FFT在功率谱密度计算中的应用

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1、FFT在功率谱密度计算中的应用一、FFT算法理论依据和编程思想FFT算法的基本思想：考察DFT与IDFT的运算发现,利用以下两个特性可减少运算量：系数是一个周期函数,它的周期性和对称性可利用来改进运算,提高计算效率.如:因此利用这些周期性和对称性,DFT运算中有些项可合并；利用WNnk的周期性和对称性,把长度为N点的大点数的DFT运算分解为若干个小点数的DFT.因为DFT的计算量正比于N2,N小计算量也就小. FFT算法正是基于这样的基本思想发展起来的.它有多种形式,下面是按时间抽取的FFTN点DFT运算的分解先从一个特殊情况开始,假定N是2的整数次方,N=2M,M：正整数 1.将N点的DFT

2、分解为两个N/2点的DFT：首先将序列xn分解为两组,一组为偶数项,一组为奇数项r=0,1,N/2-1将DFT运算也相应分为两组：其中X1k和X2k分别是x1r和x2r的N/2点DFT.可见,一个N点的DFT可以分解为两个N/2点的DFT,这两个N/2点的DFT再按照上面1式合成为一个N点DFT,注意到,X1k,X2k有N/2个点,即k=0,1,N/2-1,由1式得到Xk只有N/2点,而实际上Xk有N个点,即k=0,1,N-1,要用X1k,X2k表示全部XK值,还必须应用系数w的周期性和对称性.2.X的N-1点表示：由X= X1k+wkN X2k, k=0,1,2,N/2-1 得：,2a因为,

3、且同样.考虑到WNk对称性：.故2b2a式表示了Xk前半部分k=0N/2-1时的组成方式,2b式则表示了后半部分k=N/2N-1时的组成方式.这两式所表示的运算过程可用一个称作蝶形的信号流图来表示.3.蝶形信号流图：如图1a所示,图中左面两支为输入,中间以一个小圆圈表示加、减运算,右上支为相加输出,右下支为相减输出,如果在某一支路上信号需要进行乘法运算,则在该支路上标以箭头,并将相乘的系数标在简头边,这样2a,2b所表示的运算,可用图1b所表示的蝶形结来表示.采用这种表示法,可将以上以讨论的分解过程用计算流图来表示.图2.6所示为N=23=8的例子.通过这样分解以后,每一个N/2点DFT只需要

4、图2.6 N点DFT分解为2个N/2点DFTN=82= N2/4次复数乘法运算,两个N/2点的DFT需要22= N2/2 次复乘,再加上将两个N/2点DFT合成为N点DFT时,在蝶形结前的N/2次复乘,共需要2+N/2 N2/2次复乘,由此可见,经过这样的分解处理,运算量差不多节省了一倍.4.将N/2点的DFT分解为两个N/4点的DFT：既然这样分解是有效的,由于N=2M,N/2仍然是偶数,因此可对两个N/2点的DFT再分别作进一步分解,例如对x1r和x2r可以再按其偶数部分与奇数部分分解为两个N/4点的DFT,既然这样分解是有效的,由于N=2M,N/2仍然是偶数,因此可对两个N/2点的DFT

5、再分别作进一步分解,例如对x1r和x2r可以再按其偶数部分与奇数部分分解为两个N/4点的DFT, l=0,1,N/4-1而同样X2也可这样分解,并且将系数统一为 ,这样一个8点DFT就可分解为四个2点的DFT,如图2.7所示. 图2.7 N点DFT分解为4个N/4点的DFT5.2个点DFT的表示：最后剩下的是2点DFT,它可以用一个蝶形结表示,例如,x0,x4所组成的2点DFT就可表示式：这样,一个8点的完整的按时间抽取运算的流图如图2.8所示. 由于这样的方法每一步分解都是按输入时间序列是属于偶数还是奇数来抽取的,所以称为按时间抽取法或时间抽取法.6.时间抽取法FFT运算特点：1蝶形运算对任

6、何一2的整数幂N=2M,总是可以通过M次分解最后完全成为2点的DFT运算.这样的M次分解,就构成从x到X的M级运算过程.从上面的流图可看到,每一级运算都由N/2个蝶形运算构成.因此每一级运算都需要 N/2次复乘和N次复加,这样,经过时间抽取后M级运算总共需要的运算：复乘复加 NM=Nlog2N实际运算量与这个数字稍有出入,因为W这几个系数实际上都不用乘法运算,因此在上面N=8的例子中,实际上只有两个系数WN1与WN3是需要乘法运算的.用时间抽取法所需的计算量,不论是复乘还是复加都与Nlog2N成正比,而直接运算时则与N2成正比.例N=2048,N2=4194304,log2N=11264,N2

7、/log2N=392.4倍.FFT显然要比直接法快得多.2原位计算当数据输入到存储器中以后,每一级运算的结果仍然储存在同一组存储器中,直到最后输出,中间无需其它存储器,这叫原位计算.例如,N=8的FFT运算,输入x,x,x,x,x可分别存入A,A,A这9个存储单元中,在第一级运算中,首先是存储单元A,A中x,x进入蝶形运算,x,x输入运算器后,其数值不再需要保存,因此蝶形运算的结果可仍然送回存储单元A,A中保存,然后A,A中x,x再进入蝶形运算,其结果再送回A,A,一直到算完A,A,则完成了第一级运算过程.第二级运算仍可采用这种原位的方式,但是进入蝶形结的组合关系不同,首先进入蝶形结的是A、A

8、存储单元中的数据,运算结果仍可送回A、A保存,然后进入蝶形结的是A、A,依此类推,每一级运算均可在原位进行,这种原位运算结构可节省存储单元,降低设备成本,还可节省找地址的时间.3序数重排对按时间抽取FFT的原位运算结构,当运算完毕时,这种结构存储单元A、A,A中正好顺序存放着X,X,X,X,因此可直接按顺序输出,但这种原位运算的输入x却不能按这种自然顺序存入存储单元中,而是按X,X,X,X,X的顺序存入存储单元,这种顺序看起来相当杂乱,然而它也是有规律的.当用二进制表示这个顺序时,它正好是码位倒置的顺序.例如,原来的自然顺序应是x的地方,现在放着x,用二进制码表示这一规律时,则是在x处放着x,

9、x处放着x.即将自然顺序的二进制码位倒置过来,第一位码变成最末位码,这样倒置以后的顺序正是输入所需要的顺序.下表列出N=8时按码位倒置规律所得的顺序,其结果与按时间抽取FFT流图中的输入顺序是一致的.表一 码位倒置顺序自然顺序二进码表示码位倒置码位倒置顺序0000000010011004201001023011110641000011510110156110011371111117在实际运算中,一般直接将输入数据xn按码位倒置的顺序排好输入很不方便,总是先按自然顺序输入存储单元,然后再通过变址运算将自然顺序的存储转换成码位倒置顺序的存储,然后进行FFT的原位计算.目前有许多通用DSP芯片支持这

10、种码位倒置的寻址功能.4蝶形类型随迭代次数成倍增加观察8点FFT的三次迭代运算第一级迭代,只有一种类型的蝶形运算系数W08第二级迭代,有二种类型的蝶形运算系数W08、W28,参加运算的两个数据点间隔为2.第三级迭代,有四类蝶形运算系数W08、W18、W28、W38,参加运算的两个数据点间隔为4. 所以,每次迭代的蝶形类型比上一次蝶代增加一倍,数据点间隔也增大一倍.7.功率谱密度的计算根据相关定理与维纳-辛钦关系式可得随机信号序列x的功率谱密度 其估计值 如果观察到序列x的N个值,即x, x,x,就可以通过FFT直接求得X,再按式求得Sx,其计算过程如图2.9所示.xFFT平方平方加法除法SxX

11、2/NX2X2RXI2XR图2.9 周期图法计算功率谱密度流程XI二、程序设计本设计在主程序中分别调用了输入get_in、倒序re_order和蝶式运算butterfly,功率谱密度计算power,绘图on_draw五个子函数.1.程序框图（1） 运算主程序框图、整序程序流程图LHN/2jLHN1N-2kLHij?TxxxxTTyyyyTNNY返回i=1, N1jj-kk=k/2jj+kJk?图7 整序程序流程图3蝶形运算框图i1j0r0m12i-1m22m1m3N/ m2m4j*m2k1m4+rk2k1+m1xx-uyy-vxx+uyy+vrr+1rm1?jj+1jm3?ii+1jL?返回Y

12、YYNN图8 蝶形运算框图三、程序源代码主程序/ FFT.cpp : Defines the class behaviors for the application.#include stdafx.h#include FFT.h#include FFTDlg.h/自己添加的.#includemath.h#ifdef _DEBUG#define new DEBUG_NEW#undef THIS_FILEstatic char THIS_FILE = _FILE_;#endif/ CFftDrawingAppBEGIN_MESSAGE_MAP/AFX_MSG_MAP/ NOTE - the Cla

13、ssWizard will add and remove mapping macros here./ DO NOT EDIT what you see in these blocks of generated code!/AFX_MSGON_COMMANDEND_MESSAGE_MAP/ CFftDrawingApp constructionCFftDrawingApp:CFftDrawingApp/ TODO: add construction code here,/ Place all significant initialization in InitInstance/ The one

14、and only CFftDrawingApp objectCFftDrawingApp theApp;/ CFftDrawingApp initializationBOOL CFftDrawingApp:InitInstanceAfxEnableControlContainer;/ Standard initialization/ If you are not using these features and wish to reduce the size/ of your final executable, you should remove from the following/ the

15、 specific initialization routines you do not need.#ifdef _AFXDLLEnable3dControls;/ Call this when using MFC in a shared DLL#elseEnable3dControlsStatic;/ Call this when linking to MFC statically#endifCFftDrawingDlg dlg;m_pMainWnd = &dlg;int nResponse = dlg.DoModal;if / TODO: Place code here to handle

16、 when the dialog is/ dismissed with OKelse if / TODO: Place code here to handle when the dialog is/ dismissed with Cancel/ Since the dialog has been closed, return FALSE so that we exit the/ application, rather than start the applications message pump.return FALSE;/*/自己添加的.void CFftDrawingApp:Fftdou

17、ble PI=3.1415926535; int M,N; double x512,xr512,xi512,t512,jiaodu;M=9;N=1M;jiaodu=/N;/数据点输入:/caidian;for int i=0;i if i xi=1;elsexi=0;/数据点输入结束./将数据点重新排序,以实现fft计算./paixu;int kuaishu=1,geshu;for i=0;i ti=xi;forint pi=1;pi kuaishu=1;geshu=N/kuaishu;forint fi=0;fiifforint fk=0;fkxfi*geshu+fk=tfi*geshu+f

18、k*2;ifforint fk=0;fkxfi*geshu+fk=t*geshu+1+fk*2;forint i=0;iti=xi;/排序结束./进行fft计算:/fft;int zengzhi;double t1r,t2r,t1i,t2i;double jd; /计算的角度int wz=1M;for i=0;ixri=xi;xii=0;for=0;i1- wz=wz/2; kuaishu=1; geshu=N/kuaishu; zengzhi=kuaishu; forint i2=0;i2 forint i3=0;i3;i3+ jd=jiaodu*zengzhi*i3; t1r=xrgesh

19、u*i2+i3; t1i=xigeshu*i2+i3; t2r=xrgeshu*i2+i3+geshu/2; t2i=xigeshu*i2+i3+geshu/2; xrgeshu*i2+i3=t1r+t2r*cos-t2i*sin; xigeshu*i2+i3=t1i+t2i*cos+t2r*sin; xrgeshu*i2+i3+geshu/2=t1r-t2r*cos+t2i*sin; xigeshu*i2+i3+geshu/2=t1i-t2i*cos-t2r*sin; fori=0;ishibui=xri;xubui=xii;四、计算实例采样函数：x=A*cos+B*sin采样点数：N=51

20、2计算结果：操作过程：1. 打开生成的exe文件FFT.exe,则会出现以下窗口：2. 点击软件界面的参数导入按钮,就可以计算完成了,生成FFTdata.Txt文件保存在所在文件夹.计算结果为：X0=17.4235+0 jX1=17.4211+0.606382 jX2=17.4135+1.21888 jX3=17.4004+1.84378 jX4=17.381+2.48781 jX5=17.3543+3.15828 jX6=17.3188+3.86342 jX7=17.2725+4.61272 jX8=17.2128+5.41733 jX9=17.1364+6.2907 jX10=17.038

21、9+7.2493 jX11=16.9144+8.31383 jX12=16.7553+9.5107 jX13=16.5514+10.8745 jX14=16.2885+12.4513 jX15=15.9471+14.3045 jX16=15.4992+16.5237 jX17=14.9032+19.2398 jX18=14.0953+22.6524 jX19=12.9727+27.0821 jX20=11.3582+33.0782 jX21=8.91951+41.6675 jX22=4.95384+55.0175 jX23=-2.32497+78.6307 jX24=-19.1725+131.

22、765 jX25=-93.5606+362.07 jX26=206.277+-559.291 jX27=78.496+-164.482 jX28=57.6634+-98.7266 jX29=49.4477+-71.6923 jX30=45.2766+-56.9855 jX31=42.9308+-47.7638 jX32=41.5812+-41.462 jX33=40.8499+-36.9021 jX34=40.5424+-33.4674jX35=40.5503+-30.8044 jX36=40.8107+-28.6962 jX37=41.2871+-27.0029jX38=41.9592+-2

23、5.6305 jX39=42.8182+-24.514 jX40=43.8635+-23.6076 jX41=45.1019+-22.8788 jX42=46.5468+-22.3045 jX43=48.2183+-21.8689jX44=50.1444+-21.5618 jX45=52.362+-21.3781 jX46=54.9198+-21.3171 jX47=57.8815+-21.383 jX48=61.3308+-21.5851 jX49=65.3799+-21.9394jX50=70.1808+-22.4703 jX51=75.9449+-23.2143 jX52=82.9737

24、+-24.2255jX53=91.7124+-25.5861 jX54=102.846+-27.4244 jX55=117.487+-29.9503jX56=137.564+-33.5302 jX57=166.744+-38.8621 jX58=212.964+-47.4581jX59=297.164+-63.3121 jX60=498.461+-101.515 jX61=1615.21+-314.361jX62=-1256.78+233.677 jX63=-446.848+79.3409 jX64=-269.743+45.715 jX65=-192.181+31.0702 jX66=-148

25、.672+22.9142 jX67=-120.84+17.7421 jX68=-101.515+14.1866 jX69=-87.3207+11.604 jX70=-76.4576+9.65156 jX71=-67.8801+8.13005 jX72=-60.9384+6.91586 jX73=-55.2075+5.92821jX74=-50.3979+5.11215 jX75=-46.3055+4.42897 jX76=-42.7823+3.85067jX77=-39.7182+3.35648 jX78=-37.0299+2.93069 jX79=-34.653+2.5612 jX80=-3

26、2.537+2.23853 jX81=-30.6418+1.95519 jX82=-28.9351+1.70515jX83=-27.3904+1.48351 jX84=-25.9861+1.28629 jX85=-24.7043+1.11015jX86=-23.5298+0.952353 jX87=-22.4502+0.810567 jX88=-21.4544+0.682838jX89=-20.5334+0.567499 jX90=-19.6793+0.463127 jX91=-18.8851+0.368496jX92=-18.145+0.282548 jX93=-17.4538+0.2043

27、64 jX94=-16.8069+0.133144jX95=-16.2003+0.0681861 jX96=-15.6305+0.00887511 jX97=-15.0943+-0.0453316jX98=-14.589+-0.094914 jX99=-14.112+-0.140298 jX100=-13.6612+-0.181862jX101=-13.2344+-0.219944 jX102=-12.83+-0.254846 jX103=-12.4463+-0.28684jX104=-12.0817+-0.316168 jX105=-11.735+-0.343051 jX106=-11.40

28、49+-0.367686jX107=-11.0903+-0.390253 jX108=-10.7902+-0.410913 jX109=-10.5037+-0.429815jX110=-10.2299+-0.447092 jX111=-9.96805+-0.462866 jX112=-9.71738+-0.47725jX113=-9.47726+-0.490344 jX114=-9.24706+-0.502242 jX115=-9.02622+-0.513029jX116=-8.81422+-0.522784 jX117=-8.61056+-0.531578 jX118=-8.41479+-0

29、.539479jX119=-8.2265+-0.546547 jX120=-8.04529+-0.552838 jX121=-7.8708+-0.558404jX122=-7.70268+-0.563294 jX123=-7.54062+-0.567551 jX124=-7.38431+-0.571218jX125=-7.23349+-0.574332 jX126=-7.08789+-0.576928 jX127=-6.94726+-0.57904jX128=-6.81137+-0.580698 jX129=-6.68002+-0.58193 jX130=-6.55298+-0.582763j

30、X131=-6.43008+-0.583222 jX132=-6.31114+-0.583329 jX133=-6.19597+-0.583106jX134=-6.08443+-0.582574 jX135=-5.97635+-0.58175 jX136=-5.8716+-0.580652jX137=-5.77005+-0.579297 jX138=-5.67155+-0.577699 jX139=-5.57599+-0.575874jX140=-5.48326+-0.573835 jX141=-5.39324+-0.571594 jX142=-5.30584+-0.569163jX143=-

31、5.22094+-0.566553 jX144=-5.13847+-0.563775 jX145=-5.05832+-0.560838jX146=-4.98042+-0.557752 jX147=-4.90468+-0.554526 jX148=-4.83103+-0.551166jX149=-4.75939+-0.547682 jX150=-4.68969+-0.54408 jX151=-4.62187+-0.540368jX152=-4.55586+-0.536551 jX153=-4.49161+-0.532635 jX154=-4.42905+-0.528627jX155=-4.368

32、12+-0.524531 jX156=-4.30879+-0.520353 jX157=-4.25098+-0.516097jX158=-4.19466+-0.511769 jX159=-4.13978+-0.507371 jX160=-4.0863+-0.502909jX161=-4.03416+-0.498386 jX162=-3.98334+-0.493805 jX163=-3.93379+-0.48917jX164=-3.88547+-0.484485 jX165=-3.83835+-0.479752 jX166=-3.79239+-0.474974jX167=-3.74756+-0.

33、470153 jX168=-3.70383+-0.465294 jX169=-3.66116+-0.460396jX170=-3.61953+-0.455464 jX171=-3.57891+-0.450499 jX172=-3.53927+-0.445504jX173=-3.50058+-0.440479 jX174=-3.46283+-0.435428 jX175=-3.42598+-0.430351jX176=-3.39001+-0.425251 jX177=-3.3549+-0.420129 jX178=-3.32063+-0.414986jX179=-3.28717+-0.40982

34、4 jX180=-3.25451+-0.404644 jX181=-3.22262+-0.399448jX182=-3.1915+-0.394236 jX183=-3.16111+-0.38901 jX184=-3.13144+-0.38377jX185=-3.10248+-0.378519 jX186=-3.07421+-0.373255 jX187=-3.04661+-0.367982jX188=-3.01967+-0.362699 jX189=-2.99337+-0.357406 jX190=-2.96771+-0.352106jX191=-2.94265+-0.346799 jX192

35、=-2.9182+-0.341484 jX193=-2.89434+-0.336164jX194=-2.87106+-0.330837 jX195=-2.84834+-0.325507 jX196=-2.82618+-0.320171jX197=-2.80456+-0.314831 jX198=-2.78347+-0.309488 jX199=-2.76291+-0.304142jX200=-2.74285+-0.298794 jX201=-2.7233+-0.293442 jX202=-2.70424+-0.288089jX203=-2.68567+-0.282735 jX204=-2.66

36、757+-0.277379 jX205=-2.64994+-0.272022jX206=-2.63276+-0.266664 jX207=-2.61604+-0.261306 jX208=-2.59976+-0.255947jX209=-2.58392+-0.250588 jX210=-2.5685+-0.24523 jX211=-2.55351+-0.239872jX212=-2.53893+-0.234514 jX213=-2.52477+-0.229157 jX214=-2.511+-0.2238jX215=-2.49763+-0.218445 jX216=-2.48466+-0.213

37、091 jX217=-2.47206+-0.207737jX218=-2.45985+-0.202385 jX219=-2.44802+-0.197034 jX220=-2.43655+-0.191684jX221=-2.42545+-0.186336 jX222=-2.41471+-0.18099 jX223=-2.40433+-0.175644jX224=-2.3943+-0.170301 jX225=-2.38462+-0.164958 jX226=-2.37528+-0.159618jX227=-2.36629+-0.154279 jX228=-2.35763+-0.148941 jX

38、229=-2.3493+-0.143606jX230=-2.34131+-0.138271 jX231=-2.33364+-0.132938 jX232=-2.32629+-0.127607jX233=-2.31927+-0.122278 jX234=-2.31256+-0.116949 jX235=-2.30617+-0.111623jX236=-2.3001+-0.106297 jX237=-2.29433+-0.100973 jX238=-2.28887+-0.0956507jX239=-2.28372+-0.0903294 jX240=-2.27888+-0.0850092 jX241

39、=-2.27434+-0.0796903jX242=-2.27009+-0.0743724 jX243=-2.26615+-0.0690556 jX244=-2.26251+-0.0637397jX245=-2.25916+-0.0584248 jX246=-2.25611+-0.0531107 jX247=-2.25335+-0.0477974jX248=-2.25088+-0.0424848 jX249=-2.24871+-0.0371728 jX250=-2.24683+-0.0318614jX251=-2.24524+-0.0265504 jX252=-2.24394+-0.02123

40、99 jX253=-2.24293+-0.0159296jX254=-2.2422+-0.0106196 jX255=-2.24177+-0.00530976 jX256=-2.24163+0jX257=-2.24177+0.00530976 jX258=-2.2422+0.0106196 jX259=-2.24293+0.0159296jX260=-2.24394+0.0212399 jX261=-2.24524+0.0265504 jX262=-2.24683+0.0318614jX263=-2.24871+0.0371728 jX264=-2.25088+0.0424848 jX265=

41、-2.25335+0.0477974jX266=-2.25611+0.0531107 jX267=-2.25916+0.0584248 jX268=-2.26251+0.0637397jX269=-2.26615+0.0690556 jX270=-2.27009+0.0743724 jX271=-2.27434+0.0796903jX272=-2.27888+0.0850092 jX273=-2.28372+0.0903294 jX274=-2.28887+0.0956507jX275=-2.29433+0.100973 jX276=-2.3001+0.106297 jX277=-2.3061

42、7+0.111623jX278=-2.31256+0.116949 jX279=-2.31927+0.122278 jX280=-2.32629+0.127607jX281=-2.33364+0.132938 jX282=-2.34131+0.138271 jX283=-2.3493+0.143605jX284=-2.35763+0.148941 jX285=-2.36629+0.154279 jX286=-2.37528+0.159618jX287=-2.38462+0.164958 jX288=-2.3943+0.170301 jX289=-2.40433+0.175644jX290=-2

43、.41471+0.18099 jX291=-2.42545+0.186336 jX292=-2.43655+0.191684jX293=-2.44802+0.197034 jX294=-2.45985+0.202385 jX295=-2.47206+0.20773X296=-2.48466+0.213091 jX297=-2.49763+0.218445 jX298=-2.511+0.2238jX299=-2.52477+0.229157jX300=-2.53893+0.234514 jX301=-2.55351+0.239872 jX302=-2.5685+0.24523 jX303=-2.

44、58392+0.250588 jX304=-2.59976+0.255947 jX305=-2.61604+0.261306 jX306=-2.63276+0.266664 jX307=-2.64994+0.272022 jX308=-2.66757+0.277379 jX309=-2.68567+0.282735 jX310=-2.70424+0.288089 jX311=-2.7233+0.293442 jX312=-2.74285+0.298794 jX313=-2.76291+0.304142 jX314=-2.78347+0.309488 jX315=-2.80456+0.31483

45、1 jX316=-2.82618+0.320171 jX317=-2.84834+0.325506 jX318=-2.87106+0.330838 jX319=-2.89434+0.336164 jX320=-2.9182+0.341484 jX321=-2.94265+0.346799 jX322=-2.96771+0.352106 jX323=-2.99337+0.357406 jX324=-3.01967+0.362699 jX325=-3.04661+0.367982 jX326=-3.07421+0.373255 jX327=-3.10248+0.378519 jX328=-3.13

46、144+0.38377 jX329=-3.16111+0.38901 jX330=-3.1915+0.394236 jX331=-3.22262+0.399448 jX332=-3.25451+0.404644 jX333=-3.28717+0.409824 jX334=-3.32063+0.414986 jX335=-3.3549+0.420129 jX336=-3.39001+0.425251 jX337=-3.42598+0.430351 jX338=-3.46283+0.435428 jX339=-3.50058+0.440479 jX340=-3.53927+0.445504 jX3

47、41=-3.57891+0.450499 jX342=-3.61953+0.455464 jX343=-3.66116+0.460396 jX344=-3.70383+0.465294 jX345=-3.74756+0.470153 jX346=-3.79239+0.474974 jX347=-3.83835+0.479752 jX348=-3.88547+0.484485 jX349=-3.93379+0.48917 jX350=-3.98334+0.493805 jX351=-4.03416+0.498386 jX352=-4.0863+0.502909 jX353=-4.13978+0.

48、507371 jX354=-4.19466+0.511769 jX355=-4.25098+0.516097 jX356=-4.30879+0.520353 jX357=-4.36812+0.524531 jX358=-4.42905+0.528627 jX359=-4.49161+0.532635 jX360=-4.55586+0.536551 jX361=-4.62187+0.540368 jX362=-4.68969+0.54408 jX363=-4.75939+0.547682 jX364=-4.83103+0.551166 jX365=-4.90468+0.554526 jX366=

49、-4.98042+0.557752 jX367=-5.05832+0.560838 jX368=-5.13847+0.563775 jX369=-5.22094+0.566553 jX370=-5.30584+0.569163 jX371=-5.39324+0.571594 jX372=-5.48326+0.573835 jX373=-5.57599+0.575874 jX374=-5.67155+0.577699 jX375=-5.77005+0.579297 jX376=-5.8716+0.580652 jX377=-5.97635+0.58175 jX378=-6.08443+0.582

50、574 jX379=-6.19597+0.583106 jX380=-6.31114+0.583329 jX381=-6.43008+0.583222 jX382=-6.55298+0.582763 jX383=-6.68002+0.58193 jX384=-6.81137+0.580698 jX385=-6.94726+0.57904 jX386=-7.08789+0.576928 jX387=-7.23349+0.574332 jX388=-7.38431+0.571218 jX389=-7.54062+0.567551 jX390=-7.70268+0.563294 jX391=-7.8

51、708+0.558404 jX392=-8.04529+0.552838 jX393=-8.2265+0.546547 jX394=-8.41479+0.539479 jX395=-8.61056+0.531578 jX396=-8.81422+0.522784 jX397=-9.02622+0.513029 jX398=-9.24706+0.502242 jX399=-9.47726+0.490344 jX400=-9.71738+0.47725 jX401=-9.96805+0.462866 jX402=-10.2299+0.447092 jX403=-10.5037+0.429815 j

52、X404=-10.7902+0.410913 jX405=-11.0903+0.390253 jX406=-11.4049+0.367686 jX407=-11.735+0.343051 jX408=-12.0817+0.316168 jX409=-12.4463+0.28684 jX410=-12.83+0.254846 jX411=-13.2344+0.219944 jX412=-13.6612+0.181862 jX413=-14.112+0.140298 jX414=-14.589+0.094914 jX415=-15.0943+0.0453316 jX416=-15.6305+-0.

53、00887511 jX417=-16.2003+-0.0681861 jX418=-16.8069+-0.133144 jX419=-17.4538+-0.204364 jX420=-18.145+-0.282548 jX421=-18.8851+-0.368496 jX422=-19.6793+-0.463127 jX423=-20.5334+-0.567499 jX424=-21.4544+-0.682838 jX425=-22.4502+-0.810567 jX426=-23.5298+-0.952353 jX427=-24.7043+-1.11015 jX428=-25.9861+-1.28629 jX429=-27.3904+-1.48351 jX430=-28.9351+-1.70515 jX431=-30.6418+-1.95519 jX432=-32.537+-2.23853 jX433=-34.653+-2.5612 jX434=-37.0299+-2.93069 jX435=-39.7182+-3