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2 Contents ii (DFFT) (CIFT) DFFT i

3 1987 SN1987A 0.5 X SN1987A LaTeX ii

4 1 {x } =1,,n ( ) P = n x (1 1) =1 {x } n Norm x {x } =1,,n x = 1 n n x (1 ) =1 {x x} { x} i.e. {x x} x =0 P = x = (x x + x) = (x x) + (x x) x + x = (x x) + x (1 3) P x DC ) (AC ) {(x,y )} =1,,n n y x ŷ = a(x x)+bt (1 4) a, b n S = (y ŷ ) (1 5) =1 a, b 1 S a = n =1 1 S b = (y ŷ )(x x) = 0 (1 6) n =1 (y ŷ ) = 0 (1 7) {(y ŷ )}, {a(x x)}, {b} {y } (y ŷ )a(x x) =0, P = (y ŷ )b =0, n y = {(y ŷ )+a(x x)+b} =1 = (y ŷ ) + a (x x) + a(x x)b = 0 (1 8) b (1 9) P DC ( b ),x ( a (x x) ), 1

5 (1 6),(1 7) a = (x x)(y ȳ) (x x), b =ȳ = 1 n y (1 10) C xy = (1 9) (1 10) (y ȳ) = (y ŷ ) + { (x x)(y ȳ)} (x x) (y ŷ ) (y ȳ) =1 { (x x)(y ȳ)} (x x) (y ȳ) { (x x)(y ȳ)} (x x) (y ȳ) 0 1 y y C xy =0 1 y x 1 C xy =1 a. b. a 1 {x } =1, n x 0 {x } 0 σ n {x } x P = x = x + (x x) ( 1) σ χ

6 χ,f X i : i =1, N 0, σ i N χ N 1,N χ χ 1 χ (N 1,N ) F χ = N i=1 X i σ i ( ) F = χ 1 /N 1 χ /N ( 3) ( 1) χ X x (x x) σ = σ + σ n 1 n 1 ( 4) (1,n 1) F F = x (x x) /(n 1) ( 5) {x } F F>F(90%) ( 6) 1 90% 0 ( ) m, σ m σ m :ˆm = x = 1 x ( 7) n σ :ˆσ = 1 (x x) ( 8) n 1 m, σ n x m, σ/ n m( m = 0), ˆσ/ n x x > <x(90%) 3

7 F ( 5) σ 0 σ 0 m σ 0 σ ˆσ = 1 n n ˆσ < ˆσ >= σ < > n ˆσ σ = n χ n 1 ˆσ 90% σ χ 1,χ χ 1 < nˆσ σ <χ n =1 n =1 x x σ nˆσ χ <σ < nˆσ χ 1 0 m, σ ˆm = x = 1 n ˆσ = 1 n 1 x (x x) x m σ n χ (x m) σ = (x x) ( x m) σ + σ n n 1 1 χ ˆσ (n 1)ˆσ (x x) σ = σ n 1 χ ˆm ( x m) n(ˆm m) F = = (x x) (n 1)ˆσ (1,n 1) F 4

8 n(ˆm m) n 1ˆσ <F 1 ˆm (n 1)F1ˆσ n (n 1)F1 <m< ˆm + ˆσ n {(x,y )} =1, n n ŷ = a(x x)+b y x a, b {y } m σ n (y m) = (y ȳ) + (ȳ m) (y m) = (y ŷ ) + σ χ a (x x) + (b m) (y m) σ = (y ŷ ) σ + a (x x) σ + (b m) σ N N 1 1 F = a (x x) (y ŷ ) /(N ) (1,N ) F F a, b F {y } a, b a, b {y } y true = a true (x x)+b true σ (y y true ) = = (y ŷ ) + (ŷ y true ) (y ŷ ) + (a a true ) (x x) + (b b true ) σ (y y true ) σ = (y ŷ ) σ + (a a true ) (x x) σ + (b b true ) σ n n 1 1 χ F a = (x x) (y ŷ ) /(n ) (a atrue ) 5

9 n F b = (y ŷ ) /(n ) (b btrue ) (1,n ) F ˆσ yf 1 ˆσ a yf 1 (x x) <atrue <a+ (x x) b ˆσ y F 1 n <btrue <b+ ˆσ y F 1 n (y y true ) (y ŷ ) + (ŷ y true ) {P k } k =1, N (P k Pk true ) (ŷ ŷ true ) (ŷ y true P k P l k,l ) [ P k P (ŷ l y true ) ] (P k Pk true )(P l Pl true ) (ŷ y true ) = k A k (q k q true k ) (q k P k )A k (q k q true k ) σ F k = 1 χ A k (q k q true k ) (y ŷ ) /(n N) (1,n N) F q k P k y y true 1. χ 1 =. (y ŷ ) σ χ (y y true ) = σ = χ 1 + (ŷ y true ) σ n n N N χ χ 1 n N χ χ χ 1 χ χ χ 1 χ 6

10 1 10kΩ kω (log L X ),B (log L B ) 1. log L X log L B. log L X (log L B log L B ) log L X 3 3. log L X log L B 4. log L B log L X Table 1: EARLY-TYPE GALAXIES Name M B log L X log L B erg / s L N N N N N N N N N N N N N N

11 3 (DFFT) N {x n } n =0, 1,,N 1 ( 0 N 1 ) (discrete) (finite) (Fourier Transform) X(ω k )= n=0 x n e iω k n (3 1) ω k = π k, k=0, 1,,N 1 (3 ) N X n = 1 N k=0 X(ω k )e iω k n e iω k n (3 3) n=0 (3 3) x n 1 e iωk n (e iω k n ) = δ k,k (3 4) N n=0 x n = 1 N k=0 {x n } (3 5) (3 3) x n n n <0,n > =N x n+mn = 1 N k=0 X(ω k ) (3 5) X(ω k )e iω k(n+mn) = x n (3 6) {x n } 1 X(ω k ) π X(ω k+mn )=X(ω k +mπ) =X(ω k ) (3 7) DFFT n, k 0 N 1 DFFT n k DFFT ax n + by n ax(ω k )+by (ω k ) (3 8) x n+m e iωkm X(ω k ) (3 9) m=0 y n m x m Y (ω k )X(ω k ) (3 10) x n X ( ω k ) (3 11) x n X ( ω k )=X(ω k ) (3 1) X(ω 0 )=N x (3 5) n=0 x n = 1 N X(ω 0) + 1 N DC X(ω k ) k=1 AC 8

12 {x n } m σ N (x n ) n=0 (x k m) = n=0 ( x m) + 1 N ( N) = n=0 ( x m) + N k=1 X(ω k) (1) ( ) () n=0 ( x m) + N k=1 X(ω k) N = odd k=1 X(ω k) + 1 N X(ω N ) N = even (1) ( ) ( N 1) (1) σ χ X(w k ) { N = odd N = even, k < = N 1 X(w k ) = Nσ [ χ = = X(w k ) N σ = { N = even k = N X(w k ) k 9

13 4 (CIFT) DFFT CIFT DFFT x a (t) A. x n = tn+1 t n x a (t)dt t n = t 0 + n n =0, 1, N 1 (4 9) x(t) t n t n+1 B. x n = x a (t n ) (4 10) t = t n 10

14 DFFT x(t) CIFT A (4 9) x a (t) (4-1) x n (4-8) tn+1 x n = dt 1 dωx a (Ω)e iωt dω t n π = 1 tn+1 dωx a (Ω) dte iωt π = 1 π t n dωx a (Ω) i Ω e iωtn (e iωt 1) (4 11) Ω k X(w k )= n=0 1 dωx a (Ω) i π Ω e iωtn (e iωt 1)e iw kn (4 1) Ω k = ω k ω k n =Ω k (t n t 0 )=Ω k (n t 0 ) (4 13) n X(ω k )= 1 dωx a (Ω) i π Ω (e iω 1)X a (Ω) e i(ω Ω k)n iωt 0 (4 14) n=0 X(ω k )= 1 π e i(ω Ω k)tn = 1 ei(ω Ω k)n 1 e i(ω Ω k) n=0 = e i (Ω Ωk)() sin [ Ω Ω k N ] sin [ Ω Ω k ] dωx a (Ω)e i [(Ω Ω k)n+ω k ] sin ( ) Ω ( Ω ) sin [ Ω Ω k N ] Ω=Ω + π m m =0, ±1, ±, sin [ Ω Ω k ] e iωt0 (4 15) Ω π+πm π+πm X(ω k ) = 1 π sin π m= π [ (Ω + πm ) Ω + πm ( dω X a Ω + πm ] ) e i [(Ω Ω k )N+Ω k ] [ ] sin Ω Ω k N ] ( 1) Nm+N m e i(ω + πm )t 0 (4 16) sin [ (Ω Ω k ) g(ω) = sin ( ) Ω sin [ Ω Ω k N ] Ω (4 17), h k (Ω) = [ ] (4 18) sin (Ω Ωk ) g x n T h 11

15 (4 16) 1. g(ω) X a (Ω) g(ω). [ π, ] [ π π, π ], [ π, 3π ] 3. h k (Ω) h k (Ω) h k (Ω) N δ(ω Ω k ) Ω ( > π ) π m (aliasing) π 1 (Nyquist Frequency) h k (Ω) π N = π T h k(ω) Ω k π T ( 1 T ) B x n = x a (t n ) (4 11) A X (ω k )= 1 π m=0 π π x n = 1 dωx a (Ω)e iωtn π ( dω X a Ω + πm ) e i [(Ω Ω k )N] h k (Ω ) e i(ω + πm )t 0 (4 19) 1

16 5 x a (t) P x a (t + P )=x a (t) (5 1) X a (Ω) X a (Ω) = = = m= x a (t)e iωt dt P 0 m= 0 x a (t + Pm)e iω(t+pm) dt P e iωpm x a (t)e iωt dt m m= X a (Ω) = e iωpm =π k= k= δ(ωp πk) = π P δ(ω Ω k ) π P X(Ω k) k= δ(ω π k) (5 ) P (Ω k = π k) (5 3) P X a (Ω) P ( :Fundamental) ( :higherharmonics) δ δ 1 CFFT,X(Ω k ) (5 ) CFFT (4-3) (4-4) X(Ω k ) x a (t) = 1 T T 0 x a (t) e iω k(t t ) dt k= k= e iω k(t t ) = k= ( ) e i π π T (t t )k = Tδ(t t )=πδ T (t t ) for 0 < π T (t t ) π e ixk = πδ(x +πm) k= m= π 13

17 DFFT (5 3) (4 16) (4 19) X (33m/sec) 14

18 A P sin x a (t) =A sin(ω 1 t) X(Ω k ) k = ±1 0 Ω 1 = π P (5 4) P X(Ω k ) = A sin(ω 1 t)e iω k t dt 0 = AP i (δ k, 1 δ k,1 ) X a (Ω)=π A i [δ(ω + Ω 1) δ(ω Ω 1 )] (5 5) x a x n N x n DFFT ( <P ) (4 16) [ ]{ X(ω k )= A [ sin Ω 1 sin ωk ω N ] Ω 1 sin [ ] ω k ω e iϕ + sin [ ω k +ω N ] } sin [ ] ω k +ω e iϕ (5 6) ( ω =Ω 1 = π P ) ϕ, ϕ k N 1 ω k ω [ ] sin (ωk ω) N sin [ ω k ] ω = N sin [ π ( )] [ k N π ( )] [ ( )] P N k N P π k N P sin [ ( )] N sin [ π ( )] k N [ ( P)] π N k N P π k N P ω k ω sin [ ω k ω N ] sin [ ] ω k ω N 15

19 X(ω k ) ω k ±ω k [ X(ω k ) max = A ( sin Ω1 )] Ω 1 [ ]Ne iϕ (5 7) X(ω k ) X(ω k ) max for ω k ω ω k ω (5 8) 6 X(ω k ) max = A N sin ( ) Ω 1 (1 0.64) (5 9) 4 Ω 1 = m, = σ T s 6-1. T s T n n 1 P (ω k ) = X (l) (ω k ) l=0 (6 1) X l (ω k ) = x +ln e iω k (l =0, n 1) =0 ( T 1. M780 AES0b VP-00 N = 0 4 ( ) FFT N =10 6 M780( 10 )VP00( ). X T s (P11 (4 18), P14) π/t π/t 16

20 S/N x σ Nσ P (ω k ) n χ i n 1 P (ω k ) σp σ P 4n Nσ (n 1 σ P P ) (6 ) x (5 9) P (ω k) max σ P P (ω k ) max = A 4 nn sin ( ) Ω 1 (1 0.64) (6 3) = A 4 Ω 1 1 σ nnδ() A nn A TK s T S = A TT s (6 4) T s T T = T s ) 6-. {x } σ nn ( σ known ) n χ χ χ (n : Q) k P k P k Nσ χ (n : Q) (6 5) 1 Q k =1,, ( N k ) 1 or ( ii N P k > Nσ χ (n : Q) ) 1or k 1 Q N 1 1 Q 1 Q N (6 6) C( C =0.95) C = Q N (6 7) Q χ P k Nσ χ (n : Q) 6-3. ( ) ( π T k 1 ν π T k + 1 ) Z σ w i k = N ii (N=,N= ) x = z + w (6 8) 17

21 X(ω k )=Z(ω k )+W(ω k ) (6 9) X(ω k ) = Z(ω k ) + W (ω k ) + Z(ω k )W (ω k )+W(ω k )Z (ω k ) (6 10) w W (ω k ) =0 ( ) X(ω k ) = Z(ω k ) + W (ω k ) (6 11) z Ẑ(ω k) l l = l X (l) (ω k ) l W (l) (ω k ) X (l) (ω k ) Nσ n (6 1) Ẑ(l) (ω k ) Z (l) (ω k ) = W (l) (ω k ) Nσn+ Z(ω k )W (ω k )+W(ω k )Z (ω k ) (6 13) l l l folding ( folding ) 6-4. ( ) C (6 7) (6 5) P k Z l (ω k ) P k 1 C Z(ω k ) Z X(ω k ) Z(ω k ) W (ω k ) (6 10), X (l) (ω k ) l l l Z (l) (ω k ) + l W (l) (ω k ) < Nσ χ (n : Q) Z (l) (ω k ) < Nσ χ (n : Q) Nσ χ (n :1 C) (6 14) l χ (n :1 C) χ sin A upperlimit (6 3),(6 14) 4 nn sin ( Ω 1 ) A Ω 1 (1 0.64) (1 0.64) < Nσ χ (n : Q) Nσ χ (n :1 C) 1 N h k(ω) = sin[(ω Ω k)n/] N sin[(ω Ω k )/] 18

22 A < Ω 1 sin ( Ω 1 ) ( ) 1 N h k(ω) σ nn [χ (n : Q) χ (n :1 C)] (A 1 Ω 1 ) 19

23 3 DFFT (p7, (3-8) (3-1) 4 m σ nn {x ; =0,nN 1} N DFFT n X (L) (ω k )= =0 x +LN P (ω k )= X (L) (ω k ) e iω k (L=,...,n-1) P (ω k ) 5 n χ n 1 n n n 1 P (ω k ) 6 x(t) (t =, ) t n (n =0,...,N 1) x n = x(t n ) DFFT t =0 T (T = N) x(t) CFFT 0

main.dvi

main.dvi 3 Discrete Fourie Transform: DFT DFT 3.1 3.1.1 x(n) X(e jω ) X(e jω )= x(n)e jωnt (3.1) n= X(e jω ) N X(k) ωt f 2π f s N X(k) =X(e j2πk/n )= x(n)e j2πnk/n, k N 1 (3.2) n= X(k) δ X(e jω )= X(k)δ(ωT 2πk