: /5 ( ) gnuplot x i x[i] () x(t) =, π < t t, < t < π (2) cos (3) sin (4) Fourier Shigeki Sagayama, FourierTrans26nov.tex

Transcription

1 sagayama/enshu2/fouriertrans26nov.pdf Shigeki Sagayama, FourierTrans26nov.tex, November 8, 26/

2 : /5 ( ) gnuplot x i x[i] () x(t) =, π < t t, < t < π (2) cos (3) sin (4) Fourier Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/2

3 2: Fourier /5 ( ) gnuplot x i x[i] () : (2) : (3) () (2) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/3

4 3: Fourier /5 ( ) gnuplot x i x[i] () δ(t) (2) U(t) πδ(ω) + jω (3) e at2 π 4a (4) n= a e ω2 δ(t nt ) 2π T n= δ(ω 2π T ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/4

5 4: /5 ( ) gnuplot x i x[i] () x(t)(t ) T {x i } x H (t) x L (t) (2) ( ) ( h H (t) h L (t)) h L (t) (3) x(t) X(ω)) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/5

6 5: /5 ( ) gnuplot x i x[i] () x(t), y(t)(t ) T {x i } {y i } z i z(t) = x(t) y(t) T (2) x(t) y(t) (3) Parseval Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/6

7 (Rademacher, Haar, Walsh) Gram-Schmidt Shigeki Sagayama, FourierTrans26nov.tex, November 8, 26/7

8 *? [ ][ ] ( (signal) (symbol) ) : x(t) : I(x, y) : I(x, y, t) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/8

9 * 2 (2 ) : ( ) : a = (a, a 2 ) : x = c a + c 2 b ( ) : a = a 2 + a2 2 (Euclid) : d(a, b) = a b = Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/9 d O (a b ) 2 + (a 2 b 2 ) 2 : a b = a b + a 2 b 2 = a b cos θ : a b =, (a, b ): x = c a + c 2 b = i.e., (θ = ±π/2) (x, a) (x, b) a + (a, a) (b, b) b b a c = (a, a ) 2

10 * n : v = (v, v 2,, v n ) : x = c u + c 2 v ( ) : v = n i= v2 i (Euclid) : d(u, v) = u v = : u v = n i= u iv i : u v =, i.e., (θ = ±π/2) (v i ): x = n i= c iv i = n i= x v i v i 2v i n (u i v i ) 2 i= Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/

11 * {u i }: u i u j =, i = j, i j x x = c u + c 2 u c n u n (m, m < n) x = m i= c iu i + ε ε 2 c i = x u i u i 2 Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/

12 * t, t 2,, t n x(t) x = {x(t ), x(t 2 ), x(t 3 ),, x(t n )} = {x, x 2, x 3,, x n } n x = (x, x 2, x 3,, x n )... t n t t t 2 3 Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/2

13 * ( ) : (x, y) = x(t )y(t ) + x(t 2 )y(t 2 ) + + x(t n )y(t n ) = n i= x(t i)y(t i ) (u i ): x = n i= c iu i = n i= (x, u i ) (u i, u i ) u i ( ) : d 2 (x, y) = n i= (x(t i) y(t i )) 2 Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/3

14 * (Cf. Riemann ) ( ) Hilbert : (x(t), y(t)) = t 2 t x(t)y(t)dt Cf. ( f(t) > ) (x(t), y(t)) = t 2 t x(t) y(t) f(t)dt : = t 2 t x(t) y(t) df (t) (Stieltjes ) d 2 (x(t), y(t)) = t 2 t (x(t) y(t)) 2 dt Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/4

15 * (u i (t) ): x(t)? = i= c iu i (t) = n i= (x(t), u i (t)) (u i (t), u i (t)) u i(t) ( ) {u i (t)} (complete, ) Cf. Hilbert David Hibert: Euclid (99) F. Riesz: L 2 J. von Neumann: Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/5

16 * /2 ( 2 ) : (a, b) L 2 f(x), g(x) (f, g) = b a f(x) g(x)dx f = (f, f) /2 (f, g) = f, g ( Lebesgue ) : f = f {f i (x)}: (a, b) {f n (x)} 2 (orthogonal system) {f n } O(a, b) : f n (x) (orthonormal system, ON ) {f n } ON(a, b) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/6

17 * 2/2 ( 2 ) : R (a, b) {f n } (a, b) R {f n } {f n } (closed) : R (a, b) L 2 {f n } O(a, b) n (ϕ, f n ) = ϕ(x) R {f n } R (complete) ( R L 2 {f n } ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/7

18 * /2 f(t) v i (t)? f(t) = c v (t) + c 2 v 2 (t) + + c N v N (t) + e(t) = N n= c iv i (t) + e(t) ε = t 2 t e 2 (t)dt = t 2 t f(t) N {v i } c iv i (t) n= 2 dt ε = t 2 t f 2 (t)dt 2 N i= c i t2 t f(t)v i (t)dt + N i= c2 i t2 t v 2 i (t)dt Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/8

19 * 2/2 ε {c i }? ε = 2 t 2 t2 t c f(t)v i (t)dt + 2c i t vi 2 (t)dt = i t2 t c i = f(t)v i (t)dt t2 t vi 2 = (f, v i) (t)dt (v i, v i ) ε? min ε = t 2 t f 2 (t)dt N i= c2 i Bessel t2 t v 2 i (t)dt t2 t f 2 (t)dt N i= c2 t2 i t vi 2 (t)dt {v i } N Parseval t2 t f 2 (t)dt = i= c2 i t2 t v 2 i (t)dt Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/9

20 * : Rademacher (922) Hans Rademacher (Born: 3 April 892 in Wandsbeck, Schleswig-Holstein, Germany; Died: 7 Feb 969 in Haverford, Pennsylvania, USA) Rademacher (922): (, ) +, t > r n (t) = sign(sin 2 n πx), sign(t) =, t =, t < r (t) r 2 (t) r 3 (t) r 4 (t) r 5 (t) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/2

21 * : Haar (9) Alfréd Haar ( Born: Oct 885 in Budapest, Hungary; Died: 6 March 933 in Szeged, Hungary) Haar (9): (, ) χ () (t) =, 2 n, χ (k) n (t) = k 2 n < t < k /2 2 n 2 n, k /2 2 n < t < 2 k n, l 2 n < t < 2 l n, l k, l 2 n k 2 n, n, integer. χ () (t) χ () (t) χ () (t) χ (2) (t) χ () 2 (t) χ (2) 2 (t) χ (3) 2 (t) χ (4) 2 (t) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/2

22 * : Walsh (923) Walsh (923): (, ) Paley (932) : (r k (t) Rademacher ) n = m n k 2 k (n 2 ) k= w n (t) = m (r k+(t)) n k k= : Walsh Paley Hadamard w (t) = wal(, t) = cal(, t) - w (t) = wal(, t) = sal(, t) w 2 (t) = wal(3, t) = sal(2, t) w 3 (t) = wal(2, t) = cal(, t) w 4 (t) = wal(7, t) = sal(4, t) w 5 (t) = wal(6, t) = cal(3, t) w 6 (t) = wal(4, t) = cal(2, t) w 7 (t) = wal(5, t) = sal(3, t) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/22

23 * Gram-Schmidt {f i (x)} {φ i (x)} φ (x) = ψ (x)/ ψ (x), ψ (x) = f (x) φ 2 (x) = ψ 2 (x)/ ψ 2 (x), ψ 2 (x) = f 2 (x) (f 2, φ )φ (x) φ 3 (x) = ψ 3 (x)/ ψ 3 (x), ψ 3 (x) = f 3 (x) (f 3, φ )φ (x) (f 3, φ 2 )φ 2 (x) φ n (x) = ψ n (x)/ ψ n (x), ψ n (x) = f n (x) n n, φ i )φ i (x) i= {, x, x 2, x 3, x 4, } Gram-Schmidt (, ) Legendre (, ) e x Laguerre : z = e jω {, z, z 2, z 3, z 4, } Gram-Schmidt LPC ( ) [ 966] Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/23

24 * Legendre.8.6 Legendre Prthogonal Polynomials Legendre(x) Legendre(x) Legendre2(x) Legendre3(x) Legendre4(x) Legendre5(x).4.2 P(x) x Legendre, m n P m (x)p n (x)dx = 2 2n+, m = n P (x) =, P (x) = x, P 2 (x) = (3x 2 )/2, P 3 (x) = (5x 3 3x)/2, P 4 (x) = (35x 4 3x 2 + 3)/8, P 5 (x) = (63x 5 7x 3 + 5x)/8, Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/24

25 * Legendre P n (x) (, +) Legendre P m n (x) (, +) Gegenbauer C ν n(x) (, +) ( x 2 ) ν 2 Tchebycheff T n (x) (, +) ( x 2 ) 2 Hermite H n (x) (, ) e x2 2 Jacobi G n (α, γ; x) (, ) x γ ( x) α γ Laguerre L (α) n (x) (, ) e x x α b a f m (x)f n (x)w(x)dx =, m n >, m = n Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/25

26 Amplitude time (sec) (a) /i/ ( ) Hamming 2 Spectrum Density (db) frequency (khz) (b) ( ) (LPC) ( ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/26

27 * 2 5 Spectrum Density (linear) frequency (khz) (c) ( ) CSM CSM ( ) 2 Spectrum Density (db) frequency (khz) (d) CSM (db)( ) Christoffel LPC ( ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/27

28 * CDMA/One : Walsh : [ 98] LPC( ) z PARCOR z LSP x = cos ω TwinVQ Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/28

29 Gibbs Dirichlet Shigeki Sagayama, FourierTrans26nov.tex, November 8, 26/29

30 Jean Baptiste Fourier Jean Baptiste Joseph Fourier (March 2, May 6, 83) Shigeki Sagayama, sagayama@hil.t.

31 : Fourier Jean Baptiste Joseph Fourier ( Born: 2 March 768 in Auxerre, Bourgogne, France; Died: 6 May 83 in Paris, France) 9 8 Auxerre 4 Bézout 9 22 Auxerre ( 795 ) Lagrange Laplace Lagrange 798 Isère Grenoble Fourier Lagrange, Laplace, Monge, Lacroix Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/3

32 * Marc-Antoine Parseval des Chênes (27 Apr 755 in Rosières-aux-Saline, France 6 Aug 836 Paris) 5 2 Parseval Hans Rademacher (Born: 3 April 892 in Wandsbeck, Schleswig-Holstein, Germany; Died: 7 Feb 969 in Haverford, Pennsylvania, USA) Alfréd Haar ( Born: Oct 885 in Budapest, Hungary; Died: 6 March 933 in Szeged, Hungary) Jacques Salomon Hadamard ( Born: 8 Dec 865 in Versailles, France; Died: 7 Oct 963 in Paris, France) Raymond Edward Alan Christopher Paley ( Born: 7 Jan 97 in England; Died: 7 April 933 in Banff, Alberta, Canada ) Adrien-Marie Legendre (Born: 8 Sept 752 in Paris, France; Died: Jan 833 in Paris, France) Edmond Nicolas Laguerre (Born: 9 April 834 in Bar-le-Duc, France; Died: 4 Aug 886 in Bar-le-Duc, France) Pafnuty Lvovich Chebyshev (Born: 6 May 82 in Okatovo, Russia; Died: 8 Dec 894 in St Petersburg, Russia) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/32

33 * Laplace - la place Lagrange - la grange Legendre - le gendre Laguerre - la guerre Rademacher - Rad ; Rade Haar - Haar Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/33

34 /2 : {cos kωt} ( ) e.g., : {sin kωt} ( ) e.g., : {, cos Ωt, sin Ωt, cos 2Ωt, sin 2Ωt, } Trigonometric Functions.5 cos(t) sin(t) cos(2*t) sin(2*t) cos(3*t) sin(3*t) cos(4*t) sin(4*t) Function Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/34 Time

35 2/2 f(t) (t, t + T ) ( ) Ω = 2π/T ( : ) f(t) = a + (a k cos kωt + b k sin kωt), t < t < t + T k= a k = b k = t +2π/Ω t t +2π/Ω t t +2π/Ω t t +2π/Ω t f(t) cos kωtdt cos 2 kωtdt f(t) sin kωtdt sin 2 kωtdt = 2 T = 2 T t +T t t +T t f(t) cos kωtdt f(t) sin kωtdt cos sin c k = f(t) = k= c k cos(kωt + φ k ) a 2 k + b2 k, φ k = tan (b k /a k ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/35

36 f(t)+f( t) + f(t) f( t) f(t) = 2 f(t) + f( t) f(t) f( t) = }{{}}{{} f e (t) f o (t) f e (t) = f e ( t) f o (t) = f o ( t) x(t) =, t < t, t Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/36

37 . f(x) = x 3 + 5x 2 2x 2 + x 7 f o (x) f e (x) f(x) = f o (x) + f e (x) 2. g(x) = 3 sin(x π 4 ) g o(x) g e (x) g(x) = g o (x)+ g e (x) 3. h(x) = e (x )2 h o (x) h e (x) h(x) = h o (x) + h e (x) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/37

38 f(t) ( T/2, T/2) (Ω = 2π/T ) f(t) = a + f(t) : f(t) = f( t) a k = = 2 T b k = k= (a k cos kωt + b k sin kωt) T/2 T/2 f(t) cos kωtdt = 4 T f(t) : f(t) = f( t) a k = b k = = 2 T T/2 T/2 f(t) sin kωtdt = 4 T (cos) T/2 f(t) cos kωtdt (sin) T/2 T/2 f(t) sin kωtdt Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/38

39 ( ) Euler : e jωt = cos Ωt + j sin Ωt cos Ωt = ejωt + e jωt, sin Ωt = ejωt e jωt 2 2j f(t) = a + = a + 2 c = a c k = (a k jb k )/2 c k = (a k + jb k )/2 k= (a k cos Ωt + b k sin Ωt) k= {(a k jb k )e jkωt + (a k + jb k )e jkωt } ( : ) f(t) = c k = T k= c ke jkωt T 2 T 2 f(t)e jkωt dt Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/39

40 f(t) = c ke jkωt k= f(t) R c k = R c k I c k = I c k f(t) ( f(t) = f( t) ) c n f(t) ( f(t) = f( t) ) c n Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/4

41 Fourier Gibbs Phenomenon (Fourier approximation of a rectangular function) PartialFourier(t) PartialFourier3(t) PartialFourier7(t) PartialFourier23(t) PartialFourier(t).5 Amplitude Time * omega f(t) = (t π) (t =, t = ±π) ( π t ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/4

42 Fourier Gibbs Phenomenon (Fourier approximation of a rectangular function) PartialFourier(t) PartialFourier3(t) PartialFourier7(t) PartialFourier23(t) PartialFourier(t).5 Amplitude Time * omega c k = π 2π π sgn t ( j) sin ktdt = 2j π 2π sin ktdt = j π [cos kt/k]π (k = ) = 2j k (k = ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/42

43 Gibbs Gibbs Phenomenon (Fourier approximation of a rectangular function) PartialFourier(t) PartialFourier3(t) PartialFourier7(t) PartialFourier23(t) PartialFourier(t).5 Amplitude Time * omega ( ) (Gibbs = π 2 Si π) ( 9% ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/43

44 * Gibbs ( ) (Gibbs = 2 π Siπ) ( 9% ) 898 Michelson + Stratton : 8 Gibbs Josiah Willard Gibbs (839 93) Nature J. W. Gibbs: Fourier s series, Nature, Vol. 59, No. 2, pp. 66-, (96) ( ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/44

45 * (Dirichlet) 2π f(t) f n (t) = n c ke jωt = n k= n 2π k= n e jkt π π f(t)e jkτ dτ = π n 2π π k= n ejk(τ t) f(τ)e jkτ dτ = 2π = π π D n (τ t)f(τ)dτ D n (t) Dirichlet : D n (t) = 2π sin(n + 2 )t sin 2 t π π n k= n ejk(τ t) f(τ)dτ Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/45

46 * (Dirichlet) ( ) n Dirichlet D n (t)! Gibbs DirichletKernel(t,) DirichletKernel(t,2) DirichletKernel(t,3) DirichletKernel(t,5) DirichletKernel(t,) DirichletKernel(t,2) 4 function t Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/46

47 * : Dirichlet (Dirichlet ) (Hsu p. 8) f(t) () : t lim f(t e + e), lim f(t e e) (2) : f(t) f (t)) f (t) f (t + ), f (t ) * Fourier Dirichlet f(t) T/2 T/2 f(t) dt < Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/47

48 * (Lipót - Fejér) f(t) π < t < π t f(t ) f(t+) ( ) {f(t ) + f(t+)} 2 t f(t) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/48

49 : 4 3 Fourier Expansion of a Saw Waveform SawWave(t) Fourier(t) Fourier3(t) Fourier(t) 2 function t 2π Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/49

50 ( ) 2π : f(t) = t ( π < t < π), f(t + 2π) = f(t) Fourier (k ): c k = 2π π π te jkt dt d dt [te jkt ] = e jkt jkte jkt c k = 2jkπ te jkt jk e jkt π π = f(t) = ( ) k j e jkt = 2 k= k k= sin 2t sin 3t = 2(sin t jkπ 2π( )k = ( )k j k ( ) k sin kt k sin 4t + ) 4 ( ) : sin sin Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/5

51 ( ) 4 3 Fourier Expansion of a Saw Waveform SawWave(t) Fourier(t) Fourier3(t) Fourier(t) 2 function t f(t) = ( ) k e jkt = 2 ( ) k sin kt ( ) k= jk k= k sin 2t sin 3t sin 4t sin 5t = 2(sin t + + ) ( ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/5

52 Fourier Coefficients of a Saw Waveform 2 SawWaveSpectrum(omega).5 Imaginary part of coefficient k (Fourier coefficient number) ( ) ( =, =( ) k /k) : f(t) = c ke jkωt k= ( ) kω c k Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/52

53 Shigeki Sagayama, FourierTrans26nov.tex, November 8, 26/53

54 ( /T, /T ) f(t) ( ) f(t) = k= c k e jkωt, Ω = 2π T f(t) ( T t T ) (T f(t) ( < t < ) ) {c k } ( ) c k = T T/2 T/2 f(t)e jkωt dt f(t) ( 2 ) f(t) = k= T T/2 T/2 f(τ)e jkωτ dτ e jkωt T? Fourier ( f(t) ) Ω = 2π ω ω T Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/54

55 Fourier Fourier f(t) = 2π f(τ)e jω(t τ) dτdω 2 c k f(t) f(t) f(t) = T/2 T/2 k= T f(τ)e jk ωτ dτ e jk ωt, = [ T/2 T/2 2π f(τ)ejk ω(t τ) dτ ] ω k= T (k ω ω, ω dω) f(t) = 2π f(τ)ejω(t τ) dτdω ω = 2π T QED Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/55

56 * Fourier f(t) = 2π f(τ)e jω(t τ) dτdω = 2π e jωt F (ω) f(τ)e jωτ dτ dω = 2π e jωt F (ω)dω F (ω) = f(τ)e jωτ dτ f(t) ( < t < ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/56

57 ( ) f(t) Fourier F (ω) = F [ f(t) ] = f(t)e jωt dt ( ) F (ω) Fourier f(t) = F [ F (ω) ] = 2π F (ω)e jωt dω ( ) ω dω F (ω)e jωt dω ( ) ( ) f(t) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/57

58 * Fourier ( ) f(t) F (ω) (t) (ω) Cf. Laplace ( ) f(t) F (s) (t) (s) Laplace Fourier? Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/58

59 Fourier ( ) X(ω) = x(t)e jωt dt, x(t) = 2π X(ω)ejωt dω T T 2π X(ω) = 2π x(t)e jωt dt, x(t) = X(ω)ejωt dω X(ω) ( ) X(ω) x()( τ = ) X(ω) = 2π x(t)e jωt dt, x(t) = 2π X(ω)ejωt dω f X(f) = x(t)e j2πft dt, x(t) = X(f)ej2πft df ω Hz Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/59

60 sinc : ( ) f(t) = /2, t <, t > sinc F (ω) = f(t)e jωt dt = 2 Waveform Rectangular Pulse -2-2 Time e jωt dt = 2 [je jωt ] = sin ω ω RectPulse(t) = sinc ω SincFunction(omega).8.6 Fourier Transform Frequency (* pi) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/6

61 F [ f(t) ] = F (ω) = f(t)e jωt dt < e jωt = f(t) < t < f(t) f(t) dt < f(t + ) + f(t ) 2 = F { F (ω) } Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/6

62 (/3) F (ω) = F [ f(t) ], F (ω) = F [ f (t) ], F 2 (ω) = F [ f 2 (t) ] a, a, a 2 (linearity) F [ a f (t) + a 2 f 2 (t) ] = a F (ω) + a 2 F 2 (ω) (scaling) F [ f(at) ] = a F ω a (time inversion) (time shift) F [ f( t) ] = F ( ω) F [ f(t t ) ] = F (ω)e jωt (frequency shift) F f(t)e jωt = F (ω Ω) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/62

63 F [ f(t τ i ) ] = F (ω)e jωτ i m i (t) = s(t) + n(t + τ i ) M i (ω) = S(ω) + N(ω)e jωτ i target signal s(t) n(t) noise signal τ2 τ3 time delays m (t) m 2(t) m 3(t) m (t) 4 τ4. 2. CSCC [ 22] Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/63

64 (2/3) (symmetry) (differentiation) F [ F (t) ] = 2πf( ω) F [ f (t) ] = jωf (ω) = jωf [ f(t) ] (integration) f(t)dt = F () = F (differentiation) t f(τ)dτ = jω F (ω) = jω F [ f(t) ] F ( jt) k f(t) d k = dωkf (ω) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/64

65 (3/3) (convolution theorem) F [ f (t) f 2 (t) ] = F (ω)f 2 (ω) (, convolution)... f (t) f 2 (t) = f (τ)f 2 (t τ)dτ (commutative law) (associative law) δ f (t) f 2 (t) = f 2 (t) f (t) (f (t) f 2 (t)) f 3 (t) = f (t) (f 2 (t) f 3 (t)) f(t) δ(t) = f(t) f(t) δ(t τ) = f(t τ) (δ δ ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/65

66 F {F (ω) F 2 (ω)} = 2πf (t)f 2 (t) F [ f (t)f 2 (t) ] = 2π F (ω) F 2 (ω) : Lag Window LPC LPC LPC F (ω) : f (t) : F 2 (ω) : f 2 (t) : F (ω) F 2 (ω) : 2πf (t)f 2 (t) : Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/66

67 : sinc : sinc? Sinc Function sinc t sinc t =? Sinc Function SincFunction(omega) SincFunction(omega).8.8 Fourier Transform * Fourier Transform =? Frequency (* pi) Frequency (* pi) (?) sinc sinc at sinc bt =? Sinc Function SincFunction(omega) SincFunction(omega/2).8 Fourier Transform Frequency (* pi) / Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/67

68 : sinc : sinc? ( ) Squared Rectangular Function Sinc Function RectWave(t) SincFunction(omega) Waveform Fourier Transform * Time Squared Rectangular Function Frequency (* pi) Sinc Function RectWave(t) SincFunction(omega) Waveform Fourier Transform Time -2 Frequency (* pi) π π Squared Rectangular Function Sinc Function RectWave(t) SincFunction(omega) Waveform Fourier Transform Time Frequency (* pi) sinc ω sinc ω = 2πF [ rect (t)/2 rect (t)/2 ] = πf [ rect (t)/2 ] = sinc ω Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/68

69 : :? ( ) Trianglar Sinusoid TriangSin(t).5 Waveform Time Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/69

70 Squared Rectangular Function RectWave(t) Time Squared Rectangular Function Time Trianglar Sinusoid RectWave(t) Time Sinusoid Triang(t) Time Trianglar Sinusoid cos(2*pi*t) TriangSin(t) Time Sinc Function SincFunction(omega) Frequency (* pi) Sinc Function SincFunction(omega) Frequency (* pi) Squared Sinc Function SincFunction(omega)** Frequency (* pi) Spectrum of Sinusoidal Signal Frequency (* pi) Squared Sinc Function SinSpec(omega) TriangSinSpec(omega) Frequency (* pi) ( ) : :? ( ) Trianglar Sinusoid Squared Sinc Function TriangSin(t) TriangSinSpec(omega) Waveform.5 Fourier Transform Time Frequency (* pi) ( ) Waveform Waveform Waveform * Fourier Transform Fourier Transform Fourier Transform ( ) Waveform Waveform * Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/7 Fourier Transform Fourier Transform

71 Dirac ( ) φ(t)δ(t)dt = φ(), (t ) δ(t) =, (t = ) δ(t)dt = ɛ ɛ δ(t)dt =, (ɛ > ) (distribution) (δ() = ) φ(t) ( ) : t φ() = : Heaviside d u(t) = δ(t) dt Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/7

72 * Gaussian δ(t) = (sinc ) δ(t) = lim e πt2 τ τ τ 2 lim t, τ τ τ t τ, t > τ Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/72 δ(t) = lim e t τ τ 2τ sin(kt) δ(t) = lim k πt 2 (sinc 2 ) k sin 2 (t) δ(t) = lim k π t 2 k = lim k π sinc(kt) k = lim k π sinc2 (t)

73 (/3) F [ δ(t) ] = e jωt δ(t)dt = A( ) : R τ (t) = (τ < /2) or (τ > /2) F [ A ] = τ lim F [ A R τ (t) ] = τ lim Aτ sinc ωτ 2 signum : sgn(t) = ( ) τ/2 = 2πA τ lim π sinc ωτ = 2πAδ(ω) 2, t >, t < F [ sgn(t) ] = 2 jω Heaviside [ sgn(t) = 2 u(t) = lim e at ()(t) e at ()( t) ] a F [ sgn(t) ] = lim a [ e at e jωt dt e at e jωt ] dt = lim 2jω a a 2 + ω 2 Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/73 = 2 jω

74 (2/3) Heaviside ( ) u(t) u(t) = 2 { + sgn(t)} = {F [ ] + F [ sgn(t) ]} 2 F [ u() ] = πδ(ω) + jω cos(ωt), sin(ωt) ( ) F [ cos(ωt) ] = π[δ(ω Ω) + δ(ω + Ω)] F [ sin(ωt) ] = jπ[δ(ω Ω) δ(ω + Ω)] F [ cos(ωt) ] = lim = lim τ τ = τ lim 2 e j(ω Ω)t τ τ/2 τ/2 2j(ω Ω) + 2j(ω + Ω) sin(ω Ω)τ/2 (ω Ω)τ/2 e j(ω Ω)t + e j(ω+ω)t dt 2 e j(ω+ω)t τ/2 τ/2 + sin(ω + Ω)τ/2 (ω + Ω)τ/2 Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/74 sin(ω Ω)τ/2 sin(ω + Ω)τ/2 = τ lim + (ω Ω) (ω + Ω) = lim τ τ 2 sinc (ω Ω)τ + τ 2 2 sinc (ω + Ω)τ 2

75 (3/3) e jωt ( < t < ) F e jωt = 2πδ(ω Ω) ( ) = F [ cos(ωt) + j sin(ωt) ] = π [δ(ω Ω) + δ(ω + Ω) δ(ω + Ω) + δ(ω Ω)] f(t) ( F k ) f(t) = F ke jkωt k= F [ f(t) ] = F kf e jkωt = 2π F kδ(ω kω) k= k= Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/75

76 : ( ) δ(t) T :.2 Sampling of a Signal Sampling(t) δ T (t) = k= δ(t kt ) Time Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/76

77 ( ) δ(t) T :.2 Sampling of a Signal Sampling(t) δ T (t) = k= δ(t kt ) Ω = 2π/T δ T (t) = c k e jkωt, c k = k= T F [ δ T (t) ] = T/2 T/2 = T k= T ejkωt e jωt dt = T k= F [ ] (ω kω) = T Time T/2 T/2 δ T(t)e jkωt dt = T k= k= T/2 T/2 e j(ω kω)t dt 2πδ(ω kω) = Ω Signal Spectrum Sampling Sampling(omega) k= δ(ω kω) = Ωδ Ω (ω) Frequency Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/77

78 ( ) t( ) ω( ) δ(t) Dirac 2πδ(ω) t 2 ω 2 cos Ωt π[δ(ω Ω) + δ(ω + Ω)] sin Ωt jπ[δ(ω Ω) δ(ω + Ω)] W 2π sincw 2 t, ω < W/2, ω > W/2 sinc Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/78

79 (t > ) t( ) ω( ), t < τ/2, t > τ/2 t τ, t < τ, t > τ e a t τsinc τ 2 ω τsinc 2τ 2 ω 2a a 2 + ω 2 (= ) e t2 2σ 2 σ 2πe σ2 ω 2 2 δ T (t) Ωδ Ω (ω) (Ω = 2π T ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/79

80 (t > ) t( ) ω( ) u(t) πδ(ω) + jω u(t) e at a + jω u(t) te at (a + jω) 2 u(t) cos Ωt u(t) sin Ωt u(t) e at sin Ωt π [δ(ω Ω) + δ(ω + Ω)] + jω 2j π Ω [δ(ω Ω) δ(ω + Ω)] + 2j Ω (a + jω) 2 + Ω 2 Ω 2 ω 2 Ω 2 ω 2 Heaviside (= ) (= u(t) = ω ) (= u(t) = ω ) (= = ω ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/8

81 Nyquist Shigeki Sagayama, FourierTrans26nov.tex, November 8, 26/8

82 ( ) : etc. f(t) ( ) 2.5 Sampling a Signal signal samples Time (* sampling period) ( ) 2 Sampling a Signal samples Time (* sampling period) 2 Sampling a Signal Time (* sampling period) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/82

83 ( ) ( )? Ambiguity of a Sample Sequence low band signal high band signal samples Time (* sampling period) 2.5 Ambiguity of a Sample Sequence low band signal high band signal samples Time (* sampling period) : 2 Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/83

84 ( ) f m [Hz] f(t) /2f m [ ] ( ) F [ f(t) ] = F (ω) =, ω > ω m = 2πf m any, ω < ω m = 2πf m Satisfying the Sampling Theorem low freq signal samples Time (* sampling period) Violating the Sampling Theorem high freq signal samples Time (* sampling period) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/84

85 ( ) ( ) Signal Spectrum Distribution.2.2 Waveform(t) ( ) Signal Spectrum Distribution Spectrum(omega) Frequency (* pi) Frequency (* pi) Sampling of a Signal Signal Spectrum Sampling.2 Sampling(t) Sampling(omega) Time Frequency Sampling of a Signal Signal Spectrum Distribution.2 Waveform(t) Spectrum(omega) Samples(t) Time Frequency (* pi) Sample Interpolation Function Low Pass Filter.2 Interpolation(t) LowPassFilter(freq) Time Frequency (sinc) Sampling of a Signal Signal Spectrum Distribution.2 Waveform(t) Spectrum(omega) Samples(t) Interpolation(t,.) Interpolation(t,.) Interpolation(t,2.) Interpolation(t,3.) Interpolation(t,4.) Interpolation(t,5.) Time Frequency (* pi) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/85

86 f(t) Ω[rad/S] : F (ω) = F [ f(t) ] F (ω) = where ω > Ω Signal Spectrum Distribution Spectrum(omega) Frequency (* pi) f(t) T Ω 2π f T (t) f T (t) = f(t)δ T (t) Sampling of a Signal.2 Waveform(t) Samples(t) Time F Ω (ω) = F [ f T (t) ], Ω = 2π/T F Ω (ω) = F [ f(t)δ T (t) ] = 2π F (ω) F [ δ T (t) ] = 2π F (ω) {Ωδ Ω(ω)} = T F (ω)δ Ω (ω σ)dσ F Ω (ω) F (ω) Ω Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/86

87 ( ): B.2 Signal Spectrum Distribution Spectrum(omega) Frequency (* pi) B f m f m (aliasing).2 Signal Spectrum Distribution Spectrum(omega) Frequency (* pi) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/87

88 Nyquist ( ) : f m = 2B (Nyquist : T = 2f m ) Nyquist.2 Signal Spectrum Distribution Spectrum(omega) Frequency (* pi) Nyquist.2 Signal Spectrum Distribution Spectrum(omega) Spectrum(omega-) Spectrum(omega+) Spectrum(omega-2) Spectrum(omega+2) Frequency (* pi) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/88

89 f(t) (f m = Ω/2π ) f T (t) ( T = /2f m )? (rect W W ) f(t) = F Ω [T F T (ω) rect 2Ω (ω)] = T f T (t) π sinc(ωt) = f T (t) sinc(ωt) k f k f T (t) = f kδ(t kt ) f(t) = f T (t) sinc(ωt) = f kδ(s kt )sinc(ω(t s))ds k= = f ksinc(ω(t kt )) = f sin Ω(t kt ) k k= k= Ω(t kt ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/89

90 Interpolation of Samples signal samples Time (* sampling period) ( ) Interpolation of Samples signal samples Time (* sampling period) ( ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/9

91 : t > T [ ] f(t) /2T [Hz](= π/t [rad/s]) ( ) t > T f(t) = T = π/ω f(t) = c k e jkω ot c k = 2T T T f(t)e jkω t dt = 2T f(t)e jkω t dt = 2T F [ f(t) ] (kω ) F [ f(t) ] (kω ) = F (kω ) ω = π/t [rad/s] F (ω) = f(t)e jωt dt = T T k= = F (kω ) T k= 2T = T T F (kω ) sin(ω kω )T (ω kω )T dt T e jk(ω ω )t dt 2T F (kω )e jkωt e jωt dt Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/9

92 Ω x(t) ω < ω 2 etc. ω < ω 2 (low-pass filter: LPF) ω /2 ω (low-pass filter: LPF) ω /2 Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/92

93 x(t):, x k :, Ω: ω m / : : CD(44.kHz) DAT(48kHz) t x(t) = k= x k sinc Ω(t kt ) (T ) ω m > Ω ( ) ( T = 2π/ω m) x(it ) = k= x k sinc Ω(iT kt ) ω m < Ω ( ) ( T = 2π/ω m) x(it ) = k= x k sinc ω m(it kt ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/93

94 4 Type Type 2 Type 3 Type 4 Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/94

95 :. ( :? [ ]) 2. ( ) : type 4 (DFT) FFT 2/ 2: ( ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/95

96 (type ) (type 2) (type 3) (type 4) t( ) x(t), < t < x(t) = 2π X(ω) ejωt dt x(t), T 2 < t < T 2 x(t) = ( ) k= 2πt jk X k e T {x t }, t =,, x t = π 2π X(ω) π ejωt dω ( ) {x t }, t =,, N x t = N N k= X k e j 2πk N t ( ) ω( ) X(ω), < ω < X(ω) = x(t) e jωt dω {X k }, k =,, X k = T T 2 T 2 ( ) x(t) e 2πt jk T X(ω), π < ω < π ( ) X(ω) = t= x t e jωt {X k }, k =,, N ( ) X k = N t= 2πk j x t e N t dt Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/96

97 ... type : f(t) F (ω) type 2: f(t) = f(t) δ T (t) F (ω) δ Ω (ω) = F (ω) ( ) type 3: f(t) = f(t) δ T (t) F (ω) δ Ω (ω) = F (ω) ( ) ( π, π) type 4: f(t) = (f(t) δ T (t)) δ S (t) (F (ω) δ Ω (ω)) δ Σ (ω) = F (ω) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/97

98 4 type : type 2: type 3: type 4: : ( ): ( ) ( ) ( ) Parseval Wiener-Khintchine : ( ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/98

99 (/2) (type ) (type 2) t( ) x(t) = ξ(t), <t< x(t) = 2π X(ω) ejωt dt x(t) = ξ(t), t <T/2 x(t ± T ) = x(t) x(t), T 2 <t< T 2 x(t) = k= jk 2πt X k e T ω( ) X(ω) = Ξ(ω), <ω < X(ω) = x(t) e jωt dω X(ω) = Ξ(ω) δ Ω (ω) {X k }, k =,, X k = T T 2 T 2 jk 2πt x(t) e T dt Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/99

100 (2/2) (type 3) (type 4) t( ) x(t) = ξ(t) δ T (t) {x t }, t =,, x t = 2π π π X(ω) ejωt dω ζ(t) = ξ(t), t < NT/2 ζ(t) = ζ(t ± NT ) x(t) = ζ(t)δ T (t) {x t }, t =,, N x t = N N k= X k e j 2πk N t ω( ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/ X(ω) = Ξ(ω), ω < Ω/2 X(ω) = X(ω ± Ω) X(ω), π <ω <π X(ω) = t= x t e jωt Z(ω) = Ξ(ω), ω < NΩ/2 Z(ω) = Z(ω ± NΩ) X(ω) = Z(ω)δ Ω (ω) {X k }, k =,, N X k = N x t e t= j 2πk N t

101 Parseval Wiener-Khinchine 4 Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/

102 Parseval Parseval f(t) 2 dt = t 2π F (ω) 2 dω ω = f(t)f (t)e jωt dt ω= = F [ f(t)f (t) ] ω= = ( ) ω= = 2π = 2π = 2π 2π F (ω) F (ω) F (σ)f (ω σ)dσ F (σ)f ( σ)dσ ω= F (σ)f (σ)dσ = QED Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/2

103 Parseval : t ω f(t) g(t) 2 dt = 2π F (ω) G(ω) 2 dω t ω : F (ω) 2 ( ) f(t) ω ω + dω F (ω) 2 dω 2π Squared Rectangular Function Squared Sinc Function RectWave(t) SincFunction(omega)**2.8.8 Waveform.6.4 Fourier Transform Time Frequency (* pi) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/3

104 (cross-correlation function) φ xy (τ) φ xy (τ) = x(t)y(t τ)dt y(t) τ x(t) x(t) y(t) (= ) τ τ φ(τ) = x(t) y(t) : φ xy (τ) = [x(t) y( t)](τ) : φ xy (τ) = φ yx ( τ) (austocorrelation function) φ xx (τ) x(t) = y(t) τ : φ xx (τ) = φ xx ( τ) : δ(τ) : y(t) = x(t) h(t) φ xx (τ) h(τ) = φ xy (τ), φ xy (τ) h(τ) = φ yy (τ) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/4

105 f(t) /koshiraeru/ /a/ (.52sec ) 2 5 Amplitude 5-5 x(t) Amplitude 5-5 x(t τ) dt autocorrelation φ xx (τ) = x(t)x(t τ)dt lag Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/5

106 Wiener-Khintchine Wiener-Khintchine ( ) : Φ xx (ω) = F [ φ xx (τ) ] ( ) F [ F (ω) 2 ] = 2π = = { F (ω) F (ω) 2 e jωt dω = 2π f(τ) 2π F ( ω)e jω(τ t) dω f(τ)f(τ t)dτ Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/6 f(τ)e jωτ dτ } e jωt dω f(τ) 2π dτ = (N. Wiener) ( ) F (ω )e jω (τ

107 (power spectrum): Φ xx (ω) = F [ φ xx (τ) ] (cross spectrum): Φ xy (ω) = F [ φ xy (τ) ] >, t t x(t) = w(t)x(t), : w(t) = < T/2 =, otherwise φ xx (τ) = φ x x (τ) Φ xx (ω) = F φxx (τ) : Hamming Hamming Window HammingWindow(t).8 amplitude (scaled) time (* time constant) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/7

108 x(t) /koshiraeru/ Amplitude /a/ (at.52sec) time (sec) 2 Spectrum Density (db) /sh/ 2 (at.3sec) Spectrum Density (db) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/8

109 f(t) /koshiraeru/ /a/ (.52sec ) Amplitude ( ) autocorrelation lag 2 Spectrum Density (db) Shigeki Sagayama, sagayama@hil.t.u-tokyo.ac.jp FourierTrans26nov.tex, November 8, 26/9