ࡱ>   rtG ebjbjَ @_y]$`L-#    ! ! ! !3@!""$#%  #!oN #  _ L !@& ! @  ,Z+p BLIND SEPARATION OF OPTICAL TRACKER RESPONSES INTO INDEPENDENT COMPONENTS DISCRIMINATES OPTICAL SOURCES Ivica Kopriva(, Antun Perain( (Institute for Defense Studies, R&D, Bijeni ka 46, 10000 Zagreb, Croatia (Institute "Ruer Boakovi", Laser R&D Department, Bijeni ka 56, 10000 Zagreb, Croatia e-mail: ivica.kopriva(public.srce.hr ABSTRACT An optical system based on rotating reticle is used to determine polar co-ordinates of the primarily infrared optical source. Such system fails to discriminate simultaneously several optical sources. We demonstrate experimentally in a case of two optical sources that this drawback can principally be overcome by application of the blind signal separation (BSS) algorithms on data recorded at the outputs of an originally modified optical system. Separation of modified optical system responses into independent components yields modulating functions that carry information about polar co-ordinates of the corresponding single optical sources. 1. INTRODUCTION Potential application of the BSS theory lies in many areas such as: speaker separation in speech recognition (7(, communication signal processing by performing adaptive blind equalization (1(, (6(, recovery of biomedical signals (EEG, ECG, MEG) (13(, etc. We believe that for the first time the BSS theory is applied to the new problem: discrimination of optical sources. An optical system based on rotating reticle is used to detect and determine the position of an object from which some form of the primarily infrared energy is emanated (5(, (9(, see figure 1. Figure 1: Optical tracker basic construction From the reasons of convenience optical system shown on figure 1 will be called optical tracker in the rest of the paper. The rotating reticle, also called modulating disk, modulates incidental optical flux  EMBED Equation.2  and is located at the focal plane of an optical imaging system. Here  EMBED Equation.2 stands for wavelength and  EMBED Equation.2 means time. Depending on the shape of clear and opaque segments of the reticle the optical flux after reticle i.e. on the output of the photodetector is modulated on the appropriated way. It has been pointed out that optical tracker fails to discriminate several optical sources simultaneously (14(. The subject of this paper is to show that such serious limitation of the optical tracker can principally be overcome by combined use of the BSS algorithms and appropriated modification in the optical tracker construction. In section two the qualitative mathematical model of the optical tracker output signal according to figure 1, and also of the output signals of the modified optical tracker, figure 3, is derived. In section three a brief overview of the fundaments of the BSS theory is given. Three main assumptions on which all BSS algorithms are based (statistical independence of the source signals, non-Guassianity of the source signals and non-singularity of the mixing matrix) are interpreted in the context of the optical source discrimination problem. In the fourth section experimental results are described. It has been demonstrated that in a case of two optical sources discrimination between them is possible by using modified optical tracker and adaptive blind separation algorithms. The statistical independence assumption of the source signals is experimentally in that section also verified. Conclusion is given in section five while quoted references are listed in section six. 2. DERIVATION OF THE SIGNAL MODEL The polar coordinates  EMBED Equation.2  of the optical source projection on the reticle area are defined relative to the center of the reticle that corresponds with the center of the optical tracker field of view. It can be shown (10( that speed of rotation of the optical source projection around the center of the reticle is given with:  EMBED Equation.2  (1) where  EMBED Equation.2  is relative distance between center of the circle with radius  EMBED Equation.2  and center of the reticle, and  EMBED Equation.2  is speed of rotation of the reticle, see figure 1. For reticle with fan-bladed pattern, such as shown on figure 2, with  EMBED Equation.2  pairs of clear and opaque segments the instantaneous frequency of the optical source radiating flux after the reticle is given with (9(, (10(:  EMBED Equation.2  (2) where:  EMBED Equation.2  (3) Figure 2: Reticle with fan-bladed pattern The time function the instantaneous frequency of which is respecting rule (2) has the form:  EMBED Equation.2  (4) what is canonical representation of the frequency modulated (FM) signal (18(. This waveform is actually the fundamental term of the photodetector response on the incidental optical flux. The spectral terms around the frequencies  EMBED Equation.2  exist but the selective amplifier, figure 1, removes those terms (9(, (14(. Consequently the incidental optical flux at the detector area can be approximately described with:  EMBED Equation.2  (5) The deviation of the FM signal (4) i.e. the magnitude of the modulation is given with:  EMBED Equation.2  (6) where  EMBED Equation.2  i.e.  EMBED Equation.2  and  EMBED Equation.2  are construction constants. Deviation (6) is directly proportional to the radial distance of the image projection from the axis of rotation (14(. By using elementary semiconductor theory (16( it can be shown that current response of the PIN photodiode on the incidental optical flux (5) is given with:  EMBED Equation.2  (7) where  EMBED Equation.2  is detector sensing area, c is speed of light, h is Planck's constant,  EMBED Equation.2  is photodiode responsivity and  EMBED Equation.2  is wavelength. The overall photocurrent is obtained integrating expression (7) over detector sensitivity range:  EMBED Equation.2  (8) The selective amplifier output signal x(t), see figure 1, is obtained by convolution of the selective amplifier impulse response, g(t), with the photocurrent i(t). Assuming that photon flux  EMBED Equation.2  is slow varying function relative to  EMBED Equation.2  the following expression for x(t) is obtained:  EMBED Equation.2  (9) where  EMBED Equation.2  is given with:  EMBED Equation.2  (10) and * means convolution. When two optical sources S1 and S2 are present simultaneously in the optical tracker filed of view on coordinates  EMBED Equation.2 and  EMBED Equation.2 respectively the selective amplifier output signal will be of the form:  EMBED Equation.2  (11) where:  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2  (12) Expression (12) shows that, when photo-detector linearity is assumed, the optical tracker output signal is convolved combination of the modulating functions, that carry information about polar co-ordinates of the corresponding optical source, and non-stationary impulse responses of the form (12). It has been analytically shown in (14( that optical tracker in such case follows the centroid the coordinates of which are functions of the effective brightness of the two sources. In a case of two collinear sources located at points  EMBED Equation.2 and  EMBED Equation.2  with equal brightness the optical tracker will see the fictitious optical source with coordinates  EMBED Equation.2 . The point is that optical tracker fails to determine the accurate coordinates of any of the two sources. In order to resolve this drawback the modification of the optical tracker construction was performed such as shown on figure 3. Figure 3: Modified optical tracker The main point is in introducing beam splitter after modulating disk and then detecting modulated optical flux with two photodetectors. This is in agreement with the BSS theory that requires for blind separation of N sources  EMBED Equation.2 detectors to be used (2(, (17(, (19-21(. In accordance with the exposed material the selective amplifiers output signals are:  EMBED Equation.2   EMBED Equation.2  (13) where impulse responses g11(t), g12(t), g21(t), g22(t) are given with:  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2   EMBED Equation.2  (14) Expressions (13) and (14) represent mathematical model of the modified optical tracker output signals. In (14)  EMBED Equation.2  and EMBED Equation.2  are beam splitter transmission and reflection coefficients respectively, while  EMBED Equation.2  and  EMBED Equation.2  are photo-detector spectral responsivities. Convolutive signal model (13), suitable for application of the BSS algorithms, is illustrated with figure 4, where G11(z,k), G12(z,k), G21(z,k) and G22(z,k) are Z transforms of the related impulse responses (14).  Figure 4: Convolutive signal model By using on-line BSS algorithms it should be possible to recover the source signals  EMBED Equation.2  and  EMBED Equation.2  on the basis of the observed signals x1(t) and x2(t) only. 3. INTERPRETATION OF THE BSS THEORY REQUIREMENTS The real-world situations that often include time delays between the sources in the mixture are described with convolutive mixing model x(t)=G*s(t), where s(t) is vector of source signals, G is mixing matrix that contains channel impulse responses, x(t) is vector of the observed signals and * means convolution. The goal is to reconstruct the source signals from the observed signals only, i.e. the mixing matrix G is assumed to be unknown. Solutions for such problem are given in (1(, (4(, (7(, (8(, (11(, (12(, (15(, (17(, (19(-(21(. However, some of the proposed solutions are iterative by nature, (2(, (7(, (12(, (21(. For real time source separation adaptive i.e. sequential version of the algorithms is necessary. Such algorithms are described in (1(, (4(, (8(, (11(, (15(, (17(, (19(, (20(. This kind of algorithms is of interest for discrimination of optical sources that is a real-time problem. There are three fundamental assumptions on which all BSS algorithms are based: statistical independence of the source signals, non-Gaussianity of the source signals and non-singularity of the mixing matrix in the model of the observed signals. Here, it will be briefly examined whether these assumptions are fulfilled for the model of the modified optical tracker output signals (13). The statistical independence assumption of the source signals  EMBED Equation.2  and  EMBED Equation.2  is logical since they are generated by two different (independent) optical sources. It will be demonstrated experimentally in the fourth section that this assumption holds. The second assumption, non-Gaussianity of the source signals, is for signals  EMBED Equation.2  and  EMBED Equation.2  also fulfilled. As shown in expression (4) they are frequency modulated. This type of signals, as most of the communication signals, belongs to the sub-Gaussian class of signals having negative kurtosis. The third assumption is non-singularity of the mixing matrix when convolutive model (13) is transformed into frequency domain. Then it can be written as:  EMBED Equation.2  (15) where  EMBED Equation.2  and  EMBED Equation.2  are vectors the components of which are Discrete Fourier Transforms (DFTs) of the observed and source signals respectively, while  EMBED Equation.2  is matrix the components of which are DFTs of the related impulse responses given with (14). The non-singularity requirement of the mixing matrix over some frequency region of interest is formally expressed as:  EMBED Equation.2   EMBED Equation.2  (16) where  EMBED Equation.2  and  EMBED Equation.2  are frequencies determined by the design of the selective amplifiers. Satisfaction of the condition (16) is important since it ensures that we have benefit from using two sensors. Otherwise, both observed signals would deliver to us the same information and one of them would be useless. Assuming that optical flux  EMBED Equation.2  is piecewise stationary at least over the time interval determined by the length of DFT, the DFTs of the impulse responses (14) can be written as:  EMBED Equation.2  (17) where  EMBED Equation.2  is DFT of the related selective amplifier impulse response gi(t), while  EMBED Equation.2  can be easily identified from (14). Then condition (16) can be rewritten as:  EMBED Equation.2  (18) and is transformed in:  EMBED Equation.2  (19) since in (18)  EMBED Equation.2  and  EMBED Equation.2  represent frequency responses of the selective amplifiers and their product is always different from zero. Assuming for photo-detector spectral responsivities in (14)  EMBED Equation.2 and taking into account  EMBED Equation.2  expression (19) is transformed in:  EMBED Equation.2  (20) Condition (20) and consequently condition (16) will be fulfilled when EMBED Equation.2  over region of interest. When real beam splitter devices are taking into consideration this is always true. Hence, the role of the beam splitter device is twofold. Firstly, it ensures that both detectors see the optical sources in the same system of co-ordinates and secondly the beam splitter device ensures non-singularity of the mixing matrix in the frequency domain. The above physical based reasoning gives us framework for performing experiment with modified optical tracker. For reconstruction of the source signals the feedback separation network shown on figure 5 is applied. Specifically, it has been adopted  EMBED Equation.2  The reasons for using feedback network with W11(z) and W22(z) set to unity is to avoid whitening effect explained nicely in (20(.  Figure 5: Feedback separation network To reconstruct modified optical tracker source signals two time domain on-line separation algorithms will be used: the algorithm based on infomax criterion (12(, (20( and algorithm that minimizes instantaneous (8( or generalized instantaneous energy (11(. As pointed out in (12(, (20( separation based on infomax criteria is performed by minimizing the mutual information between components of random vector z(t)=g(y(t)), where g is a nonlinear function approximating the cumulative density function (cdf) of the sources. Sigmoids that are most often in use (g(y)=tanh(y), g(y)=1/(1+e-y)) approximate well the cdfs of the positively kurtotic signals like speech. Nevertheless, we managed to separate optical tracker signals that belong to sub-Gaussian class of signals. As it will be reported in the next section this was probably due to the fact that one source signal had negative kurtosis very close to zero. The infomax learning rules with z=g(y)=tanh(y) for feedback separation network, figure 5, are given with:  EMBED Equation.2  (21) Minimization of the energy based criterion (11( for odd value of p:  EMBED Equation.2  (22) yields the following learning rules:  EMBED Equation.2  (23) where in (21) and (23)  EMBED Equation.2  and  EMBED Equation.2 , where L is order of the cross-filters in the feedback separation network. 4. EXPERIMENTAL RESULTS Two real-world signals were recorded at the output of the modified optical modulator, see figure 3, with sampling frequency of 100 kHz. Figure 6 shows power spectrum of the observed signal x1(t) recorded at the output of one photoamplifier when both optical sources were present in the optical modulator field of view. The spectrum of the second measured signal looks similarly. The information about position of both optical sources is present in the recorded signal. First light source, with smaller radius coordinate, has deviation of approximately 20 Hz and was located in the center of the filed of view that corresponds with carrier frequency of 22.5 kHz.The second signal, with larger radius, has deviation of approximately 4 kHz and occupies spectrum from 18 to 27.5 kHz. The length of the recorded blocks of observed signals x1(t) and x2(t) was 32768 data samples. Figure 6: Power spectrum of the observed signal x1(t) Power spectrums of the separated signals y1(t) and y2(t) are shown on figures 7 and 8 respectively. Compared with figure 6 the separation of two signals is obvious. After separation determination of the co-ordinates of the optical sources is the matter of demodulation of the recovered source signals that are of the form (4). In order to measure the separation quality and compare different separation algorithms the separated signals y1(t) and y2(t) as well as source signals s1(t) and s2(t) were demodulated by means of quadrature FM demodulator. Due to the expression (4) and (6) amplitude of the demodulated signal is directly proportional with the polar coordinate r of the corresponding optical source. Figure 7: Power spectrum of signal y1(t)  Figure 8: Power spectrum of signal y2(t) Since optical sources in our experiment were not moving it was possible to record and demodulate recorded signals when each optical source was present in the modified optical modulator field of view alone. On that way we have obtained referenced data in relation to which accuracy of the blind separation approach can be verified. Results obtained with infomax separators (12(, (20( (21), with deccorelator (8( as well as those obtained by minimizing criteria (23)-(11( are reported in table 1. Kurtosis of the source signals is  EMBED Equation.2 and  EMBED Equation.2 . Since  EMBED Equation.2 is relatively close to zero source signal s1 is not far from Gaussian and infomax separator with g(yi)=tanh(yi) has not diverged. Amplitudes of the demodulated signals were expressed in dBs and measured when demodulated signals were transformed in frequency domain. First optical source alone has distance of -33.2dB and second optical source 65dB. Direct demodulation of the observed signals x1(t) and x2(t) discriminates only second optical source giving amplitudes of 64dB. The cross-filters in the feedback separation network have order 61. Table 1. Separation results AlgorithmFirst sourceSecond sourceDeccorelation (8(-36.4 dB-64.2 dBp=3, (22),(23)-36.4 dB-64.2 dBInfomax (21)-35 dB-64.4 dBNo separation-63.4-64.6 When infomax separation is employed 2dB error in relation to the reference position is obtained. Compared with direct demodulation of the observed signals the infomax separator gives 29dB more accurate estimate of the first optical source position. That clearly justifies our approach to discriminate optical sources by performing blind separation of optical tracker responses into independent components. Signals obtained by demodulation of the source signal s1(t) and reconstructed signal y1(t) are shown on figures 9 and 10 respectively. Figure 9: Demodulation of the source signal s1(t) Figure 10: Demodulation of the separated signal y1(t) The statistical independence of the source signals is of essential importance for the success of the BSS theory. We have measured the level of the second and fourth order statistical (in)dependence between observed and between separated signals. Three fourth order cross-cumulants C31, C22 and C13 were computed in order to measure fourth-order statistical (in)dependence. Extreme values for each statistics are given in table 2. Table 2. Level of statistical (in)dependence StatisticsExtreme for x1,x2Extreme for y1,y2C110.262 -0.0067C31 -0.10974.669e-5C22 -0.1123.885e-4C13 -0.05338.545e-4 In table 2 from 39 to even 2349 times less level of statistical dependence between separated signals relative to measured signals can be observed. That can be considered as experimental verification of the statistical independence assumption. The cross-correlation C11 between observed and separated signals are shown on figures 11 and 12 respectively when time lag was moving from 500 to 500. Figure 11: Cross-correlation between x1 and x2 Figure 12: Cross-correlation between y1 and y2 5. CONCLUSION AND FUTURE WORK  EMBED Equation.2  A solution for discrimination of several optical sources by means of modified optical tracker and BSS algorithms is proposed. Experiment performed with modified optical tracker and two optical sources confirmed the principal possibility to discriminate two optical sources. Future work will be directed toward on-line BSS algorithms working in frequency domain (15(, (17( where more accurate results are expected. Here, due to the orthogonal basis and natural gradient learning rules (1(, (3( faster convergence and consequently better separation quality is expected. The permutation indeterminacy inherent to all BSS algorithms, which makes especially serious problems when BSS is performed in frequency domain, should be successfully resolved by using newly proposed spectral kurtosis criterion (15(. 6. REFERENCES (1( S. Amari, et. all, "Multichannel Blind Deconvolution and Equalization Using the Natural Gradient", IEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications, Paris, France, pp. 101-104, April, 1997. (2( S. Amari, T. P. Chen, A. Cichocki, "Stability Analysis of Adaptive Blind Source Separation", Neural Networks, 1997. (3( S. Amari, S. C. 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Torkkola, "Blind Separation of Convolved Sources Based on Information Maximization", IEEE Workshop on Neural Networks for Signal Processing, Kyoto, Japan, September, 1996. (21( D. Yellin, E. Weinstein, "Criteria for Multichannel Signal Separation", IEEE Transactions on Signal Processing, Vol. 42, No. 8, pp. 2158-2168, August, 1994.  EMBED Visio.Drawing.4   EMBED Visio.Drawing.4   EMBED Word.Picture.8   EMBED Visio.Drawing.4  ~ b c d e         + , ) * = > ? @ ̽jCJEHUj[A9 UVmH jCJU j]CJ j[CJ5CJ j@CJ j! 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F$HCompObjNativepr*fObjInfo2tivesF,Equation Native F-(stem-ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ; lJqJmJ kࡱࡱ> L_960358581tiveQvF + +fOlenfo8 F.PICion Native ux/LMETA8558tive F1Hࡱ> n   .1  &: & MathTypeTimes New Roman(- 2 n & "System-ࡱ>  FMicrosoft Equation 2.0 DS EqCompObjwy7fObjInfoz9Equation Native :(_960358589}F +7+uation Equation.29qࡱ>ࡱ> ; lJqJmJ nࡱࡱ> L=ࡱ> =   .1   &z &OleObj8 F;PICnfo |<LMETAon Native >CompObj6~FEf MathType@Times New Roman(- 2 r| Times New Roman- 2 0p & "System- nࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ObjInfo8FGEquation Native H4_960358600tive{F7++(Ole58516 FI;lJqJmJ r 0ࡱࡱ> LT0ࡱ> T&<  .1   & Z & MathType-OTimes New Roman(- 2 @ iY PICObj JLMETAfo LCompObjNative\fObjInfo9F^2 @)tY 2 @SA 2 @ech 2 @p tY 2 @t RTimes New Roman- 2 @z(i 2 @,P 2 @)i Arial- 2 $Times New Roman(- 2 @(i 2 @,P 2 @ )i 2 @F (i 2 @| )iSymbol- 2 @l 2 ^l 2 @Al 2 @ lSymbol- 2 @A= 2 @F & "System-ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;\lJqJmJ i(l,t)=Equation Native _x_960358613F++Oleion Native aPIC11945 FbLAchl"F (l,t)R(l)ࡱࡱ> L==ࡱ> ==2  .1    & & MathTypeTimes New Roman(- 2 >A & "SyMETAon Native dHCompObj8FjfObjInfolEquation Native m(stem-Roman(- 2 @ iY 2 @)ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ; lJqJmJ Aࡱࡱ> L_960358608F'˝+2+Ole nPIC oLMETA qࡱ>    .1  `&Z & MathType0Times New Roman- 2 &RTimes New Roman_- 2 (i 2 ()iSymbol- 2 vl & "System- % 6 X ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;lJqJmJ R(l)ࡱࡱ> L=ࡱ> CompObjNativeyfObjInfo8F{Equation Native |4_960358629F2++*LOleion Native }PIC11978 F~LMETA (CompObjf=0  .1   & & MathTypeSymbol- 2  l & "System-(i 2 (ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ObjInfoEquation Native (_960358645F+`#+Ole ; lJqJmJ lࡱࡱ> LD 0ࡱ>  F .1  &Z & MathType- Times New Roman- 2  iY PICEntry<@EEлE LMETAEVhH  *CompObjenttE yHȽEfObjInfol<@E2 tY 2 Ahc 2 R 2  tY 2  d 2 1s| 2 *r| 2 tYTimes New Roman_- 2 z(i 2 i)i 2 (i 2 )i 2 9 (i 2 l ,P 2 ^ )i 2 (i 2 ,P 2  ,P 2 )iSymbol- 2  = 2 3 2 ISymbol- 2 .JSymbol- 2 4l 2  l 2 >~ l 2 u l 2 .j Symbol- 2 Jl{Symbol- 2 R F & "System-NOOPPQQRRSSTTUࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;ЈlJqJmJ i(t)=AhcR(l)F(l,t)l l  dls(r,j,t)ࡱ>ࡱ> LDࡱ> Equation Native _960358652F`#+ &+Ole PIC LMETAbj (CompObjfObjInfoNativeEquation Native F85  .1  `&Z & MathType0Symbol1- 2  FTimes New Roman- 2 (i 2 (,P 2 )iSymbol- 2 tlTimes New Roman- 2 tY & "System-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;lJqJmJ F(l,t)ࡱࡱ> Lࡱ> _960358655tivetMF &+ ?+Ole58652 FPIC LMETA (5  .1  `&Z & MathType0Times New Roman- 2  s| 2 r| 2 tYTimes New Roman- 2 (i 2 ,P 2 ,P 2 )iSymbol- 2  j & "System-493<*  ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;$lJqJmJ s(r,j,t)ࡱ> L ࡱ> CompObj8FfObjInfoNativeEquation Native F@_960358667F ?+O+IOlenfo8 FPICion Native LMETA8600tive FCompObj6Ff :  .1   & J & MathType@Times New Roman- 2 *x 2 BtY 2 Xg 2 tY 2 Cs| 2 =r| 2  tYTimes New Roman- 2 (i 2 )i Arial- 2 &]$Times New Roman- 2 (i 2  )i 2 (i 2 ,P 2 3 ,P 2 % )iSymbol- 2 \= 2 o*Symbol- 2 Cj & "System-ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;HlJqJmJ x(t)=2g (t)*s(r,j,t)ࡱ>ࡱ> Lࡱ> 5  .ObjInfoNativeEquation Native Fd_960359121FO+O+Ole PICnfo8 FLMETAon Native CompObj0tiveFfObjInfo6F1  &`J & MathType@ Arial- 2 &/$Times New Romanl- 2 (i 2 )iTimes New Roman- 2 *g 2 etY & "System-2 )iࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;lJqJmJ 2g (t)ࡱࡱ> Lh 0ࡱ> 4 C .1  @& Z &Equation Native 4_960359115FY+B+Oleion Native PIC11945 FLMETA8581tive FCompObj8FfObjInfoNativeEquation Native F     !"#$%'*-/0123456789:;<=>?@BEFGJLMNPUWXYZ[\]^_`abcdefghiknortuvwxyz{|}~ MathType-b  ArialP- 2 F/$Times New Romanl- 2 (i 2 )i 2 /(i 2 )i 2 ^ (i 2  )i 2  (i 2 ,P 2 )iTimes New Roman- 2 *g 2 etY 2 {g 2 tY 2 Ahc 2  R 2 tY 2 dSymbol- 2 = 2  2 Symbol- 2  JSymbol- 2  l 2 hl 2 >) l 2 pl Symbol- 2 J l{Symbol- 2  F & "System-"System-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;ЌlJqJmJ 2g (t)=g(t)AhcR(l)F(l,t)ldl l ࡱࡱ> _960359134F +@3+Ole PIC LMETA  HLhࡱ> h6  .1  &z & MathType@Times New Romanl- 2 (i 2 ,P 2 a)iTimes New Roman- 2 r| Times New Romanl- 2 1p 2 1pSymbol- 2 j & "System-)iࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;4lJqJmJ (r 1 ,jCompObjNativefObjInfo8FEquation Native P_960359130F@3+@3+L 1 )ࡱ> ࡱ> Lࡱ> V:  .1  @& z & MathType@Times New Romanl- 2 (i 2 ,P 2 )iOle58655tive FPIC58652 FLMETA HCompObj&fTimes New Roman- 2 r| Times New Romanl- 2 2p 2 2pSymbol- 2 j & "System-)iࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ObjInfoNative(Equation Native F)P_960359139tiveF@3+@+aOle11945 F+ࡱ> ;4lJqJmJ (r 2 ,j 2 )ࡱ> ࡱ> LI3 ࡱ> IF/ / .1   &z &PICnfo ,LMETAon Native .CompObj4FAfObjInfoC MathType@Times New Roman+- 2 &x 2 6tY 2 Dg 2 tY 2 s| 2  r| 2  tY 2  g 2 gtY 2 s| 2 r| 2 tYTimes New Roman5- 2 (i 2 )i 2 (i 2 l)i 2 (i 2 ,P 2  ,P 2  )i 2  (i 2 )i 2 <(i 2 E,P 2 ,P 2 )iSymbol- 2 L= 2 * 2  + 2 @* Times New Roman+- 2 1p 2 1p 2 G 2p 2 2pSymbol- 2 " j 2 j & "System-PEd````nPEࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;ЬlJqJmJ x(t)=g 1 (t)*s 1 (r,j,t)+g 2 (t)*s 2 (r,j,t)ࡱ> Equation Native FD_952698126tive;F@+Ξ+Oleion Native FHPIC58667 FILࡱ> Lࡱ> : P .1  &z & MathType@ & pe@Tiࡱ>  FMicrosoft Equation 2.0 DS EqMETA1931 FKCompObjOZObjInfoQEquation Native R<uation Equation.2ࡱ> ࡱ>  1PP7@T7@ ࡱ> L\ Dࡱ> \G Q .1  `&@d &_960359136<@EEлEFΞ+/֞+OleEVhH  *SPICocumenttE yHȽE TLMETAPool<@E V MathType- Times New RomanP- 2 @*g 2 @tY 2 @g 2 @#tY 2 @9Ahc 2 6 R 2 6tY 2 @d Times New Roman`- 2 1p 2 k1pTimes New RomanP- 2 @i(i 2 @Y)i 2 @(i 2 @)i 2 6 (i 2 6 )i 2 6(i 2 69,P 2 6+)iSymbol- 2 @= 2 @3 2 @R Symbol- 2 N JSymbol- 2 6\ l 2 6l 2 ^ l 2 @l Symbol- 2 j8 l{Symbol- 2 6} F & "System-ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;ФlJqJmJ g 1 (t)=g(t)AhcR(l)F 1 (l,t)ldl l CompObj1FjfObjInfolEquation Native m_960359149FVߞ++(Oleion Native pPIC58613 FqLMETAon Native sCompObj5Ffࡱ> LD Dࡱ> G Q .1  &d & MathType- Times New Roman@- 2 @*g 2 @tY 2 @g 2 @FtY 2 @\Ahc 2 6+ R 2 6tY 2 @3d Times New Roman`- 2 2p 2 2pTimes New Roman@- 2 @(i 2 @|)i 2 @(i 2 @)i 2 6 (i 2 63 )i 2 6I(i 2 6,P 2 6q)iSymbol- 2 @= 2 @V 2 @u Symbol- 2 q JSymbol- 2 6 l 2 6l 2 ^)l 2 @l Symbol- 2 j[ l{Symbol- 2 6 F & "System-ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;ФlJqJmJ g 2 (t)=g(t)AhcR(l)F 2 (l,t)ldl l ࡱ> Lࡱ> ObjInfoNativeEquation Native F_960358788F++Ole PICion Native LMETA8613 F(CompObjNativefObjInfo5F/  .1  @& z & MathType@Times New Roman+- 2 (i 2 ,PTimes New Roman- 2 r| Times New Roman+- 2 1pTimes New Roman- 2 0)i & "System-)i 2 6*ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;(lJqJmJ (r 1 ,0)ࡱ> ࡱ> LqEquation Native D_960358792F+`+Oleion Native PIC11945 FLࡱ> qn3  .1   &z & MathType@Times New Roman+- 2 (i 2 ,PTimes New Roman- 2 r| Times New Roman+- 2 2pTimes New Roman- 2 METAon Native (CompObj8FfObjInfoEquation Native D0)i & "System-)i 2 6*ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;(lJqJmJ (r 2 ,0)ࡱ> ࡱ> _960371334F`+@z"+Ole PIC LMETA HLgDࡱ> gF/  .1  &z & MathType-aTimes New Roman+- 2 @1 2 ^2 2 @Z0 Times New Roman- 2 u1p 2 2pTimes New Roman+- 2 @(i 2 @0),iPTimes New Roman- 2 @r| 2 @r|Symbol- 2 @5+ 2 4y 2 .y 2 Ny 2 4y 2 .y 2 Ny & "System-ࡱ> CompObjNativefObjInfoNativeFEquation Native F|_960359361TF F@z"+ C3+ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;`lJqJmJ 12(r 1 +r 2 ),0()ࡱ> Lh`ࡱ> OlenfoNative PICion Native  FLMETA9121 FCompObj  fh`FH  .1  @&: & MathType Times New Roman+- 2 >M  2 NSymbol- 2  & "System-Times New Romaࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;lJqJmJ MNࡱࡱ> Lࡱ> FH : .1  &`z &ObjInfoNative Equation Native F0_960359211tive8F C3+@:+aOle11945 FPICion Native LMETA1978 FCompObjfObjInfo MathType@Times New Roman+- 2 &x 2 tY 2 g 2 tY 2 s| 2  r| 2  tY 2  g 2 tY 2 os| 2 r| 2 vtY Times New Roman- 2 1p 2 u11pp 2 1p 2 512pp 2 2pTimes New Roman+- 2 K(i 2 7)i 2 {(i 2 g)i 2 (i 2  ,P 2  ,P 2  )i 2 O(i 2 ;)i 2 (i 2 ,P 2 ,P 2 )iSymbol- 2 = 2 * 2  + 2 *Symbol- 2  j 2 j & "System--aࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;мlJqJmJ x 1 (t)=g 11 (t)*s 1 (r,j,t)+g 12 (t)*s 2 (r,j,t)ࡱ> Equation Native F_960359206F@:+v\+Ole PIC Lࡱ> L8ࡱ> FH : .1  &z & MathType@Times New Roman+- 2 &x 2 tY 2 g 2 tY 2 s| 2 I r| 2  tY /META8581tive FCompObj8FfObjInfoNativeEquation Native F     !"#$%'*+,/123456789:;<=>?@ABCDEGJKLOQRSTUVWXYZ[\]^_`abcdegjkloqrstuvwxyz{|}~2  g 2 tY 2 s| 2 Kr| 2 tY Times New Roman- 2 2p 2 21pp 2 =1p 2 p22pp 2 +2pTimes New Roman+- 2 l(i 2 X)i 2 (i 2 )i 2 (i 2  ,P 2 6 ,P 2 % )i 2 (i 2 v)i 2 (i 2 ,P 2 8,P 2 ')iSymbol- 2 = 2 * 2  + 2 *Symbol- 2 K j 2 Kj & "System--aࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;мlJqJmJ x 2 (t)=g 21 (t)*s 1 (r,j,t)+g 22 (t)*s 2 (r,j,t)ࡱ> ࡱ> L.Dࡱ> _960359209Fv\+fv+Ole  PIC  LMETA h.F6  .1  &d & MathType-@ Times New Roman@- 2 @*g 2 @QtY 2 @gg 2 @tY 2 @4A 2 @^ hc 2 6R 2 6 tY 2 @Fd Times New Roman`- 2 11pp 2 1p 2 1p 2 1p 2 1pTimes New Roman@- 2 @(i 2 @)i 2 @(i 2 @)i 2 6 (i 2 6T)i 2 63(i 2 6i)i 2 6\(i 2 6,P 2 6)iSymbol- 2 @k= 2 @. 2 @ Symbol- 2  JSymbol- 2 6B t 2 6 l 2 6l 2 6l 2 ^l 2 @l Symbol- 2 j l{Symbol- 2 6F & "System-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>CompObjNative!&fObjInfoNative"F(Equation Native F)_960359223%Ffv+/+ࡱ> ;lJqJmJ g 11 (t)=g 1 (t)A 1 hct(l)R 1 (l)F 1 (l,t)ldl l ࡱࡱ> LRDࡱ> Ole71334 F-PIC $'.LMETA 0hCompObj&(FfRF6  .1  &d & MathType-U @Times New Roman@- 2 @*g 2 @ftY 2 @|g 2 @3tY 2 @IA 2 @s hc 2 6R 2 6DtY 2 @~d Times New Roman`- 2 12pp 2 #1p 2 1p 2 1p 2 2pTimes New Roman@- 2 @(i 2 @)i 2 @(i 2 @)i 2 63 (i 2 6i)i 2 6H(i 2 6~)i 2 6(i 2 6,P 2 6)iSymbol- 2 @= 2 @C 2 @ Symbol- 2  JSymbol- 2 6W t 2 6 l 2 6l 2 6l 2 ^l 2 @l Symbol- 2 j l{Symbol- 2 6F & "System-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ObjInfoNative)HEquation Native FI_960359227tive#1,FЎ++aOle11945 FMࡱ> ;lJqJmJ g 12 (t)=g 1 (t)A 1 hct(l)R 1 (l)F 2 (l,t)ldl l ࡱࡱ> L<Dࡱ> PICnfoNative +.NLMETAon Native FPhCompObj1-/FffObjInfo0h   .1  `&@d & MathType- Times New Roman@- 2 @*g 2 @_tY 2 @ug 2 @OtY 2 @eA 2 @ hc 2 6pR 2 6tY 2 @d Times New Roman`- 2 21pp 2 *2p 2 # 2p 2 92p 2 w1pTimes New Roman@- 2 @(i 2 @)i 2 @(i 2 @)i 2 6 (i 2 6)i 2 6(i 2 6)i 2 6(i 2 6E,P 2 67)iSymbol- 2 @y= 2 @_ 2 @ Symbol- 2  JSymbol- 2 6 r 2 6.l 2 6fl 2 6l 2 ^Ul 2 @l Symbol- 2 j l{Symbol- 2 6F & "System-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;lJqJmJ g 21 (t)=g 2 (t)A 2 hcr(l)R 2 (l)F 1 (l,t)ldl l ࡱࡱ> L%dDࡱ> Equation Native i_960359233tive3F+ +)POle59139tive FmPIC11945 25FnLMETAfoNative phCompObjNative46FfObjInfo1tive7FEquation Native F%6   .1  &d & MathType- Times New Roman@- 2 @*g 2 @ttY 2 @g 2 @dtY 2 @zA 2 @ hc 2 6R 2 6tY 2 @1d Times New Roman`- 2 22pp 2 ?2p 2 8 2p 2 N2p 2 2pTimes New Roman@- 2 @(i 2 @)i 2 @(i 2 @)i 2 6 (i 2 6)i 2 6(i 2 61)i 2 6G(i 2 6},P 2 6o)iSymbol- 2 @= 2 @t 2 @ Symbol- 2  JSymbol- 2 6 r 2 6Cl 2 6{l 2 6l 2 ^|l 2 @l Symbol- 2 j l{Symbol- 2 6F & "System-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;lJqJmJ g 22 (t)=g 2 (t)A 2 hcr(l)R 2 (l)F 2 (l,t)ldl l ࡱࡱ> Lࡱ> _960359256*:F + џ+Ole PIC 9<LMETA n   .1  `&Z & MathType0Symbol*- 2 t 2 \lTimes New Roman+- 2 (i 2 )i & "System-.DRVStyle for Next Paragraph: ࡱ FMicrosoft Equation 2.0 DS EqCompObj     /;=f!ObjInfo'()*+-,0.1I>5698;CEquation NativeMeKLNPOQ VWZXY4`_960359260ghkjmlnopqAF џ+ џ+uation Equation.29qࡱ>ࡱ> ;lJqJmJ t(l)ࡱࡱ> L+ࡱ> +n   .1  `&Z &Ole59134 FPIC @CLMETA CompObjBDf MathType0Symbol- 2 Hr 2 lTimes New Roman+- 2 (i 2 N)i & "System-.DRVStyle for Next Paragraph: ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ObjInfoNativeEEquation Native F4_960359789tiveHF۟++aOle11945 F;lJqJmJ r(l)ࡱࡱ> Lࡱ> n   .1  @& z & MathType@Times New Roman- 2 *R Times New Roman+PIC     / GJL!META&%'()*+-,0.1I 5698;(CCompObjFJHMeKLNPOQIKVWZXYf`ObjInfoghkjmlnopqLvwyx{- 2 1pTimes New Roman- 2 }(i 2 )iSymbol- 2 l & "System-`ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> Equation Native @_960359793tiveOF+N+)POle59139tive FPIC11945 NQFL;$lJqJmJ R 1 (l)ࡱ> Lࡱ> n   .1  `&@z & MathType@Times New Roman- 2 *R Times New Roman+METAbjNative (CompObjNativePRFfObjInfoNativeSFEquation Native F@- 2 2pTimes New Roman- 2 (i 2 )iSymbol- 2 l & "System-`ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;$lJqJmJ R 2 (l)ࡱ> L4ࡱ> ^= $ .1   &z & MathType@Times New Roman- 2  s| 2 r| 2 tY _960359321?[VF ++Ole PIC UXLMETA hTimes New Roman`- 2 1pTimes New Roman- 2 -(i 2 ;,P 2 ,P 2 )iSymbol- 2 j & "System-w Roman`-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>CompObj6WYFfObjInfoZEquation Native L_960359325]F%+%+ࡱ> ;0lJqJmJ s 1 (r,j,t)ࡱ> ࡱ> LHࡱ> 2 $ .1  @& z &Ole58581tive FPICnfo8 \_FLMETAon Native hCompObj8tive^`Ff MathType@Times New Roman- 2  s| 2 r| 2 8tY Times New Roman`- 2 2pTimes New Roman- 2 P(i 2 ^,P 2 ,P 2 )iSymbol- 2 j & "System-w Roman`-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;0lJqJmJ s 2 (r,j,t)ࡱ> ࡱ> LHࡱ> ObjInfo4aFEquation Native L_960359883ipdF%+@rG+LOle PIC59256 cfFLMETA hCompObjegfObjInfoh2 $ .1  @& z & MathType@Times New Roman- 2  s| 2 r| 2 8tY Times New Roman`- 2 2pTimes New Roman- 2 P(i 2 ^,P 2 ,P 2 )iSymbol- 2 j & "System-w Roman`-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;0lJqJmJ s 1 (r,j,t)ࡱ> ࡱ> Equation Native    / L!_960359867()*+-,0.1IkF@rG+ ;X+@?COleDGFJHMeKLNPOQ VWZXY`PICifghkjmlnopq jmvwyx{L    "%'()*+,-./1479:;<=>?@ABCEHKMNOPQRSUZ\]^_`abdiklmnopqsxz{|~LHࡱ> 2 $ .1  @& z & MathType@Times New Roman- 2  s| 2 r| 2 8tY Times New Roman`- 2 2pTimes New Roman- METAbjNative hCompObjNativelnF fObjInfoNativeoFEquation Native FL2 P(i 2 ^,P 2 ,P 2 )iSymbol- 2 j & "System-w Roman`-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;0lJqJmJ s 2 (r,j,t)ࡱ> ࡱ> L4ࡱ> ^= $ .1   &z & MathType@Times New Roman- 2  s| 2 r| 2 tY _960359895    rF ;X+i+Ole PIC qtLMETA hTimes New Roman`- 2 1pTimes New Roman- 2 -(i 2 ;,P 2 ,P 2 )iSymbol- 2 j & "System-w Roman`-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>CompObj4suFfObjInfov Equation Native !L_960359900byF p+ p+ Hࡱ> ;0lJqJmJ s 1 (r,j,t)ࡱ> ࡱ> LHࡱ> 2 $ .1  @& z &Ole59134 F#PIC x{$LMETA &hCompObjz|0f MathType@Times New Roman- 2  s| 2 r| 2 8tY Times New Roman`- 2 2pTimes New Roman- 2 P(i 2 ^,P 2 ,P 2 )iSymbol- 2 j & "System-w Roman`-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;0lJqJmJ s 2 (r,j,t)ࡱ> ࡱ> LC ࡱ> ObjInfo6}F2Equation Native 3L_960359965F p++LOle 5PIC59256 F6LMETA 8CompObjDfObjInfoFC ~  L .1   & z & MathType@Times New Roman- 2 >X 2 G 2 STimes New Roman+- 2 8(i 2 )i 2 (i 2 n)i 2 (i 2  )iSymbol- 2 w 2 rw 2  wSymbol- 2 2= 2  & "System-Times New Roman-ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;@lJqJmJ X(w)=G(Equation Native G\_960359970tive~F+6+IOle59227tive FIPIC11945 FJLw)S(w)ࡱ> ࡱ> Lqࡱ> q  .1  ` &Z & MathType0Times New Roman- 2 >XTimes New Roman+METAon Native LCompObj3tiveFTfObjInfo9tiveFVEquation Native FW4- 2 8(i 2 )iSymbol- 2 w & "System-Symbol-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;lJqJmJ X(w)ࡱ_960359974tiveF6+'+Ole58652 FXPIC YLMETA [ࡱ> Lࡱ> v   .1  `&Z & MathType0Times New Roman- 2  STimes New Roman+- 2 (i 2 :)iSymbol- 2 @w & "System-Symbol-ࡱ FMicrosoft 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: .1   & z & MathType@Times New Roman(- 2  k 2 k 2 k 2 !k Times New Romanx7- 2 11pp 2 U22pp 2 /21pp 2 12ppTimes New Roman(METAbjNative !CompObjNativeF,fObjInfoNativeF.Equation Native F/- 2 0Symbol- 2 - 2  & "System-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;llJqJmJ k 11 k 22 -k 21 k 12 0ࡱ> ࡱ> L0ࡱ> ,  .1  &`z & MathType@Times New Roman-_960360119tiveFZ+#+Ole58652 F2PIC 3LMETA 5( 2 G Times New Roman- 2 1pTimes New Roman- 2 (i 2 )iSymbol- 2 w & "System-Symbolwࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>CompObjNative>fObjInfoNativeF@Equation Native FA@_960360123F#+짡+ࡱ> ;$lJqJmJ G 1 (w)ࡱ> LDࡱ> V3  .1  &z & MathType@Times New Roman-Ole BPIC CLMETA /E(CompObjNf 2 G Times New Roman - 2 2pTimes New Roman- 2 (i 2 )iSymbol- 2 w & "System-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ObjInfo4FPEquation Native Q@_960360162tiveF++!LOle59900 FRࡱ> ;$lJqJmJ G 2 (w)ࡱ> Lg ࡱ> g &? { .1  @ & z & MathType@Times New Roman(-PIC SLMETA U(CompObj/bfObjInfod 2 *R 2 vR 2 R Times New Romanx7- 2 1p 2 ?2pTimes New Roman(- 2 |(i 2 )i 2 (i 2 )i 2  (i 2  )iSymbol- 2 l 2 il 2  lSymbol- 2 j@ 2 @ & "System-Fࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;TlJqJmJ R 1 (l)@R 2 (l)@R(l)ࡱࡱ> Equation Native ep_960360171tiveF+ѡ+4Ole60030 FgPIC hLLࡱ> 3 1 .1  &z & MathType@Symbol- 2 Hr 2 l 2 Ht 2 lTimes New Roman(- 2 (i 2 P)i 2 &(i 2 \)iMETA0003tive FjCompObj3tiveFufObjInfo9tiveFwEquation Native FxTSymbol- 2 = 2 z-Times New Roman(- 2 1 & "System-l 2  lࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;8lJqJmJ r(l)=1-t(l)ࡱ> ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;hlJqJmJ t(l)R(l)F 1 (l)l l  dl_960360199tiveFѡ+ѡ+Ole58652 FzCompObj{fObjInfo}Equation Native F~_960359652F@Gڡ+@+ Oleion Native PIC59900 FLR(l)F 2 (l)ldl l  t(l)R(l)F 2 (l)ldl l  R(l)F 1 (l)ldl l ࡱ> ࡱ> Lࡱ> METAbjNative (CompObjNativeFfObjInfoNativeFEquation Native FLf   .1  `&@z & MathType@Symbol- 2 t 2 ^lTimes New Roman(- 2 (i 2 )iSymbol- 2 Times New Roman(- 2 0 & "System-2 1 &ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;0lJqJmJ t(l)const.ࡱ> ࡱ> L _960360485tive F@++Ole58652 FPIC   LMETA ࡱ>  ; e .1   & a & MathType@Times New Roman@- 2 W 2 Hzp 2 W 2 ^zp Times New Romanc- 2 11pp 2 22ppTimes New Roman@- 2 1 2 0 2 (^ 2 )^ 2 (^ 2 )^ 2 .H 2 U .HSymbol- 2 J= 2 `= & "System- 2 hc~ ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> CompObjNative  fObjInfoNativeFEquation Native F|_960612329X'FC+j+;`lJqJmJ W 11 (z)=W 22 (z)=1.0.ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> װlJqJmJ w ij (k+1,l)=w ij (k,l)+OleObjNative CompObjNativeFfObjInfoNativeFEquation Native Fm2tanh(y i )y j (k-l)ࡱ> ࡱ> LJ ࡱ> J; . .1  &`Z & MathTypepSymbolw-2 _960360636tive"Fj+ %+Ole58652 FPIC LMETA [Symbolw-2 ] ]Symbol3- 2 x Times New RomanM- 2 i> 2 i> 2 pp 2  i>Times New Roman- 2 k~ 2 E 2 +y~ 2 k~ 2 signpP 2  y~ 2  k~ 2 iPTimes New RomanM- 2 (^ 2 7)^ 2 y(^ 2 )^ 2  (^ 2 y (^ 2  ))^^ 2 s,HSymbol- 2 = 2 !=Times New RomanM- 2 1 2 2 & "System-`ࡱ> CompObjNativefObjInfoNativeFEquation Native F_960612352F3.+u=+ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;ШlJqJmJ x i (k)=Ey ip (k)sign(y i (k))[]i=1,2ࡱ> ࡱ> Ole PIC LMETA /|CompObj fL{ hࡱ> {  .1  @`&  & MathType`Times New Romanp- 2 @w 2 kp 2 m 2 w 2 Kkp 2 &m 2  yp 2  kp 2  signcG 2 yp 2 kp 2 Oyp 2 kp 2 m@Times New Romanp- 2 ij55 2 _ij55 2  i5 2 + p` 2 Ii5 2 j5Times New Romanp- 2 f(T 2 ,@ 2 )T 2 (T 2 ,@ 2 )T 2 ] (T 2 Q )T 2 U(T 2 (T 2 ))TT 2 9(T 2 )TSymbol- 2 + 2  = 2 L + 2 S-@Symbol- 2 -iTimes New Romanp- 2 %1@Times New Romanp- 2 1`Symbol- 2  m &  "System- FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> lJqJmJ w ij (k+1,l)=w ij (k,l)+my ip-1 (k)sign(y i (k))y j (k-l)ࡱࡱ> ObjInfo!Equation Native _960360750tiveP$Fu=+u=+QOle11994 FCompObjNative#%FfObjInfo0&FEquation Native T_9606123711)FF+F+L FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> ;8qJlJrJ i,j1,2{}ࡱ> ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>Ole11988 FCompObj(*fObjInfo+Equation Native Lࡱ> 0lJqJmJ l=1,2,...,Lࡱ> ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> <lJqJmJ k(s 1 )_960612978tive.FF+`/h+Ole58652 FCompObj-/fObjInfo0Equation Native X_960613004,63F`/h+o+Ole CompObj24f !%&(,013456789;<=>?@ABCDEFGHIJKLN=-0.487ࡱ> ࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> 4qJlJrJ k(s 2 )=-1.38ࡱ> ࡱ> ObjInfo5Equation Native P_960613046tive8Fo+o+QOle11994 FCompObjNative79fObjInfoNative:F Equation Native F @_938176900=Fo+@x+ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ>ࡱ> $qJlJrJ k(s 1 )ࡱ> Lࡱ> < P .OleObjNative  PICnfoNative <?F LMETAon Native FCompObj0>@FZ1  &A & MathType0 & A & MathTyࡱ>  FMicrosoft Equation 2.0 DS Equation Equation.2ࡱ> ࡱ> 8 (tK4Q4 <*9o6&ࡱ> ObjInfo8AFEquation Native <_960186540tiveCDF@x+R+(Ole58516 FCompObjNativeCFiObjInfo8FVisioDocumentEGsVisioInformation"H FVISIO 4 DrawingVISIO 4.0 ShapesVisio.Drawing.49qࡱࡱ> ࡱ> Oh+'08DP\htࡱࡱ> RIVisio (TM) Drawing RT%ЄBd !fffMMM333iq0iT d Arial)-": -3Times New Roman5T?? Y@-1,,J/T  hTJbOSS0{Gz?@CPjV]uVbUUO OS? T666 NP U    UvPaC G#G#G#G#G#G#xG &#O`$$!N=4!>7i>}p'0U2?E! 67gjP>603206&0B!4?6@27Tg2k=\- g2 VT @9ܯ6L^pu )5,2?Op??ZX F&m0#P@w?l_?VCG MO_N9{OOOOOo2qG`?FoXf.Xf RrjrjrgF_ _) ]__qgaopsoon# h0T_o?u5e%wv4R&|q@"];@ B `U ;@ !%U)15UC@=AUIMQUUpa%@U i!m"q#$ %}&'W=(`)* ~ yJB ȏڏ}h1  -?Q ~P ɟ~` /A~ ~p  1(~ s(~  D#|  ewX?DxjtCU&KPJ5ψϚ }  i{P%m?Vߋߝ~  k}(~ ~  [m~r   ~  K]o(~w ~w ;M_q~w ~w +/=/Z`#aO贁Nk?>A&& XT /////f?0e#?52;?M?_?q??Q?2-E?52??? OO/?!?4kO}OZa(OOGCpOK\ PR _XNah+5 c(otavwTisPaϯs?& ,贁N @|>ϯs?* ,贁N @|>s? ,贁N @|>ϯs?A  ,贁N @|>ϯs?B ,贁N @|>s? ~ ,贁N @|>ϯs?1  @Bۭ+-í 8i,3>W,+ܭj|W.ڭU,3W'+\3W&*?A B , 1 F hTA@* ?<I?NXfx** <Ebb{b b NeH u 38 ')343P70((15<33;mE 4 nZ Z38Mid-arrow of variable direction.to other mb߿?6i6`i?8?xV4ҿ H &' hTo(@?@JIw4?0?@0JPu `u `buu `u ~!|y@Ku` ?u @?b .} \"'4-"J".U [_ "rla  tN`s@ .I: 8@y?9K# -UE#%'IK*ols(b TPOÿB;I1A[#?f uL`u`.@  u"`]rB `Qps-8R?W4@`@hO$I$h3.h hU1L@y0I 2 0;A`9Copyright 1995 Visio Corporation. All 2s reserved.`Shape.hlp!#15065glBFlUDyakCk@0݊J%#$1H @3EW"0zAB y0U@U@|N@r13 Q(1.JqP1 S G`n1D-JVPyR1@P‹P}*lXAkeDmMLj ;{Np VrAB_RT#|eUlXA_oK%_7VA)3\-I!1oRQKhq$8MkGPEFZa'0p!%{hrr*$GFCoFQ!#uK[B`Arrowhead SizeQ5r)dC$[G&V0q$0 Ep!qC$113E" x1s}1?uP$0A@IAy5rn1$i&n8E@  (`_ Poin @R1` ?Left   TcD#`r p to theC.u!ee l'.*3A5r,TBcC-Ǒ%Ca-%}'2 FT #).( y )<׭C6GD_dC}+)C4H+D_ GF hTA@* ?<I?NXfx** <Ebb{b b NeH u w wp&8J\nw QGeneric box with text and connection points. Can be stretched toy dimwenss.mb?7io68?k?4 H"'}hTM@?,544?P +qu u Gu` Zu qbb (J* **%L@_ A`9Copyright 1995 Visio Corporation. All |s reserved.r`Shape.hlp!#150002lUXBa 5*Z&!  C  *B  /-,>b/t-!-/&// rrC 01_#0#0h:qr 4% W//S `/r?(*/'.P??/O/k@tk@<k@}Z'$Fu#)wjQU )4?C\GDv+)|zc3H+ oKcF hTA@* ?<Iv?N,fx* <Ebbb b NNeG r dww w}p wx ww9wwwwpxwpxp} wAxQpwQp+pndžw 8D pKGxlVhA gv! -/?/Q/c/}/g]E0////?/$0U?g?y???.2m@?? 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All 2s reserved.`Shape.hlp!#15065glBFlUDyakCk@0݊J%#$1H @3EW"0zAB y0U@U@|N@r13 Q(1.JqP1 S G`n1D-JVPyR1@P‹P}*lXAkeDmMLj ;{Np VrAB_RT#|eUlXA_oK%_7VA)3\-I!1oRQKhq$8MkGPEFZa'0p!%{hrr*$GFCoFQ!#uK[B`Arrowhead SizeQ5r)dC$[G&V0q$0 Ep!qC$113E" x1s}1?uP$0A@IAy5rn1$i&n8E@  (`_ Poin @R1` ?Left   TcD#`r p to theC.u!ee l'.*3A5r,TBcC-Ǒ%Ca-%}'2 F4 #)m y )װC6GD| +)$c4H+t{QcGF hTA@* ?<I?NXfx** <Ebb{b b NeH u w wp&8J\nw QGeneric box with text and connection points. Can be stretched toy dimwenss.mb?7io68?k?4 H"'}hTM@?,544?P +qu u Gu` Zu qbb (J* **%L@_ A`9Copyright 1995 Visio Corporation. All |s reserved.r`Shape.hlp!#150002lUXBa 5*Z&!  C  *B  /-,>b/t-!-/&// rrC 01_#0#0h:qr 4% W//S `/r?(*/'.P??/O/k@tk@<k@}v$F#)4W3Q )_ܓ3\GDT[+)\c3H+d/cF hTA@* ?<Iv?N,fx* <Ebbb b NNeG r dww w}p wx ww9wwwwpxwpxp} wAxQpwQp+pndžw 8D pKGxlVhA gv! -/?/Q/c/}/g]E0////?/$0U?g?y???.2m@?? 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