分?jǐn)?shù)傅里葉變換域的調(diào)頻信號稀疏性研究

王 碩，郭 勇，楊立東

分?jǐn)?shù)傅里葉變換域的調(diào)頻信號稀疏性研究

王 碩1，郭 勇2*，楊立東1

1內(nèi)蒙古科技大學(xué)信息工程學(xué)院，內(nèi)蒙古 包頭 014010；2內(nèi)蒙古科技大學(xué)理學(xué)院，內(nèi)蒙古 包頭 014010

調(diào)頻信號廣泛應(yīng)用于聲納、雷達(dá)、激光和新興光學(xué)交叉研究領(lǐng)域，其緊致性(稀疏性)是調(diào)頻信號采樣、去噪、壓縮等研究中面臨的共性基礎(chǔ)問題。本文致力于研究調(diào)頻信號在分?jǐn)?shù)傅里葉變換域的稀疏性，提出了一種最大奇異值法來估計調(diào)頻信號的緊致分?jǐn)?shù)傅里葉變換域。該方法利用調(diào)頻信號幅度譜的最大奇異值來度量其緊致域，并應(yīng)用鯨魚優(yōu)化算法來搜尋緊致域，有效改善了現(xiàn)有方法的不足。與MNM和MACF方法相比，本文方法給出了調(diào)頻信號在分?jǐn)?shù)傅里葉變換域更加稀疏的表征，具有更少的重要振幅數(shù)。最后，給出了該方法在調(diào)頻信號濾波中的初步應(yīng)用。

調(diào)頻信號；稀疏性；分?jǐn)?shù)傅里葉變換；奇異值分解

1 引 言

調(diào)頻信號是一種典型的非平穩(wěn)信號，廣泛應(yīng)用于聲納、雷達(dá)、激光等傳統(tǒng)領(lǐng)域[1-3]。近年來，它開始應(yīng)用于新興的光學(xué)交叉領(lǐng)域，如微波與光學(xué)交叉的微波光子學(xué)[4-6]、傳統(tǒng)光學(xué)測量與信號處理交叉的新興光學(xué)測量領(lǐng)域[7-10]。調(diào)頻信號的稀疏表示是調(diào)頻信號采樣、濾波、壓縮等研究中面臨的共性基礎(chǔ)問題，開展其稀疏性研究對于聲納、雷達(dá)和激光等傳統(tǒng)領(lǐng)域以及新興的光學(xué)交叉領(lǐng)域具有重要的理論和實際意義。

因此，針對現(xiàn)有研究中存在的問題，本文提出一種最大奇異值法(maximum singular value method, MSVM)來估計調(diào)頻信號在分?jǐn)?shù)傅里葉變換域的稀疏性。MSVM采用分?jǐn)?shù)傅里葉變換域幅度譜的最大奇異值來度量緊致域，具有較好的稀疏性和噪聲魯棒性。由于奇異值分解能夠?qū)⒏呔S的數(shù)據(jù)空間映射到低維的數(shù)據(jù)空間，有效降低處理數(shù)據(jù)的維度，而且奇異值分解能用于任意矩陣，適用性更廣。另外，該方法采用鯨魚優(yōu)化算法(whale optimization algorithm, WOA)[31]來搜尋緊致域。由于WOA具有靈活性強(qiáng)、無梯度限制，且可以有效避免陷入局部最優(yōu)點(diǎn)等優(yōu)勢，因此有效改善了從粗到細(xì)的網(wǎng)格搜索和遍歷搜索的不足。最后，本文給出了MSVM在線性調(diào)頻信號濾波中的初步應(yīng)用，顯示了該方法潛在的應(yīng)用價值。

本文第2節(jié)簡單介紹了分?jǐn)?shù)傅里葉變換的定義和基本性質(zhì)；第3節(jié)探索研究了調(diào)頻信號在分?jǐn)?shù)傅里葉變換域的稀疏性；第4節(jié)基于線性調(diào)頻信號和二次調(diào)頻信號進(jìn)行了仿真分析；第5節(jié)介紹了該方法在線性調(diào)頻信號濾波中的初步應(yīng)用；第6節(jié)對本文的工作進(jìn)行了簡要總結(jié)。

2 分?jǐn)?shù)傅里葉變換

分?jǐn)?shù)傅里葉變換不僅具備傅里葉變換的性質(zhì)，同時還具備一些其他性質(zhì)[11]：

其中WVD的定義為[11]

因此，從時頻分析的角度來看，分?jǐn)?shù)傅里葉變換可以看作時頻平面上的旋轉(zhuǎn)變換，它可以提供時域和頻域之間關(guān)于旋轉(zhuǎn)角度的連續(xù)映射。因此，如何尋找最佳的旋轉(zhuǎn)角度，使得調(diào)頻信號在相應(yīng)的分?jǐn)?shù)傅里葉變換域中具有最好的稀疏性是本文的關(guān)鍵問題之一。

3 最大奇異值法

探索調(diào)頻信號在分?jǐn)?shù)傅里葉變換域的稀疏表示，就是尋找調(diào)頻信號的緊致分?jǐn)?shù)傅里葉變換域，使得調(diào)頻信號在該域可以使用最少的系數(shù)來覆蓋信號的絕大部分能量。因此，如何度量和搜尋調(diào)頻信號的緊致分?jǐn)?shù)傅里葉變換域是本文要解決的兩個關(guān)鍵問題。現(xiàn)有文獻(xiàn)都是直接基于調(diào)頻信號在分?jǐn)?shù)傅里葉變換域的幅度譜進(jìn)行稀疏性估計，這些方法的計算量大且容易受到噪聲的影響。

本文提出了一種最大奇異值法(簡記為MSVM)來估計調(diào)頻信號的緊致分?jǐn)?shù)傅里葉變換域。首先，對調(diào)頻信號在分?jǐn)?shù)傅里葉變換域的幅度譜進(jìn)行奇異值分解。奇異值分解(singular value decomposition, SVD)[33]是一種重要的矩陣分解，廣泛應(yīng)用于信號處理領(lǐng)域，其定義為

針對緊致域的搜尋問題，文獻(xiàn)[29]和[30]使用的是由粗到細(xì)的網(wǎng)格搜索方法，但該方法計算量大且容易受到搜索步長的影響。WOA模仿座頭鯨對獵物的搜索、圍捕和覓食等行為，是一種新型啟發(fā)式仿生算法。WOA不僅具有靈活性強(qiáng)，無梯度限制，可以有效避免陷入局部最優(yōu)點(diǎn)等特點(diǎn)，而且它還不受搜索步長的影響。正是由于以上優(yōu)勢，本文應(yīng)用WOA來搜尋調(diào)頻信號的緊致域，有效改善了現(xiàn)有方法的不足。

本文所提出的調(diào)頻信號稀疏性估計方法主要包括以下4個步驟：

步驟3：建立如下的優(yōu)化模型

4 仿真結(jié)果和分析

本節(jié)將利用MATLAB軟件分別基于線性調(diào)頻信號和二次調(diào)頻信號進(jìn)行仿真實驗和分析。WOA的初始參數(shù)設(shè)定如下：

其中：，max，im，b，b分別表示種群數(shù)、最大迭代次數(shù)、變量個數(shù)、變量的下界和上界。為了更好地分析MSVM的性能，本文比較了基于MSVM，MACF[29]和MNM[30]得到的調(diào)頻信號的緊致分?jǐn)?shù)傅里葉變換域，其中MNM和MACF的參數(shù)設(shè)置與文獻(xiàn)[29-30]中的一致。

4.1 線性調(diào)頻信號

首先，利用雙分量線性調(diào)頻信號對MSVM的性能進(jìn)行仿真分析。雙分量線性調(diào)頻信號可表示為

其中：表示振幅，表示調(diào)頻率，表示初始頻率。對此連續(xù)信號進(jìn)行均勻采樣，采樣參數(shù)為

為了充分分析MSVM的性能，本文采用MNM、MACF和MSVM方法對多組線性調(diào)頻信號進(jìn)行仿真實驗，得到的結(jié)果列于表1。分析表中數(shù)據(jù)可得：1) 對于單分量線性調(diào)頻信號來說，三種方法得到的最優(yōu)角度和NSA基本一致，這是由單分量線性調(diào)頻信號在分?jǐn)?shù)傅里葉變換域的特性所決定[18]；2) 對于雙分量線性調(diào)頻信號來說，MSVM得到最佳角度不同于其它兩種方法，相應(yīng)的NSA是最小的?？傊噍^于MNM和MACF，MSVM給出的線性調(diào)頻信號在分?jǐn)?shù)傅里葉變換域的稀疏性是較好的。

4.2 二次調(diào)頻信號

本節(jié)基于二次調(diào)頻信號對MSVM的性能進(jìn)行仿真分析，二次調(diào)頻信號可表示為

其中：A表示振幅，表示二次調(diào)頻率，k表示調(diào)頻率，f0是初始頻率。取一組二次調(diào)頻信號為測試信號，其參數(shù)為

圖2 三種方法給出的線性調(diào)頻信號在緊致 分?jǐn)?shù)傅里葉變換域中的表征

表1 線性調(diào)頻信號的估計結(jié)果比較

圖3 三種方法得到的二次調(diào)頻信號在緊致 分?jǐn)?shù)傅里葉變換域中的表征

5 應(yīng) 用

調(diào)頻信號在信道中傳輸時會不可避免地混入噪聲，影響后續(xù)的調(diào)頻信號檢測和參數(shù)估計。因此，如何濾除噪聲是調(diào)頻信號處理中的基礎(chǔ)問題之一。MSVM可以將信號能量集中到更加緊湊的緊致域，所以可進(jìn)一步用于調(diào)頻信號的濾波。選一組單分量線性調(diào)頻信號(見圖4(a))，其表達(dá)式為

另外，選取另一組雙分量線性調(diào)頻信號來說明其濾波效果，其參數(shù)為

將SNR=5 dB的高斯白噪聲添加到該信號中。按照與單分量線性調(diào)頻信號相同的濾波方法，所得結(jié)果圖5所示。從圖5(c)可得，MSVM可以將這個雙分量線性調(diào)頻信號壓縮在緊致分?jǐn)?shù)傅里葉變換域內(nèi)的一個窄帶內(nèi)，而噪聲仍均勻分布在整個帶寬中。比較濾波前后的信號波形圖，應(yīng)用MSVM可以濾掉雙分量線性調(diào)頻信號中大部分的噪聲，且可以基本恢復(fù)原始信號。

表2 二次調(diào)頻信號的估計結(jié)果比較

圖4 單分量線性調(diào)頻信號的濾波。(a) 原始信號；(b) 加噪信號(SNR=5 dB)；(c) 加噪信號在緊致分?jǐn)?shù)傅里葉變換域的幅度譜；(d) 濾波之后的信號

圖5 雙分量線性調(diào)頻信號的濾波。(a) 原始信號；(b) 加噪信號(SNR=5 dB)；(c) 加噪信號在緊致分?jǐn)?shù)傅里葉變換域的幅度譜；(d) 濾波之后的信號

6 結(jié) 論

本文研究了調(diào)頻信號在分?jǐn)?shù)傅里葉變換域的稀疏性，提出了一種最大奇異值法(MSVM)來估計調(diào)頻信號的緊致分?jǐn)?shù)傅里葉變換域。本文采用調(diào)頻信號幅度譜的最大奇異值作為緊致域的度量，建立了一個優(yōu)化模型來估計調(diào)頻信號的緊致域。采用WOA算法來搜索最佳角度，極大改善了現(xiàn)有方法中由于使用由粗到細(xì)網(wǎng)格搜索和遍歷搜索所導(dǎo)致的不足。相比較于MNM，MACF和MSVM給出的緊致域，基于MSVM得到的調(diào)頻信號的緊致域更加緊湊，且具有更少的NSA。最后，MSVM成功應(yīng)用到調(diào)頻信號的濾波中，基本實現(xiàn)了噪聲的濾除和信號性態(tài)的保持。

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Research on sparsity of frequency modulated signal in fractional Fourier transform domain

Wang Shuo1, Guo Yong2*, Yang Lidong1

1School of Information Engineering, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia 014010, China;2School of Science, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia 014010, China

Representation of bi-component LFM signal in the time domain and compact FRFT domain

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.

Citation: Wang S, Guo Y, Yang L D. Research on sparsity of frequency modulated signal in fractional Fourier transform domain[J]., 2020,47(11): 190660

Research on sparsity of frequency modulated signal in fractional Fourier transform domain

Wang Shuo1, Guo Yong2*, Yang Lidong1

1School of Information Engineering, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia 014010, China;2School of Science, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia 014010, China

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.

frequency modulated signal; sparsity; fractional Fourier transform; singular value decomposition

TN911.7

A

王碩，郭勇，楊立東. 分?jǐn)?shù)傅里葉變換域的調(diào)頻信號稀疏性研究[J]. 光電工程，2020，47(11): 190660

10.12086/oee.2020.190660

: Wang S, Guo Y, Yang L DResearch on sparsity of frequency modulated signal in fractional Fourier transform domain[J]., 2020, 47(11): 190660

2019-11-02；

2019-12-19

國家自然科學(xué)基金資助項目(11801287)；內(nèi)蒙古自然科學(xué)基金資助項目(2019BS01007)；內(nèi)蒙古科技大學(xué)創(chuàng)新基金(2019QDL-B39)

王碩(1995-)，女，碩士研究生，主要從事調(diào)頻信號處理理論研究。E-mail: wswx828211@163.com

郭勇(1988-)，男，博士，講師，研究方向為非平穩(wěn)信號處理理論與方法。E-mail: guo_yong@imust.edu.cn

Supported by National Natural Science Foundation of China (11801287), Inner Mongolia Natural Science Foundation (2019BS01007), and Inner Mongolia University of Science and Technology Innovation Fund (2019QDL-B39)

* E-mail: guo_yong@imust.edu.cn