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As frequency modulated (FM) signals widely exist in the natural world as well as in different artificial applications, it is of great practical significance to explore the ways to extract such signal components in the complex and noisy environment. To extract one component from the noisy multicomponent signal effectively, a component extraction method based on polynomial chirp Fourier transform (PCFT) is presented in this paper. First, the physical meanings of Fourier transform (FT) and fractional Fourier transform (FRFT) are analyzed and their internal relations are expounded from the perspective of signal energy accumulation. Essentially, the FT accumulates signal energy along the time-frequency beelines parallel to the time axis and obtains an energy-concentrated spectrum from the narrow-band stationary signals whose frequency does not change, whereas it fails to process non-stationary signals with changeable frequencies. By rotating the time-frequency axis, the FRFT changes the energy accumulation mode of the signal in the old time-frequency plane and achieves a more concentrated spectrum for the linear frequency modulated (LFM) signal, but with larger error or even invalidation when dealing with nonlinear frequency modulated (NLFM) signal. Using FT and FRFT, in this paper we attempt to improve the energy accumulation mode of the conventional transform method and propose the PCFT. In this transform, the beeline families in the traditional transform, independent of time (or v) axes, are replaced by a family of polynomial chirping curves in the time-frequency plane. These polynomial chirping curves are capable of approaching more closely to the instantaneous frequency curve of FM signal so as to obtain a more concentrated transform spectrum and thereby extend the application of PCFT from LFM signal to NLFM signal. When selecting the polynomial chirping curve, we build up a nonlinear optimization model guided by the principle of energy spectrum concentration and in this way convert the problem of determining the polynomial curve families into the one of optimizing the polynomial parameters. Then particle swarm optimization algorithm is employed to search for the optimal polynomial parameters so as to concentrate the energy of one component in the new transform domain, i.e., the polynomial chirp Fourier domain. After doing that, each component is separated into its concentrated spectrum with a narrow-band filter and reconstructed with the inverse PCFT. Moreover, to extract components from a noisy multicomponent signal successfully, an iteration involving parameter estimation, PCFT, filter and recovery is introduced. To verify the effectiveness of the PCFT-based method, a series of examples, including simulated and real-world signals, is chosen for simulations and experiments. The experimental results indicate that compared with FT and FRFT, the proposed method overcomes the shortcoming of distributed energy spectrum for NLFM components in the traditional transforms and obtains a concentrated energy spectrum in the polynomial chirp Fourier domain, therefore realizing component separation and time-frequency characteristic extraction. The PCFT-based method not only has the capability of dealing with the extraction of LFM components, but also performs well in the separation of crossed NLFM components, and with little extraction error.
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Keywords:
- polynomial chirp Fourier transform /
- nonlinear frequency modulated signal /
- particle swarm optimization algorithm /
- component extraction
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[23] Chen Z, Tong Q N, Zhang C C, Hu Z 2015 Chin. Phys. B 24 043303
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[25] Deng W Y, Zheng Q H, Chen L, Xu X B 2010 Chin. J. Comput. 33 279 (in Chinese) [邓万宇, 郑庆华, 陈琳, 许学斌 2010 计算机学报 32 279]
[26] Nguyen H A, Guo H, Low K S 2011 IEEE Trans. Instrum. Meas. 60 3619
[27] Liu H H, Liu Y H 2012 Chin. Phys. B 21 026102
[28] Bi G, Zeng Y 2007 J. Electron. Inform. Technol. 29 1399 (in Chinese) [毕岗, 曾宇 2007 电子与信息学报 29 1399]
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[1] Ba J 2010 Sci. Sin: Phys. Mech. Astron. 40 1398 (in Chinese) [巴晶 2010 中国科学: 物理学 力学 天文学 40 1398]
[2] Liu X Y, Pei L Q, Wang Y, Zhang S M, Gao H L, Dai Y D 2011 Chin. Phys. B 20 047401
[3] Dugnol B, Fernandez C, Galiano G, Velasco J 2008 Signal Process. 88 1817
[4] Li S M 2005 Chin. J. Appl. Mech. 22 579 (in Chinese) [李舜酩 2005 应用力学学报 22 579]
[5] Yang Y, Dong X J, Peng Z K, Zhang W M, Meng G 2015 J. Sound Vib. 335 350
[6] Lei P, Wang J, Guo P, Cai D D 2011 AEU-Int. J. Electron. Commun. 65 806
[7] Xu L J, Yang Y X, Yang L 2015 Acta Phys. Sin. 64 174304 (in Chinese) [徐灵基, 杨益新, 杨龙 2015 64 174304]
[8] Kuang W T, Morris A S 2002 IEEE Trans. Instrum. Meas. 51 440
[9] Gonzlez D, Bialasiewicz J T, Balcells J, Gago J 2008 IEEE Trans. Ind. Electron. 55 3167
[10] Alarcon V C, Daviu J A A, Guasp M R 2012 Electr. Pow. Syst. Res. 91 28
[11] Shi P, Cao G W, Li Y P 2010 Chin. Phys. B 19 074201
[12] Barkat B, Boashash B 1999 IEEE Trans. Signal Process. 47 2480
[13] Andria G, Savino M 1996 IEEE Trans. Instrum. Meas. 45 818
[14] Yin B Q, He Y G, Li B, Zuo L, Yuan L F 2015 Chin. J. Electron. 24 115
[15] Wang L, Xu L P, Zhang H, Luo N 2013 Acta Phys. Sin. 62 139702 (in Chinese) [王璐, 许录平, 张华, 罗楠 2013 62 139702]
[16] Wang X L, Wang W B 2015 Chin. Phys. B 24 080203
[17] Chen B X, Li M, Zhang A J 2007 Acta Phys. Sin. 56 4535 (in Chinese) [陈宝信, 李明, 张爱菊 2007 56 4535]
[18] Huang Y, Liu F, Wang Z Z, Xiang C W, Deng B 2013 Acta Aeronaut. Et Astronaut. Sin. 34 846 (in Chinese) [黄宇, 刘锋, 王泽众, 向崇文, 邓兵 2013 航空学报 34 846]
[19] Xu G L, Wang X T, Xu X G 2010 Chin. Phys. B 19 014203
[20] Zou H X 2002 Ph. D. Dissertation (Beijing: Tsinghua University) (in Chinese) [邹红星 2002 博士学位论文 (北京: 清华大学)]
[21] Namias V 1980 J. Inst. Maths. Appl. 25 241
[22] Yang Y, Peng Z K, Dong X J 2014 IEEE Trans. Instrum. Meas. 63 3169
[23] Chen Z, Tong Q N, Zhang C C, Hu Z 2015 Chin. Phys. B 24 043303
[24] Janeiro F M, Ramos P M 2009 IEEE Trans. Instrum. Meas. 58 383
[25] Deng W Y, Zheng Q H, Chen L, Xu X B 2010 Chin. J. Comput. 33 279 (in Chinese) [邓万宇, 郑庆华, 陈琳, 许学斌 2010 计算机学报 32 279]
[26] Nguyen H A, Guo H, Low K S 2011 IEEE Trans. Instrum. Meas. 60 3619
[27] Liu H H, Liu Y H 2012 Chin. Phys. B 21 026102
[28] Bi G, Zeng Y 2007 J. Electron. Inform. Technol. 29 1399 (in Chinese) [毕岗, 曾宇 2007 电子与信息学报 29 1399]
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