Analysis of Weighted Fractional Fourier Transform Based Hybrid Carrier Signal Characteristics

doi:10.1007/s12204-020-2156-5

Abstract

Abstract: Recently, a weighted fractional Fourier transform (WFRFT) based hybrid carrier (HC) system has been proposed, which can converge single-carrier (SC) and multi-carrier (MC) systems. The cost and power dissipation of analog components often dominate in practical HC systems. Therefore, in this paper, we analyze the baseband HC signal characteristics, including average signal power, power spectral density (PSD), probability density function (PDF), peak-to-average power ratio (PAPR), and complementary cumulative distribution function (CCDF), using a continuous-time HC signal approximation method. Through theoretical analysis and simulation validation, it is proved that the approximation method does not change the average signal power, bandwidth and out-of-band emission (OOBE). The theoretical PDF expression of HC signal with the approximation method is then proposed. Furthermore, the PDF can also explain why the PAPR of baseband continuous-time HC signal is higher than that of discrete-time HC signal. The PAPR performance and the power amplifier (PA) efficiency of HC systems in different conditions are analyzed. Through the PDF analysis of HC signal envelope and the numerical computations of PAPR, it is shown that the approximation method can offer a precise characteristic description for HC signal, which can help to improve the system PAPR performance and the PA efficiency.

Key words: weighted fractional Fourier transform (WFRFT)| power spectral density (PSD)| probability density function (PDF)| peak-to-average power ratio (PAPR)| clipping

摘要： Recently, a weighted fractional Fourier transform (WFRFT) based hybrid carrier (HC) system has been proposed, which can converge single-carrier (SC) and multi-carrier (MC) systems. The cost and power dissipation of analog components often dominate in practical HC systems. Therefore, in this paper, we analyze the baseband HC signal characteristics, including average signal power, power spectral density (PSD), probability density function (PDF), peak-to-average power ratio (PAPR), and complementary cumulative distribution function (CCDF), using a continuous-time HC signal approximation method. Through theoretical analysis and simulation validation, it is proved that the approximation method does not change the average signal power, bandwidth and out-of-band emission (OOBE). The theoretical PDF expression of HC signal with the approximation method is then proposed. Furthermore, the PDF can also explain why the PAPR of baseband continuous-time HC signal is higher than that of discrete-time HC signal. The PAPR performance and the power ampliˉer (PA) efficiency of HC systems in different conditions are analyzed. Through the PDF analysis of HC signal envelope and the numerical computations of PAPR, it is shown that the approximation method can offer a precise characteristic description for HC signal, which can help to improve the system PAPR performance and the PA efficiency.

关键词: weighted fractional Fourier transform (WFRFT)| power spectral density (PSD)| probability density function (PDF)| peak-to-average power ratio (PAPR)| clipping

CLC Number:

TN 911

WANG Xiaolu (王晓鲁), MEI lin (梅林), WANG Zhenduo (王震铎), SHA Xuejun1(沙学军). Analysis of Weighted Fractional Fourier Transform Based Hybrid Carrier Signal Characteristics[J]. Journal of Shanghai Jiao Tong University (Science), 2020, 25(1): 27-36.

References 33

[1]	NAMIAS V. The fractional order Fourier transform and its application to quantum mechanics [J]. IMA Journal of Applied Mathematics, 1980, 25(3): 241-265.
[2]	WHITE P R, LOCKE J. Performance of methods based on the fractional Fourier transform for the detection of linear frequency modulated signals [J]. IET Signal Processing, 2012, 6(5): 478-483.
[3]	LU G K, XIAO M L, WEI P, et al. Circularity of the fractional Fourier transform and spectrum kurtosis for LFM signal detection in Gaussian noise model [J].IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 2015, E98A(12): 2709-2712.
[4]	LU M F, ZHANG F, TAO R, et al. Parameter estimation of optical fringes with quadratic phase using the fractional Fourier transform [J]. Optics and Lasers in Engineering, 2015, 74: 1-16.
[5]	ZHAO Y B, YU H, WEI G, et al. Parameter estimation of wideband underwater acoustic multipath channels based on fractional Fourier transform [J]. IEEE Transactions on Signal Processing, 2016, 64(20): 5396-5408.
[6]	WU J M, LU M F, TAO R, et al. Improved FRFT-based method for estimating the physical parameters from Newton's rings [J]. Optics and Lasers in Engineering, 2017, 91: 178-186.
[7]	WEI D Y, LI Y M. Generalized sampling expansions with multiple sampling rates for lowpass and bandpass signals in the fractional Fourier transform domain [J].IEEE Transactions on Signal Processing, 2016, 64(18):4861-4874.
[8]	LIU N, TAO R, WANG R, et al. Signal reconstruction from recurrent samples in fractional Fourier domain and its application in multichannel SAR [J]. Signal Processing, 2017, 131: 288-299.
[9]	ZHANG F, HU Y, TAO R, et al. New fractional matrix with its applications in image encryption [J]. Optics and Laser Technology, 2014, 64(4): 82-93.
[10]	SUI L S, LU H W, NING X J, et al. Asymmetric double-image encryption method by using iterative phase retrieval algorithm in fractional Fourier transform domain [J]. Optical Engineering, 2014, 53(2):026108.
[11]	SHIH C C. Fractionalization of Fourier transform [J].Optics Communications, 1995, 118: 495-498.
[12]	MEI L, SHA X J, ZHANG N T. The approach to carrier scheme convergence based on 4-weighted fractional Fourier transform [J]. IEEE Communications Letters,2010, 14(6): 503-505.
[13]	MEI L, ZHANG Q Y, SHA X J, et al. WFRFT precoding for narrowband interference suppression in DFT-based block transmission systems [J]. IEEE Communications Letters, 2013, 17(10): 1916-1919.
[14]	WANG K, SHA X J, LI Y. Hybrid carrier modulation with time-domain windows and iterative equalization over underwater acoustic channels [J]. IEEE Communications Letters, 2013, 17(8): 1489-1492.
[15]	LI Y, SHA X J, WANG K. Hybrid carrier communication with partial FFT demodulation over underwater acoustic channels [J]. IEEE Communications Letters,2013, 17(12): 2260-2263.
[16]	CHEN Q, SUN H X, QI J, et al. Application of weighted fractional Fourier transform in hybrid system[J]. Journal of Harbin Institute of Technology, 2016,48(5): 100-104 (in Chinese).
[17]	HUI Y T, LI B B, TONG Z. 4-weighted fractional Fourier transform over doubly selective channels and optimal order selecting algorithm [J]. Electronics Letters, 2015, 51(2): 177-179.
[18]	FANG X J, ZHANG N, ZHANG S, et al. On physical layer security: Weighted fractional Fourier transform based user cooperation [J]. IEEE Transactions on Wireless Communications, 2017, 16(8): 5498-5510.
[19]	WANG Z D, MEI L, WANG X L, et al. Bit error rate analysis of generalised frequency division multiplexing with weighted-type fractional Fourier transform precoding [J]. IET Communications, 2017, 11(6): 916-924.
[20]	WANG Z D, MEI L, WANG X L, et al. On the performance of hybrid carrier system with spectrum precoding based on WFRFT [J]. EURASIP Journal on Wireless Communications and Networking, 2017,2017: 102.
[21]	JIANG T, WU Y Y. An overview: Peak-to-average power ratio reduction techniques for OFDM signals [J].IEEE Transactions on Broadcasting, 2008, 54(2): 257-268.
[22]	WANG X L, MEI L, ZHANG N T, et al. PAPR approximation of continuous-time WFRFT signals [C]//2014 IEEE International Conference on Communication Systems. Macau, China: IEEE, 2015: 303-307.
[23]	QU D M, DING J, JIANG T, et al. Detection of noncontiguous OFDM symbols for cognitive radio systems without out-of-band spectrum synchronization[J]. IEEE Transactions on Wireless Communications,2011, 10(2): 693-701.
[24]	WANG Y C, LUO Z Q. Optimized Iterative clipping and ˉltering for PAPR reduction of OFDM signals [J].IEEE Transactions on Communications, 2011, 59(1):33-37.
[25]	LI Y, SHA X J, ZHENG F C, et al. Low complexity equalization of HCM systems with DPFFT demodulation over doubly-selective Channels [J]. IEEE Signal Processing Letters, 2014, 21(7): 862-865.
[26]	HU L N, JIANG L G, HE C, et al. Watermark in the digital video broadcasting handheld transmission signal [J]. Journal of Shanghai Jiao Tong University(Science), 2009, 14(2): 149-153.
[27]	OPPENHEIM A V, SCHAFER R W, BUCK J R. Discrete-time signal processing [M]. 2nd ed. Upper Saddle River, New Jersey, USA: Prentice-Hall, 1999.
[28]	STOICA P, MOSES R L. Spectral analysis of signals [M]. Upper Saddle River, New Jersey, USA: Prentice Hall, 2005.
[29]	WANG X L, MEI L, WANG Z D, et al. On the probability density function of the real and imaginary parts in WFRFT signals [J]. China Communications, 2016,13(9): 44-52.
[30]	WEI S Q, GOECKEL D L, KELLY P A. Convergence of the complex envelope of bandlimited OFDM signals[J]. IEEE Transactions on Information Theory, 2010,56(10): 4893-4904.
[31]	TABOGA M. Lectures on probability theory and mathematical statistics [M]. 2nd ed. North Charleston,USA: CreateSpace Independent Publishing, 2012.
[32]	RAHMATALLAH Y, MOHAN S. Peak-to-average power ratio reduction in OFDM systems: A survey and taxonomy [J]. IEEE Communications Surveys & Tutorials, 2013, 15(4): 1567-1592.
[33]	CRIPPS S C. RF power ampliˉers for wireless communications [M]. 2nd ed. Norwood, USA: Artech House,2006.