact FRFT domain; (d) Filtered signal
图 5 双分量线性调频信号的滤波。(a)原始信号；(b)加噪信号(SNR=5 dB)；(c)加噪信号在紧致分数傅里叶变换域的幅度谱；(d)滤波之后的信号
Figure 5. Filter of bi-component LFM signal. (a) Original signal; (b) Noisy signal (SNR=5 dB); (c) Amplitude spectrum of the noisy signal in the compact FRFT domain; (d) Filtered signal
表 1 线性调频信号的估计结果比较
Table 1. Comparison of estimation results for LFM signal
衡量指标 α0 NSA {A0, k0, f0}, {A1, k1, f1} MNM MACF MSVM MNM MACF MSVM {0.1, 0.8, 5}, {0, 0, 0} 2.2455 2.2455 2.2454 3 3 3 {0.2, 0, 0.8}, {0.3, 0.7, 0} 2.1816 1.5708 1.9041 223 324 138 {5, 0.1, -0.3}, {8, 0.87, 0.3} 2.2777 2.2872 2.0055 217 239 140 {2, 0.1, 0.2}, {1, 0.4, -0.2} 1.6704 1.67 1.7714 100 100 70 {2, 0.5, -1/9}, {0.5, 9/5, 5/6} 2.0342 2.0345 2.0560 73 74 34 
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表 2 二次调频信号的估计结果比较
Table 2. Comparison of estimation results for QFM signal
衡量指标
{A, σ, k, f}α0 NSA MNM MACF MSVM MNM MACF MSVM {0.5, 0.04, 0.4, 0.06} 2.0415 2.3510 1.9279 122 218 116 {4, 0.2, 0.03, 0.01} 2.0722 0.9106 1.5729 353 368 309 {0.2, 05, 0, 0.02} 2.2627 0.9416 1.5705 332 339 290 {2.3, 0.07, 0.05, 0} 2.2932 2.2777 1.5707 241 241 210 {3, 0.052, 0, 1} 1.9633 0.7858 1.6653 197 260 166
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