analiza fourier
DESCRIPTION
Aplicatii pas cu pasTRANSCRIPT
CURS 2
CURS 2 RECAPITULARE ANALIZA FOURIER CLASIC
SFG:
SFT:
Simetrie par:
Simetrie impar:
SFA:
;
SFC:
;
Aplicaia 2.1 S se determine SFT, SFA i SFC pentru semnalul din fig. 2.9. Fig. 2.9
SFT
SFA
Spectrul SFT
Spectrul SFA
Spectrul SFC
Observaie:
Spectrul unor semnale se poate obine din spectrul cunoscut al unui semnal de referin, prin nsumarea unei constante i/sau modificarea scrii i/sau operaie de ntrziere. De exemplu, considernd u(t) din fig. 2.9 ca semnal de referin, semnalul periodic din fig. 2.14 se scrie:
Fig. 2.14 Semnal derivat din semnalul u(t), dat n fig. 2.9
Avnd n vedere modelul spectral al semnalului u(t) este
modelul spectral SFA al semnalului este:
,
deci:
; ;
Aplicaia 2.2Fie x(t) un tren de impulsuri de arie unitar, reprezentat n fig.2.15. S se modeleze semnalul prin seria Fourier, tiind c T=1 ms i =0.2ms.
Fig. 2.15: Tren de impulsuri de arie unitar
Se utilizeaz SFC . Parametrii se calculeaz astfel:
(2.35)
=
unde sinc(() este funcia sinus cardinal. Rezult parametrii i :(2.36)
(2.37)
Graficul funciei sinc(x) este dat n fig. 2.16, a). Modelul semnalului cu ajutorul SFC este:(2.38)
Fig. 2.16 Funcia sinus cardinal a) i spectrul unui tren de impulsuri b)Funcia sinus cardinal se anuleaz pentru sau, altfel, , k=1,2,.... Deci pentru frecvena , unde , ct i pentru frecvenele . In cazul aplicaiei, T=1 ms i =0.2ms, deci ,
Spectrul SFC este reprezentat n fig. 2.16, b). Se observ c .Pornind de la relaia (2.29) se poate construi spectrul de amplitudini (fig. 2.17, a)) i de faze (fig. 2.17, b)), care caracterizeaz seria Fourier armonic.
Fig. 2.17: Spectrul de amplitudini i de faze al unui tren de impulsuriAplicaia 2.3Fiind dat un tren de impulsuri cu arie unitar, avnd perioada i durata , s se realizeze programul care efectueaz urmtoarele operaii:
calculul componentelor spectrale aferente SFC;
reprezentarea n acelai grafic a semnalului dat (pe o perioad a acestuia), ct i a semnalelor calculate pe baza unui numr finit de armonici din spectrul determinat. Se vor considera 9, 19, i 29 armonici n spectru.
SHAPE \* MERGEFORMAT
Fig. 2.18 Spectrele de amplitudini i de faze
Lista comentat a programului Matlab este urmtoarea:
clear all; clg;
%parametrii trenului de impulsuri
T=2;w0=2*pi/T;tau=0.2;Amplit=1/tau;
%calcului parametrilor modelului spectral
A=zeros(1,50);phi=zeros(1,50);
for i=1:50,
alf=(i-1)*w0*tau/2;
alf=alf/pi;
A(1,i)=abs(sinc(alf)/T);
phi(1,i)=-angle(sinc(alf));
end;
%se calculeaz vectorul ind, necesar n reprezentarea grafic a spectrului
for i=1:50,
ind(i)=i-1;
end;
%reprezentarea spectrului SFC (numai pentru frecvene pozitive)
figure(1)
stem(ind,A(1,:));grid;
figure(2)
stem(ind,phi(1,:));grid;
%generarea trenului de impulsuri i reprezentarea lui graficx1=zeros(1,900);x2=Amplit*ones(1,200);
x3=zeros(1,900);x=[x1 x2 x3];
dt=0.001;t=[-T/2+dt:dt:T/2];
figure(3);
h=plot(t,x,'k');set(h,'LineWidth',2);
axis([-1 1 -1.5 7]);hold on;
%calculul semnalelor deduse pe baza spectrului determinat%se utilizeaz 9, 19 i 29 armonici n spectru; se reprezint aceste
%semnale pe un grafic comun cu cel al trenului de impulsuri
for j=9:10:29,
xy=A(1)*ones(1,2000);
for i=1:j,
xy=xy+2*A(1,i+1)*cos(i*w0*t+phi(1,i+1));
end;
plot(t,xy,'k');axis([-1 1 -1.5 7]);end;grid;
Fig. 2.19 Semnalul x(t) i aproximarea acestuia printr-un numr finit de armonici
Fig. 2.20 Detalierea zonei centrale din fig. 2.19
n fig. 2.18 sunt date comnponentele spectrale i din SFC, aferente frecvenelor pozitive. n fig. 2.19 este reprezentat semnalul x(t) i aproximrile acestuia prin considerarea a 9, 19, i 29 armonici. Pentru a discerne mai bine calitatea aproximrilor, n fig. 2.20 s-a reprezentat, la scar de timp, zona central din fig. 2.19.2.3 Utilizarea sistemelor de funcii binare ortogonale n modelarea semnalelor periodice
n acest caz, funciile ortogonale din sistemul , utilizat n dezvoltarea , pot lua doar dou valori. Principalele funcii din aceast categorie sunt: funciile Walsh, funciile Rademacher, funciile Hadamard i funciile Haar.
2.3.1 Analiza Fourier - Walsh
Funciile Walsh sunt funcii ortogonale, definite pe o baz de timp, numit i suport, [0; T], T fiind perioada. Frecvent se utilizeaz ca suport domeniul [-T/2;T/2]. De asemenea, se poate utiliza un domeniu de timp normat: = t/T. n acest caz, suportul este [0; 1] sau .
Reprezentarea grafic a primelor N=8 funcii Walsh este dat n fig. 2.21.
Fig. 2.21 Primele 8 funcii Walsh
Intervalul s-a mprit n N subintervale egale, de lime (, numrul subintervalului fiind o putere a lui 2: ; n fig. 2.21 p=3.
Funciile Walsh se noteaz prin wal(i,), i=0,1,...N-1. Prin analogie cu funciile trigonometrice, funciile Walsh pare se noteaz:
(2.39) , iar (2.40) , cele impareIndicele k din funcia cal(k, ) arat numrul de intersecii ale abscisei din jumtatea bazei de timp i se numete secvena funciei.
Norma funciilor Walsh, definite n raport cu timpul normat , este 1. Dac se utilizeaz timpul fizic, t, norma este .
Dezvoltarea unei funcii periodice u(t) n sistemul de funcii Walsh este:
(2.41) , n care:(2.42)
Fig. 2.22 Spectre n analiza Fourier-Walsh, conform modelelor (2.41) a) i (2.43) b)Expresia (2.41) se poate pune sub forma:
(2.43) , n care(2.44)
(2.45)
Generarea funciilor Walsh se face, de regul, prin relaii iterative. Vom ilustra dou astfel de proceduri.
Procedura 1. Se iniializeaz wal(0,(), cu valoare unitar pentru i zero n rest. n continuare, wal(i+1,(), se calculeaz pe baza funciei anterioare, wal(i,(), i=0,1,2, ..., utiliznd ecuaia:
(2.46)
cu , iar este partea ntreag a lui .
Fig. 2.23 Generarea funciei wal(1,()
Calculul funciei wal(1,() pe baza funciei wal(0,(). n cazul funciei wal(1,(), indicii j i l sunt: i . Aplicnd relaia (2.46), rezult:
(2.47)
n fig. 2.23 este ilustrat construcia funciei wal(1,(), pe baza celor 2 termeni din partea dreapt a expresiei (2.47).
Calculul funciei wal(5,() se face punnd i . Rezult:
(2.48)
Generarea funciei wal(5,(), este ilustrat n fig. 2.24.
Fig. 2.24 Generarea funciei wal(5,()
Procedura 2. Se noteaz prin intervalul discret n care se mparte baza de timp a funciei. Dup iniializarea funciei wal(0,r), calculul iterativ al celorlalte funcii se face cu relaia:
(2.49)
cu ; ; .
Deducerea funciei wal(1,r) din wal(0,r) se face cu relaia (2.49), n care se pune i . Considernd N=4, rezult:
Construcia funciei este utilizat n fig. 2.25.
Fig. 2.25 Construcia funciei wal(1,r)
Aplicaia 2.4 S se efectueze analiza Fourier-Walsh a semnalului , , considernd numai primele 6 funcii Walsh.
Fig. 2.26 Semnal sinusoidal
Se face schimbarea de variabil: ; rezult (fig.2.26).
Se calculeaz coeficienii ai, i=0,1,...,5, din relaia (2.41), conform expresiilor (2.42). Deoarece funcia este impar, rezult . n continuare:
Fig. 2.27 Aproximarea lui prin n analiza Fourier - Walsh
n ultima relaie de mai sus, calculm fiecare termen:
Se obine:
Deci, semnalul se aproximeaz (vezi fig. 2.27) prin expresia:
2.3.2 Analiza Fourier - Rademacher
Funciile Rademacher se genereaz cu relaia:
(2.50) ,
unde este timpul normat.
Primele 3 funcii Rademacher sunt ilustrate n fig. 2.28. Din relaia de definiie (2.50), se constat c toate funciile sunt impare, deci ele nu formeaz un sistem complet de funcii ortogonale. n consecin, ele pot fi utilizate numai pentru modelarea semnalelor cu simetrie impar.Fig. 2.28 Funcii Rademacher2.3.3 Analiza Fourier - Hadamard
Funciile ortogonale Hadamard se genereaz cu relaia:
(2.51)
unde ( este timpul normat. Ele formeaz un sistem complet de funcii ortogonale, ca i funciile Walsh.
n fig. 2.29 sunt ilustrate primele 5 funcii Hadamard.Fig. 2.29 Funcii Hadamard
2.3.4 Analiza Fourier - Haar
Sistemul de funcii ortogonale Haar este definit prin relaiile:
(2.52) ,
unde i .Fig. 2.30 Funcii HaarNotnd prin gradul funciei i prin ordinul funciei (), relaia (2.52) se mai poate scrie ca n ecuaia de mai jos:
(2.53)
Norma funciilor Haar, exprimate n raport cu timpul normat, este unitar. Modelul Fourier Haar al unui semnal u(t) este:
(2.54) ,
unde .
Reprezentarea grafic a primelor 8 funcii Haar este dat n fig. 2.30.2.4 Analiza polinomial a semnalelor periodicen SFG, sistemul de funcii utilizat la modelarea semnalului u(t) se deduce pornind de la un set de funcii polinomiale, de forma . Dac acestea satisfac relaia:
(2.55) ,
atunci polinoamele se numesc ortogonale n raport cu funcia de ponderare (t). Dac relaia (2.55) este adevrat atunci sistemul de funcii: , n care , este un sistem de funcii ortogonale i se utilizeaz efectiv n SFG. Parametrii ai din SFG se calculeaz cu relaiile:
(2.56)
n cele ce urmeaz sunt prezentate principalele funcii polinomiale utilizate n modelarea semnalelor.
Polinoamele Legendre. Aceste polinoame au funcia de ponderare egal cu 1, deci sunt efectiv polinoame ortogonale.
Polinoamele Laguerre, cu funcia de ponderare . Ele se genereaz cu relaia recursiv:
Polinoamele Hermite, cu funcia de ponderare . Ecuaia utilizat pentru generarea acestor funcii este:
Polinoamele Cebev, cu funcia de ponderare
i ecuaia recursiv de generare:
Observaie:
n principiu, n SFG, putem adopta orice sistem de funcii, cu condiia ca acestea s fie liniar independente.
Dac se adopt sistemul de funcii liniar independente, este oportun ca el s se transforme ntr-un sistem de funcii ortogonale, pentru ca determinarea parametrilor SFG s se fac uor. Se tie c, dac este ndeplinit condiia , unde este simbolul lui Kronecker, atunci sistemul este ortonormal.
Trecerea de la sistemul iniial de funcii , liniar independente, dar neortogonale, la sistemul de funcii ortonormale , se face prin procedura general Gram-Schmidt de ortogonalizare.O prezentare elementar a acestei proceduri este dat n cele ce urmeaz:
Se consider 1(t) definit ca: , i se impune condiia ca norma funciei s fie unitar:
,
adic:
,
de unde se obine a1.
Utiliznd i , se definete :
Din condiia: rezult a3, iar din se obine a2.
Funcia se definete sub forma:
Condiiile i se folosesc pentru determinarea parametrilor a5 i a6, iar pentru determinarea lui a4 se folosete condiia .
Procedura se repet pn cnd sunt implicate toate funciile din sistemul iniial, .
Dintre funciile iniiale, , cel mai des apelate n modelarea semnalelor, fcnd obiectul operaiei preliminare de ortogonalizare, sunt funciile exponeniale, , cu valori impuse pentru .
Capitolul 3
MODELAREA SEMNALELOR NEPERIODICE3.1 Analiza spectral a semnalelor utiliznd transformata Fourier
Se caut modelul unui semnal neperiodic oarecare, , de modul integrabil (fig. 3.1, a)). Pentru aceasta se utilizeaz un alt semnal, periodic, notat prin , care ntr-o perioad T posed forma semnalului (vezi fig.3.2, b)).
Fig. 3.1 Semnal periodic obinut dintr-un semnal neperiodic
S modelm semnalul periodic prin SFC:
(3.1)
i, deoarece n perioada T avem u(t) = uT(t),
(3.2)
Fig. 3.2 Pulsaiile discrete din SFC
nlocuind (3.1) n (3.2), se obine , sau:
(3.3)
,
cu . Parametrii ai SFC sunt asociai pulsaiilor (fig. 3.2).
Fig. 3.3 Pulsaiile discrete pentru o perioad T foarte mare
S considerm acum o cretere important a perioadei T. S notm pulsaia care devine foarte mic prin , deci (fig.3.3). Relaia (3.3) devine:
(3.4)
n continuare se admite c perioada T, n care se ncadreaz semnalul dat iniial, tinde spre infinit, prin transferarea spre ( a limitei din stnga a perioadei i spre ( a limitei din dreapta. Dac , se obine:
deci:
(3.5)
Paranteza din partea dreapt a relaiei (3.5) este transformata Fourier a semnalului u(t), adic:
(3.6)
Transformata Fourier reprezint modelul matematic al semnalului u(t). Aceast transformat, numit funcie spectral sau caracteristic spectral a semnalului, exist i pentru frecvene negative, deci pentru tot domeniul de frecvene . Semnalul u(t) se exprim n funcie de prin transformata Fourier invers:
(3.7)
Funcia complex se poate exprima prin urmtoarea relaie:
,
unde , iar .
n concluzie, se obine:
(3.8)
,
cu:
Funciile i sunt pare, iar funciile i sunt impare.
3.2 Semnificaia fizic a funciilor spectrale
Pornind de la relaia (3.4), adic,
n ipoteza c perioada T este foarte mare i intervalul de integrare nu este infinit, putem admite:
n consecin:
(3.9)
Pentru semnalul , SFC este:
(3.10)
,
unde . Conform cu (3.9) i (3.10), rezult:
(3.11)
Fie o form oarecare a funciei spectrale (fig. 3.4, a)). Dac se discretizeaz axa frecvenelor cu un pas , se observ c produsul reprezint aria , haurat pe fig. 3.5, a), adic:
(3.12)
Pornind de la relaiile (3.11) i (3.12), se obine:
Aceast relaie d legtura ntre caracteristica spectral i spectrul de amplitudini al SFC (fig. 3.4, b)). Pentru modelele spectrale, trecerea corespunde trecerii , unde sunt date de relaia (3.13). Fiecrui interval discret i corespunde o armonic elementar avnd amplitudinea (fig. 3.4, c)):
Se obine:
Deci mrimea este proporional cu densitatea de amplitudini a armonicilor, . Dac se trece la mrimi infinitezimale, atunci nlocuim prin iar prin i rezult . Cele prezentate justific utilizarea urmtoarelor denumiri: funcie a densitii spectrale, pentru , i densitate spectral de amplitudini, pentru .Funcia spectral a unui impuls
Fie impulsul real de arie unitar (fig. 3.5). S se calculeze funcia spectral a acestui semnal.
Funcia spectral este transformata Fourier a semnalului:
,
deci funcia posed forma dat n fig. 3.6.
Fig. 3.5 Impuls de arie unitarFig. 3.6 Funcia spectral a impulsului
S considerm acum un tren de impulsuri unitare cu , notat cu x(t) (fig. 2.15). S notm , cu , i frecvena foarte mic a semnalului periodic .
n fig. 3.7, a) i b), se ilustreaz procedeul de discretizare a funciei spectrale prezentate mai sus.
Se observ c, prin discretizarea funciei spectrale a impulsului (fig. 3.7, a)), se obine spectrul semnalului periodic (fig. 3.7, b)).
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
Fig. 3.4 Procedeul de discretizare a unei funcii spectrale
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
b)
a)
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
b)
a)
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
a)
b)
EMBED Equation.3
EMBED Equation.3
0
5
10
15
20
25
30
35
40
45
50
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
5
10
15
20
25
30
35
40
45
50
-3
-2.5
-2
-1.5
-1
-0.5
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1
0
1
2
3
4
5
6
7
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0
1
2
3
4
5
6
EMBED Equation.3
9
19
29
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
a)
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
b)
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
c)
b)
a)
EMBED Equation.3
EMBED Equation.3
cal(0, )
cal(1, )
cal(2, )
sal(0, )
sal(1, )
sal(2, )
cal(3, )
sal(3, )
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
a)
b)
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
_1285076356.unknown
_1285077832.unknown
_1285151895.unknown
_1285155874.unknown
_1285481092.unknown
_1285482337.unknown
_1285484054.unknown
_1285484375.unknown
_1285484420.unknown
_1285484451.unknown
_1285484707.unknown
_1285484433.unknown
_1285484395.unknown
_1285484122.unknown
_1285483079.unknown
_1285483416.unknown
_1285482360.unknown
_1285481881.unknown
_1285482269.unknown
_1285481445.unknown
_1285481824.unknown
_1285481444.unknown
_1285156975.unknown
_1285157047.unknown
_1285231244.unknown
_1285480570.unknown
_1285157061.unknown
_1285231243.unknown
_1285157011.unknown
_1285157029.unknown
_1285156991.unknown
_1285156573.unknown
_1285156739.unknown
_1285156779.unknown
_1285156959.unknown
_1285156869.unknown
_1285156753.unknown
_1285156700.unknown
_1285156729.unknown
_1285156717.unknown
_1285156654.unknown
_1285156679.unknown
_1285156619.unknown
_1285156517.unknown
_1285156545.unknown
_1285156556.unknown
_1285156529.unknown
_1285156386.unknown
_1285156505.unknown
_1285152074.unknown
_1285155328.unknown
_1285155465.unknown
_1285155500.unknown
_1285155536.unknown
_1285155555.unknown
_1285155525.unknown
_1285155488.unknown
_1285155451.unknown
_1285155464.unknown
_1285155378.unknown
_1285155389.unknown
_1285155355.unknown
_1285155129.unknown
_1285155189.unknown
_1285155221.unknown
_1285155316.unknown
_1285155222.unknown
_1285155220.unknown
_1285155155.unknown
_1285155169.unknown
_1285155056.unknown
_1285155068.unknown
_1285152087.unknown
_1285151993.unknown
_1285152028.unknown
_1285152048.unknown
_1285152008.unknown
_1285151942.unknown
_1285151978.unknown
_1285151925.unknown
_1285145405.unknown
_1285150873.unknown
_1285151799.unknown
_1285151835.unknown
_1285151849.unknown
_1285151816.unknown
_1285151691.unknown
_1285151781.unknown
_1285151573.unknown
_1285150756.unknown
_1285150820.unknown
_1285150850.unknown
_1285150790.unknown
_1285145433.unknown
_1285150735.unknown
_1285146418.unknown
_1285145417.unknown
_1285144966.unknown
_1285145303.unknown
_1285145367.unknown
_1285145392.unknown
_1285145332.unknown
_1285145037.unknown
_1285145243.unknown
_1285145282.unknown
_1285144991.unknown
_1285145014.unknown
_1285144876.unknown
_1285144924.unknown
_1285144938.unknown
_1285144892.unknown
_1285077895.unknown
_1285144861.unknown
_1285077877.unknown
_1285076770.unknown
_1285076993.unknown
_1285077135.unknown
_1285077669.unknown
_1285077753.unknown
_1285077796.unknown
_1285077731.unknown
_1285077738.unknown
_1285077549.unknown
_1285077643.unknown
_1285077657.unknown
_1285077629.unknown
_1285077390.unknown
_1285077415.unknown
_1285077158.unknown
_1285077097.unknown
_1285077115.unknown
_1285077056.unknown
_1285077076.unknown
_1285077036.unknown
_1285076871.unknown
_1285076948.unknown
_1285076974.unknown
_1285076890.unknown
_1285076812.unknown
_1285076831.unknown
_1285076799.unknown
_1285076603.unknown
_1285076686.unknown
_1285076729.unknown
_1285076746.unknown
_1285076707.unknown
_1285076650.unknown
_1285076669.unknown
_1285076623.unknown
_1285076524.unknown
_1285076563.unknown
_1285076579.unknown
_1285076541.unknown
_1285076433.unknown
_1285076455.unknown
_1285076415.unknown
_1109501410.unknown
_1285053187.unknown
_1285060329.unknown
_1285073316.unknown
_1285074360.unknown
_1285074466.unknown
_1285076234.unknown
_1285074381.unknown
_1285073708.unknown
_1285074090.unknown
_1285073639.unknown
_1285073507.unknown
_1285060372.unknown
_1285060424.unknown
_1285073298.unknown
_1285060392.unknown
_1285060403.unknown
_1285060415.unknown
_1285060383.unknown
_1285060351.unknown
_1285060362.unknown
_1285060339.unknown
_1285056420.unknown
_1285056748.unknown
_1285059372.unknown
_1285060220.unknown
_1285060298.unknown
_1285060319.unknown
_1285060287.unknown
_1285060204.unknown
_1285056930.unknown
_1285056749.unknown
_1285056524.unknown
_1285056646.unknown
_1285056667.unknown
_1285056747.unknown
_1285056657.unknown
_1285056539.unknown
_1285056634.unknown
_1285056481.unknown
_1285056512.unknown
_1285056499.unknown
_1285056442.unknown
_1285055603.unknown
_1285056353.unknown
_1285056375.unknown
_1285056319.unknown
_1285056330.unknown
_1285054341.unknown
_1285055559.unknown
_1285053458.unknown
_1285053547.unknown
_1285053605.unknown
_1285053490.unknown
_1285053228.unknown
_1284716019.unknown
_1285052228.unknown
_1285053160.unknown
_1285052875.unknown
_1285053139.unknown
_1284988553.unknown
_1285051873.unknown
_1285051944.unknown
_1285051997.unknown
_1285051823.unknown
_1284988240.unknown
_1284988488.unknown
_1284716070.unknown
_1284987990.unknown
_1284716087.unknown
_1284716040.unknown
_1284716067.unknown
_1109501563.unknown
_1109501634.unknown
_1110097102.unknown
_1110540186.unknown
_1284709800.unknown
_1284709910.unknown
_1112096061.unknown
_1113030497.unknown
_1113030496.unknown
_1112078095.unknown
_1112096051.unknown
_1112078077.unknown
_1110539994.unknown
_1110540178.unknown
_1110098507.unknown
_1110539986.unknown
_1110097134.unknown
_1109501660.unknown
_1109501675.unknown
_1109501684.unknown
_1109501670.unknown
_1109501645.unknown
_1109501650.unknown
_1109501640.unknown
_1109501606.unknown
_1109501617.unknown
_1109501628.unknown
_1109501612.unknown
_1109501590.unknown
_1109501600.unknown
_1109501584.unknown
_1109501439.unknown
_1109501544.unknown
_1109501556.unknown
_1109501532.unknown
_1109501420.unknown
_1109501434.unknown
_1109501415.unknown
_1104901179.unknown
_1104999919.unknown
_1109429243.unknown
_1109498815.unknown
_1109498937.unknown
_1109499108.unknown
_1109499232.unknown
_1109501345.unknown
_1109499298.unknown
_1109499113.unknown
_1109498946.unknown
_1109499103.unknown
_1109498926.unknown
_1109498931.unknown
_1109498824.unknown
_1109498661.unknown
_1109498780.unknown
_1109498794.unknown
_1109498772.unknown
_1109491262.unknown
_1109498512.unknown
_1109498650.unknown
_1109496517.unknown
_1109477221.unknown
_1109490664.unknown
_1109477475.unknown
_1109429256.unknown
_1105889753.unknown
_1106123175.unknown
_1106123611.unknown
_1106123615.unknown
_1109426214.unknown
_1109426229.unknown
_1109426531.unknown
_1109426164.unknown
_1106123616.unknown
_1106123613.unknown
_1106123614.unknown
_1106123612.unknown
_1106123325.unknown
_1106123609.unknown
_1106123610.unknown
_1106123340.unknown
_1106123346.unknown
_1106123334.unknown
_1106123313.unknown
_1106123319.unknown
_1106123305.unknown
_1105895599.unknown
_1106123064.unknown
_1106123087.unknown
_1106122906.unknown
_1106122958.unknown
_1105960150.unknown
_1105895082.unknown
_1105890498.unknown
_1105358391.unknown
_1105886115.unknown
_1105886222.unknown
_1105886241.unknown
_1105886127.unknown
_1105365477.unknown
_1105365499.unknown
_1105610281.unknown
_1105362968.unknown
_1105348935.unknown
_1105355001.unknown
_1105358375.unknown
_1105348943.unknown
_1105275903.unknown
_1105275904.unknown
_1104999973.unknown
_1104901425.unknown
_1104902329.unknown
_1104903218.unknown
_1104904036.unknown
_1104904044.unknown
_1104904119.unknown
_1104903689.unknown
_1104903065.unknown
_1104903115.unknown
_1104903146.unknown
_1104903058.unknown
_1104901857.unknown
_1104902318.unknown
_1104902324.unknown
_1104902246.unknown
_1104901858.unknown
_1104901698.unknown
_1104901711.unknown
_1104901717.unknown
_1104901705.unknown
_1104901431.unknown
_1104901419.unknown
_1104901181.unknown
_1104901348.unknown
_1104901354.unknown
_1104901360.unknown
_1104901182.unknown
_1104901180.unknown
_1104827243.unknown
_1104837209.unknown
_1104846433.unknown
_1104900279.unknown
_1104900638.unknown
_1104901089.unknown
_1104901095.unknown
_1104901039.unknown
_1104901048.unknown
_1104901030.unknown
_1104900945.unknown
_1104900293.unknown
_1104900532.unknown
_1104900549.unknown
_1104900516.unknown
_1104900286.unknown
_1104899442.unknown
_1104899937.unknown
_1104900040.unknown
_1104900089.unknown
_1104900225.unknown
_1104900073.unknown
_1104899972.unknown
_1104899454.unknown
_1104899711.unknown
_1104899742.unknown
_1104899460.unknown
_1104899448.unknown
_1104899423.unknown
_1104899430.unknown
_1104899177.unknown
_1104842939.unknown
_1104845446.unknown
_1104845685.unknown
_1104846359.unknown
_1104845459.unknown
_1104845044.unknown
_1104845427.unknown
_1104845388.unknown
_1104845426.unknown
_1104845106.unknown
_1104845105.unknown
_1104844997.unknown
_1104845025.unknown
_1104845034.unknown
_1104845016.unknown
_1104844945.unknown
_1104844958.unknown
_1104844944.unknown
_1104843995.unknown
_1104841610.unknown
_1104842881.unknown
_1104842928.unknown
_1104842866.unknown
_1104842874.unknown
_1104842828.unknown
_1104841468.unknown
_1104841595.unknown
_1104841426.unknown
_1104841461.unknown
_1104830547.unknown
_1104833372.unknown
_1104833404.unknown
_1104834765.unknown
_1104834881.unknown
_1104834883.unknown
_1104834771.unknown
_1104834780.unknown
_1104834637.unknown
_1104834733.unknown
_1104834760.unknown
_1104834621.unknown
_1104834627.unknown
_1104833387.unknown
_1104833396.unknown
_1104833379.unknown
_1104830707.unknown
_1104830870.unknown
_1104830887.unknown
_1104830807.unknown
_1104830813.unknown
_1104830821.unknown
_1104830789.unknown
_1104830799.unknown
_1104830692.unknown
_1104828305.unknown
_1104830469.unknown
_1104830540.unknown
_1104830437.unknown
_1104830429.unknown
_1104827678.unknown
_1104827992.unknown
_1104827255.unknown
_1045638370.unknown
_1103465116.unknown
_1103466256.unknown
_1103466468.unknown
_1103467040.unknown
_1103467071.unknown
_1103467072.unknown
_1103467070.unknown
_1103467069.unknown
_1103467016.unknown
_1103467034.unknown
_1103467009.unknown
_1103466355.unknown
_1103466424.unknown
_1103466269.unknown
_1103465451.unknown
_1103465577.unknown
_1103466041.unknown
_1103466234.unknown
_1103465989.unknown
_1103465505.unknown
_1103465140.unknown
_1103465303.unknown
_1103465394.unknown
_1103465155.unknown
_1103465128.unknown
_1103464182.unknown
_1103464625.unknown
_1103464944.unknown
_1103465100.unknown
_1103464878.unknown
_1103464612.unknown
_1103464618.unknown
_1103464195.unknown
_1046006000.unknown
_1062841300.unknown
_1062841786.unknown
_1103462879.unknown
_1063356801.unknown
_1062841767.unknown
_1046009444.unknown
_1046079322.unknown
_1046006944.unknown
_1046007296.unknown
_1046006038.unknown
_1046005994.unknown
_1046005997.unknown
_1046005987.unknown
_1045993092.unknown
_1045576929.unknown
_1045577179.unknown
_1045577200.unknown
_1045577203.unknown
_1045577190.unknown
_1045577195.unknown
_1045577198.unknown
_1045577192.unknown
_1045577185.unknown
_1045577187.unknown
_1045577182.unknown
_1045577154.unknown
_1045577168.unknown
_1045577174.unknown
_1045577177.unknown
_1045577171.unknown
_1045577160.unknown
_1045577163.unknown
_1045577165.unknown
_1045577157.unknown
_1045577134.unknown
_1045577141.unknown
_1045577146.unknown
_1045576874.unknown
_1045576917.unknown
_1045576923.unknown
_1045576926.unknown
_1045576920.unknown
_1045576891.unknown
_1045576900.unknown
_1045576907.unknown
_1045576911.unknown
_1045576914.unknown
_1045576903.unknown
_1045576895.unknown
_1045576883.unknown
_1045576886.unknown
_1045576877.unknown
_1045576880.unknown
_1045576847.unknown
_1045576862.unknown
_1045576868.unknown
_1045576865.unknown
_1044010805.unknown
_1044010869.unknown
_1044010875.unknown
_1044003633.unknown
_1044010797.unknown
_1044003700.unknown
_1043916805.unknown
_1044003497.unknown
_1044003566.unknown
_1044003609.unknown
_1044003537.unknown
_1043916825.unknown
_1043916793.unknown
_1043916785.unknown