３

- { } / f ( ) e nπ + f( ) = Cne n= nπ / Eucld r e (= N) j = j e e = δj, δj = 0 j r e ( =, < N) r r r { } ε ε = r r r = Ce = r r r e ε = = C = r C r e + CC e j e j e = = ε = r ( r e ) + r e C C 0 r e = 0 C ε = r e r r C + C = ( = ) r ( ) r r = r e r = 0 ε ε

(, ) f ( ) f ( ), f( + ), f( + ), f( ) = + n f ( ) f ( A+ ) f f ( ) ( f0, f,, f n ) = n + f f + f 0 f f f f n 0 n -- f g = 0 fg + fg+ fg + + fg+ = 0 0 0 f g ( f ) g( ) ( f ) g( ) (, ) n f g = 0 f( ) g( ) d= 0 f( ) g( ) d= 0 e e ( ) j e ( ) e ( ) d= δ j (, ) f ( ) e ( ) f ( ) = Ce ( ) = e

N ε = N 0 ( ), f ( ) + nπ nπ f ( ) ( cos b sn ),( b 0) = n + n 0 = n= 0 ε ε = f ( ) f ( ) d ε nπ n' π nπ n' π cos cos d = sn sn = δ nπ n' π n' π sn sn d = cos = d = nn' ( nn, ' ) ( n 0, n' ) nπ αn = f ( )cos d ( n = 0,,, ) nπ βn = f ( )sn d ( n =,,, β0 = 0) ε ( ) ( αn n βn n) n= 0 = f d + b 0 n n= + + ( + b n ) α0 = f () d + ( αn + βn ) n=

α α β + + + 0 n n ( 0 ) ( n ) ( bn ) n= f ( ) 0 0 α0 αn βn 0 =, n =, bn = ( n ) α0 d= ε + + αn + β n n= f( ) ( ) α 0 + ( αn + βn ) n= r r = Bessel +, b ( n n n ) nπ nπ, cos, sn ( n )

N ε ε 0( N) ε 0 f ( ) ε 0 f ( ) { f( + 0) + f( 0) }/ α n π n π 0 ( ) = + ( αncos + βnsn ) n= f f ( ) f ( ) Fourer( ) = e cos = + sn nπ / nπ / ( ) = ( n + n ) n= 0 f A e B e nπ / An = ( αn βn) = f( ) e d nπ / Bn = ( αn βn) = f( ) e d n= f( ) = Cne π n= n / C n nπ / = f( ) e d (, ) k = nπ / n k n π k = kn+ kn nπ / k f( ) = Ce n = Ce n n= + kn π

C n kn = f( ) e d k 0 k C C( k) dk n kn k, k dk k f ( ) = C( k) e π d k Ck ( ) = f( e ) d ( π ) ( f ), C( k) f ( ) Fk ( ) k Fk ( ) = f( e ) d π f ( ) = F( k) e k dk k k e, e t F( ω) = f( t) e ω dt 0 ( ) t F( ) f( t) e σ + ω = ω dt 0 f () t Me αt (M, ) σ > α σ + ω = S

7 f () t f X ( f ) P( f) = l X( f) l X( f) X ( f) = () t = l ( t) dt f f + df P( f) df = P( f ) df

, y y, y t ( t ( ), yt ( + τ )) y y r E E y = = ry = E = E y y = P( f) df, y r = [ ] E E E y [ ] C = E [ E ] [ ] [ ] 0, y E = E y = E[ ] = 0 D,C, 0 r =, y ( = α y, α 0) [ ] E y E = α r = α / α = 0 0 (, t ω)

{ tω (, )} τ

t () t = + ( () t ( t + τ ) Ct (, τ ) E[ t () t ( τ )] (, t τ ) τ C( τ ) = ( t) ( t+ τ ) = l ( tt ) ( + τ ) dt C( τ ) C (0) R( τ ) R( τ ) = C( τ )/ c(0) = () tt ( + τ )/ () t () t = α cosω t α C( τ ) = cosωτ R( τ) = cosωτ

() C( τ ) = l ( t) ( tτ ) dt t τ = t' τ = l ( t' + τ ) ( t') d ' t τ ± τ ± () C( τ ) = C( τ ) 0 [ t ± t+ τ ] dt > ( τ 0) l ( ) ( ) 0 tdt t dt + +τ l ( ) l ( ) ± l tt ( ) ( +τ ) dt 0 > C(0) C( ) () (v) C(0) C( ) 0 C(0) C( ) 0 C( )= C( ) C ( )= C ( ) 0 C (0)= C (0) C (0)=0 C( τ ) = l ( t) ( t τ ) dt + C'( τ ) = l ( t) '( t+ τ ) dt

+ τ C'( τ) = l ( t' τ) '( ') ' t dt + τ C''( τ) =l '( tτ) '( ') ' t dt = l '( t ) '( t' +τ ) dt (v) τ τ, C( τ ) 0 C( ) () t () t 0 () t = () t () t C ( ) l ( ) υ τ = t ( t+ τ ) dt { { } = l ( t) ( t) } ( t+ τ) ( t+ τ) d t tt t t dt+ ( ( t) = ( t+ τ ) = ) = l ( ) ( + τ) l { ( ) + ( + τ) } = C( τ ) Cυ ( τ) = C( τ)

t (), t y() t C ( τ ) = ( t) y( t+ τ ) C ( τ ) C ( τ ), y R ( τ) ( t) y( t )/ = + τ = C ( τ )/ C (0) C (0) y () C ( τ ) = C ( τ ) C y ( τ ) = l ( t) y( t+ τ ) dt + τ = l ( t' τ ) y( t') d ' t (' t = t+ τ ) + τ ( = C τ ) () τ, C ( τ ) 0

[ ] αt () ± βyt ( + τ) 0 l [ αt ( ) ± βyt ( + τ) ] dt 0 l ( tdt ) l tyt ( ) ( ) dt ± + α αβ τ l y ( t τ) dt 0 + β + α C(0) ± αβc ( τ) + β C (0) 0 y α β C ( τ ) C (0) C (0) 0 y C ( τ ) C (0) C (0) y R ( τ ) C ( τ ) C(0) Cy (0) y + = + C ( τ ) f () t = Asnωt C( τ ) = l Asnωt Asn ω( t+ τ ) dt cos( A+ B) = cos Acos Bsn Asn B cos( A B) = cos Acos B+ sn Asn B A = l { cosωτ cos ω( t τ )} + dt A = l t cosωτ sn ω( t + τ ) ω sn Asn B= cos( A B ) cos( A+B )

A l cos sn ( ) sn ( ) ωτ A A 4ω ω τ 4ω ω = + τ = A cosωτ

nt () 0 ( τ ) = l ( ) ( τ) n δ( ) + = τ C n t n t dt ( ) ( ) b δτ ( ) = 0, τ 0, δτ ( ) dτ = f () t δτ ( ) dτ= f(0) b f () t = Asn ωt+ n() t C( τ) = l { Asn ωt+ n( t) }{ Asn( ωt+ τ) + n( t )} + τ dt n(t) Asn ωt n( t+ τ) Asn ω( t+ τ) n( t) 0 ntnt () ( + τ ) n δ ( τ ) A C( τ ) = cos ωτ + n δ ( τ ) τ 0 A cosωτ

C (j) () = 0 N N C ( j) = ( j) ( j+ ) N = 0 N N C ( j)

S( ) f ω - C( τ ) = l ( t) ( t+ τ ) dt () t X ( ω) ω t () t X( ω) e d = ω ω t C( τ ) = l X( ω) e dω( t+ τ) dt + ω t = l X ( ) t ( + ) e dtd ω τ ω ωτ ω ( t+ τ ) = l X ( ) e ( t+ ) e dtd = l ω τ ω π ( ω) ( ω) X X e ωτ dω 0 C(0) l X X π ( ω) ( ω) = dω S( ) = S ( ω ) dω π π X ( ω) S( ω) = l X( ω) X ( ω) = l S( ) ( t) ( X ω) π X ( ω) ω

f( π f = ω) X ( ω) π π l X ( ω) = l X( ω) X ( ω) = S( ω) S( ) S( ) S( ) C( τ ) = S( ω) e ωτ dω = S( ω) e ωτ d ω

S( ) C( ) S( ω) = C( τ) e ωτ dτ π C( ) S( ) Wener-Khntchne S( ) const ωτ C( τ) = e dω const const ( ) 3 C( τ ) S ( ω) S ( ω) = C ( τ) e ωτ dτ π C ( ) ( ) τ = S ω e ωτ dω, y C ( τ ) = l ( τ) y( t+ τ) dt τ ω( t+ τ) = l ( t) Y( ω) e dωdt ωt ωτ = l ( te ) Y( ω) e dωdt = ωt ωτ l ( te ) dt Y( ω) e dω

= + 4 π ωτ X ω Y ω e dω l ( ) ( ) π ( ω) l X ( ω) Y( ω) S = S ( ω) X( ω) = X( ω) e Y( ω) = Y( ω) e ω t+ α ω t+ β ( X ω) Y ( ω) X,Y X Y X ( ω) Y( ω) = X( ω) Y( ω) cos( β α) 5 ) S ( ω) = S ( ω) y ) S S ω) ( ω) = ( 3) S ( ω) = S ( ω) y C ( τ ) C ( τ ) = C ( τ ) y ωτ S ( ω) dω = S ( ω) ωτ dω y ω ω ( = S ) ω ωτ dω

(co-spectru) (qud-spectru) S ( ω) = K ( ω) Q ( ω) S K Q ( ω) = ( ω) + ( ω) S = S ω) ( ω) ( K ( ω) = K ( ω) S ( ω) coh ( ω) S ( ) ω coh ( ω) S ( ω) S ( ω) y yy S θ ( ω) Q ( ) ω θ ( ω) tn K K ( ω) Q Q, y C C C ( τ ) (0) y (0) C (0) C (0) C (0) y ω S ( ω) dω S ( ω) dω Sy( ω) d = S( ) Sy( ) d d ω ω ω ω = S ( ω) dω S ( ω) dω = S ( ω ) S ( ω ) dωdω

All ω, ω, y S, S, S S ω S ω S ω Sy ω ( ) ( ) ( ) ( ) ω = ω =ω S ( ω) S ( ω) S ( ω) y y S ( ) ω 0 ωh ( ω) = S ( ω) S ( ω) y