Prolate Spheroidal Wave Functions Generation Method Based on IDFT

Abstract

Abstract:

Aimed at the issue of the generation of prolate spheroidal wave function(PSWF) in real time, based on the properties of PSWFs, and starting from the Fourier series expansion, the Fourier expansion formulas of PSWFs was proposed, the finite Fourier expansion was investigated, and the error of the finite Fourier expansion was analysed. A method to reconstruct PSWFs based on IDFT was presented, which established a link between the PSWF signal and the sine or cosine series, introduced the efficient and mature Fourier transform technology to the generation of PSWFs, and improved hardware realizability of PSWFs, providing a practical way to generate other complex signal waveforms in real time.

Key words: prolate spheroidal wave functions (PSWF), inverse discrete Fourier transform (IDFT), inverse fast Fourier transform (IFFT), reconstruct

CLC Number:

TN 911.3

KANG Jiafanga,WANG Hongxinga,b,LIU Chuanhuia,ZHAO Zhiyonga,LIU Xiguoa. Prolate Spheroidal Wave Functions Generation Method Based on IDFT[J]. Journal of Shanghai Jiaotong University, 2014, 48(03): 357-362.

References

［1］Moore I C, Cada M. Prolate spheroidal wave functions, an introduction to the Slepian series and its properties［J］. Appl Comput Harmon Anal, 2004, 16: 208230.

［2］Slepian D, Pollak H. Prolate spheroidal wave functions, Fourier analysis and uncertainty, I［J］. System Technical Journal,1961,40(1):4364.

［3］Usuda K, Zhang H G, Nakagawa M. Mary pulse shape modulation for PSWFbased UWB systems in multipath fading environment［C］∥Global Telecommunications Conference. Dallas, USA:［s.n.］,2004:39453504.

［4］Sacchi C, Rossi T, Ruggieri M, et al. Efficient waveform design for highbitrate Wband satellite transmissions［J］. Aerospace and Electronic Systems, IEEE Transactions on, 2011, 47(2): 974995.

［5］Dullaert W, Rogier H, Boeykens F. Advantages of PSWFbased models for UWB systems［C］∥IEEEAPS Topical Conference on Antennas and Propagation in Wireless Communications(APWC). Cape Town, South Africa:［s.n.］, 2012.

［6］Feng Y, Debes C, Zoubir A. Parametric waveform design using discrete prolate spheroidal sequences for enhanced detection of extended targets［J］. Signal Processing, IEEE Transactions on, 2012, 60(9): 45254536.

［7］Wei L Y, Kennedy R, Lamahewa T. An optimal basis of bandlimited functions for signal analysis and design［J］. Signal Processing, IEEE Transactions on, 2010, 58(11): 57445755.

［8］Lindquist M, Zhang C H, Glover G, et al. A generalization of the twodimensional prolate spheroidal wave function method for nonrectilinear MRI data acquisition methods［J］. Image Processing, IEEE Transactions on, 2006, 15(9): 27922804.

［9］王红星，赵志勇，刘锡国等. 非正弦时域正交调制方法［P］. 中国：ZL 2008 10159238.3, 2011.

［10］Walter G, Soleski T. A new friendly method of computing prolate spheroidal wave functions and wavelets［J］. Applied and Computational Harmonic Analysis, 2005, 19(3): 432443.

［11］Abderrazek K, Tahar M. New efficient methods of computing the prolate spheroidal wave functions and their corresponding eigenvalues［J］. Applied and Computational Harmonic Analysis, 2008,24(3): 269289.

［12］Volkmer H. Spheroidal wave functions, in: Handbook of Mathematical Functions［Z］. In: Applied Math Series, Nat Bureau of Standards. London: Combridge University Press, 2004.

［13］陆音，朱洪波. 基于近似扁长椭球波函数的超宽带脉冲设计［J］. 通信学报. 2005, 6(10): 6064.

LU Yin, ZHU Hongbo. UWB pulse design method based on approximate prolate spheroidal wave functions［J］. Journal on Communications, 2005, 26(10): 6064.

［14］Xiao H, Rokhlin V. Highfrequency asymptotic expansions for certain prolate spheroidal wave functions［J］. Journal of Fourier Analysis and Applications, 2003, 9(6): 575.

［15］Khare K, George N. Sampling theory approach to prolate spheroidal wavefunctions［J］. Journal of Physics, 2003(36): 10011100021.

［16］Kedar K. Bandpass sampling and bandpass analogues of prolate spheroidal functions［J］. Signal Processing, 2006(86): 15501558.

［17］王红星，刘锡国，赵志勇，等. 椭圆球面波函数的快速重构算法［J］. 电波科学学报. 2011(4): 765770.

WANG Hongxing, LIU Xiguo, ZHAO Zhiyong, et al. Fast method of reconstructing prolate spheroidal wave function［J］.Chinese Journal of Radio Science, 2011(4): 765770.

［18］Xiao H. Prolate spheroidal wave functions, quadrature, interpolation, and asymptotic formulae［D］. New Haven: Yale University, Department of Computing Science, 2001.

[1]	LIU Mengge, LIU Hao, HE Xin, JIN Shaohui, CHEN Pengyun, XU Mingliang. Research Advances on Non-Line-of-Sight Imaging Technology [J]. J Shanghai Jiaotong Univ Sci, 2025, 30(5): 833-854.
[2]	WANG Hongxin, XU Degang, ZHOU Kaiwen, LI Linwen, WEN Xin. Data-Driven Method of Modeling Sparse Flow Field Data [J]. Journal of Shanghai Jiao Tong University, 2025, 59(5): 684-690.
[3]	WANG Chaochen, JIANG Hongru, WANG Buli, XIA Qiaowei, ZHANG Xianchun. From Kill Chain to Kill Web： A Survey on Modeling， Evaluation， and Optimization [J]. Air & Space Defense, 2025, 8(4): 1-8.
[4]	Duolin, Xu Boyu, Ren Yong, Yang Xin. Magnetic Resonance Imaging Reconstruction Based on Butterfly Dilated Geometric Distillation [J]. J Shanghai Jiaotong Univ Sci, 2025, 30(3): 591-599.
[5]	GUO Li, YUAN Yuchao, TANG Wenyong. Inversion of Displacement Field of Marine Slender Pipelines Under Three-Dimensional Background Ocean Currents [J]. Journal of Shanghai Jiao Tong University, 2025, 59(12): 1815-1823.
[6]	LI Jinfeng, CHEN Wuguang, ZHANG Zhengchuan, XU Yongliang, LI Kaiying, YIN Junlian, WANG Dezhong. Measurement of Gas-Liquid Two-Phase Flow in Draft Tube [J]. Journal of Shanghai Jiao Tong University, 2024, 58(8): 1188-1200.
[7]	ZHOU Lei1, SU Xin2, ZHANG Qi2, HUANG Yi2. Anomaly Detection of Environmental Features Based on LSTM-Reconstructed Data [J]. Ocean Engineering Equipment and Technology, 2024, 11(4): 1-7.
[8]	HU Guofeng, DOU Qiwei. 3D Enhanced Imaging Technology for Offshore Wind Power Pile Foundation [J]. Ocean Engineering Equipment and Technology, 2024, 11(3): 42-48.
[9]	LI Xu, XIAO Longfei, WEI Handi, WU Wencheng, ZHU Ziyang, LI Yan. Inverse Reconstruction of Environmental Loads and Virtual Model Test [J]. Journal of Shanghai Jiao Tong University, 2024, 58(2): 141-146.
[10]	HOU Xianrui, ZHOU Xingyu, HUANG Xiaocheng. Reconstruction of Ship Propeller Wake Field Based on Physics-Informed Neural Networks [J]. Journal of Shanghai Jiao Tong University, 2024, 58(11): 1654-1664.
[11]	DUAN Jizhong, QIAN Qingqing. Fast Parallel Imaging Reconstruction Method Based on SIDWT and Iterative Self-Consistency [J]. Journal of Shanghai Jiao Tong University, 2023, 57(5): 582-592.
[12]	CHEN Jichi, WEI Guohua, CHAI Juanfang . A Method for Evaluating the Effect and Accuracy of 3D Reconstruction Based on Infrared Image [J]. Air & Space Defense, 2023, 6(4): 42-50.
[13]	LI Jie, SI Weijian, LIU Jianing. DOA Estimation of Nested Array Under Non-Uniform Noise [J]. Air & Space Defense, 2023, 6(4): 86-92.
[14]	LI Dingjia1,2,3,4(黎定佳)，WANG Chongang1,2,3(王重阳)，GUO Wei5(郭伟)，WANG Zhidong6(王志东)，ZHANG Zhongtao5(张忠涛)，LIU Hao1,2,3*(刘浩). Shape Sensing for Single-Port Continuum Surgical Robot Using Few Multicore Fiber Bragg Grating Sensors [J]. J Shanghai Jiaotong Univ Sci, 2023, 28(3): 312-322.
[15]	LI Jianjun, ZHU Wenfeng, SUN Haitao, LI Yuanhui, WANG Shunchao. Method of Curing Deformation Prediction and Surface Reconstruction Compensation for Roller-Hemming Structures with Dissimilar Materials [J]. Journal of Shanghai Jiao Tong University, 2022, 56(7): 965-976.