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SCIENTIA SINICA Informationis
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SCIENTIA SINICA Informationis, Volume 50 , Issue 10 : 1574(2020) https://doi.org/10.1360/SSI-2019-0092

An explicit asymptotic expression of large time-bandwidth product PSWF

Lei SAI 1,2, Hongxing WANG 1,2,*, Faping LU 1,2, Chuanhui LIU 1,2
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  • ReceivedMay 5, 2019
  • AcceptedAug 7, 2019
  • PublishedOct 14, 2020
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Abstract

To address the problems that currently solve large time-bandwidth product prolate spheroidal wave functions (PSWF) with no uniform expression, an explicit asymptotic expression of the large time-bandwidth product PSWF and its differential operator eigenvalues is proposed. The expression is proposed using the theoretical and numerical analyses of the solution errors of the explicit asymptotic expressions based on the Hermite function and Legendre polynomial. The applicable conditions and calculation methods of the large time-bandwidth product ($c>10\pi$) PSWF and its differential operator eigenvalues are accurately and efficiently solved using the Hermite function and Legendre polynomial. Results of the performance comparison analysis indicate that the proposed expression can ensure PSWF and its differential operator eigenvalues of all orders always meet the error requirements. Moreover, the orthogonality and energy concentration of PSWF signals have significant advantages.


Funded by

国家自然科学基金(61701518)

山东省“泰山学者"建设工程专项经费基金项目(20081130)


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Appendix

显式渐近表达式的具体形式[12,21]

\begin{equation}\psi _{m,\text{hermite}}^{c,5}(x)=\sum\limits_{i=0}^{5}{\alpha _{i}^{5}}\phi _{m+4i}^{\sqrt{c}}(x)+\sum\limits_{j=1}^{\min \{ [ m/4 ],5 \}}{\beta _{j}^{5}}\phi _{m-4j}^{\sqrt{c}}(x).\tag{A1}\end{equation}

式(A1) [12]中, PSWF阶数$m\geqslant0$, $c$为时间带宽积, $x\in[-1,1]$, $\phi~_{n}^{a}(x)$为Hermite函数; 式中其他参数见式(A2)$\sim$(A12): \begin{align}\alpha _{0}^{5}=& 1-\frac{12 + 22m + 23{{m}^{2}} + 2{{m}^{3}} + {{m}^{4}}}{{{2}^{10}}{{c}^{2}}} \\ & -\frac{60+158m + 115{{m}^{2}} + 80{{m}^{3}} + 5{{m}^{4}} + 2{{m}^{5}}}{{{2}^{11}}{{c}^{3}}} \\ & -\frac{328032+891024m + 1127140{{m}^{2}} + 476156{{m}^{3}}}{{{2}^{22}}{{c}^{4}}} \\ & -\frac{247887{{m}^{4}}+11768{{m}^{5}} + 3918{{m}^{6}}-4{{m}^{7}}-{{m}^{8}}}{{{2}^{22}}{{c}^{4}}} \\ & -\frac{993120+3161552m + 3698884{{m}^{2}} + 3044356{{m}^{3}}}{{{2}^{22}}{{c}^{5}}} \\ & -\frac{874439{{m}^{4}}+363350{{m}^{5}} + 13566{{m}^{6}}\text{+}3864{{m}^{7}}-9{{m}^{8}}-2{{m}^{9}}}{{{2}^{22}}{{c}^{5}}},\tag{A2} \end{align} \begin{align}\alpha _{1}^{5}=&-\frac{1}{{{2}^{5}}c}\left( \sqrt{\frac{(m+4)!}{m!}} \right)\left(1+\frac{5+2m}{4c}+\frac{4808+3470m + 669{{m}^{2}}-10{{m}^{3}}-{{m}^{4}}}{{{2}^{11}}{{c}^{2}}} \right. \\ & +\frac{46840 + 46762m+16499{{m}^{2}}\text{+}1920{{m}^{3}}-71{{m}^{4}}-6{{m}^{5}}\text{ }}{{{2}^{13}}{{c}^{3}}} \\ & +\frac{212454624+263405280m+128877012{{m}^{2}} + 29276108{{m}^{3}}\text{ }}{3\times {{2}^{22}}{{c}^{4}}} \\ & \left.+\frac{2118049{{m}^{4}}-151072{{m}^{5}}-1030{{m}^{6}} + 20{{m}^{7}} + {{m}^{8}}}{3\times {{2}^{22}}{{c}^{4}}}\right),\tag{A3} \end{align} \begin{align}\alpha _{2}^{5}=& \frac{1}{{{2}^{11}}{{c}^{2}}}\left( \sqrt{\frac{(m+8)!}{m!}} \right)\left(1+\frac{7+2m}{2c}+\frac{37308+19698m + 2833{{m}^{2}}-18{{m}^{3}}-{{m}^{4}}}{3\times {{2}^{10}}{{c}^{2}}}\right. \\ & \left.+\frac{70716+52218m + 13869{{m}^{2}} + 1291{{m}^{3}}-21{{m}^{4}}-{{m}^{5}}}{3\times {{2}^{9}}{{c}^{3}}}\right),\tag{A4} \end{align} \begin{equation}\alpha _{3}^{5}=\frac{-1}{3\times {{2}^{16}}{{c}^{3}}}\left( \sqrt{\frac{(m+12)!}{m!}} \right)\left(1+\frac{3(9+2m)}{4c}+\frac{154128 + 64022m + 7237{{m}^{2}}-26{{m}^{3}}-{{m}^{4}}}{{{2}^{12}}{{c}^{2}}}\right),\tag{A5}\end{equation} \begin{equation}\alpha _{4}^{5}=\frac{1}{3\times {{2}^{23}}{{c}^{4}}}\left( \sqrt{\frac{(m+16)!}{m!}} \right)\left(1+\frac{11+2m}{c}\right),\tag{A6}\end{equation} \begin{equation}\alpha _{5}^{5}=-\frac{1}{15\times {{2}^{28}}{{c}^{5}}}\left( \sqrt{\frac{(m+20)!}{m!}} \right),\tag{A7}\end{equation} \begin{align}\beta _{1}^{5}=& \frac{1}{{{2}^{5}}c}\left( \sqrt{\frac{m!}{(m-4)!}} \right)\left(1-\frac{3-2m}{4c}+\frac{2016-2106m + 693{{m}^{2}} + 6{{m}^{3}}-{{m}^{4}}}{{{2}^{11}}{{c}^{2}}} \right. \\ & -\frac{14592-19778m+10373{{m}^{2}}-2144{{m}^{3}}-41{{m}^{4}} + 6{{m}^{5}}}{{{2}^{13}}{{c}^{3}}} \\ & +\frac{50908320-84318336m+55101860{{m}^{2}}-19514436{{m}^{3}}\text{ }}{3\times {{2}^{22}}{{c}^{4}}} \\ & \left.+\frac{2707329{{m}^{4}}+84528{{m}^{5}}-11142{{m}^{6}}-12{{m}^{7}} + {{m}^{8}}}{3\times {{2}^{22}}{{c}^{4}}}\right),\tag{A8} \end{align} \begin{align}\beta _{2}^{5} =& \frac{1}{{{2}^{11}}{{c}^{2}}}\left( \sqrt{\frac{m!}{(m-8)!}} \right)\left(1-\frac{5-2m}{2c}+\frac{20460-13982m + 2881{{m}^{2}} + 14{{m}^{3}}-{{m}^{4}}}{3\times {{2}^{10}}{{c}^{2}}} \right. \\ & \left.-\frac{31056-28432m + 9880{{m}^{2}}-1365{{m}^{3}}-16{{m}^{4}} + {{m}^{5}}}{3\times {{2}^{9}}{{c}^{3}}}\right),\tag{A9} \end{align} \begin{equation}\beta _{3}^{5}=\frac{1}{3\times {{2}^{16}}{{c}^{3}}}\left( \sqrt{\frac{m!}{(m-12)!}} \right)\left(1-\frac{3(7-2m)}{4c}+\frac{97368-49474m + 7309{{m}^{2}} + 22{{m}^{3}}-{{m}^{4}}}{{{2}^{12}}{{c}^{2}}}\right),\tag{A10}\end{equation} \begin{equation}\beta _{4}^{5}=\frac{1}{3\times {{2}^{23}}{{c}^{4}}}\left( \sqrt{\frac{m!}{(m-16)!}} \right)\left(1-\frac{9-2m}{c}\right),\tag{A11}\end{equation} \begin{equation}\beta _{5}^{5}=\frac{1}{15\times {{2}^{28}}{{c}^{5}}}\left( \sqrt{\frac{m!}{(m-20)!}} \right).\tag{A12}\end{equation} \begin{align}\chi _{m,\text{hermite}}^{c,14}=& c-\frac{3}{4}-\frac{3}{16c}-\frac{15}{64{{c}^{2}}}-\frac{453}{1024{{c}^{3}}}-\frac{4425}{{{2}^{12}}{{c}^{4}}}-\frac{104613}{{{2}^{15}}{{c}^{5}}}-\frac{1442595}{{{2}^{17}}{{c}^{6}}} \\ & -\frac{181431165}{{{2}^{22}}{{c}^{7}}}-\frac{3200304885}{{{2}^{24}}{{c}^{8}}}-\frac{125185972551}{{{2}^{27}}{{c}^{9}}} \\ & -\frac{2689647087045}{{{2}^{29}}{{c}^{10}}}-\frac{251987915369193}{{{2}^{33}}{{c}^{11}}}-\frac{6392700476893245}{{{2}^{35}}{{c}^{12}}} \\ & -\frac{349366400286979629}{{{2}^{38}}{{c}^{13}}}-\frac{40950465047128293315}{{{2}^{42}}{{c}^{14}}}.\tag{A14} \end{align} \begin{align}\chi _{m,\text{hermite}}^{c,6}=& c(1+2m)-\frac{3+2m+2{{m}^{2}}}{4}-\frac{3+7m+3{{m}^{2}}+2{{m}^{3}}}{{{2}^{4}}c} \\ & -\frac{15 + 35m + 40{{m}^{2}} + 10{{m}^{3}} + 5{{m}^{4}}}{{{2}^{6}}{{c}^{2}}} \\ & -\frac{453 + 1321m + 1278{{m}^{2}} + 962{{m}^{3}} + 165{{m}^{4}} + 66{{m}^{5}}}{{{2}^{10}}{{c}^{3}}} \\ & -\frac{4425 + 13349m + 18478{{m}^{2}} + 10510{{m}^{3}}}{{{2}^{12}}{{c}^{4}}} \\ & -\frac{5885{{m}^{4}}+756{{m}^{5}} + 252{{m}^{6}}}{{{2}^{12}}{{c}^{4}}} \\ & -\frac{104613 + 355301m+469780{{m}^{2}} + 419424{{m}^{3}}}{{{2}^{15}}{{c}^{5}}} \\ & -\frac{163045{{m}^{4}} + 72596{{m}^{5}}+7378{{m}^{6}} + 2108{{m}^{7}}}{{{2}^{15}}{{c}^{5}}} \\ & -\frac{1442595 + 5046979m+8070552{{m}^{2}} + 6440672{{m}^{3}}}{{{2}^{17}}{{c}^{6}}} \\ & -\frac{4213538{{m}^{4}} + 1218126{{m}^{5}}+449848{{m}^{6}}+37548{{m}^{7}} + 9387{{m}^{8}}}{{{2}^{17}}{{c}^{6}}}.\tag{A15} \end{align} \begin{align}\chi _{m,\text{legendre}}^{c}=& m(m+1)+\frac{{{c}^{2}}}{2}+\frac{{{c}^{2}}(4+{{c}^{2}})}{32{{m}^{2}}}-\frac{{{c}^{2}}(4+{{c}^{2}})}{32{{m}^{3}}}+\frac{{{c}^{2}}(28+13{{c}^{2}})}{128{{m}^{4}}} \\ & -\frac{{{c}^{2}}(20+11{{c}^{2}})}{64{{m}^{5}}}+\frac{{{c}^{2}}(3904+3936{{c}^{2}}+160{{c}^{4}}+5{{c}^{6}})}{8192{{m}^{6}}} \\ & -\frac{{{c}^{2}}(5824+8416{{c}^{2}}+480{{c}^{4}}+15{{c}^{6}})}{8192{{m}^{7}}}+{{c}^{2}}O\left( \frac{{{c}^{8}}}{{{m}^{8}}} \right).\tag{A16} \end{align} 式(A13) [21]中, PSWF阶数$m>0$, $c$为时间带宽积, $x\in[-1,1]$, ${{\bar{P}}_{n}}(x)$表示归一化Legendre多项式; 式(A14) [12]中, PSWF阶数$m=0$, $c$为时间带宽积; 式(A15) [12]和(A16) [21]中, PSWF的阶数$m>0$, $c$为时间带宽积.


References

[1] Slepian D, Pollak H O. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I. Bell Syst Technical J, 1961, 40: 43-63 CrossRef Google Scholar

[2] Zhao Z Y, Wang H X, Li H L, et al. Designing method of orthogonal pulse in time domain based on prolate spheroidal wave functions for nonsinusoidal wave communication. J Electron Inform Technol, 2009, 31: 2912--2916. Google Scholar

[3] Gosse L. Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions. Ann Univ Ferrara, 2013, 59: 81-116 CrossRef Google Scholar

[4] De Sanctis M, Cianca E, Rossi T. Waveform design solutions for EHF broadband satellite communications. IEEE Commun Mag, 2015, 53: 18-23 CrossRef Google Scholar

[5] Majhi S, Richardson P. Capacity Analysis of Orthogonal Pulse-Based TH-UWB Signals. Wireless Pers Commun, 2012, 64: 255-272 CrossRef Google Scholar

[6] Djordjevic I B. On the Irregular Nonbinary QC-LDPC-Coded Hybrid Multidimensional OSCD-Modulation Enabling Beyond 100 Tb/s Optical Transport. J Lightwave Technol, 2013, 31: 2669-2675 CrossRef ADS Google Scholar

[7] Wang H X, Zhao Z Y, Liu X G, et al. China Patent, ZL200810159238.3, 2011-02-02. Google Scholar

[8] Wang H X, Chen Z N, Liu C H, et al. The design of orthogonal prolate spheroidal wave function pulse waveform based on Karhunen-Loeve transform. J Electron Inform Technol, 2012, 34: 2415--2420. Google Scholar

[9] Shu G C, Wang H X. Nonsinusoidal demodulation method based on prolate spheroidal wave functions and its performance analysis. J Circ Syst, 2011, 16: 90--93. Google Scholar

[10] Bouwkamp C J. On Spheroidal Wave Functions of Order Zero. J Math Phys, 1947, 26: 79-92 CrossRef Google Scholar

[11] Hodge D B. Eigenvalues and Eigenfunctions of the Spheroidal Wave Equation. J Math Phys, 1970, 11: 2308-2312 CrossRef ADS Google Scholar

[12] Xiao H, Rokhlin V. High-Frequency Asymptotic Expansions for Certain Prolate Spheroidal Wave Functions. J Fourier Anal Appl, 2003, 9: 575-596 CrossRef Google Scholar

[13] Khare K, George N. Sampling theory approach to prolate spheroidal wavefunctions. J Phys A-Math Gen, 2003, 36: 10011-10021 CrossRef ADS Google Scholar

[14] Khare K. Bandpass sampling and bandpass analogues of prolate spheroidal functions. Signal Processing, 2006, 86: 1550-1558 CrossRef Google Scholar

[15] Liu C H, Wang H X, Zhang L, et al. Fast convergent algorithm of reconstructing bandpass prolate spheroidal wave function. J Jilin Univ (Eng Technol Edition), 2013, 43: 1091--1097. Google Scholar

[16] Mu J, Wang H X. A reconstructing method of PSWF pulse waveform using walsh functions. J China Acad of Electron Inform Technol, 2012, 7: 623--626. Google Scholar

[17] Parr B, Cho B, Wallace K. A novel ultra-wideband pulse design algorithm. IEEE Commun Lett, 2003, 7: 219-221 CrossRef Google Scholar

[18] Shu G C, Wang H X, Zhao Z Y, et al. Numerical solution of prolate spheroidal wave functions based on bandpass sampling. J Circ Syst, 2011, 16: 132--136. Google Scholar

[19] Zhong P L, Wang H X, Zhao Z Y. Resolution and simulation of differential system for prolate spherical wave functions. J Jilin Univ (Eng Technol Edition), 2013, 43: 1675--1679. Google Scholar

[20] Liu C H, Wang H X, Zhang L, et al. Design of PSWF pulse generator based on difference quantitation. J Appl Sci, 2013, 31: 375--380. Google Scholar

[21] Rokhlin V, Xiao H. Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit. Appl Comput Harmonic Anal, 2007, 22: 105-123 CrossRef Google Scholar

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    Figure 1

    (Color online) Solution errors of PSWF $\mathrm{log}E_1$

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    Figure 2

    (Color online) Solution errors of eigenvalues of differential operators of PSWF $\mathrm{log}E_2$

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    Figure 3

    (Color online) The solution errors of PSWF

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    Figure 4

    (Color online) The frequency domain energy concentration of PSWF

  • Table 1   The values of the abscissas corresponding to the intersections in Figure
    $c$ (rad$\cdot$s) 11$\pi$ 20$\pi$ 40$\pi$ 60$\pi$ 80$\pi$ 100$\pi$ 101$\pi$
    The values of the abscissas 7.40 13.75 26.41 40.08 53.33 66.06 66.92
  • Table 2   The values of the abscissas corresponding to the intersections in Figure
    $c$ (rad$\cdot$s) 11$\pi$ 20$\pi$ 40$\pi$ 60$\pi$ 80$\pi$ 100$\pi$ 101$\pi$
    The values of the abscissas 7.32 13.17 26.02 39.54 53.31 65.98 66.15
  • Table 3   The distribution range of the cross-correlation values between PSWFs when $c=20\pi$
    Method Order The 0th$\sim$13th PSWF The 14th$\sim$19th PSWF
    This paper The 0th$\sim$13th PSWF $2.56~\times~10^{-32}~\sim~7.44~\times~10^{-15}$ $5.14~\times~10^{-9}~\sim~8.08~\times~10^{-3}$
    Ref. [12] $2.56~\times~10^{-32}~\sim~7.44~\times~10^{-15}$ $1.39~\times~10^{-3}~\sim~4.62~\times~10^{+2}$
    Ref. [21] $6.35~\times~10^{-2}~\sim~2.83~\times~10^{+2}$ $3.76~\times~10^{-2}~\sim~4.90~\times~10^{+1}$
    This paper The 14th$\sim$19th PSWF Symmetry $1.74~\times~10^{-34}~\sim~9.29~\times~10^{-19}$
    Ref. [12] $7.52~\times~10^{-2}~\sim~5.07~\times~10^{+2}$
    Ref. [21] $1.74~\times~10^{-34}~\sim~9.29~\times~10^{-19}$
  • Table 4   The distribution range of the cross-correlation values between PSWFs when $c=100\pi$
    Method Order The 0th$\sim$66th PSWF The 67th$\sim$99th PSWF
    This paper The 0th$\sim$66th PSWF $1.07~\times~10^{-30}~\sim~7.66~\times~10^{-14}$ $7.51~\times~10^{-7}~\sim~2.33\times~10^{-2}$
    Ref. [12] $1.07\times~10^{-30}~\sim~7.66~\times~10^{-14}$ $3.54~\times~10^{-2}~\sim~9.85\times~10^{+1}$
    Ref. [21] $5.43~\times~10^{-2}~\sim~1.24~\times~10^{+3}$ $6.30~\times~10^{-1}~\sim~7.12~\times~10^{+2}$
    This paper The 67th$\sim$99th PSWF Symmetry $4.35~\times~10^{-33}~\sim~5.48~\times~10^{-17}$
    Ref. [12] $6.94~\times~10^{-2}~\sim~3.07~\times~10^{+2}$
    Ref. [21] $4.35\times~10^{-33}~\sim~5.48~\times~10^{-17}$