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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


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 $\mathrm{log}E_1$

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