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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

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  • ReceivedMay 5, 2019
  • AcceptedAug 7, 2019
  • PublishedOct 14, 2020

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.


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  • Figure 1

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

  • Figure 2

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

  • Figure 3

    (Color online) The solution errors of PSWF

  • 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] $4.35\times~10^{-33}~\sim~5.48~\times~10^{-17}$